Why scalar-tensor equivalent theories are not physically equivalent?

TL;DR

This paper argues that scalar-tensor theories in different conformal frames are not physically equivalent because their symmetries and quantum equations cannot be translated between frames, challenging the common assumption of equivalence.

Contribution

It demonstrates that conformal transformations lead to fundamentally different systems with distinct symmetries and quantum behaviors, questioning the mathematical and physical equivalence of frames.

Findings

Noether symmetries differ between frames

Quantum equations cannot be translated between frames

Mathematical equivalence is only superficial

Abstract

Whether Jordan's and Einstein's frame descriptions of F(R) theory of gravity are physically equivalent, is a long standing debate. However, practically none questioned on true mathematical equivalence, since classical field equations may be translated from one frame to the other following a transformation relation. Here we show that neither Noether symmetries, Noether equations, nor may quantum equations be translated from one to the other. The reason being, - conformal transformation results in a completely different system, with a different Lagrangian. Field equations match only due to the presence of diffeomorphic invariance. Unless a symmetry generator is found which involves Hamiltonian constraint equation, the mathematical equivalence between the two frames appears to be vulnerable. In any case, in quantum domain Mathematical and therefore physical equivalence can't be established.