Quantum Field Theory – Why Isn’t the Path Integral Rigorous?

Question

I've recently been reading Path Integrals and Quantum Processes by Mark Swanson; it's an excellent and pedagogical introduction to the Path Integral formulation.
He derives the path integral and shows it to be: $$\int_{q_a}^{q_b} \mathcal{D}p\mathcal{D}q\exp\{\frac{i}{\hbar}\int_{t_a}^{t_b} \mathcal{L}(p, q)\}$$

This is clear to me. He then likens it to a discrete sum
$$\sum_\limits{\text{paths}}\exp\left(\frac{iS}{\hbar}\right)$$ where $S$ is the action functional of a particular path.

Now, this is where I get confused. He claims that, because some of these paths are discontinuous or non-differentiable and that these "un-mathematical"¹ paths cannot be disregarded, the sum is not mathematically rigorous, and, thus, that the transition amplitude described by the path integral is not rigorous either. Please correct me if I am incorrect here.

Furthermore, he claims that this can be alleviated through the development of a suitable measure. There are two things that I don't understand about this. First, why isn't the integral rigorous? Though some of the paths might be difficult to handle mathematically, they aren't explicitly mentioned at all in the integral. Why isn't the answer that it spits out rigorous? And, second, why would a measure fix this problem?

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¹ Note: this is not the term he uses

Answer 1

There are several points:

• The first is that for usual self-adjoint Hamiltonians of the form $H=-\Delta +V(x)$, with a common densely defined domain (and I am being very pedantic here mathematically, you may just ignore that remark) the limit process is well defined and it gives a meaning to the formal expression

  $\int_{q_a}^{q_b} \mathcal{D}p\mathcal{D}q\exp\{\frac{i}{\hbar}\int_{t_a}^{t_b} \mathcal{L}(p, q)\}$

  by means of trotter product formula and the corresponding limit of discrete sums. So the object has most of the time meaning, as long as we see it as a limit. Nevertheless, it would be suitable to give a more direct mathematical interpretation as a true integral on paths. This would allow for generalizations and flexibility in its utilization.

• It turns out that a suitable notion of measure on the space of paths can be given, using stochastic processes such as brownian motion (there is a whole branch of probability theory that deals with such stochastic integration, called Itô integral). To relate this notion with our situation at hand there is however a necessary modification to make: the factor $-it$ in the quantum evolution has to be replaced by $-\tau$ (i.e. it is necessary to pass to "imaginary time"). This enables to single out the correct gaussian factors that come now from the free part of the Hamiltonian, and to recognize the correct Wiener measure on the space of paths. On a mathematical standpoint, the rotation back to real time is possible only in few special situations, nevertheless this procedure gives a satisfying way to mathematically define euclidean time path integrals of quantum mechanics and field theory (at least the free ones, and also in some interacting case). There are recent works of very renowned mathematicians on this context, most notably the work of the fields medal Martin Hairer (see e.g this paper and this one, or the recent work by A. Jaffe that gives an interesting overview; a more physical approach is given by Lorinczi, Gubinelli and Hiroshima among others).

• The precise mathematical formulation of path integral in QM is called Feynman-Kac formula, and the precise statement is the following:

  Let $V$ be a real-valued function in $L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, $H=H_0+V$ where $H_0=-\Delta$ (the Laplacian). Then for any $f\in L^2(\mathbb{R}^3)$, for any $t\geq 0$: $$(e^{-tH}f)(x)=\int_\Omega f(\omega(t))e^{-\int_0^t V(\omega(s))ds}d\mu_x(\omega)\; ;$$ where $\Omega$ is the set of paths (with suitable endpoints, I don't want to give a rigorous definition), and $\mu_x$ is the corresponding Wiener measure w.r.t. $x\in\mathbb{R}^3$.