Off-shell equivalence in quantum field theory and gravity
arXiv:2607.12644
The paper proposes an operational criterion based on the Vilkovisky–DeWitt effective action to determine when two quantum field theory or gravity formulations are equivalent off shell, and illustrates it with metric f(R) gravity and scalar‑field examples.
Abstract
Field redefinitions connect many formulations of the same physics, but the standard equivalence theorem is an on-shell result and cannot be used to decide when two quantum descriptions are equivalent off shell. This paper develops an operational criterion for that problem in terms of the Vilkovisky--DeWitt effective action. The central idea is that equivalence should be tested on scalar observables built by pairing configuration-space tensors with admissible probes. This makes the criterion sensitive to the observable class under consideration and separates the usual on-shell notion of equivalence from the stronger off-shell notions needed in gravity, cosmology and non-equilibrium quantum field theory. Metric gravity and a scalar field theory example serve as prototypes, showing how the formal criterion distinguishes local, branchwise and genuinely global equivalence. In particular, we show that metric gravity and its auxiliary-field reformulation yield the same quantum theory when the auxiliary constraint is enforced in the path integral. This is distinct from quantizing the corresponding scalar--tensor action with the metric and scalar treated as independent integration variables, which defines a different quantum theory. Apparent quantum inequivalences can then be traced to comparisons between different quantum objects, rather than to a failure of actual equivalences. This leads to general and precise notions of local and global equivalence under both field redefinitions and auxiliary-variable extensions.