Quantum Space, Quantum Time, and Relativistic Quantum Mechanics

Abstract

We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being treated on an equal footing in Hilbert space, we show symmetry transformations to be implemented by unitary basis changes in Hilbert space, giving them a stronger quantum mechanical footing. Global symmetries, such as Lorentz transformations, modify the decomposition of Hilbert space; and local symmetries, such as $U(1)$ gauge symmetry are diagonal in coordinate basis and do not alter the decomposition of Hilbert space. We briefly discuss extensions of this paradigm to quantum field theory and quantum gravity.

23 pages, comments welcome. v2 contains version accepted by Quantum Studies: Mathematics and Foundations

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Cited by in corpus (4)

• Equivalence of approaches to relational quantum dynamics in relativistic settings
• Massless polarized particle and Faraday rotation of light in the Schwarzschild spacetime
• Robustness of the Page-Wootters construction across different pictures, states of the universe and system-clock interactions
• The quantum theory of time: a calculus for q-numbers