14 Renormalization 1: Basics

Transcription

1 4 Renormalization : Basics 4. Introduction Functional integrals over fields have been mentioned briefly in Part I devoted to path integrals. In brief, time ordering properties and Gaussian properties generalize immediately from paths to field integrals. The singular nature of Green s functions in field theory introduces new problems which have no counterpart in path integrals. This problem is addressed by regularization and is briefly presented in the appendix. The fundamental difference between quantum mechanics (systems with a finite number of degrees of freedom) and quantum field theory (systems with an infinite number of degrees of freedom) can be labelled radiative corrections : In quantum mechanics a particle is a particle characterized, for instance, by mass, charge, spin. In quantum field theory the concept particle is intrinsically associated to the concept field. The particle is affected by its field. Its mass and charge are modified by the surrounding fields, its own, and other fields interacting with it. An early example of this situation is Green s calculation of the motion of a pendulum in fluid media. The mass of the pendulum in its equation of motion is modified by the fluid. Nowadays we say the mass is renormalized. The remarkable fact is that the equation of motion remains valid provided one replaces the bare mass by the renormalized mass. Green s example is presented in section 4.2. Although regularization and renormalization are different concepts, Namely in section. The beginning, in section.3 The operator formalism, in section 2.3 Gaussian in Banach spaces, in section 2.5 Scaling and coarsegraining, and in section 4.3 The anharmonic oscillator.

2 2 Renormalization : Basics they are often linked together. The following handwaving argument could lead to the expectation that radiative corrections (renormalization) regularize the theory: We have noted (2.40) that the Green s functions (covariances) G are the values of the variance W evaluated at pointlike sources. In general, given a source J, W (J) = J, GJ ; (4.) it is singular only if J is pointlike. On the other hand radiative corrections surrounding a point particle can be thought of as extending the particle. Unfortunately calculations do not bear out this expectation in general. In addition to inserting radiative corrections into bare diagrams, one needs to insert counterterms regularizing the divergences. In section 6, Renormalization 3: Scaling, counterterms do not appear as a regularizing artifact, but as a consequence of working with scale dependent effective actions. 4.2 Green s example Alain onnes made us aware of a delightful paper by Green which is an excellent introduction to modern renormalization in quantum field theory. Green derived the harmonic motion of an ellipsoid in fluid media. This system can be generalized and described by the Lagrangian of an arbitrary solid in an incompressible fluid F, under the influence of an arbitrary potential V (x), x F. In vacuum (i.e. when the density ρ F of the fluid is negligible compared to the density ρ of the solid) the Lagrangian of the solid moving with velocity v under V (x) is L := d 3 x 2 ρ v 2 ρ V (x). (4.2) In the static case, v = 0, Archimedes principle can be stated as the renormalization of the potential energy of a solid immersed in a fluid; the potential energy of the system fluid plus solid is E pot := d 3 xρ F V (x) + d 3 xρ V (x) (4.3) = IR 3 \ d 3 xρ F V (x) + d 3 x (ρ ρ F ) V (x). (4.4) IR 3

3 4.2 Green s example 3 The potential energy of the immersed solid is E pot = d 3 x (ρ ρ F ) V (x). (4.5) In modern parlance ρ is the bare coupling constant, ρ ρ F is the coupling constant renormalized by the immersion. The infinite term d 3 xρ IR 3 F V (x) is irrelevant. In the dynamic case, v 0, Green s calculation can be shown to be the mass of a solid body by immersion in a fluid. The kinetic energy of the system fluid plus solid is E kin := 2 F d 3 xρ F where is the velocity potential of the fluid, If the fluid is incompressible d 3 xρ v 2 (4.6) v F =. (4.7) = 0. If the fluid is nonviscous, the relative velocity v F v is tangent to the boundary of the solid. Let n be the unit normal to, and n := n, then v F v is tangent to if n = v n on = F. (4.8) Moreover and must vanish at infinity sufficiently rapidly for 2 to be finite. Altogether satisfies a Neumann problem F d3 x = 0 on F, ( ) = 0, n = v n on F, (4.9) where n is the inward normal to F. If ρ F is constant, the Green formula d 3 x 2 = d 2 σ n (4.0) F together with the boundary condition (4.8) simplifies the calculation of E kin (F ), namely E kin (F ) = 2 ρ F d 2 σ ( ) v n. (4.) F F

