Seminar Quantum Field Theory

Transcription

1 Seminar Quantum Field Theory Institut für Theoretische Physik III Universität Stuttgart 2013

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3 Contents 1 Introduction Motivation Lorentz-transformation Necessity for field viewpoint Classical field theory introduction Lagrangian field theory Hamiltonian field theory Klein-Gordon/Dirac equation Klein-Gordon equation Dirac equation Noether Theorem Summary Canonical Quantization I - Matter Quantization of the real Klein-Gordon field Quantization The field Heisenberg picture Causality and propagators Quantization of the complex Klein-Gordon field Quantization of the Dirac field The wrong direction Finding the correct quantization Summary Propagators Spin-statistic theorem iii

4 Contents 3 Canonical Quantization II 31 4 Feynman Diagrams and Rules for QED Motivation Repitition and Notation The Interaction Hamiltonian The Scattering Matrix Scattering Experiment Dyson Series for the Time Evolution Operator First Order Terms of QED First Order S-Matrix Photon Absorption by an Electron First Order Feynman Rules and Diagrams Wick s Theorem Second Order Terms and Feynman Rules for QED Application of Wick s Theorem to the S-Matrix of 2nd Order Compton Scattering th Order Loop Diagramm for Electron-Positron Scattering 47 5 Path Integrals I Mathematical background Introduction Derivation of the path integral notation Solution for the free field Free field theory φ 4 -Theory Simplified solution of the φ 4 -Problem Simplified solution of the φ 4 -Problem II The high-dimensional propagator Perturbative Field Theory Summary Path integrals II: Fermions and Gauge Fields Motivation iv

5 Contents 6.2 Path integrals for fermions Formal motivation The Grassmann algebra The Dirac field in path integral formalism Path integrals for gauge fields The problem for the electromagnetic field Illustrative example Gauge fixing (Fadeev-Popov formalism) Application to the electromagnetic field Quantum electrodynamics in the path integral formalism More on Feynman Diagrams Correlation Functions Concepts Interacting Field Theory Feynman Rules for Correlation Functions Examples Exercise S-Matrix Concepts Contractions with Initial and Final States Amputating Legs Feynman rules for scattering processes Example Scattering Cross-section Radiative Corrections Motivation Divergence in loop integrals One-loop diagrams Superficial degree of divergence from topological analysis of Feynman diagrams Regularization Renormalization The electron propagator v

6 Contents Ward identity Analysis of the photon propagator, vertex and external lines Summary Applications Anomalous magnetic moment Lamb Shift Renormalization Motivation The Superficial Degree of Divergence and Renormalizability Renormalizability of QED Renormalizability of the ϕ n -Theory Regularization and Renormalization: Bare Perturbation Theory The Four-Point-Function: Pauli-Villars Regularization The Two-Point-Function: Dimensional Regularization Dressed Perturbation Theory The Renormalization Group Renormalization Group of the Ising Model Renormalization-Group in QFT: Wilsons Approach Talk Talk Talk Talk Bibliography 163 vi

7 Chapter 1 Introduction 1.1 Motivation In this introduction to quantum field theory the idea of this theory should become clear as well as the very basics up to the quantization of the final field. To describe processes that occur at very small (quantum-mechanical) scales and very large (relativistic) energies it is not possible to use single-particle wave equations (like the Dirac or the Klein-Gordon equation), because the Einstein relation allows the creation of particle-antiparticle pairs. We show that for a single-particle wave equation causality is violated. Therefore it is necessary to use a field theory. To begin to understand how to use the quantum field theory we start with the classical field theory and apply it to the Klein-Gordon and Dirac equation. 1.2 Lorentz-transformation To recapitulate the Lorentz-transformation necessary to understand the quantum field theory it is shown up here. Contravariant and covariant components of the local vector: x µ : x 0 = ct, x 1 = x, x 2 = y, x 3 = z x µ : x 0 = ct, x 1 = x, x 2 = y, x 3 = z The metric tensor is used to link the co- and contravariant components: x µ = g µν x ν, x µ = g µν x ν. 1

8 1 Introduction The metric tensor is given by g = (g µν ) = (g µν ) = An inertial frame of reference is a frame of reference where force-free particles move uniform. The Lorentz-transformation shows how two inertial frames of reference are transformed into each other. Therefore they are linked linear [1]: x µ = Λ µ νx ν + a µ 1.3 Necessity for field viewpoint As already introduced in the motivation the field theory is necessary to understand processes occuring at very small scales and very large energies. Besides the Einstein relation and the single-particle wave equations also the causality is violated not considering a field theory. This will be shown in a short calculation. Consider the amplitude of a free particle propagating from x 0 to x: U(t) = x e iht x0. Using the nonrelativistic quantum mechanics E = p 2 /2m, so U(t) = x e i(p2 /2m)t x0 = d 3 p x e i(p2 /2m)t p p x0 = ( ) 3 m 2 e im(x x 0) 2 /2t. 2πit As the expression is nonzero for all x and t the particle can propagate between any two points in an arbitrarily short time, what is a violation of causality referring to the relativistic theory. Using a relativistic expression E = p 2 + m 2 instead is not helping. 2

9 1.4 Classical field theory introduction U(t) = x eit p 2 +m 2 x 0 = 1 d 3 p e it p 2 +m 2 e ip(x x 0) (2π) 3 U(t) e m x 2 t 2. What is still nonzero, although it is quite small and again a violation of causality [2]. Quantum field theory will solve those problems and the very basics will be introduced in the following chapters. 1.4 Classical field theory introduction As the importance of the necessity of the field viewpoint is verified, the most important classical field theories will be introduced Lagrangian field theory The fundamental quantity of classical mechanics is the action S, the time integral of the Lagrangian L. In a local field, it can also be written with the Lagrangian density L: S = L dt = L(ϕ, µ ϕ)d 4 x. The principle of the least action allows to assume δs = 0 resulting in d 4 x { L δϕ ( ) ( L ϕ µ ( µ ϕ) δϕ + L µ ( µ ϕ) δϕ)} = 0. The last term can be turned into a surface integral over the boundary of the fourdimensional spacetime region of integration. Since initial and final field configurations are fixed, δϕ is zero at the beginning and ending of this region. Factoring out the δϕ at the remaining terms, we note, that for any arbitrary δϕ the quantity multiplied with the factor δϕ has to vanish at all points. This results in the Euler-Lagrange equation ( L µ ( µϕ) ) L ϕ = 0. 3

10 1 Introduction Hamiltonian field theory The Lagrangian formulation of field theory is indeed very nice referring to relativistic dynamics as all expressions are Lorentz invariant, but for the quantization the Hamiltonian formulation will be much easier. We show here the basics, which will be needed in later chapters for the quantization. Hence, the conjugate momentum p L/ q should be recalled allowing to write the Hamiltonian as H = p q L. The generalization to a continuous system is best understood by pretending that the spatial points x are discretely spaced. Defining p(x) L ϕ(x) = ϕ(x) L(ϕ(y), ϕ(y))d 3 y ϕ(x) L(ϕ(y), ϕ(y))d 3 y = π(x)d 3 x y where π(x) L is the momentum density conjugate to ϕ(x). That is why the ϕ(x) Hamiltonian also can be written as H = x p(x) ϕ(x) L. And passed to the coninuum, this becomes H = d 3 x [ π(x) ϕ(x) L ] = d 3 xh (1.1) with the Lagrangian density L and the Hamiltonian density H [3]. 1.5 Klein-Gordon/Dirac equation There are two relativistic equations describing the properties of spinless/spin-1/2 particles. For spinless particles the spin the Klein-Gordon equation is used. For explaining the properties of fermions (spin 1/2) the Dirac equation is used. 4

11 1.5 Klein-Gordon/Dirac equation Klein-Gordon equation Using the relativistic energy-momentum relation in its quadratic form (to avoid square roots): E 2 p 2 c 2 = m 2 c 4 and the quantum mechanical correspondence principle E = iħ t, p = ħ i the Klein-Gordon equation is immediately reached [ 1 c 2 2 t m2 c 2 ħ 2 ] ϕ(t, x) = 0. This notation can be reduced using the natural units (ħ = 1, c = 1) as well as the D Alembert operator := µ µ = 1 c 2 2 t 2 2 and x = (t, x) resulting in: ( + m 2 ) ϕ(x) = 0. Solution Using the Lagrangian [4] L = 1 2 ( µϕ)( µ ϕ) 1 2 m2 ϕ 2 the solution of the Klein-Gordon equation can be determined. The Euler-Lagrangian equation for the field ϕ is, L µ L ( µ ϕ) ϕ = 0. This is nothing else than the Klein-Gordon equation itself. [ 2 + m 2 ] ϕ(x) = 0 The general solution can be found using the approach ϕ(x) = [ d 4 k (2π) f(k)e ikx + f (k)e +ikx]. 4 5

12 1 Introduction This is obviously invariant under Lorentz-transformation, as it is anticipated for a scalar field. Substituting into the Klein-Gordon equation hands in what is satisfied if d 4 k (m 2 k 2 ) [ f(k)e ikx + f (k)e +ikx] = 0 (2π) 4 f(k) = (2π)δ(k 2 m 2 )c(k), f (k) = (2π)δ(k 2 m 2 )c (k). This leads to the general solution: ϕ(x) = Rewriting the δ function to d 4 k δ(m 2 k 2 ) [ c(k)e ikx + c (k)e ikx]. (2π) 3 δ(k 2 m 2 ) = δ(k 2 0 ω 2 k) = δ(k 0 ω k )+δ(k 0 +ω k ) 2ω k as the δ function requires k 2 = m 2 k 0 = ±ω k, with ω k = k 2 + m 2. This allows to rewrite the general solution to ϕ(x) = where a(k) = c(ω k, k) + c ( ω k, k). [ d 3 k (2π) 3 2ω k a(k)e i(ω k t kx) + a (k)e ] i(ω kt kx), Dirac equation As the Klein-Gordon equation describes spinless particles, the Dirac equation is used to describe fermions (spin 1/2). This is fullfilled by the ansatz iħβ ψ iħβα i i ψ + mcψ = 0. Here α and β cannot be scalars anymore, as the Dirac equation is asked to be Lorentz covariant. With α and β being scalars it would not even be forminvariant under rotations. They have to be matrizes. Furthermore for a breakdown the Dirac matrizes are introduced: γ 0 β and γ i βα i are defined having the properties: γ 0 : hermitian and (γ 0 ) 2 = 1 6

13 1.5 Klein-Gordon/Dirac equation (γ k ) = γ k and (γ k ) 2 = 1 Together with γ 0 γ k + γ k γ 0 = γ l γ k + γ k γ l = 0 these relations result in the basic algebraic structure This leads to the Dirac equation: γ µ γ ν + γ ν γ µ = 2g µν 1. ( ) iγ µ µ + mc ħ ψ(x) = 0. Using once again natural units (ħ = 1, c = 1) and / = γ µ µ, which is the Feynman-notation, the reduced notation of the Dirac equations is as follows ( i / + m ) ψ(x) = 0. Solution Describing a free particle to find the solution of the Dirac equation the approach ψ(x) = u(p)e ipx is done, while p 2 = m 2 and in the following only positive energy is considered p 0 = E > 0. Substituting into the Dirac equation additional restrictions for the spinor u(p) will arise, (/p m)u(p)e ipx = 0 (/p m)u(p) = 0. Using following substitution (Weyl representation) for γ-matrices, γ 0 = σ 1, γ J = σ j (iσ 2 ) m1 2 2 p σp p σp m1 2 2 ξ = 0 η 0 7

14 1 Introduction with u(p) = ξ η is obtained. As the spinor u(p) is out of four components ξ and η consists of two components each. On the upper equation one can already get solutions, but keeping in mind that the field ψ transforms as a spinor more informations can be exploited. Field transforms as a spinor: and equivalently ψ (x ) = Λ 1 ψ(x) 2 u (Λp) = Λ 1 u(p). 2 u(p) also transforms as spinor, under a Lorentz transformation p µ p µ = Λ µ νp ν. The strategy used for finding a general solution u(p) is to solve the Dirac equation for a simple value of the momentum p µ and afterwards perform a Lorentztransformation. An easy value for the momentum might be p µ = (m, 0) resulting in, m1 2 2 m1 2 2 ξ = 0 and ξ = η. m1 2 2 m1 2 2 η 0 The general solution for the spinor u in the rest frame is u (p µ = (m, 0)) = m ξ, ξ while m is a convenient normalization and the arbitrary spinor ξ is chosen to be a linear combination, ξ = s=±c s ξ s with ξ + = 1 and ξ = From this solution of the Dirac equation in the rest frame, any other momentum value can be reached by a Lorentz-boost. Redoing this calculations for an arbitrary vector p µ = (m, p 0) the spinors can be determined for ψ(x) = u(p)e ipx with p 0 > 0 and now considering the negative part ψ(x) = v(p)e +ipx with p 0 > 0 (as the exponent is now positiv) to 8

15 1.6 Noether Theorem u s (p) = v s (p) = σ pξ s σ pξ s σ pξ s σ pξ s with σ = (1, σ) and σ = (1, σ). This concludes to the general solution for the classical Dirac equation: ψ(x) = s d 3 p (2π) 2 2ω p [a s (p)u s (p)e ipx + b s(p)v s (p)e +ipx ] where a s (p) and b s (p) are arbitrary coefficients, and ω p = p 0 = p 2 + m Noether Theorem The importance of the Noether Theorem can be stated in one sentence [5]: The invariance of a Lagrangian under a continuous symmetry transformation implies the existence of a conserved quantity. Example For our classical field theory Noether s theorem can be used to derive the following relation { µ L ( µ ϕ) νϕ g µν L } = 0, where the bracket is called the energy momentum tensor Θ µν. The equation above states that the energy-momentum tensor is conserved for every component ν: Examining Θ 00 in more detail: Θ 00 = µ Θ νµ 0 Θ 0ν j Θ jν = 0 (1.2) L ( 0 ϕ) 0ϕ g 00 L = π(x)( 0 ϕ(x)) L. Remembering equation (1) in chapter 4 the Hamiltonian density can be used as H(π, ϕ) Θ 00 = π(x)( 0 ϕ(x)) L(ϕ, µ ϕ). 9

16 1 Introduction The conservation of energy now can be shown by considering t V d3 xθ 00 = V d3 x 0 Θ 00 = V d3 x j Θ j0 = S d S Θ 0 = 0, where equation (2) is used. The Hamiltonian density is a conserved quantity, provided that there is no energy flow through the surface S which encloses the volume V. 1.7 Summary It should have become clear that the field theory is necessary to understand the processes occurring at very small (quantum-mechanical) scales and very large (relativistic) energies. And therefore providing powerful tools achieving an unprecedented level of accuracy describing quantities in this range. Moreover the Klein-Gordon equation as well as the Dirac equation should be familiar by now: Klein-Gordon equation ( + m 2 ) ϕ(x) = 0 Dirac equation ( /p + m ) ψ = 0. Now it can be continuued by using the opportunities of quantization. 10

17 Chapter 2 Canonical Quantization I - Matter 2.1 Quantization of the real Klein-Gordon field Quantization We already saw the necessity for leaving the single-particle theory behind and instead using a field theory. At first we want to quantize the real Klein-Gordon field, but keep in mind that our goal is to have a quantized form of the complex Klein-Gordon field. In the case of the real Klein-Gordon field we had the Lagrangian density L = ( µ ϕ) ( µ ϕ) m 2 ϕ ϕ (2.1) by which the Klein-Gordon equation ( µ µ + m 2) ϕ = 0 (2.2) was obtained. Knowing that conjugate variables fulfill the classical commutation relations and that therefore the quantum mechanical commutators [q i, p j ] = iδ ij, [q i, q j ] = 0, [p i, p j ] = 0 (2.3) were introduced, we turn the classical fields ϕ and π into quantum mechanical operators and demand at equal time: [ϕ (x), π (y)] = iδ (3) (x y), [ϕ (x), ϕ (y)] = 0, [π (x), π (y)] = 0. (2.4) 11

18 2 Canonical Quantization I - Matter Then the Hamiltonian H = ( 1 d 3 x 2 π ( ϕ)2 + 1 ) 2 m2 ϕ 2 (2.5) also turns into an operator 1. We treat the term 1 2 ( ϕ)2 using the Fourier transform of (the classical fields) ϕ and π, i.e. Setting ω p = ϕ (x, t) = d 3 p (2π) 3 eip x ϕ (p, t). (2.6) p 2 + m 2 the Klein-Gordon equation (2.2) turns into ( ) 2 t + 2 ω2 p ϕ (p, t) = 0, (2.7) which has the same form as the equation of motion for the harmonic oscillator. The same way as ladder operators were introduced for the one dimensional harmonic oscillator 2, so called creation and annihilation operators a p, a p are now imposed for each mode in momentum space. This is the general way for the quantization: The single particle wave functions are interpreted as classical fields. The solutions of the equation (Klein-Gordon, Dirac,...) are determined. The field is then expanded in terms of those single particle solutions. The expansion coefficients are promoted to creation and annihilation operators. By setting ϕ (x) = π (x) = d 3 p (2π) 3 1 2ωp ( ap e ip x + a pe ip x) (2.8) d 3 p (2π) 3 ( i ωp 2 ) (ap e ip x a pe ip x) (2.9) and [ ap, a p ] = (2π) 3 δ (3)( p p ) (2.10) 1 At this point we already see that H is positive definite and that we will therefore have no problem ( with negative energies. 2 q = ) 1 2ω a + a, p = i ( ) ω 2 a a 12

19 2.1 Quantization of the real Klein-Gordon field we easily get [ϕ (x), π (y)] = by using δ (3) (x y) = d 3 p d 3 p ( ) i ωp (2π) 6 2 ω p ([ ] a p, a p e i(p x p y) [ a p, a ] p e i(p x p y) ) = iδ (3) (x y) (2.11) d 3 p (2π) 3 e ip (x y)3, and also H = d 3 ( p (2π) 3 ω p a pa p + 1 ), (2.12) 2 where we will drop the infinite constant since the point of zero energy can be chosen arbitrarily 4. The vacuum state 0 with a p 0 = 0 is the ground state of H. In analogy to the harmonic oscillator, by the action of a p on the vacuum we get the excited eigenstates, which fulfill H ( a pa q... 0 ) = (ω p + ω q +...) ( a pa q... 0 ), (2.13) P ( a pa q... 0 ) = (p + q +...) ( a pa q... 0 ), (2.14) where P = d 3 x π (x) ϕ (x) = is the total momentum operator. The total energy d 3 p (2π) 3 p ( ) a pa p (2.15) E = E p + E q +... = ω p + ω q +... (2.16) is always positive. We then say that a p creates particles with energy E p = p 2 + m 2 and momentum p. We could also write H = d 3 p (2π) 3 ω ( ) p a p a p = d 3 p ω p N (p). (2.17) 3 Here we started from the commutation relations for ϕ and π. Often quantization starts with annihilation and creation operators because of their relation to the oscillators describing the field s independent degrees of freedom. Therefore the form of the commutation relations for a and a is always the same. The fields may include non-independent degrees of freedom so that their commutation relations have to be modified. 4 This is formally achieved by normal-ordering. 13

20 2 Canonical Quantization I - Matter Then N (p) is the number density operator. According to a pa q 0 = +a qa p 0 (Interchanging two particles does not change the sign) and the possibility of setting more than one particle in a mode p, we see that Klein-Gordon particles are bosons. For the norm we choose the Lorentz invariant convention p q = (2π) 3 2E p δ (3) (p q), while the factor E p guarantees that the norm is Lorentz invariant. For example, a boost in p 1 -direction [p 1 = γ (p 1 + βe), E = γ (E + βp 1 )] gives 5 p q = (2π) 3 2E δ (3) (p q ) = (2π) 3 2E δ (3) (p q) dp 1 = (2π) 3 2E δ (3) (p q) γ dp 1 ( 1 β de dp 1 = (2π) 3 2E δ (3) (p q) γ E (1 βp 1) = (2π) 3 2E δ (3) (p q) = p q. (2.18) ) So we have p = 2E p a p 0, (2.19) and U (Λ) p = Λp for a Lorentz transformation Λ. The operators then transform according to U(Λ) a p U 1 (Λ) = EΛp E p a Λp. (2.20) But there is a new problem arising with this kind of normalization: The factor 2E p also appears in other places, e.g. in the identity for one-particle states, (1) 1 = = d 3 ( ) p 1 (2π) 3 p p 2E p d 4 p (2π) 3 where the p 0 -integral is over the upper mass-shell 6. ( δ ( p µ p µ m 2) p p ) p 0=Ep>0, (2.21) 5 We use δ (f (x) f (x 0 )) = 1 f (x 0 ) δ (x x 0). 6 p 0 > 0 and p µ p µ = m 2. 14

21 2.1 Quantization of the real Klein-Gordon field The field But what is now the meaning of ϕ? By calculating ϕ (x) 0 = d 3 p 1 (2π) 3 e ip x p (2.22) 2E p and p ϕ (x) 0 = e ip x (2.23) using (2.8) and (2.19) we can say that ϕ (x) creates one particle with momentum p and energy E p at the position x, since we know that in the nonrelativistic case the eigenstates x fulfilled p x = 1 2π 3 e ip x. (2.24) Heisenberg picture We know the transformation ϕ (x) = ϕ (x, t) = e iht ϕ (x) e iht, (2.25) where x is a 4-vector, and the equation of motion i A = [A, H]. (2.26) t Inserting ϕ and π we get i [ϕ t ϕ (x, t) = (x, t), d 3 x ( 1 2 π2 (x, t) ( ϕ (x, t)) )] 2 m2 ϕ 2 (x, t) (2.27) = d 3 x ( iδ (3) (x x ) π (x, t) ) = iπ (x, t), (2.28) i t π (x, t) = i ( + m 2) ϕ (x, t), (2.29) 15

22 2 Canonical Quantization I - Matter and therefore the Klein-Gordon equation Using the commutation relations we obtain 7 ( ) t + 2 m2 ϕ (x, t) = 0. (2.30) ϕ (x, t) = d 3 p 1 ( (2π) 3 ap e ipµxµ + a pe ipµxµ) p, (2.31) 0 =E 2Ep p π (x, t) = ϕ (x, t). (2.32) t On the one hand we can see the dual wave and particle interpretation: ϕ creates particles and it is a linear combination of plane waves 8. On the other hand we see: If we had wavefunctions of single-particles, e ±ip0t would stand for positive and negative energy solutions. But, since a p destroys particles and a p creates them, we only have positive excitation energies Causality and propagators We know that there was a problem of causality with the amplitude for the propagation of a particle from one point to another. So we want to investigate this amplitude, which now has the form 0 ϕ (x) ϕ (y) 0, again. We obtain D (x y) 0 ϕ (x) ϕ (y) 0 = d 3 p 1 (2π) 3 e ipµ(xµ yµ ) 2E p (2.33) and consider the cases x 0 y 0 = t, x = y (timelike separation) and x 0 = y 0, x y = r (spacelike separation). In the first case we obtain D (x y) e imt for t (2.34) and in the second D (x y) e mr for r. (2.35) 7 By the commutation relations it can be shown that the operators hold e iht a p e iht = a p e ie pt and e iht a pe iht = a pe iept. This can be inserted in the transformation rule (2.25). 8 We see that ϕ is of the same form as the quantized electromagnetic field. Therefore classical waves and classical particles are both described by quantum fields. 16

23 2.1 Quantization of the real Klein-Gordon field So we have the same problem that D (x y) does not vanish for s 2 = x 0 y 0 x y < 0 (compare figure 2.1). But causality really means that we have to investigate measurements and ask if two measurements which are separated spacelike can affect each other, that is if information can be exchanged between two such points. Then all physical observables at those points must be independent of each other and the corresponding operators must commute 9. If [ϕ (x), ϕ (y)] vanishes for s 2 < 0, causality is preserved in all cases since all commutators of observables involve functions of ϕ. We obtain d 3 p 1 ( [ϕ (x), ϕ (y)] = e ip µ(xµ yµ) (2π) 3 e ) ipµ(xµ yµ ) 2E p = D (x y) D (y x). (2.36) For s 2 < 0 we can continuously transform y x into x y and the commutator vanishes. For s 2 > 0 there is no transformation like that. So causality holds for our theory. Figure 2.1: Light cone. The commutator also has another meaning. If we want to determine the Green s function for the Klein-Gordon equation, we need the solution of ( µ µ + m 2) G (x y) = iδ (4) (x y) (2.37) 9 This principle is called microscopic causality or local commutativity. 17

