I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.


The Wiener integral is an analytic tool to define certain "integrals" that one would like to evaluate in quantum and statistical mechanics. (Hrm, that's two different mechanics-es....) More precisely, one often wants to define/compute integrals over all paths in your configuration or phase space satisfying certain boundary conditions. For example, you might want to integrate over all paths in a manifold with fixed endpoints. It's conventional to write the integrand as a pure exponential $\exp(f)$. In statistical mechanics, the function $f$ in the exponent is generally real and decays to $-\infty$ (fast enough) in all directions. If the path space were finite-dimensional, you would expect such integrals to converge absolutely in the Riemann sense. In quantum mechanics, $f$ is usually pure-imaginary, so that $\exp(f)$ is a phase, and the integral should not be absolutely convergent, but in finite-dimensional integrals may be conditionally convergent in the Riemann sense. Typically, $f$ is a local function, so that $f(\gamma) = \int L(\gamma(t))dt$, where $\gamma$ is a path and $L(\gamma(t))$ depends only on the $\infty$-jet of $\gamma$ at $t$. In fact, typically $L(\gamma(t))$ depends only on the $1$-jet of $\gamma$, so that $f(\gamma)$ is defined on, for example, continuous piece-wise smooth paths $\gamma$.

For example, one might have a Riemannian manifold $\mathcal N$, and want to define: $$U(q_0,q_1) = \int\limits_{\substack{\gamma: [0,1] \to \mathcal N \\ {\rm s.t.}\, \gamma(0) = q_0,\, \gamma(1)=q_1}} \exp\left( - \hbar^{-1} \int_0^1 \frac12 \left| \frac{\partial \gamma}{\partial t}\right|^2dt \right)$$ where $\hbar$ is a positive real number (statistical mechanics) or non-zero pure imaginary (quantum mechanics). The "measure" of integration should depend on the canonical measure on $N$ coming from the Riemannian metric, and the Wiener measure makes it precise.

On $\mathcal N = \mathbb R^n$, I believe I know how to define the Wiener integral. The intuition is to approximate paths by piecewise linears. Thus, for each $m$, consider an integral of the form: $$I_m(f) = \prod_{j=1}^{m-1} \left( \int_{\gamma_j \in \mathbb R^n} d\gamma_j \right) \exp(f(\bar\gamma)) $$ where $\bar\gamma$ is the piecewise-linear path that has corners only at $t = j/m$ for $j=0,\dots,m$, where the values are $\bar\gamma(j/m) = \gamma_j$ (and $\gamma_0 = q_0$, $\gamma_m = q_1$). Then the limit as $m\to \infty$ of this piecewise integral probably does not exist for a fixed integrand $f$, but there might be some number $a_m$ depending weakly on $f$ so that $\lim_{m\to \infty} I_m(f)/a_m$ exists and is finite. I think this is how the Wiener integral is defined on $\mathbb R^n$.

On a Riemannian manifold, the definition above does not make sense: there are generally many geodesics connecting a given pair of points. But a theorem of Whitehead says that any Riemannian manifold can be covered by "convex neighborhoods": a neighborhood is convex if any two points in it are connected by a unique geodesic that stays in the neighborhood. Then we could make the following definition. Pick a covering of $\mathcal N$ by convex neighborhoods, and try to implement the definition above, but declare that the integral is zero on tuples $\gamma_{\vec\jmath}$ for which $\gamma_j$ and $\gamma_{j+1}$ are not contained within the same convex neighborhood. This would be justified if "long, fast" paths are exponentially suppressed by $\exp(f)$. So hope that this truncated integral makes sense, and then hope that it does not depend on the choice of convex-neighborhood cover.

Of course, path integrals should also exists on manifolds with, say, indefinite "semi-"Riemannian metric. But then it's not totally clear to me that the justification in the previous paragraph is founded. Moreover, really the path integral should depend only on a choice of volume form on a manifold $\mathcal N$, not on a choice of metric. Then one would want to choose a metric compatible with the volume form (this can always be done, as I learned in this question), play the above game, and hope that the final answer is independent of the choice. A typical example: any symplectic manifold, e.g. a cotangent bundle, has a canonical volume form.

One other modification is also worth mentioning: above I was imagining imposing Dirichlet boundary conditions on the paths I wanted to integrate over, but of course you might want to impose other conditions, e.g. Neumann or some mix.


Question 0: Is my rough definition of the Wiener integral essentially correct?

Question 1: On $\mathcal N = \mathbb R^n$, for functions $f$ of the form $f(\gamma) = -\hbar^{-1}\int_0^1 L(\gamma(t))dt$, for "Lagrangians" $L$ that depend only on the $1$-jet $(\dot\gamma(t),\gamma(t))$ of $\gamma$ at $t$, when does the Wiener integral make sense? I.e.: for which Lagrangians $L$ on $\mathbb R^n$, and for which non-zero complex numbers $\hbar$, is the Wiener integral defined?

Question 2: In general, what are some large classes of functions $f$ on the path space for which the Wiener integral is defined?

By googling, the best I've found are physics papers from the 70s and 80s that try to answer Question 1 in the affirmative for, e.g., $L$ a polynomial in $\dot\gamma,\gamma$, or $L$ quadratic in $\dot\gamma,\gamma$ plus bounded, or... Of course, most physics papers only treat $L$ of the form $\frac12 |\dot\gamma|^2 + V(\gamma)$.


1 Answer 1


Hi Theo,

0) Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. One warning: the rough definition of Wiener measure is misleading in one way: Wiener measure is naturally constructed as a measure on a space of distributions which contains the continuous functions.

1) I don't think there's a general theory. Trying a Lagrangian of the form $F = (\dot{\phi})^{1000}$ will not result in happiness. But: You can define Wiener measure using the standard kinetic term $\int \frac{1}{2}|\dot{\phi}|^2dt$ for any $\hbar$, and you can safely add a potential $V$ which is bounded below and integrable to the kinetic term.

2) Any observable you can write down should give you a Wiener integrable function, in quantum mechanics. This is not true in QFT, however. Most of the work in constructive QFT is proving that the measure on the space of histories actually has moments!


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