Measure of Feynman path integral

Question

Feynman path integral for non-relativistic case is defined as: $$\int\mathcal{D}[x(t)]e^{iS/\hbar}$$ where $$\int \mathcal{D[x(t)]}=\lim_{N\rightarrow\infty}\Pi_{i=0}^{i=N}\bigg(\int_{-\infty}^{\infty}\mathrm{d}x_{i}\bigg)$$

In the course and the books, I have encountered while studying path integral the term "measure" is introduced without any previous notion. The best intuition I have about measure is, it's like a weight you assign to each term of series you're summing (I only understand Riemann integral so it kinda lean towards the idea of area, just a bit of generalization).

Also while studying a little bit about Brownian motion, I came across Wiener process the integral (propagator) used there is almost identical to Feynman's one, the $i$ is replaced by $-1$ but I couldn't understand much about it because the discussion was based on central limit theorem and Lebesgue measure. But the headache-causing thing I remember is while Wiener integrals are convergent Feynman integrals are not (the explanation I came across about it is that it's because of the oscillatory behaviour of the factor, $e^{iS/\hbar}$. The integrals are conditionally convergent, while in the case of Wiener we have decreasing function $e^{-S}$ and that's the reason why we go for $-i\epsilon$ prescription).

As you may have noticed in the above explanation the exponential factor is tugged together with the $\mathcal{D}$ factor to compare with the Wiener integrals. So is the measure $M$, equals to $\color{red}{\mathcal{D}[x(t)]e^{iS/\hbar}}$ and Feynman integrals are of form $\int M f(x(t))$? People also say that there are Feynman path integrals which don't use Wiener measure but I have never seen them. Do they exist in literature?

When the books make transition from one particle QM to QFT, the authors simply state that propagator is given by $$\int\mathcal{D}[\phi(x)]e^{iS/\hbar}$$ with no definition of $\mathcal{D}[\phi(x)]$.

I have heard that $\mathcal{D}[\phi(x)]$ can be defined in terms of $a$, $a^{\dagger}$ (annihilation and creation operator) and this has been done by Ludvig Faddeev but couldn't find it.

Answer 1

The results in this answer are taken directly from Blank, Exner and Havlíček: Hilbert space operators in quantum physics. Look there for more details.

At least in non-relativistic QM, the path integral is derived/defined using the limiting procedure of “taking finer and finer time-slicing” of the path. The precise formula for a system of $M$ particles is:

\begin{align} &U(t) \, \psi(x) \\ &= \lim_{N\to\infty} \; \prod_{k=1}^M \; \Big( \frac{m_k \, N}{2\pi\mathrm{i} \, t} \Big)^{N/2} \hspace{-0.2em} \lim_{j_1, ... j_N \to \infty} \; \int_{B_{j_1}\!\times...\times B_{j_N}} \hspace{-3em} \exp(\mathrm{i} \, S(x_1, ...x_N)) \; \psi(x_1) \; \mathrm{d}x_1 \, ... \, \mathrm{d}x_{N-1} \\ &=: \int\exp(\mathrm{i} \, S[x]) \; \mathcal{D}x \, \end{align} where $S[x]$ is the classical action over the path $x$ and $S(x_1,...x_N) := S[\gamma]$ is the same action taken over a polygonal line $\gamma$, such that $\gamma(t_i) = x_i$ are the vertices. (Actually, it is not guaranteed that the above expression converges to $U(t)\psi(x)$ for every $S$, but it does for a large class them.)

Note, that specifically for the kinetic part of $S = \int_0^t (\frac{1}{2} \sum_i m_i \dot x^2_i(t') + V(x(t') ) \, \mathrm{d}t'$ we have: $$ \int_0^t \dot{\gamma}(t_1, ... t_N) = \sum_{k=0}^N \big| x_{k+1} - x_k \big|^2 $$

From this definition, it is unclear whether $\mathcal{D} x$ is actually a measure, or not, so let us compare the integral to the Wiener integral – an integral over Wiener processes with deviation $\sigma\in\mathbb{R}$, which actually does have a proper measure, $w_\sigma$. Let a functional $F[x]$ on Wiener processes depend only on a finite number of points of $x$. Such a functional is called cylindrical and is similar in spirit to our action $S(x_1,...x_N)$ on a polygonal line. Then we get: \begin{align} &\int F[x] \, \mathrm{d}w_\sigma\\ &= \prod_{k=0}^{N-1} \Big( \frac{N}{2\pi \mathrm{i} \, t} \Big)^{\!-N/2} \!\! \int_{\mathbb{R}^{nN}} \hspace{-1em} \exp\big( \frac{-1}{2\sigma} \sum_{k=0}^{N-1} \big| x_{k+1} - x_k \big|^2 \, \frac{t}{N} \big) \, F(x_1, ... x_{N-1}) \, \mathrm{d}x_1 \, ... \, \mathrm{d}x_{N-1} \end{align} One can check that by formally choosing $m_1=...=m_M=\mathrm{i}/\sigma$, we arrive at the definition of the path integral. However, the Wiener measure $w_\sigma$ is only defined for real $\sigma$ – the question is then, can it be generalized?

Surprisingly, the answer is negative. Citing the Cameron's theorem (again from the book of Blank, Exner, Havlíček):

Let $\sigma$ be a nonzero complex number, $\operatorname{Re} \sigma \ge 0$. Afinite complex measure $w_\sigma$ such that the relation above holds for any Borel function $F$ from Wiener processes to $\mathbb{C}$, which is cylindrical and bounded, exists iff $\sigma \in (0,\infty)$.

This result is taken as a proof that there isn't a suitable measure for the Feynman path integral. Instead, we can choose one of several different approaches. Apart from using the definition we used, one can also calculate the Wiener integral, parametrized by $\sigma\in\mathbb{R}$, and then make an analytic continuation to $\mathbb{C}$. Another possible approach is to use generalized Fresnel integrals.