Finsler geometrical path integral

Gil Elgressy, Lawrence Paul Horwitz

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory.

BORA AKTAŞ

Abstract We develop a Finslerian extension of the Schrödinger equation in which the underlying geometry depends on both position and phase direction. This approach generalizes the curved-phase formalism introduced earlier, replacing the Riemannian metric gμνg_{\mu\nu}gμν with a direction-dependent Finsler metric gμνF(x,y)g^{F}_{\mu\nu}(x,y)gμνF(x,y). The resulting dynamics unify quantum probability, geometric curvature, and analytic (ζ-layer) structure under a single covariant operator framework. An explicit construction of the Finsler–Schrödinger Hamiltonian, its Madelung decomposition, and a small-anisotropy expansion based on the Randers metric are presented. We further analyze the self-adjointness, unitarity, and potential experimental signatures of the theory, highlighting the geometric origin of anisotropic quantum propagation and phase–curvature coupling.

AKTAŞ, BORA

Abstract We present a covariant generalization of the Phase–Cone Inequality to curved geometries, formulating it within both Riemannian and Finsler frameworks. In the flat limit, the inequality reduces to vϕ2≤κn(ΔΦ)2,v_{\phi}^{2} \leq \kappa_{n} (\Delta \Phi)^{2},vϕ2≤κn(ΔΦ)2, where κn\kappa_{n}κn is the multicarrier curvature coefficient characterizing interference order. Here, curvature and analytic continuation introduce two new covariant terms: the geometric curvature penalty K[u,ω]K[u,\omega]K[u,ω], governed by the Ricci or Finsler–Ricci tensor, and the analytic continuation term An[g;Φ]A_{n}[g;\Phi]An[g;Φ], carrying the transcendental constants (π,ζ(3),ζ(5),…)(\pi, \zeta(3), \zeta(5), \ldots)(π,ζ(3),ζ(5),…) that encode the parity–driven analytic hierarchy previously identified in flat CnC_{n}Cn manifolds. The resulting covariant form, vϕ2≤κn(ΔΦ)2−K[u,ω]+An[g;Φ],v_{\phi}^{2} \leq \kappa_{n} (\Delta \Phi)^{2} - K[u,\omega] + A_{n}[g;\Phi],vϕ2≤κn(ΔΦ)2−K[u,ω]+An[g;Φ], links quantum phase transport to local spacetime curvature and analytic number–theoretic structure. In the Finsler extension, anisotropic phase propagation introduces direction–dependent curvature coupling through the Finsler–Ricci scalar. These results imply that phase dynamics in curved or anisotropic backgrounds are constrained by both geometric focusing (Ricci contraction) and analytic opening (zeta hierarchy), offering a unified geometric–analytic limit for the speed of quantum evolution.