An Approximation to Wiener Measure and Quantization of the Hamiltonian\n on Manifolds with Non-positive Sectional Curvature

2012/10/12 by Thomas Laetsch, Laetsch, Thomas
Mathematics · Physics and Astronomy · #28C20 #58D30 #60H99 #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1210.3524

openalex publication_date 2012/10/12 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

This paper gives a rigorous interpretation of a Feynman path integral on a\nRiemannian manifold M with non-positive sectional curvature. A L² Riemannian\nmetric G_P is given on the space of piecewise geodesic paths H_P(M) adapted\nto the partition P of [0,1], whence a finite-dimensional approximation of\nWiener measure is developed. It is proved that, as mesh(P) \→ 0, the\napproximate Wiener measure converges in a L¹ sense to the measure\ne^{-\(2 + \√(3))/(20\√(3)) \∫₀1 Scal(\σ(s)) ds} d\ν(\σ)\non the Wiener space W(M) with Wiener measure \ν. This gives a possible\nprescription for the path integral representation of the quantized Hamiltonian,\nas well as yielding such a result for the natural geometric approximation\nschemes originating in [L. A. Andersson and B. K. Driver, J. Funct. Anal. 165\n(1999), no. 2, 430-498] and followed by [Adrian P. C. Lim, Rev. Math. Phys. 19\n(2007), no. 9, 967-1044].\n

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