24 2 Canonical Quantization I - Matter and use a Fourier transformation G (x y) = d 4 p (2π) 4 e ipµ(xµ yµ ) G (p). (2.38) By this way we obtain ( p µ p µ + m 2) G (p) = i, (2.39) G (x y) = d 4 p i yµ) (2π) 4 p µ p µ m 2 e ipµ(xµ. (2.40) On the other hand, since [ϕ (x), ϕ (y)] = 0 [ϕ (x), ϕ (y)] 0 ([ϕ (x), ϕ (y)] is simply a complex number) it is 0 [ϕ (x), ϕ (y)] 0 = = d 3 p 1 ( e ip µ (xµ yµ) (2π) 3 e ) ip µ(xµ yµ ) 2E p d 4 p i (2π) 4 p µ p µ m 2 e ip µ(xµ yµ ) (2.41) when we perform the integral over p 0 by closing the contour of figure 2.2 below for x 0 > y 0 and above for x 0 < y 0 (which gives zero). Therefore D R (x y) = Θ ( x 0 y 0) 0 [ϕ (x), ϕ (y)] 0 (2.42) is the retarded Green s function. If we choose another contour we obtain the Feynman propagator, another Green s function, which will be used in the Feynman rules: D F (x y) = D (x y) D (y x) for x 0 > y 0 for x 0 < y 0 (2.43) = 0 T ϕ (x) ϕ (y) 0, (2.44) where T is the time-ordering operator. We could also write D F (x y) = d 4 p i (2π) 4 p µ p µ m 2 + iϵ e ip µ(xµ yµ). (2.45) 18

25 2.2 Quantization of the complex Klein-Gordon field Figure 2.2: Contour for the retarded Green s function (left) and the Feynman propagator (right). [2] The poles are then removed from the real axis and the countour has to be closed the way that the integral, which is not along the real axis, vanishes. By using this method the way around the poles is always chosen correctly. 2.2 Quantization of the complex Klein-Gordon field We start from L = ( µ ϕ ) ( µ ϕ) m 2 ϕ ϕ (2.46) and treat ϕ and ϕ as independent variables 10. Since L is invariant under the transformation ϕ e iλ ϕ, ϕ e iλ ϕ, λ R (2.47) there is a conserved current j µ = i 2 (( µ ϕ ) ϕ ϕ ( µ ϕ)) and a conserved charge Q = d 3 x i 2 (ϕ π πϕ). (2.48) In analogy to the real Klein-Gordon field we demand commutation relations [ϕ (x), π (y)] = iδ (3) (x y), (2.49) 10 We could also treat the real and imaginary part of ϕ as independent variables. 19

26 2 Canonical Quantization I - Matter [ ϕ (x), π (y) ] = iδ (3) (x y), (2.50) and obtain the field operators 11 ϕ (x) = π (x) = ϕ (x) = π (x) = d 3 p 1 ( (2π) 3 ap e ip x + b 2ωp pe ip x), (2.51) d 3 p (2π) 3 ( i ωp 2 ) (ap e ip x b pe ip x), (2.52) d 3 p 1 ( (2π) 3 bp e ip x + a 2ωp pe ip x), (2.53) d 3 p (2π) 3 ( i ωp 2 ) (bp e ip x a pe ip x), (2.54) since the classical fields ϕ and ϕ fulfill 0 = 0 = ( ) t + 2 m2 ϕ, (2.55) ( ) t + 2 m2 ϕ. (2.56) The commutation relations for the creation and annihilation operators are [ ap, a p ] [ bp, b p ] = (2π) 3 δ (3)( p p ), (2.57) = (2π) 3 δ (3)( p p ), (2.58) while all of the other commutators vanish. We obtain the Hamiltonian H = = d 3 x ( π π + ( ϕ ) ( ϕ) + m 2 ϕ ϕ ) d 3 p (2π) 3 ω ( p a p a p + b pb p + 1 ) (2.59) and again neglect the infinite constant. The conserved charge is Q = d 3 x i 2 ( ϕ π πϕ ) 11 ϕ contains b p instead of b p because of two reasons: We will see that b p as well as a p, and therefore ϕ, destroys charge. Only by this way ϕ ϕ conserves charge. The other reason is that for the case (particle)=(antiparticle) ϕ and ϕ must be identical. 20

27 2.2 Quantization of the complex Klein-Gordon field = d 3 p (2π) 3 2 ( a p a p b pb p ). (2.60) We see that a p and b p create particles with opposite charge. We now change to the Heisenberg picture with ϕ (x, t) = ϕ (x, t) = d 3 p (2π) 3 1 2Ep ( ap e ipµxµ + b pe ipµxµ) p 0 =E p, (2.61) d 3 p 1 ( (2π) 3 bp e ip µx µ + a pe ip µx µ) p, (2.62) 0 =E 2Ep p and take a look at the commutator [ ϕ (x), ϕ (y) ] = d 3 p d 3 p 1 (2π) 6 2 E p E p ( [ap, a ] [ ] p e i(p µ x µ p µy µ ) + b p, b p e i(p µ x µ p µy µ ) ) = d 3 p 1 ( e ip µ(xµ yµ) (2π) 3 e ) ipµ(xµ yµ ) 2E p = D (x y) D (y x). (2.63) We can interpret the two parts as the propagation of a particle from y to x and the propagation of a particle with opposite charge from x to y (figure 2.3). Both processes exist and we have D (x y) = D (y x) for (x y) 2 < 0. Therefore we have to conclude that for every particle an antiparticle (with the same mass but opposite charge) must exist 12. After all we can say that by the quantization of the Klein-Gordon field we solved several problems of the Klein-Gordon equation for single-particle wave functions. But we have to keep in mind that in this theory there are no interactions among particles. Figure 2.3: Interpretation of the two parts of the commutator as a propagating particle and a an antiparticle propagating in the opposite direction. 12 In case of the real Klein-Gordon field antiparticles and particles are identical. 21

28 2 Canonical Quantization I - Matter 2.3 Quantization of the Dirac field The wrong direction The Lagrangian density for the Dirac field is given by L = ψ (iγ µ µ m) ψ We treating ψ and ψ = ψ γ 0 as independent variables and immediately obtain the Dirac equation ( iγ + m) ψ = γ 0 h D ψ = 0 (2.64) (2.65) h D = a p + βm (2.66) and the Hamiltonian H = d 3 x ψ ( iγ + m) ψ = d 3 x ψ h D ψ. (2.67) As we did for the Klein-Gordon field we could now demand the following commutation relations for the spinor components of the field: [ ψa (x), ψ b (y)] = δ ab δ (3) (x y). (2.68) Since we already know that the eigenfunctions of the Hamiltonian h D are u s (p) e ip x and v s (p) e ip x with energies ±E p we expand the field in this basis and get an expression similar to that of the complex Klein-Gordon field: ψ (x) = d 3 p 1 (2π) 3 e ip x 2Ep 2 ( a s p u s (p) + b s pv s ( p) ). (2.69) s=1 The index s always describes the spinor components of the operators or vectors, respectively. If we again set [ a s p, a r q ] = [ b s p, b r q ] = (2π) 3 δ ab δ (3) (p q), (2.70) 22

29 2.3 Quantization of the Dirac field while all other commutators vanish, and use the completeness relations u s (p) u s (p) = γ µ p µ + m, (2.71) s v s (p) v s (p) = γ µ p µ m, (2.72) s we can show that [ ψ (x), ψ (y) ] = δ (3) (x y) (2.73) Now we can take a look at the Hamiltonian. Inserting ψ in (2.67) and using the orthogonality relations we obtain the resulting expression 2mδ rs for m 0 u r (p) u s (p) = 2E p δ rs for m = 0, (2.74) 2mδ rs for m 0 v r (p) v s (p) =, (2.75) +2E p δ rs for m = 0 u r (p) v s ( p) = v r ( p) u s (p) = 0, (2.76) H = d 3 p (2π) 3 E p 2 s=1 ( a s p a s p b s p b s p). (2.77) The energy of particles created by b s p b p into b p and b p into b p and demand [ b s p, is always negative. If we choose to change ] = + (2π) 3 δ ab δ (3) (p q), we do not obtain the correct value for [ ψ (x), ψ (y) ]. Our conlusion has to be that we went in the wrong direction. But we want to take a look at the other results at first. By using b r q e iht a s pe iht = a s pe ie pt, (2.78) e iht b s pe iht = b s pe iept, (2.79) 23

30 2 Canonical Quantization I - Matter the field operators in the Heisenberg picture are ψ (x, t) = ψ (x, t) = d 3 p 1 (2π) 3 2Ep d 3 p 1 (2π) 3 2Ep 2 ( a s p u s (p) e ipµxµ + b s pv s (p) e ipµxµ), (2.80) s=1 2 s=1 ( a s p u s (p) e ip µx µ + b s p v s (p) e ip µx µ). (2.81) Then the commutator [ ψa (x), ψ b (y) ] = (iγ µ xµ + m) ab [ϕ (x), ϕ (y)] (2.82) vanishes for (x y) 2 < 0 and there s no problem with causality. But, according to a s p 0 = b s p 0 = 0 and the form of the field operators (2.80), (2.81) the second term in [ ψa (x), ψ b (y) ] = 0 [ ψ a (x), ψ b (y) ] 0 = 0 ψ a (x) ψ b (y) 0 0 ψ b (y) ψ a (x) 0 (2.83) vanishes. The first term then tells us that we have two particles, one with postive and one with negative energy, which both propagate in the same direction. At this point we have to deny the commutation relations (2.68), (2.70) and try to obtain reasonable results Finding the correct quantization We want the expression 0 ψ a (x) ψ b (y) 0 0 ψ b (y) ψ a (x) 0 to describe two particles with positive energy, one of which propagates from y to x and one of which propagates from x to y. Since ψ (x) and ψ (y) shall only create particles with positive energy we conclude that b p 0 = 0, a p 0 = 0 while the adjungate operators create particles. Without commutation relations the value 0 a s pa r q 0 in 0 ψ (x) ψ (y) 0 = 0 d 3 p 1 2 (2π) 3 a s pu s (p) e ip µx µ 2Ep s=1 d 3 q (2π) 3 1 2Eq 2 r=1 a r q u r (q) e iqµxµ 0 (2.84) 24

31 2.3 Quantization of the Dirac field has to be calculated in a different way. We assume that the vacuum is invariant under rotations and transllations 13. We still say that a r q momentum q. We then obtain 0 a s pa r q 0 = 0 a s pa r q e ip x 0 = e i(p q) x 0 e ip x a s pa r q 0 creates a particle with = e i(p q) x 0 a s pa r q 0, (2.85) which implies that 0 a s pa r q 0 = 0 for p q, and in a similar way 0 a s pa r q 0 = 0 for r s. We may then say 0 a s pa r q 0 = (2π) 3 δ sr δ (3) (p q) A (p) (2.86) with A (p) > 0, since the norm a r q 0 2 should always be positive. We want the resulting expression 0 ψ (x) ψ (y) 0 = = d 3 p 1 2 (2π) 3 u s (p) u s (p) A (p) e ipµ(xµ y µ ) 2E p s=1 d 3 p 1 (2π) 3 (γ µ p µ + m) A (p) e ipµ(xµ yµ ) 2E p (2.87) to be Lorentz invariant. This is only valid, if the unknown function A (p) fulfills 14 A (p) = A (p µ p µ ) = A (m 2 ) = const. By an analog calculation for the second propagation amplitude, with B > 0, we obtain 0 ψ a (x) ψ b (y) 0 = (iγ µ d 3 p µ + m) ab (2π) 3 Ae ip µ(xµ yµ), (2.88) 0 ψ a (x) ψ b (y) 0 = (iγ µ d 3 p µ + m) ab (2π) 3 Beip µ(xµ yµ). (2.89) So 0 [ ψ a (x), ψ b (y) ] 0 would vanish for (x y) 2 < 0, if we could choose A = B, which is not possible. Now we have 0 ψ a (x) ψ b (y) 0 = 0 ψ b (y) ψ a (x) 0 (2.90) 13 0 = e ip x 0 14 We use p µ p µ = m 2. 25

32 2 Canonical Quantization I - Matter for A = B = 1 and (x y) 2 < 0. The fields anticommutate in this case. But since all observables O include an even number of fields there s no problem with causality since then it is always [O 1 (x), O 2 (y)] = 0 for (x y) 2 < 0. We postulate { ψa (x), ψ b (y)} = δ ab δ (3) (x y) {ψ a (x), ψ b (y)} = { ψ a (x), ψ b (y)} = 0 (2.91) and { a s p, a r q } = { b s p, b r q } = (2π) 3 δ ab δ (3) (p q). (2.92) We are still left with H = d 3 p (2π) 3 E p 2 s=1 ( a s p a s p b s p b s p). (2.93) But now we have the chance to change b p into b p and b p into b p since the anticommutator is symmetric. After neglecting the infinite constant we obtain H = d 3 p (2π) 3 E p 2 s=1 ( ) a s p a s p + b s p b s p (2.94) where we simply removed the tilde. So in the following b s p positive energy and b s p destroys them. creates particles with The created particles obey ( Fermi-Dirac statistics since we cannot create two particles in the same mode p = 0 and interchanging particles changes the (b ) ) 2 s sign of the wave function ( a s p a s q 0 = a s q a s p 0 ) Summary In the following we use the field operators 15 ψ (x, t) = d 3 p 1 (2π) 3 2Ep 2 s=1 ( a s p u s (p) e ip µx µ + b s p v s (p) e ip µx µ), (2.95) 15 Some books write ψ and ψ in this way from the beginning. They find out that negative energies are possible and then simply demand anticommutation relations. Causality is then proved afterwards. 26

33 2.3 Quantization of the Dirac field ψ (x, t) = d 3 p 1 (2π) 3 2Ep 2 s=1 ( a s p u s (p) e ip µx µ + b s pv s (p) e ip µx µ), (2.96) ( ) ( ) where the operator a s p b s p destroys fermions (antifermions) and a s p b s p creates them. They obey anticommutation relations (2.92). Additionally to the Hamiltonian (2.94) the momentum operator is P = d 3 x ψ ( i ) ψ = d 3 p (2π) 3 2 s=1 p ( a s p a s p + b s p b s p). (2.97) Analog to the Klein-Gordon field we demand the normalization condition p, r q, s = (2π) 3 2E p δ rs δ (3) (p q) (2.98) for one-particle states p, s = 2E p a s p 0. (2.99) Again we have U (Λ) a s pu 1 (Λ) = EΛp E p a s Λp (2.100) and see that we get the right transformation rule for the field operator, U (Λ) ψ (x) U 1 (Λ) = = d 3 p 1 (2π) 3 u ( s Λ 1 p ) E p a s p e i p µλx µ E p d 3 p 1 (2π) 3 Λ 1 1 u s ( p) E p a s p e i pµλxµ E p = Λ 1 1 ψ (Λx), (2.101) 2 where Λ 1 2 = e i 2 ωµνsµν (2.102) is the spinor representation of the Lorentz transformation Λ, ω µν an antisymmetric tensor and S µν = i 4 [γµ, γ ν ], S ij = i 2 [γi, γ j ] = 1 2 ϵijk Σ k. There are two other important quantities: The angular momentum operator and the charge of the particles. The second one is obtained by integrating the current 27

34 2 Canonical Quantization I - Matter j µ = ψγ µ ψ: Q = d 3 x ψ (x) ψ (x) = d 3 p (2π) 3 2 s=1 ( ) a s p a s p + b s pb s p, (2.103) or after neglecting an infinite constant: Q = d 3 p (2π) 3 2 s=1 ( a s p a s p b s p b s p). (2.104) We see that fermions and antifermions have opposite charge. Since there s not only a translational invariance (which delivered P ) but also a rotational invariance we want to find the angular momentum operator J by investigating the behaviour of ψ under rotations. rotation about the z-axis (ω 12 = ω 21 = Θ) and find We choose an infinitesimal δψ (x) = Λ 1 ψ ( Λ 1 x ) ψ (x) 2 (1 i ) 2 ΘΣ3 ψ (t, x + Θy, y Θx, z) ψ (x) Θ (x y y x + i ) 2 Σ3 ψ (x) = Θ ψ. (2.105) With the conserved current j 0 = L ψ = ( 0 Ψ) iψγ0 ψ and analog calculations for rotations about the two other axes we obtain J = d 3 x ψ ( x ( i ) Σ ) ψ (2.106) and can show that Dirac fermions have spin ± 1. We choose fermions with momentum p = 0 such that the orbital term is 2 zero: J z = d 3 x 2 s,s =1 d 3 p d 3 p 1 (2π) 6 2 e i(p p )x E p E p ( a s p us (p ) + b s p vs ( p ) ) 1 2 Σ3 ( a s pu s (p) + b s pv s ( p) ). (2.107) 28

35 2.3 Quantization of the Dirac field Since J 0 must vanish we have J z a r 0 0 = [ J z, a r ] 0 0 and therefore j z a r 0 0 = 2 s=1 ( u s (0) 1 ) 2 Σ3 u r (0) a s 0 0 = 2 s=1 (ξ s 1 2 σ3 ξ r ) a s 0 0. (2.108) After calculating the sum we see that the state a r 0 0 with ξ r = (1 0) t, ξ r = (0 1) t fulfills J z a r 0 0 = ± 1 2 ar 0 0. (2.109) For antifermions we get a reversal of sign as we expect according to the Dirac hole theory: J z b r 0 0 = 1 2 br 0 0. (2.110) Propagators We can look for a Green s function for the Dirac equation. The same way as for the Klein-Gordon equation we have (iγ µ xµ m) G (x y) = iδ (4) (x y) 1 4 4, (2.111) iδ (4) (x y) = d 4 p (2π) 4 (γµ p µ m) e ip µ(xµ yµ ) G (p), (2.112) G (p) = G (x y) = i γ µ p µ m = i (γµ p µ + m) p µ p µ m 2, (2.113) d 4 p i (γ µ p µ + m) yµ) (2π) 4 p µ p µ m 2 e ipµ(xµ. (2.114) If we choose the contour of figure 2.2 we obtain the retarded Green s function, which is equal to S R (x y) = (iγ µ xµ + m) D R (x y) (2.115) with S ab R (x y) = Θ ( x 0 y 0) 0 { ψ a, ψ b } 0. (2.116) 29

36 2 Canonical Quantization I - Matter Analog to the Feynman propagator for the Klein-Gordon equation we have S F (x y) = = d 4 p i (γ µ p µ + m) (2π) 4 p µ p µ m 2 + iϵ e ip µ(xµ yµ ) + 0 ψ (x) ψ (y) 0 0 ψ (y) ψ (x) 0 for x 0 > y 0 (2.117) for x 0 < y 0 (2.118) = 0 T ψ (x) ψ (y) 0 (2.119) with an additional change of sign in the time-ordering. 2.4 Spin-statistic theorem Quantum statistics depends on the fact that all elementary particles are indistinguishable from one another. The quantization of the Klein-Gordon field and the Dirac field showed that particles with spin 0 obey Bose-Einstein statistics and particles with spin 1 obey Fermi-Dirac statistics. 2 In general the spin-statistic theorem states that particles of half integer spin obey Fermi-Dirac statistics and particles of integer spin obey Bose-Einstein statistics 16. This theorem is based on the demands for causality, positive norms, positive energies and Lorentz invariance and has far-reaching influences on different fields of physics. 16 For a proof see R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Benjamin/Cummings, Reading, Mass.,

37 Chapter 3 Canonical Quantization II [See website for handwritten notes.] 31

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39 Chapter 4 Feynman Diagrams and Rules for QED 4.1 Motivation So far we only considered non interacting quantum fields, namely the free Dirac field ψ(x), the Klein-Gordon field ϕ(x) and the electromagnetic field A µ (x). The Lagrange density L for a theory of interacting fields consists of the Lagrange density L 0 describing the free fields and a part L 1 describing the interaction between the fields L = L 0 + L 1 (4.1) In this chapter we will focus on quantum electrodynamics (QED), the theory of the interaction between leptons (described by the Dirac field ψ(x)) and photons (described by the electromagnetic field A µ ). A huge advantage of QED is, that we can apply perturbation theory due to the small expansion parameter, the fine structure constant α. As in the chapters before I will use heaviside units c = ħ = Repitition and Notation In QED we have to deal with the quantized Dirac field and the quantized electromagnetic field. For the Dirac field operators and anticommutation relations I used the same notation as in the book by Peskin and Schrder [2]. The field operators 33

40 4 Feynman Diagrams and Rules for QED in the Heisenberg picture are given by ψ(x) = ψ(x) = A µ (x) = d 3 p 1 (2π) 3 2Ep d 3 p 1 (2π) 3 2Ep d 3 k 1 (2π) 3 2E k 2 ( bs (p)u s (p)e ipx + d s(p)v s (p)e ipx) = ψ (+) (x) + ψ ( ) (x), s=1 (4.2) 2 ( b s (p)ū s (p)e ipx + d s (p) v s (p)e ipx) = ψ ( ) (x) + ψ (+) (x), s=1 3 λ=0 ( aλ (k)ε µ λ (p)e ikx + a λ (k)εµ λ (k)eikx) = A µ(+) + A µ( ), (4.3) (4.4) where b s (p) (b s()p) is annihilating (creating) an electron e with spin s and momentum p, d s (p) (d s(p)) is annihilating (creating) a positron e + with spin s and momentum p and a λ (k) (a λ (k)) is annihilating (creating) a photon γ with Polarisation λ and momentum k. The commutation/anticommutation relations are given by {b s1 (p 1 ), b s 2 (p 2 )} = (2π) 3 δ s1 s 2 δ (3) (p 1 p 2 ), (4.5) [a λ1 (k 1 ), a λ 2 (k 2 )] = g λ 1λ 2 (2π) 3 2E k1 δ (3) (k 1 k 2 ). (4.6) Later on we will also need the Feynman propagators for the Dirac field (S F ) and the electromagnetic field (D F,νµ ): S F (x 1 x 2 ) = i 0 T ( ψ(x 1 ) ψ(x 2 ) ) 0 (4.7) = d 4 p 1 e ip(x 1 x 2 ), (4.8) (2π) 4 /p m + i0 + }{{} S F (p) D F,νµ (x 1 x 2 ) = i 0 T (A ν (x 1 )A µ (x 2 )) 0 (4.9) = d 4 k (2π) 4 g νµ k 2 + i0 + e ik(x 1 x 2 ), (4.10) 34

41 4.2 The Interaction Hamiltonian where T is the time ordering operator. 4.2 The Interaction Hamiltonian The complete Lagrange density for QED is given by L = L Dirac 0 + L P hoton 0 + L 1 (4.11) = ψ(i/ m)ψ 1 2 ( νa µ )( ν A µ ) + L 1, (4.12) where L 1 is describing the interaction between the two fields. To derive the expression for the interaction term we take a look at the Dirac equation with given electromagnetic field A eµ iγ µ ( µ + iea eµ m)ψ = 0. (4.13) This corresponds to a Lagrange density L = ψ(i/ m)ψ e ψγ µ A eµ ψ (4.14) = L Dirac 0 + L 1, (4.15) where we now found the interaction Lagrange density L 1 to be L 1 = e ψγ µ A eµ ψ = e ψ /A e ψ. (4.16) By going from the Lagrange density to the Hamilton density and by replacing the field A eµ with the free quantized electromagnetic field A µ and calculating we get the interaction Hamilton density for QED. H 1 = L 1 = e ψ /Aψ. (4.17) Due to this form with the operators ψ, ψ which can create (annihilate) an e /e + and A µ creating (annihilating) a photon γ we already get the hint, that at a space-time point x, two fermions and one photon are involved in the interaction. To make calculations easier, we introduce that the Hamilton density is normal ordered. That means that all creation operators stand on the left side and all 35

42 4 Feynman Diagrams and Rules for QED annihilation operators on the right. The vacuum expectation value of a normal ordered Hamiltonian is then zero. H 1 = e: ψ /Aψ :. (4.18) To get the interaction Hamiltonian we have to integrate the density over space H 1 = d 3 xh 1. (4.19) 4.3 The Scattering Matrix Scattering Experiment For a scattering experiment we can make the following assumptions. At the initial time the particles are far away from each other and therefore don t interact. Then the particles move towards each other and interact during a short time interval. The particles then move away from each other again and are observed at a much later time, where they are seperated again and therefore don t interact with each other. During the interaction particles can be annihilated and new new particles can be created. Since the particles are only interacting during a very short time Figure 4.1: Illustration of a scattering process where particles can be annihilated and created during the scattering process. [1] interval we can put the initial time to t i = and the time where the scattered particles are detected to t f =. The particles at initial time are described by the initial state i = ψ(t = ). After the scattering process the particles are described by the final state f = 36

