Diffeomorphisms of quantum field variables

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ArticleMPAG (2020) 23 : 33

Non-linear field transformations

In quantum field theory, one regularly uses the fact that (quantum) field variables can be scaled or shifted by constant numbers. What is more subtle are position-independent non-linear redefinitions of the field variable \phi(x) of the form

    \begin{align*} \phi(x) = \rho(x) + a_1 \,\rho(x)^2 + a_2 \, \rho(x)^3 + \ldots .\end{align*}

Maybe a first naive guess would be that such transformations should be irrelevant altogether, but that is too simple: In the path integral, such transformations give rise to another Jacobian which is not trivially equal to unity, and the Feynman graphs of the field \rho involve (infinitely many) additional vertices with non-standard dependence on momenta.

It had been known that in the path integral, all terms introduced by non-linear field redefinitions are eventually proportional to the classical equations of motion. Similarly, in perturbation theory of free scalar fields, all the newly created terms are proportional to inverse propagators, and hence do not contribute to the onshell S-matrix. In my article MPAG (2020) 23 : 33, I gave a systematic combinatorial proof that this is also true for scalar fields with arbitrary polynomial self-interactions.

Invariance under field redefinitions is not only interesting philosophically, but it is also a fundamental assumption for the construction and classification of effective field theories: There, one uses non-linear redefinitions to transform interaction terms to a standard form suitable for systematic analysis.

Ward identities and renormalizability

A completely different perspective on non-linear field redefinitions is to take the resulting Lagrangian density as an actual new quantum field theory, and examine its properties apart from the fact that the new terms vanish onshell. A crucial finding (MPAG (2020) 23 : 33) is that a diffeomorphism produces vertices that are, in an appropriate sense, proportional to inverse propagators. In particular, it is impossible to have a field diffeomorphism between two distinct (perturbatively) renormalizable theories. For an usual bosonic scalar field, that means that the newly-generated vertices are proportional to momentum squared. This is exactly the scaling-behavior known from quantizations of the Einstein-Hilbert Lagrangian, that is, from interpreting general relativity as a quantum field theory. Hence, field diffeomorphisms can serve as a simple toy model of a theory with such scaling behavior.