Off-shell equivalence in quantum field theory and gravity

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 $f(R)$ 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 $f(R)$ 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.