Equivalence of the wave function of the universe in the Einstein and Jordan frames
Abstract
We investigate the question of equivalence between the Einstein and Jordan frames for generic scalar-tensor and f(R) gravity theories in the context of minisuperspace quantum cosmology. We consider, particularly, the minisuperspace of a homogeneous and isotropic universe with the Friedmann-Lemaître-Robertson-Walker metric. We show that the equivalence between the frames depends on the operator ordering scheme of the quantum Hamiltonian: if the quantum theory is covariant under point canonical transformations in the minisuperspace, then the equivalence between the two frames holds. We find that for a large class of operator orderings, including the Laplace-Beltrami operator in the minisuperspace, the wave functions of the universe obtained in the two frames are related by the same transformations that connect the two frames, \textit{i.e.}, the frames are quantum mechanically equivalent. Further, we show that adding a fluid clock with an arbitrary equation of state parameter w to the system breaks the equivalence, except for the case of a radiation fluid (w=1/3). All our arguments are presented at a formal level in both canonical and path integral approaches, and for any scalar-tensor, f(R) theories without assuming any particular model.
Related
- Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle
- Jordan-Block Degeneracy and Cubic-Order Bifurcating Periodic Orbits in Minimum-Energy Optimal Control of Hamiltonian Equilibria
- Classical and quantum perspectives on temperature in accelerated frames
- Classical and semiclassical stability of emergent universes in Jordan–Brans–Dicke theory
- Jordan-like invariant representations of scalar-tensor gravity and the return of imperfect-fluid thermodynamics
BibTeX
@misc{mondal2026equivalence,
title = {{Equivalence of the wave function of the universe in the Einstein and Jordan frames}},
author = {Vikramaditya Mondal and Sumanta Chakraborty and Soumitra SenGupta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23785},
eprint = {2609.23785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23785},
}
RIS
TY - UNPB
TI - Equivalence of the wave function of the universe in the Einstein and Jordan frames
AU - Vikramaditya Mondal
AU - Sumanta Chakraborty
AU - Soumitra SenGupta
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23785
UR - https://arxiv.org/abs/2609.23785
AB - We investigate the question of equivalence between the Einstein and Jordan frames for generic scalar-tensor and f(R) gravity theories in the context of minisuperspace quantum cosmology. We consider, particularly, the minisuperspace of a homogeneous and isotropic universe with the Friedmann-Lemaître-Robertson-Walker metric. We show that the equivalence between the frames depends on the operator ordering scheme of the quantum Hamiltonian: if the quantum theory is covariant under point canonical transformations in the minisuperspace, then the equivalence between the two frames holds. We find that for a large class of operator orderings, including the Laplace-Beltrami operator in the minisuperspace, the wave functions of the universe obtained in the two frames are related by the same transformations that connect the two frames, \textit{i.e.}, the frames are quantum mechanically equivalent. Further, we show that adding a fluid clock with an arbitrary equation of state parameter w to the system breaks the equivalence, except for the case of a radiation fluid (w=1/3). All our arguments are presented at a formal level in both canonical and path integral approaches, and for any scalar-tensor, f(R) theories without assuming any particular model.
ER -