Let $M$ be a Riemannian manifold, $o∈M$ be a fixed base point, $W_o(M)$ be the space of continuous paths from $[0,1]$ to $M$ starting at $o∈M$, and let $ν_x$ denote Wiener measure on $W_o(M)$ conditioned to end at $x∈M$. Finite dimensional approximation scheme is an important way to study informal path integrals. In this talk we use this method to give a rigorous interpretation of the informal path integral expression for $ν_x$:

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dν_x (σ) “ = ”δ_x (σ (1))1/Ze^{-1/2E(σ)}Dσ,  σ ∈ W_o (M).

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In this expression $E(σ)$ is the “energy” of the path $σ$, $δ_x$ is the $δ$–function based at $x$, $Dσ$ is interpreted as an infinite dimensional volume “measure” and $Z$ is a certain “normalization” constant. We will interpret the above path integral expression as a limit of measures, $ν^1_{P, x}$, indexed by partitions, $P$ of $[0,1]$. The measures $ν^1_{P, x}$ are constructed by restricting the above path integral expression to the finite dimensional manifolds, $H_{P, x}(M)$, of piecewise geodesics in $W_o(M)$ which are allowed to have jumps in their derivatives at the partition points and end at $x$. We then assert that $ν^1_{P, x}→ ν_x$ (in a weak sense) as the mesh size of $P$ tends to zero. Along the way we develop a number of integration–by–parts arguments for the approximate measures, $ν^1_{P, x}$, which are analogous to those known for the measures, $ν_x$.

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