43 4.3 The Scattering Matrix ψ(t = ). Note that i and f are describing noninteracting particles and are therefore Eigenstates of H 0. We can now calculate the transition amplitude into a certain final state f f ψ( ) = f U(, ) i = f S i = S fi, (4.20) with the time evolution operator U(t f, t i ) and the scattering matrix S = U(, ) containing all the information about the interaction. Thus we have to find an expression for the time evolution operator (and hence the S-matrix) of the QED Hamiltonian and we can calculate the probability P to find the system in a certain final state f after the sacttering process P = f S i 2. (4.21) Dyson Series for the Time Evolution Operator To find an expression for S it s conveniant to go into the interaction picture. In the interaction picture the field operators ψ(x) I ψ(x)i and A µ (x) I are equal to the field operators in the Heisenberg picture in equation (4.4). The field operators therefore carry all the information of the free noninteracting system H 0. Meanwhile the time evolution operator U (U is the time ecolution operator in the interaction picture) only depends on the interaction Hamiltonian and contains only the information of the interaction between the two fields. We will see that in the following. The transformation from the Schrdinger picture into the interaction picture is ψ, t I = e ih 0t ψ, t 0 S for states and (4.22) A I (x, t) = e ih 0t A S (x)e ih 0t for operators. (4.23) One can see that the transformation into the interaction picture for operators is the same as the transformation of operators into the Heisenberg picture. The equation of motion for U (t, t 0 ) in the interaction picture is given by i t U (t, t 0 ) = H I U (t, t 0 ), (4.24) 37

44 4 Feynman Diagrams and Rules for QED with H I = H 1I = e ih 0t H 1 e ih 0t. (4.25) The interaction Hamilton density in the interaction picture in QED takes then the form H I = ψ I (x)γ µ A Iµ (x)ψ I (x). (4.26) We can solve equation (4.24) with the initial condition U (t 0, t 0 ) = 1 by integrating from time t 0 to time t. Through iteration we get t U (t, t 0 ) = 1 i t 0 dt 1 H I (t 1 )U (t 1, t 0 ). (4.27) t t t 1 U (t, t 0 ) = 1 + ( i) dt 1 H I (t 1 ) + ( i) 2 dt 1 dt 2 H I (t 1 )H I (t 2 ) (4.28) t 0 t 0 t 0 = t + ( i) 3 n=0 t 0 dt 1 t ( i) n t 1 t 0 dt 1 t 0 dt 2 t 1 t 2 t 0 dt 3 H I (t 1 )H I (t 2 )H I (t 3 ) +... (4.29) t 0 dt 2 t n 1 t 0 dt n H I (t 1 )H I (t 2 )... H I (t n ). (4.30) This can further be simplified by using the time ordering operator T, which was already introduced for the definition of the Feynman propagator. The time ordered product of two Hamilton density operators H I (t 1 ) and H I (t 2 ) can be written with the step function θ T (AB) = θ(t 1 t 2 )H I (t 1 )H I (t 2 ) + θ(t 2 t 1 )H I (t 2 )H I (t 1 ). (4.31) We then get U ( i) n t t (t, t 0 ) = dt 1 dt 2... (4.32) n=0 n! t 0 t 0 38

45 4.3 The Scattering Matrix t t 0 dt n T (H I (t 1 )H I (t 2 )... H I (t n )). (4.33) To get the S-matrix we have to take the limes of t 0 going to and t going to. Expressed by means of the Hamilton density we get S = ( i) n n=0 n! =T exp ( i)... d 4 x 1... d 4 x n T (H I (x 1 )... H I (x n )) (4.34) dxh I (x). (4.35) The first term of the S-matrix is called the zero order term S (0) = 1. (4.36) The zero order S-matrix is not changing the initial state. the initial particles don t interact with each other in zeroth order. The first order S-matrix is given by and the second order S-matrix by S (1) = ie d 4 xh I (x) (4.37) S (2) = ( ie)2 2 d 4 x 1 d 4 x 2 T (H I (x 1 )H I (x 2 )). (4.38) One can see that in order to calculate the second order term one has to find an expression for the time ordered product of normal ordered Hamilton densities. This problem can be solved with Wick s theorem and will be done in section 4.5. But we are already able to take a closer look at the first order processes, since we only have one Hamiltonian and therefore no time ordering. 39

46 4 Feynman Diagrams and Rules for QED 4.4 First Order Terms of QED First Order S-Matrix With the expression for the first order scattering matrix (4.37) and the Hamilton density in QED we get S (1) = ie d 4 x: ψ(x) /A(x)ψ(x):. (4.39) The field operators of the Hamiltonian can be expressed by terms of positive and negative frequencies as in equation 4.4 and thus the Hamilton density can be written as follows H I = e: ( ψ (+) }{{} ann. e + + ( ) (+) ψ )( /A + /A ( ) )( ψ (+) + ψ ( ) ):. (4.40) }{{}}{{}}{{}}{{}}{{} cre. e ann. γ cre. γ ann. e cre. e + This leads to 8 terms for the first order scattering matrix S (1) ie: ψ ( ) /A (+) ψ (+) : (1) ie: ψ (+) /A (+) ψ ( ) : (2) ie: ψ (+) /A (+) ψ (+) : (3) ie: ψ ( ) /A (+) ψ ( ) : (4) ie: ψ ( ) /A ( ) ψ (+) : (5) ie: ψ ( ) /A ( ) ψ (+) : (6) ie: ψ ( ) /A ( ) ψ (+) : (7) ie: ψ ( ) /A ( ) ψ (+) :, (8) which are discussed in the next section Photon Absorption by an Electron First we take a closer look at the first term (1) ( ie: ψ ( ) γ µ A (+) µ ψ (+) : ), where ψ annihilates an electron e, A (+) µ annihilates a photo γ and ψ ( ) creates an electron again. We can refer to this special process as the absorption of a photon by an electron. The Feynman Diagram describing this first order process is shown in 40

47 4.4 First Order Terms of QED figure 4.2. Fermions are described by a normal line, where an electron gets an arrow along the time axis and a positron gets an arrow opposite to the time axis. A photon is described by a wave line. The spot where the lines meet is called a vertex. This term only contributes to the transition amplitude, if at initial time an Figure 4.2: First order Feynman diagramm for the absorption of a photon by an electron ( [1]) electron and a photon are present and the final state describes just one electron. Thus our initial and final state have to be ( [2]) i = e p 1, γk 1 = 2E p1 b s 1 (p 1 )a λ 1 (k 1 ) 0 (4.41) f = e p 2 = 2E p2 b s 2 (p 2 ) 0, (4.42) with an incoming electron with momentum p 1 and spin s 1, an incoming photon with momentum k 1 and polarisation λ 1 and an outgoing electron with momentum p 2 and spin s 2. We can now calculate the transition amplitude for this process. Note that all the other terms of S (1) don t contribute to the the transition amplitude due to the initial condition: f S (1) i = e p 2 ie (4.43) d 4 : ψ ( ) γ µ A (+) µ ψ (+) : e p 1, γk 1 (4.44) = g λ 1λ 2 (2π) 4 δ (4) (k 1 + p 1 p 2 ) ū s2 (p 2 )( ieγ µ )ε λ1 µ(k 1 )u s1 (p 1 ). (4.45) }{{} M 41

48 4 Feynman Diagrams and Rules for QED The δ (4) -function implies the conservation of the total 4-momentum, hence the conservation of momentum and energy: p 2 = p 1 + k 1 (4.46) E p2 = k 1 + E p1. (4.47) But in general this condition cannot be fulfilled in this case, where we only have one outgoing particle and two incoming particles. This is true for all 8 first order processes. Hence these 8 processes can only occure within diagrams of higher orders and are therefore called virtual processes. Taking a closer look at the so called invariant amplitude M = ū s2 (p 2 )( ieγ µ )ε λ1 µ(k 1 )u s1 (p 1 ), (4.48) we can already derive some Feynman rules for QED by comparing the expression with the corresponding Feynman diagram. For example the spinor u s1 (p 1 ) is connected to the momentum p 1 and spin s 1 of the incoming electron. Therefore we can conclude u s1 (p 1 ) incoming e ε λ1 µ(k 1 ) incoming γ ieγ µ vertex ū s2 (p 2 ) outgoing electron First Order Feynman Rules and Diagrams In figure 4.3 all the 8 first order diagrams corresponding to the 8 terms (1)-(8) are shown. By analyzing them we can gain additional Feynman rules for incoming/outgoing positrons and photons, so that at this moment we can summerize the QED Feynman rules as follows: 1. Vertex: ieγ µ and conservation of the 4-momentum 2. Incoming electron: u s (p) Outgoing electron: ū s (p) 42

49 4.5 Wick s Theorem 3. Incoming positron: v s (p) Outgoing positron: v s (p) 4. Incoming photon: ε λµ (k) Outgoing photon: ε λµ(k) Figure 4.3: 1st order Feynman diagrams [1] 4.5 Wick s Theorem To calculate processes of nth order, we need to find an expression for the time ordered product of Hamilton densities occuring in the nth order scattering matrix S (n) = ( i)n n! dt 1 dt 2 dt n T (H I (t 1 )H I (t 2 )... H I (t n )). (4.49) This can be done with the help of Wick s theorem, which states, that an arbitrary time ordered product can be expanded into a sum of normal ordered products. To get to this point we need to define the contractions between two operators A and B: AB T (AB) : AB :. (4.50) 43

50 4 Feynman Diagrams and Rules for QED Since in QED we are only dealing with field operators, where the commutators/anticommutators are a complex number, the contraction between field operators is also a complex number. Hence the vacuum expectation value is 0 AB 0 = AB = 0 T (AB) 0, (4.51) which (except for an i) is euqal to the definition of the Feynman propagator (see (4.8) and (4.10)). the important contractions in QED are ψ α (x 1 ) ψ β (x 2 ) = ψ β (x 2 )ψ α (x 1 ) = is F,αβ (x 1 x 2 ) (4.52) A µ (x 1 )A ν (x 2 ) = id µν F (x 1 x 2 ), (4.53) while a lot of contractions just give zero ψ(x 1 )ψ(x 2 ) = ψ(x 1 ) ψ(x 2 ) = 0 (4.54) ψ α (x 1 )A µ (x 2 ) = 0 (4.55).... (4.56) Now we can start to write down Wick s 1st theorem. 1st Theorem: The time ordered product T (A 1 A 2... A n ) of n field operators can be expanded into a sum of their normal ordered products, where we sum over all possible contractions As an example we can take a look at the time ordered product of 4 field operators A, B, C and D: T (ABCD) =: ABCD : (4.57) +: ABCD : +: ABCD : +: ABCD : +... (4.58) +: ABCD : +: ABCD : +: ABCD : (4.59) For the time ordered product of partly normal ordered products as it is the case for the nth order S-matrix, one can use Wick s 2nd theorem: 44

51 4.6 Second Order Terms and Feynman Rules for QED 2nd Theorem: A time ordered product of field operators, which are partly normal ordered, can be expanded into a sum of their normal ordered product as Theorem 1 states. However contractions between operators within the same normal ordered factor, don t appear in the sum. As an example we apply Wick s theorem again to the time ordered product of 4 field operators A, B, C and D, where B, C and D are now normal ordered: T (A: BCD : ) =: ABCD : +: ABCD : +: ABCD : +: ABCD :. (4.60) 4.6 Second Order Terms and Feynman Rules for QED Application of Wick s Theorem to the S-Matrix of 2nd Order With Wick s second theorem it s now possible to expand the time ordered Hamilton densities occuring in the S-matrix of higher order. Exemplary we will apply Wick s theorem to the second order S-matrix S (2) = ( ie)2 2 d 4 x 1 d 4 x 2 T (H I (x 1 )H I (x 2 )). (4.61) We can expand the time ordered product T (H I (x 1 )H I (x 2 )) into 8 terms. Note that contractions between commutating field operators are zero (see (4.56)): T (H I (x 1 )H I (x 2 )) = T (: ψ(x 1 ) /A(x 1 )ψ(x 1 ): : ψ(x 2 ) /A(x 2 )ψ(x 2 ): ) (4.62) =: ψ /Aψ ψ /Aψ : (4.63) +: ψ /Aψ ψ /Aψ : +: ψ /Aψ ψ /Aψ : +: ψ /Aψ ψ /Aψ : (4.64) +: ψ /Aψ ψ /Aψ : +: ψ /Aψ ψ /Aψ : +: ψ /Aψ ψ /Aψ : (4.65) +: ψ /Aψ ψ /Aψ :. (4.66) 45

52 4 Feynman Diagrams and Rules for QED Compton Scattering Exemplary we can examine the second order process of Compton scattering, where a photon gets sacttered by a free electron e (p) + γ(k) e (p ) + γ(k ). (4.67) Considering this initial and final state only the two terms : ψ(x 1 )γ µ A µ (x 1 )ψ(x 1 ) ψ(x 2 )γ µ A µ (x 2 )ψ(x 2 ):, (4.68) : ψ(x 1 )γ µ A µ (x 1 )ψ(x 1 ) ψ(x 2 )γ µ A µ (x 2 )ψ(x 2 ): (4.69) of equation (4.66) contribute to the transition amplitude f S (2) i. One electromagnetic fied operator is needed to annihilate the incoming photon and one is needed to create the outgoing photon. The same holds for the Dirac field operators where ψ annihilates the incoming electron and ψ creates the outgoing electron. Calculating the transition amplitude one ends up with two possible processes A and B, described by the two following Feynman diagrams: Figure 4.4: The two possibel processes A and B describing Compton scattering. In process A the incoming photon gets absorbed by the electron and then is emitted again at a later time. Whereas in process B first the outgoing photon is emitted by the electron and then at a later time the incoming photon is absorbed. The corresponding invariant amplitude M = M A + M B is the sum of the two invariant amplitudes for each process A and B M A = ū s (p )ε λ ν(k )( ieγ ν i ) /p + /k m ( ieγµ )ε λµ (k)u s (p), (4.70) 46

53 4.6 Second Order Terms and Feynman Rules for QED M B = ū s (p )ε λµ (k)( ieγ µ i ) /p /k m ( ieγν )ε λ ν(k )u s (p). (4.71) Analysing the expression of M A, we see that the process between the point where the electron absorbes the incoming photon and the point where it emits the outgoing photon (in the Feynman diagram described by the so called inner fermion line), is described by the Feynman propagator for fermions (in momentum space) S F (p + k) = 1 /p + /k m + i0 +. (4.72) The interpretation of that is, that we have a virtual fermion propagating with 4-momentum p + k. It s called a virtual fermion because it is not measurable and doesn t have a Dispersion relation as free fermions (As seen earlier, with the dispersion relation for real fermions the condition of conserved 4-momentum cannot be fulfilled). Further more we verify Feynman rule number 1, the 4-momentum is conserved at every vertex, since we have the feynman propagator with a momentum p + k. By analyzing the other possible second order processes we also find that the Feynman propagator for photons D F,µν corresponds to an inner photon line. We can formulate two additional Feynman rules 5. Inner fermion line: is F (p) = i /p m+i Inner photon line: id F,µν = ig µν k 2 +i0 +. Additionally one gets a δ (4) (P i P f )-function for all process of every order, which means that for all processes the total 4-momentum before and after the scattering process is conserved: 7. Total 4-momentum before and after the scattering process is conserved th Order Loop Diagramm for Electron-Positron Scattering In a final step we will apply the gained Feynman rules to the 4th order loop diagram 4.5 as one of 18 possible 4th order diagrams describing the scattering 47

54 4 Feynman Diagrams and Rules for QED (a) Inner fermion line: is F (p) (b) Inner photon line: id F,µν (k) process e (p 1 ) + e + (p 2 ) = e (p 1 ) + e + (p 2 ) (4.73) Figure 4.5: One of 18 possible Feynman diagrams describing electron-positron scattering. The time axis goes from left to right. [6] As can be seen, in the loop diagramm we have an undefined momentum q which can take any value. To account for all possibilities we therefore have to integrate over q. amplitude Using the previous gained Feynman rules we can derive the invariant M = d 4 q ig νµ ig ρσ (2π) 4 (p 1 q) 2 + i0 + (q + p 2 ) 2 + i0 + i /q m + i0 + ( ieγν )u s1 (p 1 ) (4.74) i /p 1 /q /p 2 m + i0 + ( ieγµ )v s 2 (p 2 ) v s2 (p 2 )( ieγ ρ ) ū s 1 (p 1 )( ieγ σ ) 48

55 4.6 Second Order Terms and Feynman Rules for QED This chapter is mainly based on chapter 15 of the book by Schwabl [1]. Additionally [2] and the script [6] was used. 49

56

57 Chapter 5 Path Integrals I 5.1 Mathematical background In this subchapter we will deal with the mathematical basics, which will be very helpful for the further understanding of the topic of the path integrals in quantum field theory. In the beginning the basic Gaussian integral in one dimension will be considered. This would be the equation, to which all further integrals will be related. The idea for evaluating the equation is to square the integral and switch to polar coordinates. In the end the square root over the result will be taken. This yields dx e 1 2 ax2 = 2π a. (5.1) Extending equation 5.1 with an additional linear function in the exponent brings the equation dx e 1 2 ax2 +Jx = 2π a e J 2 /2a. (5.2) It has been solved by completing the square in the exponent on the left side. In the next step equation 5.2 will be considered in higher dimension. Translating the single quantities to dim= D would mean scalar a matrix A variable x vector x scalar J vector J 51

58 5 Path Integrals I Considering this we gain an extended version of the Gaussian integral dx 1 dx 2... dx D e 1 2 xt A x+j T x = ( (2π) D det[a] ) e JT A 1 J 2 (5.3) Problem 5.3 has been solved by diagonalizing A by an orthogonal transformation O so that A = O 1 D O with D a diagonal matrix of A. Now the integral can be evaluated by rotating the coordinate system and integrating over the single coordinates. Considering that in the low dimensional problem, in the solution the scalar a was in the denominator, in the high dimension the result would contain A 1, or the inverse of the matrix. Analogously the prefactor contains the determinant of the matrix, because we deal with its diagonalized version and we integrate over every coordinate. The expectation value of two coordinates is given by < x i x j >= dx 1 dx 2... dx N e 1 2 x.a.x x i x j = (A 1 )... + ij. (5.4) dx 1 dx 2... dx N e 1 2 x.a.x This has been gained by differentiating with respect to J i and setting J = 0. The result gives the entries of the inverse matrix A 1. These factors are related to the so called Wick contractions. Extending the expectation value 5.4 to more than two coordinates yields < x i x j... x u x v >= permutations < x a x b > < x s x t >. (5.5) This means that in order to evaluate 5.5 one should consider all the possibilities for connecting every value with the rest of the x s and sum over all permutations. 5.2 Introduction The idea of using path integrals in quantum field theory originates from Feynman. However it was Dirac, who firstly developed this calculation. In the beginning we consider the double-slit experiment as shown in figure 5.1, which illustrates the fundamental property of quantum mechanics, namely the 52

59 5.2 Introduction Figure 5.1: Schematic representation of the double slit experiment. Aim is to illustrate the fundamental property of quantum mechanics the superposition principle. [7] superposition principle. The point S in the figure denotes a light source. A particle emitted at S goes through one of the two holes drilled on a screen A 1 or A 2. By the time t = T it is being detected at a detector, placed at O. The amplitude for detection is given by the sum over the amplitudes, that a particle is emitted, then goes through A 1 and then to O and a particle is emitted, goes through A 2 and is detected at O. Analogously the detection amplitude for more then two holes in the screen is the sum over the single amplitudes emission at S, propagation through one of the drills A i and detection at O. Here the logical question arises: is there a limitation in the number of screens and in the number of holes, drilled in them. And the answer of course is no. Considering this we take a look at figure 5.2. It represents an extension of the double-slit problem with two screens and respectively 3 and 4 slits drilled in them. If we enumerate the number of screens and the number of holes, the resulting propagation amplitude A can be written as A(detected at O) = ij... A(S A i B j O). Letting the number of screens and holes go to infinity, the result would be a sum over all paths, starting at the source S and ending at the detector O, as implied at figure 5.3. Quantum mechanically the amplitude for a propagation from an initial point q i (or the light source S) to the final point q f (the detector O) in the time T is given 53

60 5 Path Integrals I Figure 5.2: An extension of the double-slit-problem, presented on figure 5.1. A second screen is added with more holes drilled in them. [7] Figure 5.3: After letting the number of the screens between the light source S & the detector O and the number of the holes in them go to infinity, the detection amplitude can be calculated by considering all the paths between S and O. [7] 54

61 5.3 Derivation of the path integral notation by the time evolution operator. The mathematical expression is q f e iĥt q i = A. (5.6) Here it becomes obvious that the exact probability amplitude depends on the Hamiltonian of the system. The explicit calculation for a potential-free picture will be given in the next chapter. The mathematical evaluation for a Hamilton operator with an external potential source goes analogously. 5.3 Derivation of the path integral notation The aim is to calculate the probability amplitude for a propagation in space from the initial point q i to a final point q f. It is possible to do this calculation in one dimensional quantum mechanics and to translate the result into quantum field theory afterwards. As already mentioned, in quantum mechanics a propagation in space can be described by the time evolution operator U(t, t ) = e iĥt with Ĥ the Hamiltonian of the system and T a time interval. By knowing the starting and end positions of a wave function and setting the time interval T for the propagation, the probability amplitude is given by calculating A = q f e iĥt q i. (5.7) The basic idea for evaluating the probability amplitude problem is to take a single path from the sum and determine its contribution to A. The calculation for a single path can be done by dividing its amplitude in a finite number of slices, do the calculation for every slice and then sum over all of them. Graphically this is demonstrated on figure 5.4. Mathematically the slice division can be done by splitting the time T in N finite small time intervals δt. In order to gain the approximation in the end, the δt s will become infinite small and their number N will be send to infinity. q f e iht q i = q f e} ihδt e ihδt {{... e ihδt } q i (5.8) N times 55

62 5 Path Integrals I Figure 5.4: A single path from S to O, divided in smaller path-slices the basis for the calculation of the probability amplitude. [7] Now make use of the fact that the position eigenstates form a complete set of states. Mathematically this can be written as dq q q = 1. Inserting a factor 1 between the exponentials in equation 5.8 brings q f e iht q i = N 1 dq j q f e ihδt q N 1 q N 1 e ihδt q N 2... q 2 e ihδt q 1 q 1 e ihδt q i. j=1 (5.9) For the sake of simplicity observe a single factor q j+1 e iĥδt q j. The Hamilton operator for the system is generally given by Ĥ = ˆp2 + V (ˆq), so that 2m q j+1 e iĥδt ˆp2 i( q j = q j+1 e 2m +V (ˆq))δt q j. (5.10) The following calculation will be done explicitly for a free particle. For the case of a potential energy with the general form V (ˆq) the calculation is analogous. The problem with equation 5.10 is that the momentum operator ˆp should be applied on the state q j, which is not an eigenstate. This problem is solved by inserting dp 2π p p = 1 and using the relation q p = eipq. Now the operator will be applied on the eigenstate, from which the eigenvalue is known: ˆp p = p p. ˆp2 iδt q j+1 e 2m qj = = = dp 2π q j+1 e iδt ˆp 2 2m p p qj dp p 2 2m qj+1 p p q j 2π e iδt dp p 2 2π e iδt 2m e ip(q j+1 q j ) 56

63 5.3 Derivation of the path integral notation The equation that has been gained is a Gaussian integral with an extension function. Its evaluation has been already discussed. The solution can be written without any further problems. ˆp2 iδt im q j+1 e 2m qj = 2πδt e[im(q j+1 q j )2 ]/2δt im m = 2πδt eiδt 2 [(q j+1 q j )/δt] 2 Plugging this result in equation 5.9 yields ( q f e iĥt q i = im ) N/2 ( N 1 ) dq k exp iδt m 2πδt k=1 2 N 2 j=0 [(q j+1 q j )/δt] 2 (5.11) Going to the continuum limit would mean to make the time slices infinitely small (δt 0) and their number big (N ). This would also transform the sum into an integral and [(q j+1 q j )/δt] 2 into time derivative q 2. Finally the definition for the path integral can be written as ( Dq(t) = lim im ) N 2 N 2πδt ( N 1 k=1 dq k ) (5.12) and with this the path integral notation for the propagation amplitude becomes ( q f e iht q i = Dq(t) exp i T 0 dt ( 1 2 m q2 V (q)) ). (5.13) Result 5.13 has been written taking into account the potential energy V (q) in the Hamiltonian, which has been neglected in the performed calculation. Taking a precise look on the exponent of the gained equation, one would recognize the Lagrangian function, known from the classical mechanics 1m 2 q2 V (q) = L. Integrating with respect to time over L would bring the action. With this the final result for the path integral Z in one dimensional quantum mechanics is Z = Dq(t) e is[q(t)]. (5.14) Equation 5.14 means that in order to determine the propagation amplitude from an initial point q i to the final point q f in time T one should concider all possible 57

