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=lnZ 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.
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References
G. ’t Hooft, M. Veltman, One-loop divergences in the theory of gravitation. Ann. Inst. Henri Poincaré 20, 69 (1974)
M.H. Goroff, A. Sagnotti, The ultraviolet behavior of Einstein gravity. Nucl. Phys. B 266, 709 (1986)
A.E.M. van de Ven, Two loop quantum gravity. Nucl. Phys. B 378, 309 (1992)
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 2002). § 6.4 and Chap. 10
D.J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1984), §5–7
L.S. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, 1992), § 5.5
J. Collins, Renormalization (Cambridge University Press, Cambridge, 1984). Chap. 6
S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in An Einstein Centenary Survey, ed. by S. Hawking, W. Israel (Cambridge University Press, Cambridge, 1979)
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Anselmi, D. A general field-covariant formulation of quantum field theory. Eur. Phys. J. C 73, 2338 (2013). https://doi.org/10.1140/epjc/s10052-013-2338-5
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DOI: https://doi.org/10.1140/epjc/s10052-013-2338-5