Preliminaries: what you need to know

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1 January 7, 2014 Preliminaries: what you need to know Asaf Pe er 1 Quantum field theory (QFT) is the theoretical framework that forms the basis for the modern description of sub-atomic particles and their interactions. It was born by a merge of quantum mechanics and special relativity. Examples within this framework are quantum electrodynamics (QED) which describes the interactions of electrons and photons, and quantum chromodynamics (QCD) which describes the interactions between quarks and gluons. As such, quantum field theory heavily relies on a number of previous courses, both physical and mathematical. These include: (I) Lagrangian mechanics; (II) quantum mechanics; (III) special relativity; (IV) Fourier transforms; (V) Dirac delta function; and (VI) functions of complex variables. Here I summarize some key concepts that you will need throughout this course. This will also serve to set conventions and notations. I would assume that you are already familiar with these concepts, before entering this course. If you are not comfortable with any of these concepts or terms, you should review a suitable book as soon as possible. 1. Classical Dynamics We will use both the Lagrangian and Hamiltonian formulation of dynamical systems. According to Lagrangian mechanics, a physical system is described by coordinates q a, a = 1,...,n whose dynamics are governed by a Lagrangian L = L(q a, q a,t). The action is defined by t2 S = L(q a, q a,t)dt (1) t 1 The path taken by the system is an extremal of the action, i.e., δs = 0. This leads to the equations of motion, ( ) d L L = 0 (2) dt q a q a An important result from classical dynamics that will carry over to field theories is Noethers theorem (this theorem will be reviewed in class). It states that every continuous symmetry gives rise to a conserved quantity. Consider an infinitesimal transformation, δq a = X a (q, q,t) (3) 1 Physics Dep., University College Cork

2 2 This is a symmetry if δl = 0 for all paths 1. A short proof of Noethers theorem goes as follows: consider an arbitrary deformation, δq a. We can write ( L δl = δq a + L ) ( L δ q a = d ( )) L δq a + d ( ) L δq a. (4) q a q a q a dt q a dt q a One can thus define the quantity and write the last term in Equation 4 as dq dt. Q = L q a δq a, (5) When the equations of motion are obeyed, the first term in brackets in the right hand side of Equation 4 vanishes, and we have δl = Q. Since the symmetry transformation in Equation 3, δq a = X a (q, q,t) is defined to have δl = 0, we conclude that when the equations of motion are obeyed, Q = ( L/ q a )X a is a conserved quantity, namely Q = Hamiltonian formalism An equivalent way of describing a classical system is by the use of Hamilton s formalism. We define the momentum p a conjugate to q a as One can then define the Hamiltonian of the system as p a L q a (6) H(q,p,t) p a q a L, (7) where H is to be considered a function of q a and p a rather than q a and q a. The equations of motion are now given by Hamiltons equations: q a = H p a, ṗ a = H q a. (8) Finally, let us recall the Poisson bracket, an important quantity in classical mechanics which aids the leap to the quantum theory. For any functions f(q,p) and g(q,p) on phase space, the Poisson bracket is defined by {f,g} = f q a g p a f p a g q a (9) In particular, {q a,q b } = {p a,p b } = 0, and {q a,p b } = {p a,q b } = δ b a. 1 More precisely, this condition can be relaxed to δl = df/dt, so that the action remains unchanged for all paths.

3 3 2. Quantum mechanics In classical mechanics, the state of a system is fully determined by a point in phase space, specified by the set (q a,p a ). In contrast, in quantum mechanics, the state of a system is specified by a state vector ψ ( ket in Dirac s notation) which belongs to a vector space known as Hilbert space. Given a classical system, one can construct the analogue quantum system using a prescription known as canonical quantization. Every function f acting on the phase space in a classical system is promoted to an operator F acting on the state vectors in Hilbert space. The map between the two systems is given by the relationship between Poisson brackets and commutators, In particular, we have {, } classical i [, ] quantum (10) [ˆQ a, ˆQ b ] = [ˆP a, ˆP b ] = 0 [ˆQ a, ˆP b ] = [ˆP a, ˆQ b ] = i δ b a (11) This prescription is known as canonical quantization. Note that a generic classical function f(p,q) does not uniquely define a quantum operator ˆF due to ordering ambiguities. The dynamics of a quantum system is governed by the Hamiltonian operator Ĥ. There are two different ways of viewing the dynamics. In the Schrödinger picture, the states ψ evolve in time, while the operators Ô are time independent. The dynamics is governed by the Schrödinger equation, i d dt ψ S = Ĥ ψ S, (12) where the subscript on ψ S reminds us that we are in the Schrödinger picture. In contrast, in the Heisenberg picture, the states are time independent, while the operators change with time. The relationship between the two is given by and the operators now evolve by ψ H = e iĥt/ ψ S, (13) Ô H = e iĥt/ Ô S e iht/ (14) d dtôh = i [Ĥ, Ô H ] (recall that ĤS = ĤH Ĥ). The two pictures are entirely equivalent since all correlation functions φ Ô ψ agree. In quantum field theory we will jump merrily between these two different pictures. We will also employ a third viewpoint which is something of a hybrid of the first two, known as the interaction picture. (15)