64 5 Path Integrals I paths between these two points, sum over them and then weight every path with an exponential factor, depending on the action for this single path. In the next step we extend our problem to higher dimensions and translate the gained result for quantum field theory. In dim= D the vector q has D components, which can be labeled with the indices a, so that q q a. This system represents an interactive picture between a particles, which means that the energy should be the sum over the energies for every single particle 1m 2 q2 a 1m 2 aq 2 a. Going to the continuum limit would mean to make the distance between the a s infinitely small, which would transform the label a into a vector x, the sum over a into a spacial integral and the vector q a into the field φ( x, t), the new dynamical variable of the system. The variable x specifies which field variable are we corresponding to. In the continuum limit the sum over all a coordinates turns out to be a spacial integral over the action, or the Lagrange density dt a L(q a, t) dt d D 1 xl(φ, t). Doing the translation from the Lagrangian into Lagrange density we can use any of the theories mentioned in the previous chapters. Here we define a scalar field theory, describing particles without spin (or Bosons). Doing so we gain the exact form of the Lagrangue density: T ( dt ( 1 ) 0 a 2 m a q a 2 V [q a ]) ( 1 dt d D 1 x 2 (( φ)2 m 2 φ 2 ) V [φ]). In the next chapters we will see what other possibilities for the Lagrange density we have. To sum, in order to translate the result for the path integral notation we derived for dim= 1, one should apply table 5.1. Equation 5.14, translated in quantum field theory for Bosons takes the form Z = Dφ e i d D x[ 1 2 (( φ)2 m 2 φ 2 ) V (φ)]. (5.15) 58

65 5.4 Solution for the free field D-dimensional quantum mechanics Quantum field theory q φ a x q a (t) φ( x, t) a d D x dt a L(q a, t) dt d D 1 xl(φ, t) Table 5.1: The translation from quantum mechanics to quantum field theory. 5.4 Solution for the free field Generally speaking, the path integral in quantum field theory allows us to describe the creation and annihilation of particles in a field, we ve defined it for. However, equation 5.15 we gained in the previous chapter includes only a potential energy, acting on the states, which is not sufficient for the description of such events. In order to be able to describe some more complex processes, one should add an additional function to the path integral. This function would take the role of a disturbing factor to our system, which allows gaining probability amplitudes or energies different then the ground state energy. Let s call the disturbing function J(φ). It would be a function, acting on all a particles in the D-dimensional quantum mechanics. Translating this in quantum field theory we get an additional factor in the exponent and equation Z = Dφ e i d D x[ 1 2 (( φ)2 m 2 φ 2 ) V (φ)]. turns to Z = Dφ e i d 4 x{ 1 2 [( φ)2 m 2 φ 2 ] V (φ)+j(x)φ(x)}. (5.16) The path integral for a system with external sources and sinks (or generally called sources) could describe any propagation in the space we are working in and it is the tool to calculate any creation or annihilation process for Bosons. This makes it quite complicated to be evaluated exactly, so the first step would be consider a simplified version of equation 5.16 and then extend it to a form, more similar to the reality. 59

66 5 Path Integrals I Free field theory The first idea to simplify equation 5.16 is to consider it for the case of a free particle. This implies a potential energy equal to zero. In the next step 5.16 is being partially integrated with respect to space, which brings the following result Z = Dφ e i d 4 x{ 1 2 φ( 2 +m 2 )φ+jφ} where the Klein Grodon equation in the exponent is obvious. (5.17) Writing 5.17 in this form has the advantage, that the Gaussian form of the integral becomes clear and with this the solution can be directly deduced. The result of Gaussian integral of this type reads as follows dx 1 dx 2... dx D e 1 2 xt A x+j T x = ( (2π) N det[a] ) e 2 JT A 1 J. (5.18) On the right side of equation 5.18 in the exponent the inverse of the matrix A is multiplied by the vectors J T and J. In the path integral notation the matrix is being replaced by the differential operator ( 2 + m 2 ) and knowing the characteristic equation for an inverse matrix A ij A 1 jk = δ ik, where A ij and A jk are matrix entries, we see that the solution should be obtained by solving ( 2 + m 2 )D(x y) = δ 4 (x y). In the continuum limit the δ ik turns into a delta function δ 4 (x y). The solution of such a differential equation is known from the electrodynamics and is very similar to the so called Green s function. In the quantum field theory it plays the essential role of a propagator and mathematically it takes the form D(x y) = d 4 k e ik(x y) (2π) 4 k 2 m 2 + iϵ. (5.19) Here it is important to mention, that the result 5.19 for the propagator was gained from the solution for the free field theory. Later a more general expression will be obtained. After finding an expression for A 1 the complete solution of equation 5.17 in 60

67 5.5 φ 4 -Theory terms of quantum field theory can be written. It has the form Z(J) = Ce i 2 d 4 x d 4 y J(x)D(x y)j(y). (5.20) Here the overall factor C is given from C = Z(J = 0). At this point the power of the path integral notation becomes visible. Just by doing Gaussian integrals and deliberating about the meaning of their results for the quantum field theory, one obtains expressions, which previously were derived from tedious calculations. In the upcoming chapters we will continue making use of this and will obtain results of more sophisticated expressions. 5.5 φ 4 -Theory Now the model, described in chapter 5.4 will be extended by introducing an interaction between the particles. An interacting anharmonic term will be added to the Lagrangian, however its form should be chosen carefully. On the one hand the aim is to be able to solve the resulting path integral. On the other hand, to create a theory, complex enough, so that it can describe real scattering events. A term in the order four is added and the resulting path integral has the form Z(J) = Dφ e i d 4 x{ 1 2 [( φ)2 m 2 φ 2 ] λ 4! φ4 +Jφ. (5.21) The theory, which deals with finding a solution of 5.21 and interpreting the physical events corresponding to it, is the so called φ 4 -theory. Our approach for dealing with it will be similar to what we did with the free field. Firstly we will consider an extremely simplified version of 5.21 and after this we will consider what can extend the gained result to the quantum field. 61

68 5 Path Integrals I Simplified solution of the φ 4 -Problem Let s evaluate the integral Z(J) = + dq e 1 2 m2 q 2 λ 4! q4 +Jq. (5.22) Equation 5.22 is a 0 dimensional time independent version of equation In order to bring this expression to a more familiar form, the term λ 4! q4 should be removed from the exponent. The most convenient way to do so is to write e λ 4! q4 Z(J) = + dq e 1 2 m2 q 2 +Jq [ in its power series. 1 λ 4! q ( λ 4! )2 q ]. (5.23) In the next step q 4 can be written as a 4 n -order derivative of e 1 2 m2 q 2 +Jq with respect to J: Z(J) = ( 1 λ d 4 4! dj ! ( λ ) + d8 )2 4! dj +... dq e m2 q 2 +Jq. (5.24) The power series in J does not depend explicitly on q anymore, so it has been taken out of the integral. What remains is the extended Gaussian integral, which solution can be written by comparing the terms with the familiar equation 5.2 from the mathematical background. Considering that the power series in J can be also written as an exponential function, the result of 5.22 is Z(J) = e λ 4! d 4 dj 4 e 1 2m 2 J 2 (5.25) with Z = Z(J) Z(J=0,λ=0). In order to understand the meaning of the derived equation let s calculate the term of order λ and J 4 from equation For this we write Z(J) in its power series: Z(J) = [ d 4 1 λ 4! dj ! λ 2 4! d 8 ] [ dj J 2 8 J 4 2m ! (2m 2 ) ]. (5.26) 62

69 5.5 φ 4 -Theory The term, which is in the first order of λ is λ. It carries the fourth derivative 4! dj 4 with respect to J with it, which should be considered for the calculation of J 4. Now which term in equation 5.26 has J 4 after differentiating it four times with respect to J? (J 2 ) 4 This is 1. Now writhing these terms together we gain 4! (2m 2 ) 4 [ λ d 4 ] [ 1 (J 2 ) 4 ]. = λ 1 1 8! 4! dj 4 4! (2m 2 ) 4 4! 4! (2m 2 ) 4 4! J 4. (5.27) Thinking in which connection the variables J and λ were introduced, the general meaning of the calculated term can be deduces. The function J was introduced in chapter 5.4 and its task was to describe the positions in the quantum field, where a particle can be created or annihilated the sources of the field. The variable λ plays the role of coefficient in the interacting φ 4 -Theory, so it should describe some sort of an interaction between the particles. So the solution of the simplified path integral for an interacting theory (equation 5.22) should provide us with terms, which describe the creation and annihilation of particles in the quantum field φ and their interactions, the so called scattering events. What is still missing, is how the propagation from the source of the particle to the spot of interaction and then to the sink where it vanishes can be mathematically described with the results we gained till now. The mathematical tool for this process we have already deduced in chapter 5.4 this was the so called Feynman Propagator. Being in dim= 0 expression 5.19 for the propagator collapses to 1/m 2. In equation 5.27 the term 1/m 2 is found to be to the power of 4. d 4 63

70 5 Path Integrals I Figure 5.5: The Feynman diagrams for the first order of λ and fourth order in J and 1/m 2. The quantity iλ denotes the scattering amplitude. [7] We sum up: λ Vertex 1 Propagator m 2 J external source/sink A convenient way to deal with the scattering events is to illustrate them. This was done by Richard Feynman and the result are the so called Feynman diagrams. For the example we have explicitly calculated, we have one vertex (λ), four sources (J) and four propagators (1/m 2 ) or lines. The resulting diagrams are given in figure 5.5. As one can see on the graphic, for one set of variables, more then one Feynman diagram can be written. The diagrams shown in (b) and (c) are contained in the term Z(0, 0) by which we divide the the solution Simplified solution of the φ 4 -Problem II The simplified 0-dimensional equation 5.22, solved in section 5.5.1, can be evaluated also by expanding it in powers of J, analogous to the power expansion in λ. This would bring Z(J) = s=0 1 s! J s dq e 1 2 m2 q 2 λ 4! q4 q s Z(0, 0) s=0 1 s! J s G (s) (5.28) with G (s) = dq e 1 2 m2 q 2 λ 4! q4 q s. (5.29) 64

71 5.5 φ 4 -Theory The coefficient G (s) can be also written as a power series of λ and its analogous is known as a Green s function. However the meaning of this observation will become obvious in the next subchapter The high-dimensional propagator In the next step of the evaluation of the φ 4 -Theory, we would consider the equation 5.22 we started with, in higher dimension. Z(J) = It can be expanded in powers of J, which leads to N N 1 Z(J) = s=0 i 1 =1 i s =1 s! J i 1... J is dq 1 dq 2... dq D e 1 2 qt A q λ 4! q4 +J T q ( N N 1 = s=0 i 1 =1 i s=1 s! J i 1... J is G (s) i 1...i s. l dq l ) e 1 2 qt A q λ 4! q4 q i1... q is With this result, the D-dimensional Feynman propagator can be written as G (s) i 1...i s = ( ) N N dq l e 1 2 qt A q λ 4! q4 q i1... q is (5.30) i 1 =1 i s =1 l and the coefficients G (s) i 1...i s are Wick contractions. The latter have been discussed in the mathematical background. Due to the fact that this time we are dealing with path integrals of higher dimension, an important property emerges from the last equation the propagation. In order to convince ourselves in the identity Wick coefficients propagation, we calculate the the 2-point Green s function G (2) ij. G (2) ij (λ = 0) = 1 Z(0, 0) ( l dq l ) e 1 2 qt A q q i q j = (A 1 ) ij (5.31) The inverse matrix element (A 1 ) ij describes a propagation starting at the spacial point i and ending in j. 65

72 5 Path Integrals I The coefficient G (4) ijkl is the 4-point Green s function and its calculation is identical to the one for 2 points. The order of λ we are interested in is one. G (2) ijkl = 1 Z(0, 0) ( m ) [ dq m e 1 2 q.a.q q i q j q k q l 1 λ ] qn 4 + O(λ 2 ) 4! = (A 1 ) ij (A 1 ) kl + (A 1 ) ik (A 1 ) jl + (A 1 ) il (A 1 ) jk λ n (A 1 ) in (A 1 ) jn (A 1 ) kn (A 1 ) in + + O(λ 2 ). n The interpretation of this result is the following: the first three terms stand for a propagation from the point i, ending at point j and another one from k to l and the rest are the possible permutations. The term with λ describes the interaction between the four excitations. They propagate from their source i, j, k or l to the point n and interact with the amplitude λ. The sum over n denotes that this point can be anywhere in the space Perturbative Field Theory The most interesting part of course is when we translate every result in the quantum field theory and interpret its meaning. This is what we are going to do in this chapter. The equation in the φ 4 -Theory we want to evaluate is Z(J) = Dφ e i d 4 x { 1 2 [( φ)2 m 2 φ 2 ] λ 4! φ4 +Jφ}. (5.32) Analogously to the high dimensional problem discussed in chapter 5.5.3, expanding it in J and solving the Gaussian integral would bring Z(J) = s=0 i s s! =Z(0, 0) s=0 dx 1... dx s J(x 1 )... J(x s ) i s s! dx 1... dx s J(x 1 )... J(x s ) G (s) (x 1... x s ) and the 4-point Greens function G (4) would be written as Dφ e i d 4 x { 1 2 [( φ)2 m 2 φ 2 ] λ 4! φ4} φ(x 1 )... φ(x s ) G (4) (x 1, x 2, x 3, x 4 ) 1 Z(0, 0) Dφe i d 4 x { 1 2 [( φ)2 m 2 φ 2 ] λ 4! φ4} φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 ). (5.33) 66

73 5.5 φ 4 -Theory Figure 5.6: Scattering of two bosons emerging at x 1 and x 2, interacting at w and propagating to their sinks at x 3 and x 4 [7] Now we want to compare the results we gained from the simplified version of the interacting path integral with the quantum field theory result. For this we are interested in extracting the first order in λ from equation This would bring G(x 1, x 2, x 3, x 4 ) = 1 ( iλ ) d 4 w Z(0, 0) 4! φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 )φ(w) 4 Dφ e i d 4 x { 1 2 φ( 2 +m 2 )φ} Evaluating this integral brings G(x 1, x 2, x 3, x 4 ) ( iλ) d 4 w D(x 1 w)d(x 2 w)d(x 3 w)d(x 4 w). (5.34) Here some terms have been neglected. Some of them come from the integration and others originate form the term φ(w) 4 and its contraction with itself.// From the Gaussian integration there are a lot of factors coming to the result, however they don t bring any physical information along, so for simplicity they will be neglected. The result 5.34 is graphically represented on 5.6. Two bosons emerge in the spacial points x 1 and x 2 and propagate to the point w with the amplitude D(x 1 w)d(x 2 w). There they scatter with the amplitude iλ and propagate from w with D(x 3 w)d(x 4 w) to the sinks at x 3 and x 4. The integration over w denotes that it has being chosen arbitrary in the space and the scattering process can take place everywhere. With this we have deduced an approach for the evaluation of scattering processes 67

74 5 Path Integrals I in the quantum field theory. It becomes obvious, that writing the process in the form of a diagram is a very convenient way to illustrate such an event. 5.6 Summary In this chapter we have seen that some very important QFT results can be deduced by evaluating path integrals. The exact derivation of the latter was shown in ordinary quantum mechanics and afterwards it was translated to quantum field theory with the help of the Lagrangian density L. It is very interesting that for most of the calculations mainly Gaussian integrals were used. The path integral notation can also be used for the evaluation of fermionic fields and gauge theories, which will be shown in the next chapters. 68

75 Chapter 6 Path integrals II: Fermions and Gauge Fields 6.1 Motivation As was discussed in the previous chapter, the mathematical concept of path integrals provides a different access to quantum field theory which is equivalent to the canonical formalism. Many problems can be discussed more illustratively within the framework of the path integral formalism. Furthermore it extends our toolbox for solving such problems. The generating functional of the path integral formalism is defined as Z(J) = Dφ e i d 4 x (L(φ)+Jφ). (6.1) In this definition, φ stands for an arbitrary field φ(x) defined on the four dimensional spacetime. The Lagrangian density L(φ) determines the physical behavior of the fields and its integral over the four-dimensional space-time S[φ] = d 4 x L(φ) is called action. The above path integral stands for the summation of the phase-factors in the integrand over the whole space of configurations φ(x) and may be interpreted as the amplitude for the evolution of the field under the presence of a given source or sink term J(x). A central result of the previous chapter was the path integral for a bosonic, noninteracting and scalar field theory. For such a theory the action integral S is a bilinear form. It can therefore be represented as the continuum limit of the discrete vector-notation S = 1φ 2 ik ij φ j =: 1 φ Kφ. In this notation, (6.1) becomes 2 69

76 6 Path integrals II: Fermions and Gauge Fields a Gaussian integral that can be evaluated as [7] Z(J) = Dφ e i( 1 2 φ Kφ+J φ) = C det[k] e (i/2)j K 1J. (6.2) This simple formula already shows a subtlety that has up to now been neglected: The operator K generating the bilinear form might be singular (i.e. have 0 as an eigenvalue). Since such a matrix is not invertible, the right hand side of (6.2) is no longer defined. As will be demonstrated in section 6.3, this problem does indeed arise in the naïve ansatz for the path integral of the electromagnetic field. The formal solution to this problem leads us to a general procedure that can be applied to arbitrary gauge fields. The other topic of this chapter deals with the application of the path integral formalism to fermionic fields. In (6.1) there has been no distinction between bosons and fermions, but we know from the canonical formalism that there are crucial differences between these types of particles. A formulation of the path integral that accounts for the anticommutating character of fermionic field operators will be presented in section 6.2. Once these two problems are solved, the whole theory of quantum electrodynamics can be formulated within the path integral formalism. Section 6.4 will give an outlook to this important application. 6.2 Path integrals for fermions Formal motivation In order to work out the path integral formalism for fermionic fields we consider the relation between the generating functional (6.1) and the canonical notion of transition amplitudes as described in ref. [8, chap. 43]. (For an alternative motivation for the need of a different formalism for fermions, see ref. [7, chap. II.5], where the negative vacuum energy for fermionic fields is discussed.) Green s functions for scalar fields φ(x) are in the canonical formalism expressed via time ordered products of field operators acting on the vacuum. An equivalent 70

77 6.2 Path integrals for fermions quantity can be found in the path integral formalism as 0 Tφ(x 1 )φ(x 2 )... 0 = 1 δ 1 δ i δj(x 1 ) i δj(x 2 )... Z(J) J=0. (6.3) Therein T stands for the time-ordering operator; the field operators φ(x i ) take four-dimensional arguments 1. On the right hand side, the functional derivative δ/δj(x i ) shows up, which has to be considered as the continuous generalization of the partial derivative / J i in the vector notation in expressions like (6.2). The formal definition of the functional derivative is given by [8, chap. 7] δ δf(t 1 ) f(t 2) = δ(t 1 t 2 ) (6.4) for arbitrary functions f(t). From eq. (6.2) one can easily see, that (6.3) reproduces indeed the free propagator K 1 as two point correlation function. We now wish to calculate correlation functions also for fermionic fields. For example the Feynman propagator requires the calculation of 2 S(x y) := i 0 Tψ(x) ψ(y) 0. (6.5) We would like to represent this propagator also in the form of functional derivatives acting on a path integral as in eq. (6.3). However, we know from the anticommutation relations of the fermionic field operators ψ and ψ, that Tψ(x) ψ(y) = T ψ(y)ψ(x). (6.6) Since partial derivatives always commute in ordinary calculus, the analogy of eq. (6.3) will not reproduce this property. However, we can use a different algebraic structure for the description of fields and source-terms. It will be defined in a way that ensures the required anticommutating behavior. 1 The link to the time-independent definition of the field operators in the canonical formalism is given by time evolution operators connecting the field operators, which are notationally suppressed in eq. (6.3). 2 Spinor indices have been omitted for a clearer notation. 71

78 6 Path integrals II: Fermions and Gauge Fields The Grassmann algebra An algebraic structure that exhibits exactly the properties required for the description of fermionic fields was invented by Grassmann in the 19 th century 3. The defining property of two arbitrary Grassmann (or anticommutating ) numbers η, ξ is [7, chap. II.5] ηξ = ξη. (6.7) One can immediately see that any Grassmann number η to its square vanishes, i.e. η 2 = 0. This property is especially useful for the calculation of Taylor expansions of functions of Grassmann numbers: All the powers greater than one vanish. Thus we can write the most general analytic function of a single Grassmann variable as f(η) = a + bη (e.g. e η = 1 + η). (6.8) Ordinary numbers are defined to commute with Grassmann numbers. As products ηξ of two Grassmann numbers do also commute with any Grassmann number χ χ(ηξ) = ηχξ = (ηξ)χ, (6.9) they are equally considered as ordinary numbers. We also want to differentiate functions with respect to Grassmann numbers. For this purpose, differential operators are set up which are defined to be anticommutating with both Grassmann numbers and other differential operators of this type. Some examples for clarification (Greek letters: Grassmann numbers, Latin letters: ordinary numbers): ξ (a + bξ) = b ξ ξ = b, ξ (a + ηξ) = η ξ ξ = η η ξ ηξ = ξ η ηξ = ξ ξ = 1. (6.10) 3 A big part of the foundations of vector calculus we are using today is due to Hermann Grassmann, but - having only been an ordinary teacher - he is very little known for it. Of course Grassmann was not worrying about fermionic fields and path integrals: Anticommutating numbers also arise as a generalization of the vector product a b = b a. 72

79 6.2 Path integrals for fermions We now also want to do integrals with Grassmann numbers. For this purpose it is sufficient to introduce two integration rules: dη := 0, (6.11) dη η := 1. (6.12) Note that no specific integration ranges are given. Grassmann integrals always run over the whole Grassmann algebra. Above rules emerge quite naturally from the postulation of familiar properties of integrals: The value of the integral of an arbitrary function f(η) should not be affected by a shift in the integration variable.. Using the most general function (6.8) we find dη f(η + ξ) = dη(a + bη + bξ) dηf(η) = dη(a + bη + bξ) bξ dη = 0.! = This can only be satisfied if (6.11) holds. The second integration rule arises from the notion that the product dη η and thus the integral should be an ordinary number. This number can in principle be chosen freely (of course it should not be zero, otherwise every integral would give zero), the choice 1 is the most convenient one. Gaussian integrals play a central role in QFT, so we will discuss their calculation for Grassmann variables explicitly. Since squared Grassmann numbers give zero, we specify the exponent in a Gaussian function as a bilinear form of two independent variables η and η. The one-dimensional Gaussian integral is now calculated as dη d η e ηaη = dη d η (1 + ηaη) = dη aη = a. (6.13) Note that a can be any real number (including a < 0 in contrast to the case of ordinary Gaussian integrals). calculate dη 1 d η 1... dη N d η N e η ia ij η j For the N-dimensional Gaussian integral we 73

80 6 Path integrals II: Fermions and Gauge Fields 1 = dη 1 d η 1... dη N d η N k=0 k! ( η ia ij η j ) k 1 = dη 1 d η 1... dη N d η N N! ( η ia ij η j ) N 1 N = dη 1 d η 1... dη N d η N η π(i) A π(i)π N! (j)η π (j) π,π S N i=1 = 1 N ( 1) π ( 1) π A π(i)π N! (j) = 1 ( 1) π det{a π(i),j } = det A (6.14) π,π S N i=1 N! π The idea behind steps three and four of this calculation is, that every distinct η i (or η i ) must show up exactly once in each term. Every term with higher powers of η i vanishes as well as any term in which one of the η i s is missing (because of the first integration rule). The symmetric group of the permutations of N indices is denoted by S N, ( 1) π is the sign of π S N. It arises upon reordering the differentials for the application of integration rule (6.12). determinant of a matrix A is det A = π S N ( 1) π i A i,π(i). Recall that the The Dirac field in path integral formalism We can now set up the path integral for the free fermionic field starting from the general path integral (6.1) via the specification L L D = ψ(i/ m)ψ φ ψ, ψ : Dirac spinors J η, η : Fermion sources/sinks. (6.15) Both the fields and the source/sink terms have to be treated as Grassmann valued Dirac-spinors. The field ψ and the adjoint ψ = ψ γ 0 (as well as η, η) are considered as independent components, expressing the fact that fermionic fields are complex (which can be mapped on two real components). Using above substitutions we end up with the generating functional Z(η, η) = Dψ D ψ e i d 4 x [ ψ(i/ m)ψ+ ηψ+ ψη]. (6.16) For the evaluation of this integral we use again the vector notation for the fields (as in eq. (6.2)) such that the operator i/ m is identified with a matrix K. The exponent of (6.16) then becomes a scalar product that can be brought to a form 74

81 6.2 Path integrals for fermions suitable for integration by completion of the square: ψkψ + ηψ + ψη = ( ψ + ηk 1 ) K (ψ + K 1 η) ηk 1 η. (6.17) }{{}}{{} ψ ψ Now the path integral (6.16) can be calculated as Z(η, η) = Dψ D ψ e i[ ψkψ+ ηψ+ ψη] = (i det K) e i ηk 1 η = C e i d 4 x d 4 y η(x)s(x y)η(y). (6.18) In the second line, the continuous notation has been restored such that the inverse matrix becomes the propagator S(x y). Its defining equation is (in analogy to KK 1 = 1) (i/ m)s(x) = δ (4) (x), (6.19) which reads in momentum space (/p m)s(p) = 1. (6.20) Finally we obtain S(x) = d 4 p (2π) 4 S(p)e ipx = d 4 p e ipx (2π) 4 /p m + iε, (6.21) where the usual iε was inserted in the denominator in order to keep the poles off the real axis. The matrix in the denominator can be avoided by using /p 2 = γ µ p µ γ ν p ν = g µν p µ p ν = p 2, the propagator then becomes S(p) = /p + m p 2 m 2 + iε. (6.22) The same propagator was calculated using the canonical formalism in eq. (2.117). Green s functions can now be calculated for Grassmann valued fields similarly to eq. (6.3) as 0 Tψ(x 1 )... ψ(y 1 )... 0 = 1 δ i δ η(x 1 )... i δ δη(y 1 )... Z(η, η). (6.23) η, η=0 75

