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Quantum space, quantum time, and relativistic quantum mechanics

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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.

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Notes

  1. Often referred to as first quantization, though we will refrain from using this terminology in this paper.

  2. A similar entangled state \(\left| \Psi \right\rangle \bigr \rangle \) can be written in the Page-Wootters formulation too with entanglement between states in \({\mathcal {H}}_{t}\) and \({{{\mathcal {H}}}}_S\) [21].

  3. The function \(\lambda (t,\vec {x})\) is typically taken to be continuous and sufficiently differentiable in its variables and dies off rapidly enough as \(\vec {x} \rightarrow \pm \infty \).

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Acknowledgements

I would like to thank Sean Carroll, Charles Cao, Aidan Chatwin-Davies, Swati Chaudhary and Frank Porter for helpful discussions during the course of this project. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, as well as by the Walter Burke Institute for Theoretical Physics at Caltech and the Foundational Questions Institute.

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    Ashmeet Singh

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Singh, A. Quantum space, quantum time, and relativistic quantum mechanics. Quantum Stud.: Math. Found. 9, 35–53 (2022). https://doi.org/10.1007/s40509-021-00255-9

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