Beta functions in quantum electrodynamics

Transcription

1 Beta functions in quantum electrodynamics based on S-66 Let s calculate the beta function in QED: the dictionary: Note!

2 following the usual procedure: we find:

3 or equivalently: For a theory with N Dirac fields with charges : = 1 we find:

4 For completeness, let s calculate the beta functions in scalar ED: the dictionary: needed for consistency of two different relations between renormalized and bared couplings Note!

5 following the usual procedure we find: Generalizing to the case of arbitrary number of complex scalar and Dirac fields:

6 The path integral Schwinger-Dyson equations based on S-22 doesn t change if we change variables thus we have: assuming the measure is invariant under the change of variables taking n functional derivatives with respect to and setting we get:

7 A comment on functional derivative: δϕ b (y) = δ ba δ 4 (y x)δϕ a (x) δϕ b (y) = a dxδ ba δ 4 (y x)δϕ a (x) Similarly: δf (ϕ a (x)) δϕ b (y) = F ϕ a (x) δf (ϕ a (x)) = b or, for the action: δϕ a (x) δϕ b (y) = δs = dy 4 F ϕ a (x) δ baδ 4 (y x) F ϕ a (x) δ baδ 4 (y x)δϕ b (y) = dx 4 δs δϕ a (x) δϕ a(x) F ϕ a (x) δϕ a(x)

8 since is arbitrary, we can drop it together with the integral over. since the path integral computes vacuum expectation values of T-ordered products, we have: for a free field theory of one scalar field we have: SD eq. for n=1: Schwinger-Dyson equations is a Green s function for the Klein-Gordon operator as we already know for a free field theory of one scalar field we have:

9 in general Schwinger-Dyson equations imply thus the classical equation of motion is satisfied by a quantum field inside a correlation function, as far as its spacetime argument differs from those of all other fields. if this is not the case we get extra contact terms

10 Ward-Takahashi identity: For a theory with a continuous symmetry we can consider transformations that result in : summing over a and dropping the integral over Ward-Takahashi identity thus, conservation of the Noether current holds in the quantum theory, with the current inside a correlation function, up to contact terms (that depend on the infinitesimal transformation).

11 Ward identities in QED based on S-67,68 When discussing QED we used the result that a scattering amplitude for a process that includes an external photon with momentum should satisfy (as a result of Ward identity): we used it, for example, to obtain a simple formula for the photon polarization sum that we needed for calculations of cross sections: now we are going to prove it!

12 To simplify the discussion let s treat all particles in the LSZ formula as outgoing (incoming particles have ): it can be also written as: we do not fix must include an overall energymomentum delta function

13 near we can write: residue of the pole multivariable pole contributions that do not have this multivariable pole do not contribute to! (for simplicity we work with scalar fields, but the same applies to fields of any spin)

14 near we can write: residue of the pole multivariable pole this means that in Schwinger-Dyson equations: contributions that do not have this multivariable pole do not contribute to! classical eq. of motion contact terms: in the momentum space a contact term is a function of ; it doesn t have the right pole structure to contribute! contact terms in a correlation function F do not contribute to the scattering amplitude!

15 Let s consider a scattering process in QED with an external photon: with momentum. the LSZ formula: the classical equation of motion in the Lorentz gauge: thus we find: contact terms cannot generate the proper singularities and thus these do not contribute to the scattering amplitude!

16 replace by integrate by parts now we use Ward-Takahashi identity: and thus we find: contact terms do not contribute to the scattering amplitude!

17 Finally, we will derive another consequence of Ward identity that we used: consider the correlation function: adds a vertex in the momentum space: exact propagator later we will use: exact 1PI vertex function

18 multiply by integrate by parts Z1 The Ward identity: becomes:

19 Z1 before we found: thus we get: or:

20 finite in the scheme since the divergent pieces must cancel, and there are only divergent pieces In the OS scheme, near,, we have: and so we get! The Ward identity means that the kinetic term and the interaction term are renormalized in the same way and so they can be combined into covariant derivative term (which we could have guessed from gauge invariance)!

21 Formal development of fermionic PI based on S-44 We want to derive the fermionic path integral formula (that we previously postulated by analogy with the path integral for a scalar field): Feynman propagator inverse of the Dirac wave operator

22 Let s define a set of anticommuting numbers or Grassmann variables: for we have just one number with. We define a function of by a Taylor expansion: the order is important! if f itself is commuting then b has to be an anticommuting number: and we have:

23 Let s define the left derivative of with respect to as: Similarly, let s define the right derivative of with respect to as: We define the definite integral with the same properties as those of an integral over a real variable; namely linearity and invariance under shifts: The only possible nontrivial definition (up to an overall numerical factor) is:

24 Let s generalize this to, we have: all indices summed over completely antisymmetric on exchange of any two indices Let s define the left derivative of with respect to as: and similarly for the right derivative...

25 To define (linear and shift invariant) integral note that: just a number (if n is even) the only consistent definition of the integral is: Levi-Civita symbol, alternatively we could write the differential in terms of individual differentials: and use to derive the result above.

26 Consider a linear change of variable: matrix of commuting numbers then we have: integrating over we get and thus: Recall, for integrals over real numbers with we have:

27 We are interested in gaussian integrals of the form: antisymmetric matrix of (complex) commuting numbers for we have: expanding the exponential: we find:

28 For larger (even) n we can bring a complex antisymmetric matrix to a block-diagonal form: a unitary matrix we will later need: taking: we have: we drop primes represents 2x2 blocks

29 using the result for n= 2 we get: we finally get: Recall, for integrals over real numbers we have: a complex symmetric matrix

30 Let s define complex Grassmann variables: we can invert this to get: thus we have: determinant = also since we have:

31 A function can be again defined by a Taylor expansion: the integral is: in particular:

32 Let s consider n complex Grassmann variables and their conjugates: define: under a change of variable: we have not important (the integral doesn t care whether is the complex conjugate of ) we want to evaluate: a general complex matrix can be brought to a diagonal form with all entries positive by a bi-unitary transformation

33 under such a change of variable we get: positive we drop primes Analogous integral for commuting complex variable

34 using shift invariance of integrals: we get: generalization for continuous spacetime argument and spin index the determinant does not depend on fields or sources and can be absorbed into the overall normalization of the path integral