Path integral formulation

Definition

Let $d \ge 1$, $m > 0$ and $\hbar > 0$, and let $V \colon \R^d \to \R$ be measurable. Let $H_0 = -\frac{\hbar^2}{2m}\Delta$ on $L^2(\R^d)$, and assume that $H_0 + V$ is essentially self-adjoint on $D(H_0) \cap D(V)$, with closure $H$. For a path through the points $x_0, x_1, \dots, x_N \in \R^d$ at the times $0, \varepsilon, \dots, N\varepsilon$, the time-sliced action is $$S_N(x_0, \dots, x_N) = \sum_{k=1}^{N} \varepsilon \left( \frac{m}{2} \abs{\frac{x_k - x_{k-1}}{\varepsilon}}^2 - V(x_{k-1}) \right).$$ The path integral formulation expresses the solution $\psi(t) = e^{-itH/\hbar}\psi_0$ of the Schrödinger equation as a limit. $$\psi(t)(x) = \lim_{N \to \infty} \left( \frac{m}{2\pi i \hbar \varepsilon} \right)^{Nd/2} \int_{\R^d} \cdots \int_{\R^d} e^{i S_N(x_0, \dots, x_{N-1}, x)/\hbar}\, \psi_0(x_0)\, dx_0 \cdots dx_{N-1}.$$ Here $\varepsilon = t/N$ and $\psi_0 \in L^1 \cap L^2(\R^d)$. The integrals are iterated integrals, each read as an $L^2$ limit of integrals over balls when needed. The limit in $N$ is taken in $L^2(\R^d)$. Formally this is written $\psi(t)(x) = \int e^{iS[q]/\hbar}\, \psi_0(q(0))\, \mathcal{D}q$, an integral over all paths $q$ with $q(t) = x$, where $S[q] = \int_0^t \left( \frac{m}{2}\abs{\dot q}^2 - V(q) \right) ds$ is the classical action.