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Path integral via discretization

So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with coordinates $q_a$ and momenta $p_b$ be given satisfying commutation relations $$[q_a,p_b]=i\delta_{ab}.$$

Further suppose the system has a Hamiltonian $H$ which is time-independent. Usually in that setting the path integral is introduced as a means to compute the transition amplitude

$$\langle q',t'|q,t\rangle=\langle q'|e^{-iH(t'-t)}|q\rangle=\int\mathfrak{D}x(t) \exp\left\{iS[x(t)]\right\}$$

This is usually defined by a discretization procedure allied to a Wick rotation to Euclidean time $\tau = it$ to deal with convergence. The right discretization seems to be derived by slicing the time interval, evaluating $\langle q',t'|q,t\rangle$ to first order in $t'-t$, and imposing some ordering convention. So, for example with a Lagrangian $L = T - V$ the above integral would be defined as something of the form

$$\int\mathfrak{D}x(t) \exp\left\{iS[x(t)]\right\}=\lim_{N\to \infty} C_N\int \prod_{k=1}^N dx_k \exp \left\{i\sum_{k=1}^N \frac{m}{2}\frac{(x_k-x_{k-1})^2}{\epsilon_N^2}-V(x_k)\right\}\tag{1}$$

So: a path integral in Physics is defined by the continuum limit of these aforementioned discretizations.

Cylinder set measures

Now there's the mathematicians point of view on which one studies integration over locally convex vector spaces which are infinite dimensional. In that case, if $E$ is such a space we perform two definitions:

Definition: Let $E$ be a locally convex vector space. A cylinder set is defined to be a subset $C\subset E$ of the form $$C=\{x\in E : (\ell_1(x),\dots,\ell_n(x))\in C_0\}$$ where $C_0\subset \mathbb{R}^n$ is a Borel subset and $\ell_k\in E^\ast$ are continuous linear functionals. Equivalently, it is a preimage $C = P^{-1}(C_0)$ of a Borel set under a continuous linear map $P : E\to \mathbb{R}^n$.

Definition: Let $E$ be a locally convex vector space. A cylinder set measure $\nu$ is a nonnegative additive set function defined on the $\sigma$-algebra generated by cylinder sets of $E$ such that for any continuous linear $P : E\to \mathbb{R}^n$ the set function $$\nu\circ P^{-1} : B\mapsto \nu(P^{-1}(B))$$ is countably additive.

Comparison

If we now compare there are a few points to mention:

  1. It seems that cylinder sets capture discretizations. If $x$ is a continuous path, $(\ell_1(x),\dots, \ell_n(x))$ is an $n$-point discretization. To be even more precise in the case of paths we could take $\ell_k(x) = x(t_k)$ for some $t_1,\dots, t_n$ in the interval. In the same way we could take $\ell_k(x) = a_k$ some Fourier coefficient of $x(t)$. I've seem both things done in Physics.

  2. It seems cylinder set measures are in fact a way to define "a measure per discretization". So for each discretization we give a measure - integrate over $n$ points, integrate over $n$ Fourier coefficients, so forth.

Still, the connection doesn't feel complete for me. The issue is that to define a cylinder set measure we must define $\nu$ on the whole algebra generated by cylinder sets.

The Physicist approach seems to do this exactly for a specific collection of cylinder sets. Either for the ones with $\ell_k(x) = x(t_k)$ or for the ones with $\ell(x_k)=a_k$ a Fourier coefficient.

But there are infinitely many other choices of the $\ell_k$ which give rise to many more cylinder sets. And one would need still to define $\nu$ on the $\sigma$-algebra itself.

The question: is there really a relation between cylinder set measures and the Physicist discretization of a path integral? If so, how the relation can be made more precise? If not, why not, considering the similarities?

For this discussion, please let us consider the Euclidean path integral. So the issue here is not the imaginary exponent.

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  • $\begingroup$ The Wiener measure in the space of paths, used to rigorously define the euclidean path integrals (the so-called Feynman-Kac formula) originates from a cylindrical measure. Further details on the connection between path integrals and cylindrical measures (or promeasures, using french terminology) can be found in this book and this one $\endgroup$ – yuggib Jul 12 at 9:45
  • $\begingroup$ The main obstruction to simply using cylindrical measures to define path integrals is that since in general they are only finitely additive, it is not possible to integrate general functions with respect to them, but only cylindrical ones. Thus one looks for a probability measure that suitably extends, usually in a "bigger space", the cylindrical measure. With respect of such bigger measure it is then possible to integrate non-cylindrical functions such as the exponential of the action (or of the interaction part, since the free part is used to define the measure itself). $\endgroup$ – yuggib Jul 12 at 9:53
  • $\begingroup$ Thanks for the references @yuggib! I've obtained DeWitt's book and it seems very helpful, I'll certainly read it in detail. In fact I was aware that the Wiener (Gaussian) measure can be obtained out of cylindrical measures (CM) and this "bigger space" extension. In that case one starts with the canonical Gaussian CM on a Hilbert space, if I understood. What I'm wondering is if discretizations like (1) in the post, give rise to CMs, even ones which cannot later be turned into real measures, and if so, how we bridge the approaches. Is it really the case as I suspect? $\endgroup$ – user1620696 Jul 12 at 14:38
  • $\begingroup$ One would need to prove that such discretizations give rise to cylindrical measures, although it seems possible. One could maybe see the procedure of fixing one time as a continuous linear functional on the space of (everywhere defined) paths (the "delta" distribution, associating to a path its value at a given time), and build the cylinders using only these distributions. However, I do not know if this is an idea that has been explored. $\endgroup$ – yuggib Jul 12 at 16:08
  • $\begingroup$ Let me also comment that every cylindrical measure can be seen as a true measure on a bigger space, although such big space in some cases can become extremely "big" and nasty to characterize. For the Gaussian measure, the bigger space is a Banach space, but in general it is the product, over all weakly closed subspaces of finite codimension, of the Cech compactifications of the quotients of the original LCV space w.r.t. the aforementioned subspaces. $\endgroup$ – yuggib Jul 12 at 16:13

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