4 Note on units In this course our notation will differ slightly from that above. We will use natural units, with = c = 1. In this system, [length] = [time] = [energy] 1 = [mass] 1. Thus, the mass m of a particle is equal to its rest mass energy (mc 2 ) and also to the inverse of its Compton wavelength (mc/ ). The mass of the electron is thus m e = MeV. 3. Special relativity Note. while special relativity is taught here at 1 st and 2 nd years, we will need a more generalized treatment than done at that level. While I give here a brief summary of the subject, I strongly recommend that you read the section on special relativity in the GR lecture notes, as this is the level that is needed for QFT. 2 In special relativity, space and time are connected into one 4-dimensional space-time, known as Minkowski space. Each event is specified by 4 coordinates, x µ = (x 0,x i ), where it is understood that µ = 0,1,2,3 and i = 1,2,3 represents the spatial part of the coordinates. It is agreed that x 0 denotes the time coordinate t and the three vector x i, which will also be written as x is the spatial component. The key is that the interval, defined by (interval) 2 = ( s) 2 (c t) 2 ( r) 2 = c 2 ( t 2 ) ( ( x) 2 +( y) 2 +( z) 2), (16) is invariant, namely two observers always agree on it. Here, t is the time interval between two events, and r is the space interval between the same events. We can write the interval in a more compact form, ( s) 2 = η µν x µ x ν = η µν x µ x ν = ( dx 0) 2 d x d x, (17) µν=0...3 where in the last two equalities we used Einstein summation convention, according to which indices that appear both as superscripts and subscripts are summed over. Here, η µν is a 4 4 matrix, known as the metric, which is given by η µν = = ηµν (18) There is no need to know other parts of the GR course when taking QFT.

5 5 Important note. The sign convention here follows the sign convention in many (all that I saw) textbooks on QFT, and has the opposite sign to that used in astrophysics / general relativity. During this course, I will stick to this notation, for ease of comparison with textbooks. Furthermore, recall that we will work in units where c = 1. One immediate result is η µν η νρ = δ ρ µ. We use η µν (η µν ) to raise (lower) indices on vectors and tensors. Lorentz transformations will be represented by 4 4 matrices, denoted by Λ µ ν. For example, boost in the +x direction is written as γ γβ 0 0 γβ γ 0 0 Λ µ ν(boost along x axis) = (19) Using this notation, coordinate transformations is simple: x µ = Λ µ νx ν. Furthermore, this can be generalize, e.g., for rotation around the z-axis: cosθ sinθ 0 Λ µ ν(rotation around z axis) = 0 sinθ cosθ 0. (20) By definition, the interval is invariant under Lorentz transformation. This implies that a Lorentz transformation must fulfill: η ρσ = Λ µ ρλ ν ση µ ν. (21) A relativistic particle of mass m traces out a path x µ (s) in Minkowski space. The 4-momentum p µ is defined by p µ m dxµ (22) ds which satisfies p 2 p µ p µ = η µν (dx µ /ds)(dx ν /ds) = m 2. So writing the 4-momentum p µ = (E, p) in terms of the energy E and the 3-momentum p we arrive at the relativistic dispersion relationship for a massive particle, E 2 p 2 = m 2. (23) We will denote the energy of a particle of mass m and 3-momentum p as E p = p 2 +m 2 (24)

6 6 where we take the + square root. This quantity will feature a lot in this course! Another quantity that features prominently is the scalar product p x η µν p µ x ν = Et p x (25) A note on indices: throughout this course, we will employ Einstein s summation convention in which repeated indices are summed over. For the spacetime indices µ it will be crucial to keep track of whether they are up or down: you should never encounter expressions that look like a µ b µ!. (For the other indices, such as the a = 1,...,n index which appear on the q a, we dont need to be as careful, although its good practice to try!). Also, its worth stressing that repeated indices are dummy indices it doesnt matter what you call them. But its very important that you dont use the same pairs of dummy indices twice. For example, the expression (a µ b µ )(c µ d µ ) makes no sense even with the brackets! Avoid rampant confusion by writing a µ b µ c ν d ν to show which pairs are summed over. Mistakenly denoting multiple pairs of dummy indices with the same label will be a very easy trap to fall into in this course. 4. Fourier transforms We will be frequently changing from position space to momentum space using the Fourier transform. As always, we have to decide where the factors of 2π sit. Our convention will be d n k f(x) = (2π) f(k)e ik x (26) n so that the inverse reads f(k) = d n xf(x)e ik x (27) There is now no excuse for losing track of factors of 2π: they always accompany momentum integrals, never position integrals. Remember also that if we are doing a Fourier transform over spacetime (as opposed to just space) then k x = k 0 x 0 k x. 5. Dirac delta functions The Dirac delta function is defined by δ(x) = 0 for all x 0, and δ(x)dx = 1. (28)

7 7 The Fourier transform of the delta-function provides a useful representation that we will make extensive use of: dk δ(x) = 2π eikx. (29) In an n-dimensional space, the delta-function is given by d δ (n) n k (x) = neik x. (30) (2π) 6. Complex functions There are a few places in the course where we will need to make use of the properties of complex functions. In particular, we will need the residue theorem. Let Γ be a positively oriented (i.e. anticlockwise) simple closed contour within (and on) which a function f(z) is analytic except for a finite number of singular points z 1,...,z n in the interior of Γ. The residue theorem states that n f(z)dz = 2πi b i, (31) Γ where b i is the residue of f(z) at the singular point z i. The residue of a function with an isolated singular point z 0 is defined as the coefficient c 1 of the Laurent expansion of f(z) about z 0, f(z) = n=0 i=1 (z z 0 ) n + c 1 z z 0 + for 0 < z z 0 < R, the radius of convergence. c (32) (z z 0 ) 2