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Wheeler–DeWitt quantization for point-particles

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Abstract

We present the Hamiltonian formulation of a relativistic point-particle coupled to Einstein gravity and its canonical quantization à la Wheeler–DeWitt. In the resulting quantum theory, the wave functional is a function of the particle coordinates and the 3-metric. It satisfies a particular Hamiltonian and diffeomorphism constraint, together with a Klein–Gordon-type equation. As usual in the Wheeler–DeWitt theory, the wave function is time-independent. This is also reflected in the Klein–Gordon-type equation, where the time derivative is absent. Before considering gravity, we consider the coupling of a particle with electromagnetism, which is treated similarly, but simpler.

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Notes

  1. Units are assumed such that \(\hbar =c=1\).

  2. We have chosen the Laplace-Beltrami operator ordering for the particle but not for gravity [19].

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Acknowledgements

This work is supported by the Research Foundation Flanders (Fonds Wetenschappelijk Onderzoek, FWO), Grant No. G066918N. It is a pleasure to thank Christian Maes and Kasper Meerts for useful discussions.

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  1. Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium

    Ward Struyve

  2. Centrum voor Logica en Filosofie van de Wetenschappen, KU Leuven, Leuven, Belgium

    Ward Struyve

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  1. Ward Struyve
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Correspondence to Ward Struyve.

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Struyve, W. Wheeler–DeWitt quantization for point-particles. Gen Relativ Gravit 53, 32 (2021). https://doi.org/10.1007/s10714-021-02802-6

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