A master functional for quantum field theory

Abstract

We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The usual functional Γ does not behave as a scalar under the transformation law inherited from its very definition as the Legendre transform of W=lnZ, although it does behave as a scalar under an unusual transformation law. The master functional, on the other hand, is the Legendre transform of an improved functional W with respect to the sources coupled to both elementary and composite fields. The inclusion of certain improvement terms in W and Z is necessary to make this new Legendre transform well defined. The master functional behaves as a scalar under the transformation law inherited from its very definition. Moreover, it admits a proper formulation, obtained extending the set of integrated fields to so-called proper fields, which allows us to work without passing through Z, W or Γ. In the proper formulation the classical action coincides with the classical limit of the master functional, and correlation functions and renormalization are calculated applying the usual diagrammatic rules to the proper fields. Finally, the most general change of field variables, including the map relating bare and renormalized fields, is a linear redefinition of the proper fields.

Appendix: Field redefinitions and field equations

We know that if we perturb the action adding a local term proportional to the field equations, we can reabsorb such a term inside the action by means a local field redefinition to the first order of the Taylor expansion. It is interesting to know that if we perturb the action adding a local term quadratically proportional to the field equations, we can perturbatively reabsorb it inside the action to all orders by means of a local field redefinition. In this appendix we briefly rederive this result and its generalization to non-local functionals and non-local field redefinitions. The theorem was proved in Ref. [5], where a number of applications and explicit examples can be found.

Theorem 1

Consider an action S depending on fields ϕ _i , where the index i labels both the field type, the component and the spacetime point. Add a term quadratically proportional to the field equations S _i ≡δS/δϕ _i and define the modified action

$$ S^{\prime }(\phi _{i})=S(\phi _{i})+S_{i}F_{ij}S_{j}, $$

(A.1)

where F _ij is symmetric and can contain derivatives acting to its left and to its right. Summation over repeated indices (including the integration over spacetime points) is understood. Then there exists a field redefinition

$$ \phi _{i}^{\prime }=\phi _{i}+\varDelta _{ij}S_{j}, $$

(A.2)

with Δ _ij symmetric, such that, perturbatively in F and to all orders in powers of F,

$$ S^{\prime }(\phi _{i})=S\bigl(\phi _{i}^{\prime } \bigr). $$

(A.3)

Proof

The condition (A.3) can be written as

after a Taylor expansion, where $S_{k_{1}\cdots k_{n}}\equiv \delta ^{n}S/(\delta \phi _{k_{1}}\cdots \allowbreak \delta \phi _{k_{n}})$. This equality is verified if

$$ \varDelta _{ij}=F_{ij}-\varDelta _{ik_{1}} \Biggl[ \sum _{n=2}^{\infty }\frac{1}{n!}S_{k_{1}k_{2}k_{3}\cdots k_{n}}\prod_{l=3}^{n}(\varDelta _{k_{l}m_{l}}S_{m_{l}}) \Biggr] \varDelta _{k_{2}j}, $$

(A.4)

where the product is meant to be equal to unity when n=2. Equation (A.4) can be solved recursively for Δ in powers of F. The first terms of the solution are

$$ \varDelta _{ij}=F_{ij}-\frac{1}{2}F_{ik_{1}}S_{k_{1}k_{2}}F_{k_{2}j}+ \cdots. $$

(A.5)

This result is very general. It works both for local and non-local theories. If S(ϕ _i ) and F _ij are perturbatively local, namely they can be perturbatively expanded so that every order of the expansion is local, the field redefinition (A.2) and the action S′(ϕ _i ) are perturbatively local. If both S(ϕ _i ) and F _ij are local, in general (A.2) and S′(ϕ _i ) are only perturbatively local. Actually, the resummation of the expansion can produce a non-local field redefinition. Finally, if S(ϕ _i ) and F _ij are local or perturbatively local at the classical level, then (A.2) and S′(ϕ _i ) are perturbatively local at the classical level. □