Recovering classical gauge geometry from quantum master constraints

Abstract:Exact normalizable solutions of quantum constraints form a symplectic submanifold of the space of rays, whereas classical first-class constraints define a surface on which the symplectic form is degenerate. We ask how the degenerate directions can be recovered when quantum theory is regarded as fundamental and the constraints are imposed through a single positive master operator. For a nonnegative master function whose zero set is a Morse--Bott critical manifold, the image of the linearized Hamiltonian vector field is the symplectic orthogonal of the zero set, and coisotropy is equivalent to this linearization having square zero. The image is fixed by the zero set, but the theorem identifies it with the response of the master dynamics, which makes it accessible to quantum states. Three realizations follow. The exact quantum physical space is the opposite, rotational case of the same dichotomy. For quadratic masters, the bottom of the spectrum is $(\hbar/2)\Tr_+L$, so first-class zero sets carry no offset of order $\hbar$; within the harmonic approximation this explains the $\hbar$ versus $\hbar^2$ offsets observed in tests of the master-constraint programme. Finally, the second time derivative of the limiting covariance of a semiclassical packet, which can be determined by bounded probes, has the gauge distribution as its image. Two oscillators at fixed total energy give exact master evolution, an exact realization of this covariance shear, and the reduced symplectic limit. For the non-Abelian SL(2,$\R$) model we compute the coherent and Weyl master symbols, prove essential self-adjointness and a finite-time propagation estimate, and derive the spectral bottom $\hbar^2/2$.

arXiv:2610.37493 [quant-ph]

(or arXiv:2610.37493v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.37493 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.