82 6 Path integrals II: Fermions and Gauge Fields Note that the functional derivatives for η have been assigned an extra minus-sign (1/i i). This is because the differential operators have to pass through ψ in the current term ηψ + ψη of eq. (6.16) before acting on η (c.f. the derivation rules in (6.10)). The anticommutation property required in (6.6) is reflected by the anticommutating differential operators in (6.23). By inserting the explicit form of the generating functional (6.18) into (6.23), we obtain the correct two-point correlation function 0 Tψ(x) ψ(y) = is(x y). (6.24) 6.3 Path integrals for gauge fields The problem for the electromagnetic field As was already mentioned in section 6.1, the operator generating the bilinear form of the non-interacting Lagrangian can not be singular, otherwise the path integral (6.1) will not converge. Now one might wonder whether this is actually a physical problem: Are there any fields in nature for which this operator is singular? It turns out that this is indeed the case for one of the most important fields, namely the electromagnetic field, as will be shown in the following. The Lagrangian density of the electromagnetic field - represented by the fourpotential A µ - is given by L = 1F 4 µνf µν (6.25) with the faraday tensor F µν = µ A ν ν A µ. (6.26) The corresponding action integral can be brought to a bilinear form in the field A via partial integration S[A] = = = = d 4 x L = d 4 x [ 1( 4 µa ν ν A µ )( µ A ν ν A µ ) ] d 4 x [ 1 (( 2 µa ν )( µ A ν ) ( ν A µ )( µ A ν )) ] d 4 x 1 2 (A ν µ µ A ν A µ ν µ A ν ) d 4 x 1A 2 µ ( 2 g µν µ ν ) }{{} A ν (6.27) =:K µν 76

83 6.3 Path integrals for gauge fields The arising operator K µν has zero as an eigenvalue!the corresponding eigenvectors can be constructed as the gradient ν Λ of an arbitrary scalar field Λ(x) as is easily verified by calculating K µν ν Λ(x) = 2 g µν }{{ ν Λ µ ν } ν Λ = 0. (6.28) }{{} µ 2 The above eigenfunctions are the same as the generators of gauge transformations in classical electrodynamics. This already points out that the occurrence of singular matrices in the Lagrangian is closely connected to the concept of gauge invariance. This connection will be demonstrated in the following section in terms of a simplified example Illustrative example Consider a theory in a zero-dimensional spacetime - i.e. a world that consists of only one point both in space and time - and a two component field A = (A 1, A 2 ) T. The physics of this world is defined by defining an action 4 S(A). In analogy to the free field theories discussed above we choose a bilinear form S(A) = A T KA. In order to study the problem mentioned above, we choose K to be non-inverible, for example K = 1 0, (6.29) 0 0 such that S(A) = A 2 1 [7, chap. III.4]. If we now set up the path integral (which in this case only is an integral over R 2 ) naïvely, we obtain the obviously divergent expression Z = d 2 A e is(a) = da 1 da 2 e iatka = da 1 e ia2 1 }{{} π/i da 2. (6.30) }{{} The origin of this problem can easily be seen in this minimal example: Since the Lagrangian does not depend on the component A 2, the integration over it has to 4 Because of the lack of space and time dimensions, there is no distinction between Lagrangian, Lagrangian density and action. 77

84 6 Path integrals II: Fermions and Gauge Fields Figure 6.1: Illustration of gauge fixing for our simple gauge theory. give infinity. But as physics does not depend on it, there is absolutely no need for the second component in the description of the field - the formulation of the theory is therefore redundant. In mathematical terms, we may consider this redundancy as a gauge freedom. Generally, gauge transformations are defined as transformations of a field A under which the action S(A) is invariant. A gauge transformation therefore identifies physically equivalent fields. In our simple example, gauge transformations have the form A A = A + λ 0. (6.31) 1 In figure 6.1, the set of possible field configurations is depicted as a two dimensional plane. Physically equivalent fields with same components A 1 have been assigned the same color. The solution of the problem of the diverging path integral can easily be found in our simple example: Just cancel the second component of the field in the description of the theory, such that the divergent integration over A 2 in (6.30) vanishes. However, in more complex gauge theories, the redundancy might not show up as a simple extra component that can be discarded. If for some reason the redundant description of the field has to be kept, one can still restrict the integration range in (6.30) to only a subset of the possible fields, such that the integral meets only one representative of every class of physically equivalent fields. This procedure is called gauge fixing. Such a restriction of the integration range can happen via 78

85 6.3 Path integrals for gauge fields multiplication of the integrand with a delta function δ(a 2 ξ), such that only representatives with A 2 = ξ are counted. This leads to the corrected path integral Z = d 2 A e is(a) δ(a 2 ξ) = da 1 e ia2 1 }{{} π/i da 2 δ(a 2 ξ) <. (6.32) } {{ } Gauge fixing (Fadeev-Popov formalism) We now want to transfer the idea of gauge fixing to general gauge theories. So we consider again fields A(x) on the whole four dimensional spacetime. A characteristic property of gauge fields is the existence of gauge groups operating on the set of fields via g : A A g. (6.33) Usually the elements of the gauge group are themselves fields g(x), whose elements act locally on the field A(x), for example via representations of the Lie groups U(1) or SU(2) (c.f. chapters 10 and 11). The defining property of the gauge transformations (6.33) is that they leave the action invariant, i.e. S(A) = S(A g ) (6.34) for every group element g. Using the mapping (6.33), one can therefore identify every field configuration A with all its physically equivalent partners A g. For a redundancy-free description it is helpful not to consider A itself but the corresponding equivalence class [A] as the physically relevant object characterizing the field. From this point of view it is natural that the path integral must not run over all the fields A like in (6.1), but over all the classes of physically equivalent fields. What makes the naïve path integral actually diverge is the integration over the unmeasurable set of the representatives of the classes [A] (the vertical lines in figure 6.1). We therefore want to restrict the path integral to only a measurable selection of representatives of every distinct class [A]. These representatives can be defined 79

86 6 Path integrals II: Fermions and Gauge Fields as the zeros of an arbitrary 5 function f(a). The formal way (following ref. [7]) of setting up the correct path integral starts with the definition of the so called Fadeev-Popov determinant ( (A) := Dg δ[f(a g )]) 1. (6.35) This quantity is gauge invariant (i.e. constant on every class [A]), because ( (A g ) = ) 1 ( Dg g δ[f(a gg )] :=gg = Dg δ[f(a g )]) 1 = (A). For the validity of this calculation one has to make sure that the group is parametrized in such a way that the measure is invariant under group transformations, i.e. Dg = Dg. The yet diverging path integral DA exp(is) can be split into a part running over the different classes and a part running over the redundant gauge degrees of freedom. For this purpose we insert a one, consisting of (A) and its inverse definition DA e is(a) = DA e is(a) (A) Dg δ[f(a g )] = Dg DA e is(a) (A) δ[f(a g )] = Dg DA g 1 e is[a g 1 ] (A g 1) δ[f(a)] = ( Dg ) DA e is(a) (A) δ[f(a)]. In the last step we used the fact, that both S(A) (by definition) and (A) (as proven above) are gauge invariant, a property that was also assumed for the measure DA. Now we can simply throw away the redundant (and therefore unphysical) integration over the group elements g and end up with the definition Z = DA e is(a) (A) δ[f(a)] (6.36) of the path integral for gauge fields in the Fadeev-Popov formalism. An illustrative argument for this procedure is given in fig Of course f(a) is not completely arbitrary: for every equivalence class [A] there has to be at least one representative A for which f(a) = 0, on the other hand, the subset of [A] for which f(a) = 0 has to be measurable. 80

87 6.3 Path integrals for gauge fields Figure 6.2: Illustration of the Fadeev-Popov formalism: The chart corresponds to the (unphysical but easily understandable) case of discrete fields and groups. Every configuration of the field is drawn as a dot, physically equivalent configuration with equal S(A) are connected by lines. The number of fields on such a ring is to be considered infinite and therefore the path-integral A e is(a) diverges. We restrict the path integral to only the finite subset on the blue stripe distinguished by f(a) = 0. Fields may still be overcounted and therefore contribute with different weights. The integrand is therefore multiplied by (A) = ( g δ f(ag),0) 1 which counts the blue points for every class. The continuum limit of Z = A e is(a) δ f(a),0 (A) is exactly the path integral (6.36). 81

88 6 Path integrals II: Fermions and Gauge Fields Application to the electromagnetic field We now want to apply the Fadeev-Popov formalism to the important example of the electromagnetic field. The corresponding action integral was already calculated in (6.27) as S[A] = d 4 x 1A 2 µ( 2 g µν µ ν )A ν. Gauge transformations (the group elements g in the previous section) can now be expressed in terms of the scalar field Λ(x) as Λ : A µ (x) A µ (x) + µ Λ(x) = A µ (x) + 1 i e iλ(x) µ e iλ(x). (6.37) As required, the action S[A ] = S[A + Λ] = S[A] + S[ Λ] +... = S[A] }{{} =0 is invariant under the gauge transformation (c.f. sec , the operator K µν acting from the right or left makes all terms with the eigenfunction Λ vanish). The right hand representation of the transformed field in (6.37) shows that the gauge transformation acts locally as the compact Lie group U(1) = {e iφ φ [0, 2π)}. (6.38) The process of gauge fixing is also known in classical electrodynamics. There the most prominent gauges are the Lorentz gauge µ A µ = 0 and the Coulomb gauge A = i A i = 0. We will now use a more general approach by fixing µ A µ = σ, i.e. we set the function 6 f(a) to f(a µ ) = µ A µ σ. (6.39) with an arbitrary scalar field σ(x). The Lorentz gauge is recovered for σ = 0 and the Coulomb gauge for σ = 0 A 0. Plugging this into equations (6.36) and (6.35) 6 Strictly speaking, f(a) is in this case rather an operator than a function. This is still covered by the discussion of the previous section, where f(a) is considered as some arbitrary mapping of the abstract elements A. Note however, that the delta-function δ[f(a)] then becomes a functional, so to say the generalization of δ (n) (x) for n. 82

89 6.3 Path integrals for gauge fields yields Z = DA e is[a] δ[ A σ] (A) (6.40) ( 1 with (A) := DΛ δ[ (A + Λ) σ]) (6.41) for the path integral and the Fadeev-Popov determinant, respectively. Exploiting the first delta-function, one obtains Z = ( DA e is[a] δ[ A σ] ) 1 DΛ δ[ 2 Λ] } {{ } const.. (6.42) The constant factor is of no interest and will be neglected in the further discussion. The occurrence of the delta function in the path integral is still rather inconvenient for applications and can be suppressed by using a continuous distribution of different σ s. This comes up to simply inserting a 1 in (6.42) because Z was constructed in a way that it does not depend on the specific gauge σ. The path integral assumes a particularly compact form if we use a kind of Gaussian distribution with variance ξ/i as weighting for σ: Z = Dσ e (i/2ξ) d 4 x σ 2 (x) DA e is[a] δ[ A σ] = DA e is[a] (i/2ξ) d 4 x ( A) 2. (6.43) The advantage of this expression is that we can concatenate the original action and the extra term, defining the effective action 7 S eff [A] = d 4 x 1A ( 2 µ 2 g µν ( ) 1 1 ξ µ ν) A ν. (6.44) }{{} K µν eff 7 We are lucky that there are only minor modifications to the action for the U(1) gauge group. For more complicated (non-abelian) gauge groups the additional terms may be much more complex and behave like additional fields called Fadeev-Popov ghosts (see chapter 11) 83

90 6 Path integrals II: Fermions and Gauge Fields The extra term with 1/ξ can also be incorporated into the Lagrangian density (6.25): With one integration by parts we obtain 8 L eff = 1 4 F µνf µν 1 2ξ ( µa µ ) 2. (6.45) It can easily be verified that the two terms in eq. (6.28) no longer annihilate each other if K µν eff is inserted instead of K µν. The operator K µν eff is indeed non-singular and hence the path integral Z = DA e iseff[a] becomes convergent. The propagator D µν can now be calculated by requiring K µν eff D νλ (x) = δ µ λ δ(4) (x). In momentum space this reduces to ( k 2 g µν + ( 1 1 ξ ) k µ k ν) D νλ = δ µ λ, which is the simpler problem of finding the inverse of a 4 4-matrix. The final result is D νλ (k) = 1 ( (1 ξ) k ) νk λ g k 2 k 2 νλ. (6.46) The parameter ξ can be chosen freely without affecting any physical results. Convenient choices are ξ = 0 (known as Landau gauge) and the Feynman gauge ξ = 1. In the latter, the propagator becomes particularly compact D νλ (k) = g νλ k 2, (6.47) which is convenient for calculations with Feynman diagrams. We have already gained the same result using the Gupta-Bleuler procedure in the framework of the canonical formalism in chapter Quantum electrodynamics in the path integral formalism Having found formalisms to treat both fermionic and the electromagnetic fields, we can now try to formulate quantum electrodynamics (QED) in terms of path 8 This is similar to the Lagrangian one would get for a massive photon [7]. Thus, such a massive photon would not exhibit gauge invariance, but instead additional degrees of freedom corresponding to longitudinal polarization. 84

91 6.4 Quantum electrodynamics in the path integral formalism integrals. For this purpose we have to introduce interaction terms in the Lagrangian. This can be done by minimal coupling, which leads to the complete Lagrangian L(ψ, ψ, A) = ψ[iγ µ ( µ iea µ ) m]ψ 1 }{{} F 4 µνf µν 1 ( 2ξ µa µ ) 2 }{{} Dirac field Electromagnetic field = ψ[i / m]ψ 1F 4 µνf µν 1 ( 2ξ µa µ ) 2 + e ψγ µ A µ ψ. (6.48) }{{}}{{} L 0 L 1 The fist part of the Lagrangian describes the Dirac field L D (6.15) in which the operator has been replaced by the covariant derivative iea (e is the charge of the particle). The second part is the electromagnetic field with the extra term for gauge fixing, as was derived in (6.45). In the second line the Lagrangian was split into terms that are quadratic in the fields (L 0 ) and a higher order term L 1. This splitting is crucial, as it allows for perturbation theory for small coupling constants e to be applied. Fortunately this constant is small enough, namely e = α with the fine structure constant α 1/137 (finally only terms with powers of e 2 = α will show up). The path integral corresponding to L 0 is easy to handle because of the decoupling of the two fields. We calculate Z 0 (η, η, J) = DψD ψda e i d 4 x(l 0 (ψ, ψ,a)+ ηψ+ ψη+a µj µ ) [ ] = C exp i d 4 x d 4 y η(x)s(x y)η(y) [ exp i ] d 4 x d 4 y J(x) µ D µν (x y)j ν (y) 2 (6.49) using the propagators S for the fermions (6.21) and D µν for the photons (6.46) (see also (6.2) and (6.18)). The nonlinear term L 1 couples the two fields and makes the complete path integral hard to calculate. However, we can write it as Z(η, η, J) = DψD ψda e i d 4 x(l 1 (ψ, ψ,a)+l 0 (ψ, ψ,a)+ ηψ+ ψη+a µ J µ ) [ ( ) ( ) ( )] 1 = exp ie d 4 δ δ 1 x i γ µ δ Z i δj µ 0 (η, η, J) (x) δη(x) i δ η(x) (6.50) 85

92 6 Path integrals II: Fermions and Gauge Fields where the functional derivatives for the currents create the corresponding fields from the exponent of Z 0 (see also chapter 5). This form of the path integral is appropriate for perturbation theory, since we can simply do the Taylor expansion of the exponential function. By inserting (6.49) for Z 0, the Feynman rules for quantum electrodynamics - as listed in chapter 4 - can be set up. The characteristic vertices of QED consisting of two fermion lines and a single photon line arise from the combination of the three functional derivatives for η, η and J. For instance, the four-point Green s function (cf. (6.23)) 0 Tψ(x 1 ) ψ(y 1 )ψ(x 2 ) ψ(y 2 ) 0 = 1 δ i δ η(x 1 ) i δ δη(y 1 ) 1 i δ δ η(x 2 ) i δ δη(y 2 ) Z(η, η, J) η, η,j=0 (6.51) can be evaluated using this technique. In the zeroth order of the expansion of the exponential function in (6.50) the functional derivatives act immediately on Z 0. Therefore they simply return the propagators is(x 1 y 1 ) is(x 2 y 2 ) is(x 1 y 2 ) is(x 2 y 1 ) (6.52) In order to understand the necessity for the minus sign in the second term, consider the anticommutation rules for interchanging the functional derivatives for η(y 1 ) and η(y 2 ) in (6.51). The first order of the expansion of the exponential function does not give any contribution to the amplitude (6.51). This is obvious because the functional derivative for J µ (x) is creating the corresponding J ν (y) from the second exponential in (6.49), which will eventually be set to zero. Thus, the first non-trivial term of the expansion is of the order e 2. The corresponding amplitude reads 1 i δ δ η(x 1 ) i δ 1 δ δη(y 1 ) i δ η(x 2 ) i δ δη(y 2 ) ( ) ( ) ( 1 1 2! (ie)2 d 4 v d 4 δ δ 1 w i γ µ i δj µ (v) δη(v) i ( ) ( ) ( 1 δ δ 1 i γ ν i δj ν (w) δη(w) i ) δ δ η(v) ) δ δ η(w) Z 0 (η, η, J). η, η,j=0 This looks like a horrible mess of functional derivatives, each of which is creating a propagator and a source term from the exponentials in Z 0. However, most 86

93 6.4 Quantum electrodynamics in the path integral formalism Figure 6.3: Feynman diagram corresponding to eq. (6.53) (Møller scattering). of these terms vanish upon setting η, η, J = 0, except those in which all of the source terms created from the exponential are later on derived for by another functional derivative. These surviving terms can be illustrated by a set of Feynman-diagrams 9. For example, one of the second order terms will be (ie) 2 ( 1 i ) 4 d 4 v d 4 w S(y 1 w)γ µ S(w x 1 )D µν (w v)s(y 2 v)γ ν S(v x 2 ), (6.53) it corresponds to the Feynman-diagram given in fig A further term can immediately be constructed by interchanging x 1 and x 2 and flipping the sign of the prefactor (this way we have already found all possible connected Feynmandiagrams). For practical applications the integral is usually evaluated in momentum space, i.e. we specify initial and final momenta p 1, p 2, q 1, q 2 instead of the positions x 1, x 2, y 1, y 2. Furthermore we associate the Ψ s in eq. (6.51) with the spinors u(p, s) and ū(q, s ) respectively (see chapter 1). The explicit forms the propagators S and D µν are given in (6.21) and (6.46) (Feynman gauge), so we want to specify in (6.53) for example S(y 1 w) = d 4 q 1 (2π) 4 e iq1(y1 w) /q 1 m + iε. (6.54) As we are interested in a specific momentum for the external lines, we skip the integration over q i and p i. In order to keep the notation short, we also discard the propagators corresponding to the external lines, thus we keep only the term 9 In order to avoid disconnected diagrams, one usually considers W (η, η, J) defined by Z = exp(iw ) as generating function (see chapter 4). 87

94 6 Path integrals II: Fermions and Gauge Fields e iq 1( w) from (6.54). Finally we end up with the amplitude (ie) 2 d 4 v d 4 w ū(q 1, s 1)e iq 1( w) γ µ u(p 1, s 1 )e ip 1w d 4 k (2π) 4 e ik(w v) g µν k 2 ū(q 2, s 2)e iq 2( v) γ µ u(p 2, s 2 )e ip 2v. (6.55) The integration over w and v simply returns d 4 w e iw( q 1+p 1 +k) d 4 v e iv( k q 2+p 2 ) = (2π) 4 δ (4) ( q 1 + p 1 + k) (2π) 4 δ (4) ( k q 2 + p 2 ), (6.56) which fixes the momentum of the internal line to k = p 1 + q 1 = p 2 q 1. This also reflects total momentum conservation p 1 + p 2 = q 1 + q 2. Exploiting the deltafunctions, we obtain as final result a scattering amplitude that is proportional to g µν (ie) 2 ū(q 1, s 1)γ µ u(p 1, s 1 ) (q 1 p 1 ) ū(q 2, s 2)γ µ u(p 2 2, s 2 ) [p 1 p 2, s 1 s 2 ]. (6.57) In the second term p i and s i are interchanged, accounting for the second possible connected Feynman-diagram. The process for which we have just derived the amplitude in second order is known as Møller-scattering [6]. 88

95 Chapter 7 More on Feynman Diagrams 7.1 Correlation Functions Concepts A correlation function asks whether several quantities, say fields describing particles, are linked to one another. In the following we will concentrate on only two quantities, in which case the correlation function is usually called the Green s function of the system. In the context of quantum field theory, a typical Green s function takes the form T ψ(x)ψ(y) (7.1) and yields the probability amplitude to encounter a specific field configuration at spacetime-position y as well as another field configuration at a (different) spacetime-position x. The ominous T in this formula represents the time-evolution operator, and ensures causality (for an in-depth discussion, see Chapter 4 ). For the non-interacting theory, the time-evolution operator U takes a closed-form expression, and we are able to calculate the time-evolution of any field configuration and thus the Green s function (as well as any other quantity) explicitly. In turn, we do call the Green s function the (free) propagator of the particular field in question (see Chapter 4 ) Interacting Field Theory However, the interesting case in quantum field theory is the interacting one, in which we are often no longer able to calculate an exact analytic expression for 89

96 7 More on Feynman Diagrams U. If the interaction is small, we can apply perturbation theory to the timeevolution operator. In particular, if we assume that the Hamiltonian is given by H = H 0 + H int, with H 0 and H int the free and interacting part of the Hamiltonian, respectively, we can express U in terms of a Taylor-expansion, the so-called Dysonseries, as was discussed previously in chapter 4. Moreover, the vacuum in the non-interacting theory 0 need no longer be the vacuum of the interacting one, which we will call Ω. Yet we would like to express the correlation function in terms of expectations-values of the non-interacting vacuum so that we can use the analytic expressions for the free propagator. A rather lengthy calculation (see Peskin Schroeder, Chapter 4.2) allows us to do exactly that, and yields the relation Ω T ψ(x)ψ(y) Ω = 0 T [ ψ I (x)ψ I (y)u(+, )] 0 Ω U(+, ) 0 (7.2) with the time-evolution operator U defined in the interaction-picture U(t, t 0 ) = T e i t t 0 dt H I (t ), HI (t) = e ih 0(t t 0 ) H int e ih 0(t t 0 ). (7.3) Mind you, we used the fermion-operators ψ as fields in our correlation. Yet a similar expression holds for photon-fields or any other field. To understand eq. (7.2) we write down the time-ordering of the nominator explicitly, and, for the sake of convenience, assume that x 0 > y 0. We then see directly that the previous expression takes the form 0 U(+, x 0 ) ψ I (x)u(x 0, y 0 )ψ I (y)u(y 0, ) 0 (7.4) The rightmost time-evolution operator U(y 0, ) takes the non-interacting vacuum and evolves it to the interacting vacuum at time y 0. To ensure that this is actually the case, we may tilt the time-integration slightly by adding a small imaginary part, namely (1 iε). Hence only the large overlap between interacting and non-interacting vacuum 0 Ω survives the process of time-evolution, whereas all other parts are exponentially suppressed for the time going to infinity (see Peskin Schroeder, Chapter 4.2 for an explicit calculation). 90

97 7.1 Correlation Functions For the time-evolution operator between the two fields U(x 0, y 0 ) we use the Dysonseries. In QED, this leads to the expression U 1 + ( ie) d 4 z ψ(z) /A(z)ψ(z) + ( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z ) + O(e 3 ) (7.5) The first-order term always drops out due to energy-momentum conservation (see chapter 4 ). As a result, the first non-trivial contribution is of second order, and we obtain 0 T [ ψ I (x)ψ I (y)] T [ ψ I (x)ψ I (y)( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] 0 (7.6) for the expectation value (7.4). We can now evaluate that by use of Wicks theorem, which creates normal-ordered products out of time-ordered ones. Since this expectation-value is calculated with respect to the vacuum, only fully contracted terms that do no longer contain any normal-ordered operators survive. For the first term we then simply find the free Feynman propagator, 0 T [ ψ I (x)ψ I (y)] 0 = is F (x y) (7.7) The second term, however, is more involved. There are, in fact, two non-trivial ways of how to contract the time-ordered product. The first of those takes the Figure 7.1: Diagram for eq. (7.8) 91

