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Bilocal Phase Space. Geometric and Operator Structure

Tymiński, Andrzej

Published Dec 5, 2025• 0 citations
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Abstract

This work presents the construction of a bilocal phase space arising from arelational description of the state $\Psi(x_{1},x_{2})$ on an extended configurationspace. We introduce bilocal configuration and gradient variables and analyse thecorresponding symplectic form on $T^{\ast}\mathbb{R}^{3,3}$. Angle reduction leads to aneffective phase-space structure\[\omega_{\mathrm{eff}} = dt\wedge dE + dr\wedge dp,\]described by two canonically conjugate pairs: a relational pair $(t,E)$ and a localpair $(r,p)$. The first pair is associated with the moduli of bilocal configuration andgradient vectors, and its structure is a classical counterpart of the quantum relation$[t,\hat E]=2i\hbar$. The second pair reproduces the radial part of the standard phasespace. The resulting phase space is entirely relational and can be interpreted as a geometricintertwining of two Minkowski spaces, $M_{1,3}^{(x)} \times M_{1,3}^{(\nabla)}$. This suggeststhat the bilocal construction provides a natural framework for describing structuresin which relations between configuration points, rather than the points themselves,are fundamental. The analysis presented here forms the basis for formulating bilocalextensions of dynamics and quantization, which will be developed in future work.

DOI
10.5281/zenodo.17834314

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