Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a Hamiltonian $H\in \text{End} [L^2(X,\mathbb{C})]$, with endomorphisms defined in an appropriate densely defined sense. (I am not requiring that $H$ is Hermitian, at least for now.)

Now solving this evolution equation is equivalent to writing down the time-$T$ evolution matrix $U_T= e^{-iTH},$ with "matrix coefficients" $U_T(x,y) : = \langle y|U_T|x\rangle.$ Here $x, y\in X$ points and $|x\rangle$ the delta-function on $X$ (this is not in $L^2,$ but $U_T(x,y)$ can be made sense of as a function of $x, y$ in a suitable distributional sense).

The path integral formalism gives (at least in theory) another way of computing $\langle y|U_T|x\rangle$ (for Hermitian $H$). Namely, the matrix coefficient $\langle y|U_T|x\rangle$ can be written as a limit of integrals $$\int dx_1dx_2\dots dx_N \langle y| U_{\epsilon} |x_N\rangle \langle x_N|U_{\epsilon}|x_{N-1}\rangle \langle x_{N-1}|U_{\epsilon}|x_{N-2}\rangle\cdots \langle x_2|U_\epsilon|x_1\rangle\langle x_1|U_\epsilon|x\rangle$$ where $\epsilon = T/N$ and $N$ goes to $\infty$, with the integrand a multiplicative function in the consecutive pairs $(x_k, x_{k+1})$. Taking a continuous limit, this "morally" reduces to an integral $$\int D\gamma \exp \big( i S(\gamma)\big)$$ over paths $\gamma:[0,T]\to X$ from $x$ to $y$, with the "action" $S(\gamma) = \int_0^T dt L\big(\gamma(t), \dot{\gamma}(t)\big),$ and $L$ the Lagrangian, a functional on $TX$. Here $\gamma$ is the continuous limit of the sequence $x_0, x_1,\dots, x_N$ and the form of the action encodes its multiplicative structure and its dependence only on "neighboring pairs" (corresponding to the data $\gamma(t), \dot{\gamma}(t)$).

In practive it is difficult to make sense of the path integral: rigorous definitions use the Wiener measure (associated to some metric on $X$), corresponding to the Brownian random process, and concentrated on nowhere differentiable paths, which means that the action functional cannot be treated as a function. On the other hand, there are "bounded-energy" probability measures on paths which give Brownian motion in a limit but are more nicely behaved: only concentrated on differentiable (or piecewise differentiable) paths, and well-behaved with respect to the $C^1$ topology on paths (where the action functional is explicitly defined and continuous). One such approximation (if I understand correctly) is to consider piecewise-linear paths with direction changing according to a Poisson process. Another is to consider everywhere differentiable paths with derivative undergoing a more continuous random process.

Now here's my question. It should be possible to write down a (potentially non-Hermitian) Hamiltonian $H'$ on the tangent space $TX$ with the property that its evolution operator $U_T'$ has matrix coefficients $\langle x', v' | U_T'| x, v\rangle$ which compute the expectation of $\exp\big( -i S(\gamma)\big)$ in one of these bounded-energy measures over differentiable (or piecewise differentiable) paths that start at $x$ with derivative $v$ and end at $x'$ with derivative $v'.$ Indeed, all that's needed is to incorporate the Lagrangian, the standard dynamic relationship between $x$ and $v$ and some kind of a stochastic term on each tangent fiber.

This seems to me like a promising and straightforward way to replace the analytic difficulties involved in the path integral with ordinary quantum mechanics (i.e. exponentiating operators). If this works, someone must have tried it. Are there problems with this approach or references where it is done?


1 Answer 1


Path integrals over simultaneous position/velocity, or more commonly position/momentum, degrees of freedom are known as phase space path integrals. I don't know very much about the rigorous construction of path integral measures, and even less so about their phase space version. However, there does appear to be at least one classic work on the subject by Berezin, who discussed in particular the regularity of the paths on which the path integral measure is concentrated. Then there were also some follow-ups by Daubechies and Klauder. Perhaps these works could point you in the right direction.


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