98 7 More on Feynman Diagrams Figure 7.2: Diagram for eq. (7.9) form 0 T [ ψ I (x)ψ I (y)( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] 0 (7.8) Again, we encounter the bare fermion propagator is F as the contraction between ψ(x) and ψ(y), times a factor consisting of two fermion and one photon propagator. The latter describes a so called vacuum diagram, a closed loop (see Fig. 7.1) that does not couple to either field ψ(x) or ψ(y). Hence we find the same contribution in the thus far neglected denominator of eq. (7.2) as well (however without the propagator term), which directly cancels the vacuum loop of eq. (7.8). This is a general feature, and holds up to all orders in the perturbation series. In turn, we only need to take diagrams into account that are connected to the fields of which we want to find the correlations. One such diagram is shown in Fig. 7.2, with the corresponding contraction 0 T [ ψ I (x)ψ I (y)( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] 0 (7.9) The explicit expressions for the fermion- and photon-propagators are is F (x) = /p + m d 4 pi p 2 m 2 + iε e ipx (7.10) and id F αβ (x) = d 4 qi g αβ q 2 + iε e iqx (7.11) 92

99 7.1 Correlation Functions respectively, and we get the rather lengthy expression ( ie) 2 d 4 p (2π) 4 d 4 q (2π) 4 d 4 k (2π) 4 d 4 p (2π) 4 is F (p)e ip(x z ) γ α id F αβ (q)e iq(z z ) is F (k)e ik(z z ) γ β is F (p )e ip (z y) (7.12) for eq. (7.9). The physical interpretation goes as follows: The field ψ(y) connects to the first vertex at a space-time point z. From that on out, one fermion and one photon propagate to the point z, from which, ultimately, a single fermion propagates to the point x. Although the notation in configuration-space yields an intuitive picture for this particular process, it is more convenient to switch to momentum-space, especially considering that we want to have easy expressions connected to the Feynman diagrams later on. With that goal in mind, we carry out the integration over the spatial coordinates z and z and obtain two delta-functions, d z e i(p k q)z d 4 z e i(p k q)z = (2π) 4 δ(p k q)(2π) 4 δ( p + k + q) (7.13) which, in turn, take out one of the integration over the internal momentum q. As a result, we find the single delta-function δ(p p ), which ensures energy-momentum conservation at the individual vertices (q = p k) and hence energy-momentum conservation over the whole process in total. Thus, eq. (7.12) reduces to [ ( ie) 2 is F (p) and yields the final result for this particular process. d 4 kγ α id F αβ (p k)is F (k)γ β ] is F (p) (7.14) Feynman Rules for Correlation Functions Now, we can directly deduce the rules via which the mapping between the diagram and the explicit mathematical expression is to be done in momentum-space: (i) Every vertex is represented with a factor ( ieγ α ). (ii) A fermion propagator in momentum-space represents a factor /p + m is F (p) = i p 2 m 2 + iε (7.15) 93

100 7 More on Feynman Diagrams (iii) A photon propagator in momentum-space represents a factor id F αβ (q) = i g αβ q 2 + iε (7.16) (iv) We have energy-momentum conservation at every vertex. (v) Integrate over every unknown four-momentum, k = d 4 k/(2π) 4. (vi) (without proof) Every fermion loop acquires a factor of 1 due to the anticommumation relation between fermion operators. In addition, one needs to trace over every fermion loop (see the following example). (vii) The ordering of vertices and fermion propagators is such that reading from right to left, they occur in the same sequence as following the line in the direction of its arrows. The overall recipe then goes as follows: sum of all connected diagrams Ω ψ(x 1 )... ψ(x N ) Ω = with N external points (7.17) Again, ψ on the left-hand side is a placeholder for fermion- and photon-operators Examples (i) To begin, we start quite simple: The mathematical expression for a single solid line as is shown in Fig It is simply the free propagator, /p + m i p 2 m 2 + iε = is F (p) (7.18) Figure 7.3: Diagram for the free propagator. (ii) Then we consider a slightly more complicated diagram, as is shown in Fig There, the fermion emits a photon at the vertex α, and absorbs it at a 94

101 7.1 Correlation Functions second vertex β. The mathematical expression is then, due to the solid line at the beginning and the end, is F (p) [ iσ 2 (p)] is F (p) (7.19) where Σ 2 denotes the loop in the middle. According to the rules, we need to integrate over the unknown momentum k. Further, we do have a free an internal fermion and an internal photon propagator, both sandwiched in between the two vertices ( ieγ α ) and ( ieγ α ), respectively, and taking energymomentum-conservation into account, leads to iσ 2 (p) = (ieγ α )is F (k)(ieγ β )id F αβ (p k) (7.20) k This is the expression we already calculated before, but without going through the whole expansion of the time-evolution operator. Such a contribution will become important later on in chapter 8, when radiative corrections are considered. Figure 7.4: Diagram for a photon-rainbow. (iii) We may draw a diagram similar to the one considered in (ii), where we exchange the initial and final fermion-line with a photon-line (see Fig. 7.5). Since one photon-line and two fermion-lines meet at every vertex, the only possible second-order contribution may be due to a single fermion-loop in the middle. Again, we find an expression similar to the one in eq. (7.19), id F αβ (p) [ iπ βλ 2 (p) ] id F λσ (p) (7.21) where the fermion-propagators are exchanged with photon-propagators. However, the middle part now carries vertex-indices in order to match the ones at the photon-propagators. The fermion-loop in itself has an unknown mo- 95

102 7 More on Feynman Diagrams mentum k, over which we need to integrate. Further, since we do consider fermions, we have a factor of ( 1) as well as a trace over the propagators. Therefore, we obtain the expression Π βλ 2 (p) = ( 1) tr [(ieγ α )is F (k + k)(ieγ α )is F (p k)] (7.22) k Again, such a contribution is encountered when considering radiative corrections. Figure 7.5: Diagram with a single fermion-loop Exercise Now, try and find the mathematical expression which matches the diagram in Fig Figure 7.6: Diagram representing a fourth-order contribution to the photonpropagator. 96

103 7.2 S-Matrix 7.2 S-Matrix Concepts In order to understand the conceptual connection between correlation-functions and the S-Matrix, we consider the following correlation-function between the four fields ψ(x)ψ(x ) and ψ(y)ψ(y ), 0 T [ ψ(x)ψ(x )U(x 0 = x 0 +, y 0 = y 0 ) ψ(y)ψ(y )] 0 (7.23) where, for the sake of convenience, we assume that x 0, x 0 > y 0, y 0. As is implied in the notation, we do send both the creation and annihilation of the fields to the time plus and minus infinity, respectively. Furthermore, we assume that the system at times plus/minus infinity is well described by the non-interacting Hamiltonian H 0. The interaction H I (t) will start to kick in somewhere between plus and minus infinity, and therefore create the scattering phenomena in which we are interested in. Such a behavior is easy to justify from a purely physical standpoint: Consider two particles far apart from each other. Due to the large spatial separation any interactions between the two may be neglected (otherwise the notion of a free particle would be problematic in the first place). Approaching each other, they start to feel their respective presence, meaning that some form of interaction takes place that alters the trajectory of the particles. If we now assume that the particles do not form a single entity (either a bound state or a new particle), they will ultimately move away from each other and, after an infinite time, may be described within the non-interacting theory again. There we do know how the creation- /annihilation-operator works onto the vacuum, namely creating/annihilating a particle in a momentum eigenstate according to 2E p a s p 0 = p, s and an equivalent expression for the antiparticle. In general, we will denote the initial state at time y 0 = y 0 = with i and the final state at time x 0 = x 0 = + with f. The time-evolution operator in the middle is known as the S-matrix, and describes the actual scattering process of the initial states. As was the case for the correlation function in the previous section, we are usually not able to provide a closed-form expression for it, and are forced to use a perturbative approach. However, we are not interested in the process in which nothing happens at all, 97

104 7 More on Feynman Diagrams meaning that the initial and final states are the same. definition of the so called T-Matrix via This gives rise to the S = 1 + it (7.24) where we remove such processes Contractions with Initial and Final States To evaluate the (time-ordered!) product f it i, we again make use of Wick s theorem. Yet, one has to keep in mind that the initial and final states in itself contain creation-/annihilation-operators. To make that point clear, we apply a fermion operator ψ I (x) such as is encountered within the T-Matrix onto an initial fermion momentum-eigenstate 2E p a s p 0 : ψ I (x) p, s = = d 3 p 1 (2π) 3 a s p (p )e us ip x 2E p a s p 0 2Ep s d 3 p 1 (2π) 3 u s (p )e ip x 2E p (2π) 3 δ 3 (p p)δ ss 0 2Ep s (7.25) = u s (p)e ipx 0 One can directly see that the application of ψ I generates a spinor u s (p) times the (non-interacting) vacuum. It is therefore natural to define the contraction of a field operator with initial and final states, which for fermions take the form ψ p, s = u s (p), ψ k, s = v s (k), p, s ψ = ū s (p), k, s ψ = v s (k), (7.26) where p, s denotes the fermion and k, s the antifermion. Moreover we switched from configuration- to momentum-space via a trivial Fourier-transform. Note that this is in agreement with the formalism presented in the previous section. Now, the recipe is: write down all possible contractions of the field operators as well as the external states. Still, there are some diagrams that only contribute in a trivial fashion. To see this, we consider the second-order diagrams with two fermions scattering on each other: 98

105 7.2 S-Matrix (i) First, we consider the case in which the (second-order) T-Matrix only features contractions in itself, p 1 p 2 ( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] p A p B (7.27) Accordingly, the contractions between the initial and final state are only nonzero if p A = p 1 and p B = p 2 or vice versa. As such, this specific term does not involve any scattering at all, but rather describes how the vacuum fluctuates. We already encountered something similar before when we were discussing correlation functions. There, those terms dropped since vacuum fluctuations are there regardless of whether particles are present or not. Following the same argument as in the previous section, we will neglect these contributions and put them into the trivial unity-operator of the S-Matrix (or, considering higher-order processes, into lower orders). Figure 7.7: Diagram representing the T-Matrix element of eq. (7.27) (ii) Second, if we contract but one of each initial and final particles with a field of the S-Matrix in the middle, we find the expression p 1 p 2 ( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] p A p B (7.28) Rather than leading to a scattering process, the interaction merely modifies the propagator describing the transition from the state p A to p 2 in this specific case. Thus, such a term is a contribution to the unity-operator in the S-Matrix (although in disguise), and may be ignored when considering scattering. 99

106 7 More on Feynman Diagrams Figure 7.8: Diagram representing the T-Matrix element of eq. (7.28) (iii) Last we do contract all the external states with operators from the T-Matrix, and obtain p 1 p 2 ( ie) 2 d 4 z ψ(z) /A(z)ψ(z) d 4 z ψ(z ) /A(z )ψ(z )] p A p B (7.29) Similarly to the case of the correlation function, we can calculate this particular expression using Wick s Theorem. For the sake of a clear presentation, we explicitly go through the transition from configuration- to momentum-space. By use of the relations (7.10) and (7.11) we do find ( ie) 2 d 4 qd 4 zd 4 z ū(p 2 )e ip 2z γ α u(p A )e ip Az id F αβ (q)e iq(z z )ū(p 1 )e ip 1z γ β u(p B )e ip Bz (7.30) The integration over the internal spatial coordinates z and z yields two deltafunctions, namely d 4 ze i(p 2 q p B )z d 4 z e i(p 1+q p A )z (7.31) = (2π) 4 δ(p 2 q p B )(2π) 4 δ(p 1 + q p A ), which ensure energy-momentum conservation at every vertex. In combination with the integration over q we find the single delta-function δ(p A + p B p 1 p 2 ), that is to be interpreted as energy-momentum conservation of the whole scattering process, and is a general feature of the T-Matrix. This gives rise to the definition of the so-called invariant amplitude im via it = (2π) 4 δ p i i f p f im, (7.32) 100

107 7.2 S-Matrix where the sums i p i and f p f denote the total initial and final momenta, respectively. The invariant amplitude itself takes the form im = ū(p 2 )(ieγ α )u(p A )id F αβ (p A p 2 )ū(p 1 )(ieγ β )u(p B ) (7.33) It is this invariant amplitude that depends on the interaction, and whose mathematical expression directly corresponds to a specific diagram. The total energy-momentum conservation is only kinematics, and does neither depend on the interaction nor the order in which we look at a particular process. Figure 7.9: Diagram representing the T-Matrix element of eq. (7.33) Amputating Legs As a last ingredient for Feynman diagrams, we need to learn about amputation of legs, and, for that matter, a fourth-order process with four vertices. In particular we take a look at the diagram in Fig. 7.10, which basically is the combination of the diagrams from Fig 7.9 and 7.8, respectively. Although this diagram clearly exhibits four vertices and, as such, describes a fourth-order process, the actual scattering does not. The third and fourth vertex only change the propagator on one of the legs of the (second-order) scattering-process. If we are only interested in the latter, we may simply remove this part, which reduces the whole process to second order. This removal is what we call amputation of legs, and makes sure Figure 7.10: Fourth-order process with a leg to be amputated. 101

108 7 More on Feynman Diagrams that the order of a diagram is actually the same as the scattering-process we are interested in. Formally, the process of amputation may be put as follows: If you are able to cut a single line in a diagram describing a scattering process so that the separated part is an (interacting) propagator as was discussed in section 7.1 you should cut there and only consider the remaining (scattering) diagram. Of course, you might need to repeat that process several times until you reach the bare scattering Feynman rules for scattering processes In addition to the rules presented in the previous section 7.1, we find the additional rules corresponding to the external legs and their amputation. (viii) Amputate the legs according to subsection (ix) An initial fermion line represents a factor of u s (p). (x) A final fermion line represents a factor of ū s (p). (xi) An initial anti-fermion line represents a factor of v s (k). (xii) A final anti-fermion line represents a factor of v s (k). (xiii) An initial photon line represents a factor of ε r (q). (xiv) A final photon line represents a factor of ε r(q). (xv) The spinor-factors and vertices for each fermion-line are ordered so that reading from right to left, they occur in the same sequence as following the line in the direction of tis arrows. With these results, we may summarize the recipe to calculate the invariant amplitude im via sum of all fully connected im = (7.34) amputated diagrams We can easily obtain a mathematical expression for any order of the T- and, accordingly, the S-Matrix. 102

109 7.3 Scattering Cross-section Example For an example, we take a look at the electron-positron annihilation-process, in which a muon-antimuon pair is created. The corresponding diagram is shown in Fig Now, how do we go about that? First, we start at the electron-line, and Figure 7.11: Electron-positron-annihilation. follow it in the direction of the arrows. According to the rule (xv), we write down the corresponding parts from right to left. Thus we start with the spinor u(p) for the electron, then write down the first vertex ieγ α, and last, leftmost, the spinor for the positron v(p ). In total, we find v(p )( ieγ α )u(p). Then, corresponding to the photon-propagator line, we place the photon-propagator id F αβ (p + q) on the right of the previous expression. Last, and completely analogue to the first fermion line, we do have the expression ū (µ) (k)( ieγ β )v (µ) (k ) for the muons. Overall, we find im = v(p )( ieγ α )U(p)iD F αβ (p + p )ū (µ) (k)( ieγ β )v (µ) (k ) (7.35) with energy-momentum-conservation giving rise to p + p = k k, i.e. k = k p p. 7.3 Scattering Cross-section In a scattering process in high-energy physics, one is usually only interested in the presence of a particle, and not its polarization. As such, we have to account for our ignorance of the spin orientation/polarization by averaging over all initial spins/polarization vectors ( 1 2 s) and summing over all the final ones ( r). For the particular example of electron-positron annihilation in which a muon- 103

110 7 More on Feynman Diagrams Figure 7.12: Definition of momenta and the angle θ in the center-of-mass frame. antimuon pair is created, this leads to the factor s 2 s r r M 2 = 8e4 (p + p ) 4 [ (pk)(p k ) + (pk )(p k) + m 2 µ(pp ) ] (7.36) where we neglected the electron mass since m e m µ /207. The latter point is actually consistent, since the error is on the order of 1/207, whereas the fourthorder process yields correction on the order of α = 1/137. The evaluation of such an expression is not as hard as it may seem, since a number of formulas exist that ease the process (see Peskin Schroeder, Chapter 5.1, Trace Technology). If we, for convenience, choose to work in the center of mass (CoM) frame, the above expression takes the form with 1 4 [( M 2 = e 4 spins ) 1 + m2 µ E 2 + ( 1 m2 µ E 2 ) cos θ ] (7.37) (p + p ) 2 = 4E 2, pk = p k = E 2 E k cos θ pp = 2E 2, pk = p k = E 2 E k cos θ (7.38) We can use this in the general formula for the scattering cross-section (see Peskin Schroeder, eq. (4.84)), and do find dσ dω = 1 4E A E B v A v B [( = α2 4E 1 m2 µ E 2 p 1 M(p (2π) 4 A, p B p 1, p 2 ) 2 4E CoM ) ( ) ] (7.39) 1 + m2 µ + 1 m2 µ cos θ E 2 E 2 104

111 7.3 Scattering Cross-section Integration over the solid angle dω directly yields the total cross-section [ σ tot = 4πα2 1 m2 µ m 2 ] µ 3E 2 E 2 2 E 2 (7.40) 105

112

113 Chapter 8 Radiative Corrections 8.1 Motivation So far we have quantized our fields, found the Feynman propagator and developed rules to calculate the interaction of fields. To extract these Feynman rules we have assumed that the interaction is very small and we can apply perturbation theory to the time-evolution operator. So now we come to the point were we actually want to apply these rules, do the calculations for real, and receive the expected small radiative corrections (that decrease with increasing order of perturbation). This is where we encounter a problem if we do the calculation naively: instead of receiving a well-defined, small quantity, we find that loop integrals diverge and we get infinity. This will be shown in the next section. The fact, that a divergent solution cannot be physical if our calculations and assumptions up to this point have been correct, leads us to consider a process of renormalization that takes into account the nature of the parameters that we put into the equations. The process of making the integrals become finite generally involves three steps: the regularization, the renormalization and the elimination of the regularizing parameters. These steps will be shown in sections 8.3 and Divergence in loop integrals Radiative corrections are signified by loops in Feynman diagrams, which means that they all contain at least one undetermined momentum that is to be integrated over. Unfortunately the expressions for the propagators do not vanish sufficiently 107

114 8 Radiative Corrections fast enough at infinity (high energies). This corresponds to the singular behavior of (non-renormalized) Green functions at short relative distances One-loop diagrams The lowest 1 order radiative corrections diagrams are displayed in the following figure. Figure 8.1: Lowest order radiative corrections, from left to right: electron selfenergy, photon self-energy and vertex contribution We can write down the mathematical expressions for these diagrams by applying the Feynman rules that we have found in earlier chapters to study the qualitative behavior of of these interactions at the limits of very low and very high energies. The expression for the electron self-energy: f { pfffp k µ ν k Ff pff...is F (p) [ iσ (p)] }{{} iσ(p) = ( ie) 2 k+q loop contribution is F (p)... d 4 k i ( ig µν) (2π) 4 /p /k m + iϵ γµ k 2 + iϵ γν, (8.1) the expression for the photon self-energy (vacuum polarization): k g L qgg qgν...id F,µβ [iπ µν (q)] id F,να... }{{} µ loop contribution iπ µν (q) = ( 1) ( ie) 2 d 4 [ k (2π) tr 4 γ µ ] i i /k m + iϵ γν, (8.2) /k + /q m + iϵ 1 second order in e ( e 2 α ), as the correction involves the insertion of two new vertices 108

115 [ νep 8.2 Divergence in loop integrals and finally the expression for the vertex contribution: Dp k σdp k dµe Ep k ] }q=p p ieγ µ ieγ µ (p, p ) ieγ µ (p, p ) = ( ie) 2 d 4 k (2π) 4 i /p /k m + iϵ γσ ( ig σν) k 2 + iϵ γν i /p /k m + iϵ γµ. (8.3) We have to distinguish between the divergences at low and high momenta. These two effects are of a totally different nature. The terms for the electron self-energy and the vertex contribution each exhibit an infrared 2 divergence. This is due to the fact that these expressions both contain a photon propagator and we have already seen that the use of the redundant gauge field 3 causes this formal problem. The infrared divergence is usually dealt with by inserting a small counterterm µ into the denominator of the internal photon propagator. Physically, this can be interpreted as giving the photon a small, non-zero, mass 4 which allows for a longitudinal polarization of the virtual photon. The more important case is the behavior of the interaction expressions at very short distances (corresponds to very high energies in momentum space). ultraviolet divergence which we find here is not a formal problem as in the case of the infrared divergence, but poses a real physical problem that requires another consideration and which will eventually lead to the process of renormalization. The ultraviolet divergence of the one-loop diagrams can be shown more explicitly if the equations (8.1)-(8.3) are further simplified. The In the ultraviolet limit k m, p, p, q..., so that we can ignore everything but k in the denominator of the propagators; thus the photon propagator becomes proportional to 1 k 2 2 divergence at very low energies 3 a Lorentz vector field A µ with four degrees of freedom was used to describe two physical degrees of freedom of a photon 4 putting it off-shell and 109

116 8 Radiative Corrections the fermion propagator becomes proportional to 1 k. After a Wick rotation to the Euclidean space 5 and using the proportionality d 4 k E dk k 3 these three equations become: iσ(p) ( ie) 2 iπ µν (q) ( 1) ( ie) 2 ieγ µ (p, p ) ( ie) 2 dk (8.4) dk k (8.5) dk 1 k. (8.6) We can see that the electron self-energy, the photon self-energy and the vertex contribution show a linear, quadratic and logarithmic divergence 6 respectively Superficial degree of divergence from topological analysis of Feynman diagrams Topological analysis of Feynman diagrams allows us to determine the superficial degree of divergence of these interactions and thus which diagrams are actually causing a problem in the ultraviolet region. In order to do this, we have to introduce the dimension D of a diagram. As we are using natural units with ħ = c = 1, we can choose the mass m as a dimensional quantity with D (m) 1. The dimensions of the other parameters can now be gained from looking at any simple physical equation 7 or using the fact that in these natural units the action S = d 4 xl must have the dimension zero 8. The dimensions of some important parameters in the diagram equations are: D (x) = 1; D ( ) = 1; D (/k) = 1; D ( ) ( 1 = 1; D /k d 4 x ) = 4; D ( ) d 4 k (2π) 4 = 4. 5 where k 2 E = k2 0 + k 2 6 These are the superficial degrees of divergence; a closer inspection, taking into account symmetries and current conservation, yields an effective logarithmic divergence in all cases. 7 for example Heisenberg s uncertainty principle 8 this means that every term in the Lagrangian L must have dimension 4 110

117 8.2 Divergence in loop integrals A Feynman diagram diverges superficially with its dimension D as: D < 0 : convergent; D = 0, 1, 2,... : log., linear, quadratic... divergence. As every loop gives an integration over an undetermined momentum, which corresponds to a dimension of 4, and the photon and the fermion propagator have the dimensions -2 and -1 respectively, the simple formula for the superficial degree of divergence of a diagram looks like this: D = 4 (number of loops) 2 (number of internal photon propagators). (number of internal electron propagators) (8.7) This formula can be further reduced to only include the external lines of a diagram. As each internal line corresponds to an undetermined momentum and each vertex is associated with a momentum conservation delta function, it is possible to express the number of loops L by counting internal fermion lines F i and boson lines B i and subtracting the number of vertices V minus a factor 1 for the delta function that goes into the overall momentum conservation of the entire diagram: L = B i + F i (V 1). (8.8) Each external line goes into one vertex and each internal line connects two vertices. In QED there are also always two fermion and one boson line meeting at each vertex. We therefore have two different expressions for the number of vertices: 2V = F e + 2F i ; V = B e + 2B i (8.9) where F e (B e ) is the number of external fermion (boson) lines. Solving equation (8.9) for the number of internal lines and then plugging the results into equation (8.8) and finally plugging everything into equation (8.7) we finally receive the simple expression: D = F e B e (8.10) which we can use to determine all the diagrams that are divergent (D 0). The result is shown in the following table: 111

118 8 Radiative Corrections The more external lines are added, the more convergent the diagrams become. Any diagram with more external lines than listed in the above table is finite. We find that the three diagrams discussed in the previous subsection were in fact the only divergent diagrams in the lowest order of perturbation. 8.3 Regularization The task now is to make the integrals convergent. A formal way to regularize the equations, is to introduce an upper limit of the integration, a cut-off: 0 Λ 0. The regularization in QFT can of course not be performed as simply as just changing the upper limit of the integration as this is not a gauge invariant process. Instead, there are several alternative ways to do the regularization, the most commonly used being the Pauli-Villars and the dimensional regularization. The Pauli-Villars regularization basically amounts to minimally coupling the photons to additional spinor fields with very large mass Λ 10, while the dimensional regularization means that the computation of the equations is performed in a slightly lower dimension. The result in each case is no longer divergent, but dependent on the cut-off Λ. In taking the limit Λ, the original QED theory is restored. What does this cut-off signify physically? We know that quantum field theory is valid up to some energy scale. At higher energies new processes might happen, for example we see that the quantum electrodynamic theory of the interaction of electrons with photons does not hold at arbitrarily high energies, where other particles come into the interaction and we find the transition to the electroweak theory. So the cut-off simply corresponds to the energy up to which our theory holds, or in other words, our threshold of ignorance. The actual regularization is rather tedious and lengthy and shall therefore not be shown here 11, instead only qualitative results for the three divergent cases will be given. In lowest order these are: 10 which means that we get an additional propagator term of a very heavy photon in the equations 11 see textbooks for this, such as Peskin Schroeder, chapter 6 112