4 4 Renormalization : Basics Example Let be a ball of radius R centered at the origin. The dipole potential ( r) = 2 ( v r) (R/r)3 (4.2) satisfies the Neumann conditions (4.9). On the sphere = F, and (R) = R v n (4.3) 2 E kin (F ) = 2 ρ F R d 2 σ ( v n) 2. (4.4) 2 By homogeneity and rotational invariance d 2 σ ( v n) 2 = 3 v 2 d 2 σ = 3 v 2 4πR 2 and E kin (F ) = 2 ρ F vol () 2 v 2 (4.5) with vol () = (4/3) πr 3. Finally the total kinetic energy (4.6) is ( E kin = ρ + ) 2 ρ F vol () 2 v 2. (4.6) The mass density ρ is renormalized ρ + 2 ρ F by immersion. In conclusion, one can say that the bare Lagrangian d 3 x 2 ρ v 2 ρ V (x) (4.7) is renormalized by immersion to ( ( d 3 x ρ + ) ) 2 2 ρ F v 2 (ρ ρ F ) V (x). (4.8) Green s calculation for a vibrating ellipsoid in a fluid media leads to a similar conclusion. As in (4.6) the density of the vibrating body is increased by a term proportional to the density of the fluid, obviously more complicated than ρ F /2. From Archimedes principle, to Green s example, to the Lagrangian formulation, one sees the evolution of the concepts used in classical mechanics: Archimedes gives the force felt by an immersed solid, Green gives the harmonic motion of an ellipsoid in fluid media, and the Lagrangian gives the kinetic and the potential energy of the system. In quantum physics, action functionals defined

5 4.3 From path to field integrals 5 in infinite dimensional spaces of functions play a more important role than Lagrangians or Hamiltonians defined on finite dimensional spaces. Renormalization is often done by renormalizing the lagrangian. Renormalization in quantum field theory by scaling (section 6), on the other hand, uses the action functional. 4.3 From path to field integrals Before applying functional integrals to renormalization in Quantum Field Theory, we review briefly key features of field integrals. Two lessons from path integrals can be applied readily to field integrals: Time-ordering by path integrals Gaussian properties of path integrals. In quantum mechanics, one integrates functionals of functions x defined on a line T = [t a, t b ]; the integrals represent either solutions of Schrödinger s equation (see section 6) or matrix elements of operators on Hilbert spaces. For example: 2πi b, t b T (x(t)x(s)) a, t a Q = D s,q (x) exp X a,b h Q(x) x(t)x(s) (4.9) where x is a function and x(t) is an operator operating on the state a, t a. The matrix element on the l.h.s. is the probability amplitude that a system (action functional Q) in the state a at time t a be found in the state b at time t b. This path integral gives the 2-point function of the Gaussian Wiener integral (see section 3. and eq. 2.47) when s =, h =, Q(x) = 2 T dtẋ(t)ẋ(s): { t ta if t < s 0, t b T (x(t)x(s)) 0, t a = s t a if s < t. (4.20) The path integral on the r.h.s. of (4.9) has time ordered the matrix element on the l.h.s. of (4.9). In field theory time ordering becomes causal ordering dictated by light cones: if a point x j is in the future light cone of x i, written j > i,

6 6 Renormalization : Basics then the causal (time, chronological) ordering T of the field operators φ(x j )φ(x i ) is the symmetric function which satisfies the equation T (φ(x j )φ(x i )) = T (φ(x i )φ(x j )) (4.2) { φ(xj )φ(x T (φ(x j )φ(x i )) = i ) for j > i φ(x i )φ(x j ) for i > j. In general out T (F(φ)) in S = F(φ)v S (dφ) in,out (4.22) where F is a functional of the field φ (or a collection of fields) and v S (dφ) is a volume element that we shall characterize in the following. Volume Elements Recall the Gaussian volume element R dγ s,q (x) = D s,q (x) exp ( πs ) Q(x) (4.23) where D s,q (x) has the following properties D s,q is translation invariant D s,q has no physical dimension. If Q(x) = Q ij x i x j then D s,q (x) = (det Q ij ) /2 D n= dx n for x IR D. (4.24) The guiding principles for determining the volume element v S (dφ) are: Schwinger s variational principle (.) δ(a B) = 2πi A δs/h B, h = 2π. (4.25) Dirac s quantum analogue of the classical action function (a.k.a Hamilton s principal function) adds to the real classical action function an imaginary part of order [I.3]. orrespondingly the quantum action

