An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature

作者：Laetsch Thomas^*

来源：Journal of Functional Analysis, 2013, 265(8): 1667-1727.

DOI：10.1016/j.jfa.2013.05.038

• Path integrals
• Finite-dimensional approximations
• Wiener measure
• Infinite-dimensional analysis

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. An L-2 Riemannian metric G(P) is given on the space of piecewise geodesic paths Hp(M) adapted to the partition P of [0, 1], whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as mesh(P) -%26gt; 0, the approximate Wiener measure converges in an L-1 sense to the measure exp{-2+root 3/20 root 3 integral(1)(0) Scal(sigma(s)) ds} d nu (sigma) on the Wiener space W(M) with Wiener measure v. This gives a possible prescription for the path integral representation of the quantized Hamiltonian, as well as yielding such a result for the natural geometric approximation schemes originating in Andersson and Driver (1999) [3] and followed by Lim (2007) [34].

• 出版日期2013-10-15