119 8.4 Renormalization for the electron self-energy: Σ lowest (p) = e 2 A (Λ) for the photon self-energy: Π µν B (Λ) + lowest (q) = ie 2 g µν q 2 C (q = 0, Λ) + Π and the vertex contribution: Γ µ lowest (p, p ) = e 2 Σ C (p) }{{} independant of Λ, finite D (p = p, Λ) γ µ + ( ) C q 2 }{{} independant of Λ, finite Γ µ C ( p / m), (8.11) ( p, p ) }{{} independant of Λ, finite, (8.12). (8.13) We can see that while all the expressions contain a small radiative correction (signified by the lower index C) that is finite and independent of Λ, we also have in each case a term that is cutoff-dependent and diverges if we let Λ. These are the problematic terms and we have to get rid of them in order to receive the desired result: the simple radiative corrections. This is done by renormalization. 8.4 Renormalization We have found diverging integrals because our approach so far has been quite naive. We have simply put bare, that is to say, unphysical properties in our equations. We can not experimentally determine these bare properties. What we measure is a physical mass, a physical charge, a physical field, and so on. These quantities have been modified by their interaction. Already in the classical point-particle theory the physical inertial mass is obtained by adding an electromagnetic contribution to the bare mass. In QFT we can do the same thing: by re-normalizing we exchange the bare properties in the equations with the corresponding (measurable) physical properties 12. If we assume that when we use these physical properties in our equations, we should get a physical, finite result, then 12 this corresponds to adding counterterms in the Lagrangian 113

120 8 Radiative Corrections the step of re-normalization should eliminate the divergent terms in the evaluated integrals. An explicit calculation with the renormalized quantities will in fact show that these divergences disappear. Now comes the task to find the relationship between bare properties, the cutoffparameter and the physical properties The electron propagator We consider the full electron propagator 13 with all the possible electron self-energy loop contributions that can be inserted into it. To this end we first have to define a one-particle irreducible (1PI) diagram as any diagram that cannot be split into two separate diagrams by removing a single line 14. For example fkykzff is 1PI, while fkykzfffyff is not. Any such diagram signifies the interaction (of a specific order) of the propagator with the field. A sum over all possible 1PI diagrams will therefore give the complete interaction corrections to the free propagator at a specific place in configuration space. So, the electron self-energy (what was called the loop-contribution in section and there consisted of a single photon loop as we were looking at the lowest order of radiative corrections), can now be given to all orders by the following formula: iσ (p) = FcF 1PI = ffyff + fkykzff +... (8.14) (where the external lines of the 1IP diagrams are not included in the expression for Σ (p)). 13 electron two-point function: Ω T ψ (x) ψ (y) Ω 14 so basically a diagram where the loops are interconnected 114

121 8.4 Renormalization With this we can write the full electron propagator (without external points 15 ) in the following way: full electron propagator =F + = i /p m FcF 1PI + i /p m 0 ( iσ (p)) i /p m 0 ( iσ (p)) FckkcF 1PI 1PI +... i /p m 0 (8.15) i /p m 0 ( iσ (p)) i /p m where the bare mass is denoted as m 0 to distinguish it from the physical mass (from now on, all bare properties will be labeled with a subscript 0 as a distinction to the physical properties). Upon closer inspection of the infinite series in (8.15), we see that it forms a geometric series 16 and we get: i full electron propagator = /p m 0 Σ (p). (8.16) We know that a one-particle state has a simple pole at its inertial mass 17. For a free particle, the inertial mass would be the bare mass, but an interactionless particle, and therefore the bare mass, can never be observed. If we now consider the full electron propagator, we see from equation (8.16) that the pole is shifted away from the bare mass m 0 by Σ (p). The shifted location of the pole corresponds to the physical mass m. To find it, the denominator of the full propagator is simply set to zero: and the physical mass becomes: [ /p m 0 Σ (p) ] / p=m= 0 (8.17) m = m 0 + δm = m 0 + Σ ( /p = m ). (8.18) When computing the mass shift δm to higher orders, the number of diagrams to be evaluated rapidly increases while the corrections become smaller and smaller. 15 and ignoring the iϵ term in the denominator 16 as the real correctional term will be of the order α 17 a particle is on mass shell if p 2 = m 2 115

122 8 Radiative Corrections To the lowest correctional order, the mass shift is: δm = Σ lowest ( /p = m ) Σ lowest (/p = m 0 ). (8.19) This lowest correction can be calculated explicitly 18, and with this the mass shift becomes (in the limit of the cutoff going to infinity): δm (Λ) Λ ( ) 3α Λ 2 4π m 0 ln. (8.20) m 2 0 Here we see directly that this correctional term is also logarithmically divergent. This means that the bare mass is re-normalized into the physical mass via a divergent (infinite) quantity. But as the bare mass is not an accessible quantity, this causes no problem. (In classical electrodynamics, the energy shift of a charge by its interaction with its electrostatic field is also a (linearly 19 ) divergent quantity). We can now also look at the form of the denominator close to the pole: ( 1 dσ ) d/p /p=m We define another renormalization factor: (/p m ) + O ( ( /p m ) 2 ). (8.21) 1 dσ d/p /p=m Z 1 2 (8.22) and with this acquire the following expression for the full electron propagator: i ( Z2 1 /p m ) ( ( + O /p m ) 2 ) = + iϵ + iz 2 /p m + iϵ iz 2 (/p m ) [1 + Z 2 Σ Clowest (p)] + iϵ +... (8.23) where the first term on the right-hand side corresponds to that of a free field with the physical mass m that has been rescaled by the factor Z 2. The factor Z 2 is a field-strength renormalization factor, the probability for the field to create or 18 see f.ex. Peskin Schroeder, p while in QED, the quantum statistics, the anti-commuting of fermions, makes the divergence of the self energy only logarithmically divergent (Weisskopf phenomenon) 116

123 8.4 Renormalization annihilate an exact one-particle eigenstate from the vacuum. Each inner line is sandwiched between to vertices 20 at its ends that each contribute a charge e 0 to the full expression which will therefore become proportional to e 2 0Z 2 : µfν = γ µ ie 2 0Z 2 /p m + iϵ γν γ µ ie 2 /p m + iϵ γν (8.24) where in the last step the renormalization factor was used to shift the bare charge to become the physical charge. The relationship between physical and bare charge is given by: e = e 0 Z 2. (8.25) Z 2 re-normalizes the charge at both ends of the inner propagator line. The lowest order correction term to Z 2 can again be computed which gives a rather lengthy expression which is dependent on Λ 21. The inspection of the full electron propagator yields two renormalization factors, δm and Z 2, that (as infinite quantities) relate the bare properties (mass, charge) to the measurable physical properties. When the bare properties are eliminated in favor of the physical properties, the divergence of the loop-diagrams disappears (while small, correctional terms are retained). Or in other words, the divergence of the whole loop-integral equation is absorbed into the propertionality factor between the bare and physical properties and it is therefore that the expression becomes finite if the renormalized properties are used in the calculation Ward identity The Ward identity states that for any physical amplitude M (k) = ϵ µ M µ... (k,...) that involves an external photon of momentum k going into or coming out of a vertex labeled µ, the amplitude vanishes if the external photon polarization vector ϵ µ is replaced by its corresponding 4-momentum k µ : 20 where particles are created or annihilated 21 see Peskin Schroeder, page

124 8 Radiative Corrections If we look at a free photon in the Lorentz gauge: k µ M µ... (k,...) = 0. (8.26) A µ (x) ϵ µ (k, λ) exp (±ikx) (8.27) and its gauge transformation: A µ (x) A µ (x) = A µ (x) + µ f (x) (8.28) we see that the gauge transformation for ϵ µ (k, λ) becomes: ϵ µ (k, λ) ϵ µ (k, λ) = ϵ µ (k, λ) ± ik µ f (k) (8.29) and that it is by the virtue of the Ward identity that the invariance of the scattering amplitude under a gauge transformation is retained 22. A prove of the more general Ward-Takahashi identity, where the external momenta do not necessarily have to be on-shell, is shown in detail in Peskin Schroeder, chapter 7.4. For the prove the following amplitude is considered: Figure 8.2: Amplitude M with n incoming external electron lines with momenta p i and n outgoing external electron lines with momenta q i. where the shaded circle represents any diagram that contributes to an amplitude M 0 that differs from the amplitude M only by lacking the insertion of the photon with momentum k. A summation over all possible diagrams that contribute to 22 physically the Ward identity stems from current conservation: k µ J µ (k) = 0 118

125 8.4 Renormalization M 0, followed by a summation over all possible insertion points of the photon and finally the multiplication with k µ results in the Ward-Takahashi identity for correlation functions in QED: k µ M µ (k; p 1...p n ; q 1...q n ) = e i [ M 0 (p 1...p n ; q 1...(q i k )...) ] M 0 (p 1... (p i + k)...; q 1...q n ). (8.30) The Ward-Takahashi identity reduces in physical scattering processes to the simple Ward identity because the right-hand side of the above equation does not contribute to the S-matrix Analysis of the photon propagator, vertex and external lines The treatment of the full electron propagator has been done in some detail to show the workings of renormalization. A similar approach works for the photon propagator and vertex. Photon propagator As in the case with the electron propagator, we can again use 1-particle-irreducible diagrams to describe the photon self-energy: Π µν (q) = gµcνg 1PI = ( q 2 g µν q µ q ν) Π ( q 2) (8.31) (where in the last step the Ward identity was used, as well as the facts that due to Lorentz invariance g µν and q µ q ν are the only possible tensors to appear in Π µν (q) and that there can be no intermediate single-massless-particle state in any 1PI diagram). Π (q 2 ) is regular 23 at q 2 = 0. With this the full photon propagator becomes: full photon propagator = 23 vanishes linearly with q 2 at q 2 = 0 ( i g q 2 (1 Π (q 2 µν q ) µq ν + i )) q 2 q 2 ( ) qµ q ν q 2 119

126 8 Radiative Corrections = ig µν q 2 (1 Π (q 2 )) (8.32) where in the last step the Ward identity was again used to eliminate all terms proportional to q µ or q ν since the photon propagator is always connected to at least one fermion line if an interaction (physical process) is to be calculated. For Π (q 2 ) regular at q 2 = 0, the pole of equation (8.32) will always be at q 2 = 0, which means that the photon described by the full photon propagator remains massless at all orders in perturbation theory. We find another renormalization factor Z 3, the factor by which the residue of the pole in equation (8.32) is shifted away from the value one: 1 Π (0) Z 1 3. (8.33) Incorporating two vertices at the end of the photon line, we can again use Z 3 to re-normalize the charge: e = e 0 Z 3 (8.34) which physically amounts to the bare charge being screened at large distances by just that factor Z 3 through the creation of virtual electron-positron pairs between the two test particles. To the lowest correctional order, the charge shift can be found to be: δz 3 = α ( ) Λ 2 3π ln. (8.35) m 2 Vertex modification Here again, we can write a complete one-particle irreducible three-point function: Λ µ (p, p ) = γ µ + Γ µ (p, p ). (8.36) Again we find a renormalization factor Z 1 that can be absorbed into a re-normalized physical charge as follows: e = e 0 Z 1. (8.37) 120

127 8.4 Renormalization With this the vertex modification (in lowest correctional order) becomes: iγ µ (p, p ) = i [ γ µ ( Z ) + Z 1 1 Γ µ C(p, p ] + O ( e 5) (8.38) with Λ µ C (p, p ) the finite radiative correctional term. The vertex modification is closely related to the electron self-energy 24, as can be shown by inserting an external photon line of zero momentum into the internal electron propagator of an electron self-energy diagram: pffyf pf pffyf pf q=0 } so that it becomes a vertex modification diagram 25, the mathematical equivalent to the above diagrammatic representations being: iσ lowest (p) ieγ µ lowest (p, p). (8.39) These expressions simplify to: 1 /p m + iϵ 1 /p m + iϵ γµ 1 /p m + iϵ = p µ 1 /p m + iϵ (8.40) if the photon loop is not taken into account. By attaching an external photon to an electron propagator carrying the external charge flow (carrying the electron momentum flow), yet another form of the Ward identity is found (by comparing equations (8.39) and (8.40): Γ µ (p, p) = p µ Σ (p) (8.41) which can be shown to hold for all orders of perturbation theory. 24 as the electromagnetic coupling is introduced through the minimal substitution /p /p /A 25 if the electron propagators are shifted 45 upwards, the right-hand diagram looks exactly like the usual depiction of the vertex modification 121

128 8 Radiative Corrections With the help of equation (8.21) we find: p µ Σ (p) = [ Z ] γ µ (8.42) which we can plug into equation (8.41) together with equation (8.38) evaluated at p = p : Γ µ (p, p) = [ Z ] γ µ (8.43) to receive: Z 1 = Z 2. (8.44) We can now look at the combined effect of all charge renormalizations at a vertex 26 : Z3 Z 2 e = e 0 = e 0 Z 3 (8.45) Z 1 and find that the charge renormalization is caused solely by the vacuum polarization which can also be seen by rewriting the Lagrangian doing a transformation A 1 e A: L = ψ [iγ µ ( µ iea µ ) m] ψ 1 4 F µνf µν ψ [iγ µ ( µ ia µ ) m] ψ 1 4e 2 F µνf µν (8.46) where the last term corresponds to the inverse of the photon propagator and is the only term containing a factor e after the transformation. So it is only within the photon propagator that the charge can be modified. External lines The insertion of electron or photon self-energies into external lines does not lead to finite radiative correction. The only effect is the renormalization of the mass and the charge each vertex has one photon and two electron lines attached 27 for the explicit derivation see f.ex. Mandl Shaw, section

129 8.5 Applications Summary We have evaluated three diagrams and found three divergent constants. These are Z 1 = Z 2 (equal due to Ward identity 28 ), Z 3 and δm 29. The number of diagrams corresponds to the freedom of choice to select a measurable mass, coupling constant and pole residues as the quantities to be re-normalized. By re-normalizing, a shift from bare (infinite) to physically measurable (finite) properties via an infinite renormalization factor (that can theoretically be computed to any order) is performed. The re-normalized functions are small, finite corrections to the free functions. 8.5 Applications The finite radiative corrections of self-energies and vertex modification that are gained after the regularization and renormalization has been performed, can be experimentally verified. The comparison of theoretical to experimental results act as precision tests of quantum electrodynamics. Two famous examples shall be shown here Anomalous magnetic moment For this effect, only one type of diagrams has to be studied: the vertex modification which represents the electron s response to a given applied field. The effects of diagrams that are only modified in the external legs are absorbed into the mass and charge renormalization, while the photon self energy diagrams (= vacuum polarization diagrams) only correct the electromagnetic field itself. We consider scattering from a static external vector potential A µ ex (0, q): 28 as well as gauge dependent and infrared divergent, see Itzykson Zuber, chapter 7 29 both gauge independent and infrared finite 123

130 8 Radiative Corrections d { e d e de } x} which can be interpreted as the Born approximation to the scattering of an electron from a potential well which is produced by the magnetic moment interaction: V (x) = µ B x. (8.47) The vertex modification contribution can be written in the following form 30 : ( Γ µ (p, p ) = γ µ F ) 1 q 2 + iσµν q ν 2m F ( ) 2 q 2 (8.48) where the two unknown functions F 1 (q 2 ) and F 2 (q 2 ) are called form factors and contain the information about the interaction of the electromagnetic field with the electron. While F 1 (0) = 1 (in all orders of perturbation), is the charge of the electron in units of e, the form factor F 2 (0) will give the correction to the electron s magnetic moment. If we now consider the amplitude of this function for scattering from the external field A µ ex (0, q): [ ( ) ] im = +ie u (p ) γ i F 1 + iσiν q ν 2m F 2 u (p) Ã i ex (0, q), (8.49) we find after some computation 31 : im = e m [F 1 (0) + F 2 (0)] S B (0, q) (8.50) with S the electron spin. We can compare this with (8.47) to find: µ = e m [F 1 (0) + F 2 (0)] S, (8.51) 30 for derivation see Schroeder Peskin, sections 6.2 and section see Peskin Schroeder, pages 187,

131 8.5 Applications or in the standard form: so that the Land g-factor is given by: ( ) e µ = g S (8.52) 2m g = 2 [F 1 (0) + F 2 (0)] = 2 + 2F 2 (0). (8.53) We see that the anomalous magnetic moment is found by determining the form factor F 2 (0) to a specific order in perturbation theory to gain g = 2 + O (α). The leading term gives F 2 (0) = 0, so that in this order, the found g-factor is that which was predicted by the Dirac equation. The following table shows the corrections to the Dirac g-factor as a function of the order of perturbation and the number of Feynman graphs that had to be evaluated in each case to compute these shifts: The computational value of of the highest (10 th) order perturbation correction is taken from the following paper: Testing the QED contribution to the anomalous magnetic moment: (Aoyama, Hayakawa, Kinoshita, Nio, Physical Review Letters 109, (2012)) Lamb Shift As a result of the interaction between the vacuum and the electron the 2 s 1/2 and 2 p 1/2 states in the hydrogen atom are no longer degenerate as in the Dirac theory: Figure 8.3: Bound state levels in hydrogen according to different theories 125

132 8 Radiative Corrections The total level shift, in full consistency of theory and experiment, is about 1058 MHz. All types of correctional Feynman diagrams contribute to this shift, but as we are now considering bound states scattering from a vector potential (Coulomb potential), it is mainly the s-state electrons with their non-vanishing probability to be in the center, that experience the shift. The biggest contribution is a big positive energy shift of the s-state atomic electrons due to the mass renormalization by the electron self-energy graphs. The vacuum polarization (photon self-energy) on the other hand, creates a small downward shift of the s-state atomic electrons of 27 MHz. The virtual electronpositron pairs experience a polarization through the external Coulomb field and although this leads to a screening of the nuclear charge by the virtual electrons, the s-state atomic electrons can penetrate the screening and therefore exhibit a small downward energy shift as compared to the p-state atomic electrons as the unshielded coupling to the bare charge is more and more retrieved at very small relative distances. Mathematically this amounts to looking at a scattering process involving dynamical polarization with a small momentum transfer q 2 = q 2 m 2 which corresponds to a modification of the Coulomb law in Fourier space by the replacement: e 2 0 q e q 2 (1 Π (q 2 ) = e 2 [ q 2 (1 Π (q 2 ) + Π (0)) e2 1 α q 2 ] q 2 15π m 2 (in the leading order). (8.54) In configuration space, the potential of an infinitely heavy nucleus of charge Ze is therefore modified in the following way: ( V (r) = Ze2 0 4πr 1 + α ) 2 Ze 2 15πm 2 4πr = Ze2 4πr α Z 2 15π m 2 δ3 (r) (8.55) with the delta function indicating that the correction is strongly peaked at small distances r. We can now use first order perturbation theory to calculate the energy shift: E n,l = Ze2 α 15πm 2 Ψ n,l (0) 2 = Z2 α 2 m 8Z 2 α πn δ l,0, (8.56) 3 126

133 8.5 Applications and explicitly for the 2 s state in the hydrogen atom (n = 2, l = 0, Z = 1): ν = E 2,0 h = 27 MHz. (8.57) 127

134 8 Radiative Corrections F e B e type superficial degree effective degree of divergence of divergence 0 2 gcg 2 0 photon self-energy 0 3 vfku de } 1 cancels out due to Fury s theorem strongly convergent photon-photon scattering (see Itz. Zuber; sec ) 2 0 ffyff 1 0 electron self-energy 2 1 ffyff } 0 0 vertex contribution Table 8.1: Superficially divergent one-loop diagrams Dirac g = 2 Schwinger (1948) (α) g = graph (α 4 ) g = graphs (α 10 ) g = (77) graphs experiment g = (0.28) Table 8.2: Theoretical and experimental values for the Landé g-factor 128

135 Chapter 9 Renormalization 9.1 Motivation When we calculate scattering amplitudes at higher orders in perturbation theory, it turns out that the Feynman diagrams contain divergent integrals. In this chapter the method called Regularization is presented which makes these integrals finite by introducing a high momentum cutoff without violating gauge and Lorentz invariance. This high momentum cutoff is somehow arbitrary while physical predictions like scattering amplitudes or correlation functions must be cutoffindependent. The procedure to make physical predictions cutoff-independent again after Regularization is called Renormalization. 9.2 The Superficial Degree of Divergence and Renormalizability Renormalizability of QED When we want to calculate scattering amplitudes at higher order in perturbation theory we have to compute a sum of Feynman diagrams. In a Feynman diagram each loop corresponds to a potentially divergent integral over the whole momentum space. 129

136 9 Renormalization d 4 k 1 d 4 k 2 d 4 k L (/k i m) (k 2 j ) (k 2 n) (9.1) The internal lines of these loops correspond to propagators which are part of the integrand. In QED, the electron propagator carries the momentum to the power of one in the denominator and the photon propagator carries the momentum to the power of two in the denominator. These propagators weaken the strength or even compensate the divergence which occurs because of the integration. Since the divergences occur due to the integration over the high momentum degrees of freedom the divergences are called UV-divergences. The divergent integrals can be cured by introducing a high momentum cutoff as upper integration boundary. Such a cutoff can be used to measure the strength of the divergence. Simple examples of one dimensional integrals can illustrate this procedure: dk k dk k Λ dk log Λ (9.2) k Λ dk k Λ 2 (9.3) We call the first integral logarithmically divergent and the second integral quadratically divergent. The one dimensional integrals can also justify the following definition of the Superficial Degree of Divergence (SDD): D (power of k in numerator) (power of k in denominator) (9.4) = 4L P e 2P γ (9.5) As an example we calculate the SDD of an integral occurring as a so called radiative correction in QED, the fermion self-energy: In this Feynman diagram (fig. 9.1) the loop corresponds to the integral: iσ 2 (p) = ( ie) 2 d 4 k i ig 2π 4 /p /k m + iϵ γµ µν k 2 + iϵ γν d 4 k k 3 (9.6) 130

137 9.2 The Superficial Degree of Divergence and Renormalizability Figure 9.1: The fermion self-energy For the SDD we obtain: D = 4 3 = 1 (9.7) In QED the overall SDD of a Feynman diagram can be calculated just by counting the number of loops (L), the number of internal fermion propagators (P e ) and the the number of internal photon propagators (P γ ). The SDD turns out to be D = 4L P e 2P γ. (9.8) Although it seems sensible to assume that the cutoff-dependence of a Feynman diagam could be calculated in the manner Λ d 4 k 1 d 4 k 2 d 4 k L (/k i m) (k 2 j ) (k 2 n) ΛD (9.9) it turns out that this assumption is often wrong. In fig. 9.2 some simple QED diagrams illustrate this difference between the SDD and the actual divergent behavior of the diagrams. Nevertheless the SDD turns out to be a useful tool in order to estimate whether a quantum field theory is renormalizable or not or in other words whether we are able to calculate scattering amplitudes also at higher orders in perturbation theory or not. In order to determine the renormalizability of a QFT it is necessary to express the SDD not in terms of the number of loops and internal propagators like in eq. 9.5, but in terms of the external lines and the number of vertices. Since in section it will be necessary to do calculations not only in our fourdimensional spacetime, we will do this replacement in an arbitrary dimension of spacetime d. The following abbreviations will be used to determine the SDD in QED: 131

138 9 Renormalization Figure 9.2: QED diagrams which illustrate the difference between the SDD (middle column) and the actual divergent behavior (cutoff-dependence, right column): In the first diagram D = 0 but the diagram is finite since there is no integration over the momentum space. In the third diagram the divergence is weakened due to the ward identity so it is logarithmically divergent although it is D = 2. In the forth diagram D is negative which could naively considered to be a hint that the diagram is finite, but it contains a divergent sub-diagram making it logarithmically divergent. Just in the second and the fifth diagram the SDD match with the actual degree of divergence. Source [2]. 132

