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PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

机译：PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

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A typical path integral on a manifold, M is an informal expression of the form 1/Z ∫_(σ∈H(M)) f(σ)e~(-E(σ) Dσ, where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), Dσ is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret Dσ as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, P of 0, τ, let H_P be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to P. Under certain curvature restrictions on M, it is shown that 1/Z_Pe~(-1/2E(σ)) d Vol H_P(σ) →ρ(σ)dv(σ) as mesh (P) →0, where Z_P is a "normalization" constant, E : H(M) → 0,∞) is the energy functional, Vol_(H_P) is the Riemannian volume measure on , ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.

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