7 4.3 From path to field integrals 7 functional Squ is, up to terms of order 2, the sum of the classical action functional S cl and an imaginary term of order S qu = S cl + iσ + O( ) (4.26) = S cl i log µ + O( ) µ := exp( σ) + O( ).(4.27) A volume element v(dφ) convenient for integration by parts, namely, v(dφ) = exp A(φ)[dφ] (4.28) where [dφ] is translation invariant; i.e. [dφ] is characterized by δf (φ) [dφ] = 0. (4.29) δφ(x) Henceforth the quantum action functional S (φ) := S cl (φ) ı log µ (φ) + O ( 2) ; (4.30) Guided by the characterization of volume elements in path integrals, we propose the following characterization of v S (φ) for a system of quantum action functional S where ( δf (φ) δφ (x) + 2πı δs (φ) F (φ) h δφ (x) δf (φ) = ) v S (dφ) = 0. (4.3) δf (φ) δφ (x) δφ (x) dd x, x IR D. (4.32) This equation gives a characterizations of v S (dφ) in terms of the unknown functional µ (φ) as desired in the guiding principle (4.26): 2πı v S (dφ) = µ (φ) exp h S cl (φ) [dφ] (4.33) 2πı = exp h S qu (φ) [dφ]. The characterization (4.33) of the volume element v S (dφ) is similar to the characterization of the volume element v g (x) on a D dimensional Riemannian manifold with metric g given in section 7.4.

8 8 Renormalization : Basics Equation (4.3) says Application F = δs (φ) δφ (x) v S (dφ) = 0 (4.34) which translates (4.22) in terms of the matrix element of a time ordered operator to δs (φ) 0 = δφ (x) v S (dφ) = out δs (φ) T δφ (x) in. (4.35) in, out Eq. (4.35) is a particular case of Schwinger s variational principle (4.25) when the initial and final states are kept fixed. It is the quantum version of the classical equation of motion S cl φ (x) = 0 (4.36) where the classical action S cl is the limit of the quantum action when 0 S = S cl ı log µ + O ( 2). (4.37) When F, the term F can be used to account for the variation of A and B in a matrix element B M A represented by a functional integral. Most often we choose the domain of integration A,B of the functional integral to be fixed; but variations of A and B are also interesting. In which case we write appropriate characteristic functions, A,B = and insert appropriate terms for characterizing A and B. Let Application F (φ) = exp (2πı J, φ ) Z (J) := v S (dφ) exp ( 2πı J, φ ) (4.38) with v S (dφ) given by (4.6) or equivalently 2πı v S (dφ) = exp S qu (φ) [dφ] (4.39) or 2πı v S (dφ) = exp S cl + log µ (x) [dφ]. (4.40)

9 4.3 From path to field integrals 9 It follows that 0 = Z (J) φ (x) = ( 2πı S cl (φ) φ (x) + ) µ (φ) 2πıJ (x) µ φ (x) exp ( 2πı J, φ ) v S (dφ). (4.4) For an action which is the sum of a quadratic form 2 Dφ, φ and a polynomial P (φ) of order higher than 2 S cl (φ) φ (x) = D xφ + P (φ). The differential D x can be taken outside the functional integral, and the property of Fourier transforms v S (dφ) φ (x) exp ( 2πı J, φ ) = v S (dφ) exp ( 2πı J, φ ) 2πı J (x) used to derive a functional differential equation for Z (J): J (x) Z (J) + 2πı D x Z (J) + P Z (J) = 0. J (x) 2πı J (x) Z (J) can be normalized by dividing it by Z (0). Let ( Z (J) /Z (0) =: exp 2πı ) h W (J). (4.42) It has been shown [Ryder] that the diagram expansion of Z (J) contains disconnected diagrams, but that the diagram expansion of exp W (J) contains only connected diagrams. TO BE OMPLETED (SEE EILE S NOTES AND ARTIER SH 5, SH 6, SH 7.)