139 9.2 The Superficial Degree of Divergence and Renormalizability D = superficial degree of divergence d = dimension of spacetime P e, P γ = number of electron (photon) propagators N e, N γ = number of external electron (photon) lines V = number of vertices In a D-dimensional spacetime we have to take d times the number of loops in eq. 9.5: D = dl P e 2P γ (9.10) In the original Feynman rules each propagator has an integral. Each Vertex however carrys a delta function for momentum conservation which reduces the number of integrations. Since one of these delta functions enforces the overall momentum conservation, there remain a single integration for each loop. This consideration leads to the expression L = P e + P γ V + 1. (9.11) On the other hand we know that at each vertex a single photon line and two electron lines meet. Therefore we are able to express the number of vertices with the number of lines: V = 2P γ + N γ = 1 2 (2P e + N e ) (9.12) The number of the propagators counts twice in these equations since they connect two vertices. Now we can use eq.9.11 and 9.12 in order to eliminate L,P e and P γ in eq We find an expression for the SDD: ( ) ( ) ( ) d 4 d 2 d 1 D = d + V N γ N e (9.13) Now consider a QED in less than four dimensions. Then the factor in front of the number of vertices in eq becomes negative. Remember that the number of vertices correspond to the order in perturbation theory. So it turns out that in 133

140 9 Renormalization this case at sufficiently high order in perturbation theory the additional Feynman diagrams of the scattering amplitudes have a smaller and smaller (finally negative) SDD. In this case we can be sure that at sufficiently high order in perturbation theory there won t be any divergent diagrams. Such a QFT we call a Super- Renormalizable Theory. In the case of a four-dimensional spacetime the dependence on V vanishes completely in eq and we obtain: D = 4 2N γ 3 2 N e (9.14) Since there is no dependence on the number of vertices in this case, a divergent amplitude stays divergent also at higher orders in perturbation theory. But fortunately due to the dependence on the number of external legs there should be a finite number of divergent scattering amplitudes. Such a QFT we call a Renormalizable Theory. Now consider the case of a spacetime with more than four dimensions. The factor in front of V in eq is positive. This means that at higher orders in perturbation theory the SDD of the additional diagrams of all scattering amplitudes becomes larger and larger. In such a case we can be sure that finally every scattering amplitude becomes divergent at a sufficiently high order in perturbation theory. We can t cope with such a situation with the methods of regularization and renormalization. Such a theory is called a Non-Renormalizable Theory. These definitions of Renormalizability should be summarized: Super-Renormalizable theory: Only a finite number of Feynman diagrams superficially diverge. Renormalizable theory: Only a finite number of amplitudes superficially diverge, but divergence occur in all orders in perturbation theory. Non-Renormalizable theory: All amplitudes are divergent at a sufficiently high order in perturbation theory. As explained in a four-dimensional spacetime QED turns out to be renormalizable. Since the SDD is independent of the number of vertices, there is a finite number of superficially divergent amplitudes. Fig. 9.3 shows these seven superficially 134

141 9.2 The Superficial Degree of Divergence and Renormalizability Figure 9.3: The seven superficially divergent amplitudes in four-dimensional QED. Diagram (a) describes an unobservable shift in the vacuum energy. This diagram is irrelevant to scattering processes, diagrams (b) and (d) vanish because of symmetries, in diagram (e) (photon scattering) the divergent parts cancel due to the Ward-identity, the diagrams (c), (f) and (g) turn out to be logarithmically divergent (even though D > 0 for (c) and (f)). The latter three amplitudes have to be regularized and renormalized in order to compute QED-scattering processes in arbitrary high order of perturbation theory. Source [2]. divergent amplitudes as well as the three amplitudes which turn out to be actually divergent Renormalizability of the ϕ n -Theory The computations of regularization and renormalization in QED are very complicated. Since this complexity in calculations distract the effort to understand the basic ideas of the renormalization theory this chapter will just deal with the renormalization of the scalar ϕ 4 -theory (For Renormalization in QED please read [9]). But at first we investigate the renormalizability of a scalar ϕ n -theory in a d-dimensional spacetime. The Lagrangian density of such a QFT is L = 1 2 ( µϕ) m2 0ϕ 2 λ 0 n! ϕn (9.15) 135

142 9 Renormalization In the scalar ϕ n -theory just two Feynman rules, a propagator corresponding to the internal lines and a coupling constant (λ 0 ) corresponding to the vertices, have to be taken into account in order to calculate the loop integrals: pf = i p 2 m iϵ de ed = iλ 0 The vertex with for legs occurs in this manner in the ϕ 4 -theory, in a ϕ n -theory this vertex would carry accordingly n legs. In order to calculate the SDD we introduce the following abbreviations: d = dimension of spacetimep = number of internal lines/propagators (9.16) L = number of loops (9.17) N = number of external legs (9.18) V = number of vertices (9.19) Since the propagator contains the momentum squared in the denominator the SDD is calculated with the equation D = dl 2P. (9.20) Again we can replace the number of propagators with the number of vertices and external legs: L = P V + 1 (9.21) nv = N + 2P (9.22) The former equation follows the considerations we did when we calculated the SDD in QED. The latter equation is valid because at each vertex of a ϕ n -theory n lines meet. In this context the number of propagators count twice since they 136

143 9.2 The Superficial Degree of Divergence and Renormalizability connect two vertices. Replacing P and L in eq with the results of eq and 9.22 we obtain [ ( ) ] ( ) d 2 d 2 D = d + n d V N (9.23) 2 2 We are especially interested in the case of a four-dimensional spacetime. It turns out that in this dimension n < 4 would produce a negative factor in front of V (number of vertices). With the considerations we discussed in section 9.2.1, we identify this case with a super-renormalizable theory. In the case d = 4 and n = 4 the number of vertices vanish completely in eq This theory therefore turns out to be renormalizable. For d = 4 and n > 4 however the factor in front of V becomes positive and we end up with a non-renormalizable theory. This is exactly the reason why we are always talking about the ϕ 4 -theory. This theory is interesting enough (n 2-theories are non-interacting theories, a ϕ 3 -theory has a lack of symetry) while it stays renormalizable. Nevertheless we should keep in mind that the ϕ 4 -theory becomes super renormalizable in less than four dimensions. This means that our loop-integrals could become finite if we calculate them in less then four dimensions. This fact turns out to be useful for the certain procedure of regularisation (dimensional regularization sec ). Now I want to discuss another approach to the question whether a QFT is renormalizable or not. In the system of units which is usually used in QFT it is c = ħ = 1. Therefore every quantity can be expressed in the units of the mass. For example the energy has the dimension [E] = [mass]. We will count now the dimension of the quantities. Therefore we define the mass-power-exponent [mass d ] d. For the length x we obtain [x] = 1 [mass] 1 and therefore [dxd ] d. Since the action has to be dimensionless the dimension of the Lagrangian density is [L] d. The kinetic term of the Lagrangian density furthermore tells us that [ϕ] d 2 2. The coupling term in the Lagrangian density has to have the same dimension as the Lagrangian density: [λϕ n ] d. Now we can calculate the dimension of the coupling constant: ( ) d 2 [λ] = d 2 (9.24) 137

144 9 Renormalization It turns out that the dimension of the coupling constant is exactly the factor which occurs in eq in front of the number of vertices. A similar calculation can be carried out also in other QFTs. Therefore we are able to formulate another possibility to estimate the renormalizability of a QFT: Coupling constant with positive mass dimension [λ] > 0: Super-Renormalizabile Quantum Field Theory Dimensionless coupling constant [λ] = 0: The theory can either be renormalizable or not. In the case of the ϕ < n-theory we end up with a renormalizable theory Coupling constant with negative mass dimension [λ] < 0: Non-Renormalizabile Quantum Field Theory In QED the dimension of the coupling constant is the dimension of the electric charge. In our chosen system of units it is [e] = 0 and therefore we end up with a renormalizable QED. Now have a look on a quantum field theory of Gravitation. The coupling constant of such a theory would be λ grav = Gm. With G the Gravitation Constant with the dimension [G] = 2. We end up with a negative dimension of the coupling constant [Gm] = 1. Unfortunately a quantum field theory of gravitation turns out to be non-renormalizable. In order to give a more physical argument for the fact that [λ] < 0 leads to a nonrenormalizable theory consider a perturbation series of a scattering amplitude. In higher orders higher powers of the coupling constant occur: Since we have to keep the right overall dimension of the sum there has to occur a factor in each addend which compensates the dimension of the coupling constant. This factor can just be built with the cutoff of the integration which has the dimension [Λ] = 1. In the QFT of gravitation a perturbation series of a scattering-amplitude would show the following behavior. M λ grav + λ 2 gravλ + λ 3 gravλ 2 + λ 4 gravλ 3 (9.25) In higher orders the cutoff-dependence become worse and worse. Such a cutoffdependence can t be renormalized. 138

145 9.3 Regularization and Renormalization: Bare Perturbation Theory vacuum energy shift Λ 2 + p 2 log Λ + finite terms log Λ + finite terms Figure 9.4: The divergent amplitudes in ϕ 4 -theory and their actual dependence on a high momentum cutoff. Just the two- and four-point-function need to be regularized and renormalized since the vacuum energy shift is unobservable 9.3 Regularization and Renormalization: Bare Perturbation Theory In this section the whole procedure of regularization and renormalization will be carried out in the ϕ 4 -theory. As discussed in the previous section this theory is renormalizable and therefore there exists a finite number of divergent amplitudes in this theory and these amplitudes are divergent in all orders of the perturbation series. The three actually occurring divergent amplitudes are shown in fig Since one of these amplitudes describe an unobservable shift in the vacuum energy we end up with just two amplitudes which need to be regularized and renormalized: the two-point-function and the four-point function. There are two equivalent methods of Renormalization, called Bare Perturbation Theory and Dressed Perturbation Theory This section deals with the bare perturbation theory The Four-Point-Function: Pauli-Villars Regularization There are several equivalent methods to regularize a divergent integral which means an introduction of a cutoff which makes the integral finite. In order to regularize the four-point-function we will use a method called Pauli-Villars- Regularization. 139

146 9 Renormalization In second order perturbation theory the four-point-function which describes the scattering process of two scalar particles is a sum of four Feynman-diagrams. The three diagrams containing a loop turn out to be divergent. The first diagram correspond to the following integral: ( iλ)2 2 d 4 k (2π) 4 i k 2 m 2 i (k + p) 2 m 2 (9.26) But before we start the regularization we introduce a new system of coordinates which simplifies the calculations: The Mandelstam-Coordinates. s = p 2 = (p 1 + p 2 ) 2 (9.27) t = (p 1 p 3 ) 2 (9.28) u = (p 1 p 4 ) 2 (9.29) The Mandelstam coordinate s describes the center of mass energy. The coordinates t and u describe the scattering angles. In order to calculate eq.9.26 we do the Wick-rotation in order to end up in a Euclidean space in which we can easily introduce spherical coordinates. After the Wick-rotation the integral becomes M 2 = ( iλ)2 i 2 2 d 4 k 1 1 (9.30) (2π) 4 k 2 m 2 (p k) 2 m 2 Now we use the so called Feynman-trick to transform our product in the denominator of the integrand into a sum: 1 1 xy = 1 dα (9.31) 0 (αx + (1 α)y) 2 Using this equation, shifting our integration variable k k + pα and introducing the abbreviation c 2 = m 2 α(1 α)p 2 we end up with the integral 140

147 9.3 Regularization and Renormalization: Bare Perturbation Theory 1 0 dα d 4 k 1 (9.32) (2π) 4 (k 2 c 2 + iϵ) 2 So far we just simplified the integral, we haven t regularized anything. This integral is still divergent. Now we come to the important step called regularization. The integral will be made finite by introducing a cutoff. In Pauli-Villars- Regularization this cutoff occurs as a high momentum cutoff. Since the integral is divergent because of the integration over the high momentum degrees of freedom, the physical interpretation of what we actually do by regularizing an integral is quite obvious when we use the Pauli-Villars-method: We ignore the high momentum degrees of freedom. Nevertheless in the most QFTs we are not allowed to introduce the cutoff as an upper integration bound as we have done it in the simple one-dimensional examples eq This is due to the fact that such a procedure would violate gauge invariance. The method of Pauli-Villars-Regularization introduces the high-momentum cutoff Λ in an additional addend in the integrand (we ignore the α-integration for a moment): d 4 [ ] k 1 (2π) 4 (k 2 c 2 + iϵ) 1 2 (k 2 Λ 2 + iϵ) 2 R(Λ 2, c 2 ) (9.33) For Λ going to infinity we obtain again our divergent integral. For small k 2 the second addend can be neglected since Λ is of a huge value. On the other hand for k 2 even much larger than Λ the two addends cancel each other. All in all the second addend make the integrand vanish slowly when we go to high momentum. Calculating the derivative of eq one time after c 2 and one time after Λ 2 we end up with integrals which are listed in integration tables: R c = d 4 k 2 2 (2π) 4 (k 2 c 2 + iϵ) = i (9.34) 3 16π 2 c 2 R Λ = i (9.35) 2 16π 2 Λ 2 R = i ( ) Λ 2 16π log (9.36) 2 c 2 M 2 = iλ2 ( 1 Λ 2 ) dα log (9.37) 32π 2 0 m 2 α(1 α)p 2 + iϵ 141

148 9 Renormalization ( ) Λ = icλ 2 2 log = ( iλ) 2 iv (s, Λ) (9.38) s In the last step when we introduced the function V (s, Λ) as an abbreviation for the (now finite) integral we remembered that p 2 is nothing else than the Mandelstam coordinate s. For the other two loop-integrals contributing to the second order in perturbation series we obtain exactly the same result except that we have to replace s with the other Mandelstam-coordinates u and t. The whole scattering amplitude in second order perturbation theory becomes: im = iλ 0 + ( iλ 0 ) 2 [iv (s, Λ) + iv (t, Λ) + iv (u, Λ)] (9.39) Renormalization of the four-point-function By introducing a high momentum cutoff without violating either the gauge invariance or the Lorentz invariance we obtained a finite scattering amplitude for the four-point-function. This amplitude depends on a high but yet arbitrary cutoff Λ. Physical predictions should of course be independent of this cutoff. The procedure used to make the physical prediction cutoff independent is called renormalization. Let s have a look on equation As explained the cutoff Λ is not well defined. Fortunately there is another constant which is so far not well defined: the constant λ 0 is a constant which was introduced in the non-interacting theory. The coupling constant shall be a measure for the strength of the interaction between two particles. The value of the coupling constant therefore must be determined in an experiment. A suitable experiment for the determination would be a scattering process between two particles with certain initial conditions. The coupling constant can be nothing else than the scattering amplitude we measure with these initial conditions. In short: before we can calculate anything we have to determine our coupling constant in a scattering experiment. Therefore we have to choose some arbitrary initial conditions which means a certain initial set of Mandelstam coordinates s 0, u 0 and t 0. This set of Mandelstam coordinates is called the renormalization point. The value of the coupling constant depends on the renormalization point. With this considerations it turns out, that λ 0 isn t the physical coupling constant, which is from now on called λ p : 142

149 9.3 Regularization and Renormalization: Bare Perturbation Theory iλ p = iλ 0 + ( iλ 0 ) 2 [iv (s 0, Λ) + iv (t 0, Λ) + iv (u 0, Λ)] (9.40) This equation we change after λ 0 and take just the terms of first and second order in λ P into account (we want to calculate the second order in perturbation series). iλ 0 = iλ P + ( iλ P ) 2 [iv (s 0, Λ) + iv (t 0, Λ) + iv (u 0, Λ)] = iλ p Z λ (Λ) (9.41) Equation 9.39 isn t a physical equation because it contains an unphysical constant Λ. Since λ 0, the so called bare coupling constant, wasn t well defined it can also be regarded as an unphysical constant. So far we have outsourced these two unphysical constants into an unphysical equation Although this equation is not physical it redefines λ 0. If we use this equation in order to replace λ 0 in eq we end up with a scattering amplitude which contains neither Λ nor λ 0 but the physical coupling constant λ P defined in a scattering experiment: M = iλ P + icλ 2 P [ log ( ) ( ) s0 t0 + log + log s t ( )] u0 u (9.42) Since we are calculating scattering amplitudes in perturbation series the predictions of equation 9.42 are just valid if s, u and t do not differ too much from the renormalization point s 0, t 0, u 0 we used to define the value of the coupling constant. For low energy scattering processes a suitable renormalization point is t 0 = u 0 = 0 (the four external legs are of equal length) and s 0 = 4m 2 (center of mass in in rest) The Two-Point-Function: Dimensional Regularization In first order perturbation theory there occurs a correction to the propagator: fpf = kf + + kf = i k 2 m 2 + iϵ + i ( iσ(k2 )) + k 2 m 2 + iϵ (9.43) The loop corresponds to a divergent integral. As explained in the previous sections the ϕ 4 -theory is super-renormalizable in less than four dimensions. That s 143

150 9 Renormalization why we can calculate the integral in a spacetimedimension d < 4 in which it is not divergent at all. This consideration leads to a procedure called dimensional regularization: iσ(p 2 ) = i λ 0 2 d d p 1 (2π) d p 2 m iϵ (9.44) After a Wick-rotation we obtain a Euclidean metric and are able to introduce spherical coordinates. The integral we want to calculate is of the form: I = d d p p 2 m = 2 dω d 1 pd 1 dp (9.45) p 2 m 2 Without going through the calculations (see also [10]) in detail I present the result for equation 9.44: iσ(p 2 ) = iλ 0m 2 0Γ(1 (d/2)) iλ 0m 2 ( ) (9.46) 32π 2 16π 2 4 d So far d was the dimension of spacetime and therefore a natural value. But now we end up with an equation which is not defined just for natural values but for all real values d < 4. For d = 4 we obtain again an infinite result. In this procedure of dimensional regularization we introduced our cutoff by choosing a real-valued spacetime dimension very close to four. Since the physical meaning of the Pauli-Villars-cutoff is much more obvious from now on we consider the two point function also to be Pauli-Villars-regularized. If we do the two-point-calculation more in detail taking also repeated loops into account we end up with the expression: = i iz ϕ (Λ) (p) = p 2 (m Σ(p 2, Λ)) + iϵ (9.47) In this equation Σ(p 2, Λ) and Z ϕ (Λ) are cutoff dependent constants which diverge when Λ goes to infinity. Renormalization of the two-point function After the regularization we are again confronted with a cutoff-dependent object. To renormalize this object we have to redefine a constant which was so far not well defined: the bare mass m 0. In order to introduce the physical mass of the particle 144

151 9.4 Dressed Perturbation Theory m P in the calculation we have again to choose something like the renormalization point. In QFT the mass of a particle is defined in its propagator. The pole of the propagator has to occure at the value of the particle mass. This physical mass can be determined in an experiment. Therefore we can identify: m 2 0 = m 2 p Σ(m 2, Λ) (9.48) The cutoff-dependent constant Z ϕ (Λ) is determined by choosing the residue of the pole to be equal one. In order to achieve that the propagator appears completely invariant after renormalization Z ϕ (Λ) is usually referred to the vertices or to the field functions in the case of the external legs (field functions are not observable). We end up with renormalized field-functions, a renormalized mass, a renormalized coupling constant and an invariant propagator: m 2 0 = m 2 p Σ(m 2, Λ) (9.49) i (p) = p 2 m 2 P + iϵ (9.50) iλ 0 = iz λ(λ)λ p (9.51) (Z ϕ (Λ)) 2 ϕ 0 = Z ϕ (Λ)ϕ P = Zϕ P (9.52) If we replace with these equations the bare constants and the bare fields in the regularized integrals in second order perturbation theory, all scattering amplitudes become finite and cutoff-independent. In many QFTs the cutoff-dependent constants (Z ϕ, Z λ, Σ) are not independent from each other. In the ϕ 4 -theory for example it turns out that Z ϕ = Z λ = Z 9.4 Dressed Perturbation Theory So far we started the calculations with the bare constant λ 0, ϕ 0 and m 0 of the noninteracting theory and redefined them in the procedure of renormalization. This method of renormalization is called bare perturbation theory. It is also possible to start with the physical constants λ P, m P and ϕ P. For this method called dressed perturbation theory we have to introduce the physical constants in 145

152 9 Renormalization the Lagrangian density. We replace ϕ 0 with equation 9.52 and introduce the so called counterterms: δ Z = Z 1 (9.53) δ m = m 2 0Z m 2 p (9.54) δ λ = λ 0 Z 2 λ P (9.55) Now the Lagrangian density depends on physical constants.. L = 1 2 ( µϕ P ) m2 P ϕ 2 P λ P 4! ϕ4 P δ Z ( µ ϕ P ) δ mϕ 2 P δ λ 4! ϕ4 P (9.56)...but contains counterterms. Feynman diagrams: These counterterms lead to new elements in the Now λ and m stand for the physical constants. Now the two particle scattering process in second order perturbation theory contains an additional diagram: + Of course the loop integrals are still divergent and we have regularize them. Now we obtain im = iλ P + ( iλ P ) 2 [iv (s, Λ) + iv (t, Λ) + iv (u, Λ)] + iδ Z. (9.57) 146

153 9.5 The Renormalization Group Again we choose a renormalization point at which we want the scattering amplitude to be equal the coupling constant (im = iλ P ). Again a natural choice for the renormalization point would be s = 4m 2, t = u = 0. In order to achieve im = iλ P the counterterm must be [ δ Z (Λ) = λ 2 P V (4m 2, Λ) + 2V (0, Λ) ] (9.58) Now the cutoff-dependence of δ Z cancels the other cutoff-dependent parts in eq and we obtain again the finite cutoff-independent scattering amplitudes for arbitrary values of s, u and t (eq The Renormalization Group When we calculate scattering amplitudes in QFT we are working in Fourier space. A high momentum cutoff in the integration therefore ignores the high momentum Fourier components. In position space these components cause fluctuations of the field amplitudes over small distances. Therefore the high momentum cutoff smoothen the field functions. The small distance fluctuations occur due to the Heisenberg relation E t ħ which allows a violation of the energy conservation over small periods of time resulting in a continuous production and destruction of virtual particles in the vacuum. All in all the cutoff ignores quantum processes occurring on small length- and timescales at high energy regions. The idea to ignore the high energy region or the atomic length scale is useful in many physical theories in order to simplify the calculations (just consider classical mechanics), however in QFT the situation is slightly different: In QFT we are not just allowed to ignore the high energy processes we even must ignore them to be able to calculate any physical prediction. The reason must be that at high energies, on small length- and timescales the so far developed QFTs loose their validity and must be replaced by a more comprehensive theory (maybe string theory or quantum loop theories). Theories in which essential interactions take place on small length scales therefore must turn out to be non-renormalizable. A deeper insight into the nature of the cutoff and in what we are doing when we choose our renormalization point gives the so called Renormalization-Group-Theory. 147

154 9 Renormalization Renormalization Group of the Ising Model Renormalization is a procedure not only useful in QFTs but also in other parts of physics for example in condensed matter physics. In order to illustrate the idea of the renormalization group I will introduce the concept using the example of the Ising model. Consider a 2D-lattice of spin- 1 -particles in which the spin 2 can just choose between two directions: up or down. An interaction between the spins occurs just between the next neighbors. Nevertheless there will of course occur a correlation also between spins separated over larger distances. From a macroscopic point of view the spin system appears as a continuum and also the correlation function can be treated as depending continuously on the position. Nevertheless the correlation function contains a natural cutoff in position space: It does not make sense to study correlations on length scales smaller than the lattice constant. If we are not interested in spin fluctuations on small distances but want to analyze the macroscopic behavior of the system we could try to increase the cutoff by building block-spins (fig.9.5): We replace nine spins in a square by just a single spin by just counting the up-spins in the square. If their are five or more of them the whole block is replaced by an up-spin or in the other case of four or less up-spins by a down-spin. In a second step we have to reduce the size of our lattice. This procedure changes the cutoff L (smallest distance on which we want to take fluctuations into account). The blockspin-transformation ignores all spin fluctuations within a spin-block. When we want to study the macroscopic behavior of the 2D-ferromagnet described by the Ising-model this procedure can simplify the calculations. What we have changed by doing the transformation is the Hamiltonian describing the lattice. The original Hamiltonian of the Ising-model is: H = J n s n s n+e + B e n s n (9.59) In this Hamiltonian n counts the spins in the lattice, e counts the next neighbors of each spin, J is the coupling constant in this model. The blockspin-transformation changes this Hamiltonian: H = R L (H) = J 1 s n s n+e + B 1 s n +... (9.60) n e n 148

155 9.5 The Renormalization Group Figure 9.5: Blockspin-transformation in the Ising-model. R L is an operator leading to a new cutoff-length L. J 1 and B 1 are now of different values than J and B. There occur also further terms in the Hamiltonian for example addends proportional to s 3. The blockspin-transformation can be continued iteratively in order to achieve larger cutoff-length: R L : H H (9.61) R L : H H (9.62) (R L R L ) : H H (9.63) Eq.9.63 gives a hint that the operation R L representing the blockspin-transformation builds a group-like structure: R L R L = R L. Since R L R L = R L there exists also a neutral element. But there are no inverse elements because with each blockspintransformation we loose informations about fluctuations on small length scales. These informations can t be regained. Therefore what we call the renormalization group is actually a semi-group. Now consider the blockspin transformation to be a transformation continuous in L. When you change L you obtain a trajectory of the effective Hamiltonian in 149