A general field-covariant formulation of quantum field theory
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Regular Article - Theoretical Physics
A general field-covariant formulation of quantum field theory Damiano Anselmia Dipartimento di Fisica “Enrico Fermi”, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
Received: 4 January 2013 / Published online: 8 March 2013 © Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013
Abstract In all nontrivial cases renormalization, as it is usually formulated, is not a change of integration variables in the functional integral, plus parameter redefinitions, but a set of replacements, of actions and/or field variables and parameters. Because of this, we cannot write simple identities relating bare and renormalized generating functionals, or generating functionals before and after nonlinear changes of field variables. In this paper we investigate this issue and work out a general field-covariant approach to quantum field theory, which allows us to treat all perturbative changes of field variables, including the relation between bare and renormalized fields, as true changes of variables in the functional integral, under which the functionals Z and W = ln Z behave as scalars. We investigate the relation between composite fields and changes of field variables, and we show that, if J are the sources coupled to the elementary fields, all changes of field variables can be expressed as J -dependent redefinitions of the sources L coupled to the composite fields. We also work out the relation between the renormalization of variable-changes and the renormalization of composite fields. Using our transformation rules it is possible to derive the renormalization of a theory in a new variable frame from the renormalization in the old variable frame, without having to calculate it anew. We define several approaches, useful for different purposes, in particular a linear approach where all variable changes are described as linear source redefinitions. We include a number of explicit examples.
1 Introduction The present formulation of quantum field theory is not sufficiently general. Several properties we are interested in depend on the variables we use to formulate and quantize the theory. For example, power-counting renormalizability requires that the action should contain no parameters of negative dimensions in units of mass, but this property is spoiled a e-mail:
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by a general change of field variables. If we work in a generic field-variable setting, the only way we have to state the power-counting criterion is to demand that there should exist a field-variable frame where the theory becomes renormalizable according to the usual rules. We do not have a field-covariant formulation of quantum field theory, and we lack efficient variable-independent criteria to identify theories belonging to special classes, such as the renormalizable, conformal and finite theories. We can state, for example, that a theory is finite if all divergences can be reabsorbed by means of field redefinitions, but this is just the definition of finite theory, not a
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