Quantum models a la Gabor for space-time metric

Abstract

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The $x=\left(x^{\mu }\right)$'s are space-time variables and the $k=\left(k^{\mu }\right)$'s are As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The $x=\left(x^{\mu }\right)$'s are space-time variables and the $k=\left(k^{\mu }\right)$'s are their conjugate wave vector-frequency variables. The procedure is first applied to the variables $\left(x,k\right)$ and produces canonically conjugate essentially self-adjoint operators. It is next applied to the metric field $g_{\mu \nu }(x)$ of general relativity and yields regularised semi-classical phase space portraits ${\check{g}}_{\mu \nu }(x)$. The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed.

22 pages

References in corpus (3)

• Quantum Mixmaster as a model of the Primordial Universe
• Singularity avoidance in quantum gravity
• Two-mode squeezed state quantisation and semiclassical portraits