Is this actually the rigorous definition of the path integral in Quantum Mechanics?

Question

Let a quantum system with a single degree of freedom be given. We want to define the path integral so that we get the representation for the propagator as

$$\langle q' |e^{-iHT}|q\rangle=\int_{x(a)=q, x(b)=q'} \mathcal{D}x(t) e^{iS[x(t)]}.$$

Now, I want to make sure I understand how this integration is defined in more rigorous terms. My issue is that most texts seem to just define that particular integral above (of $e^{iS[x(t)]}$) but for me, to define a path integral would mean define what we mean for a functional to be integrable and define how it would be integrated, the above being just one case.

So I've tried to extract the actual definition from Peskin's QFT book and I want to verify if I got it right.

So in summary we want to define the integral of a functional $\mathfrak{F}[x(t)]$

$$\int \mathfrak{F}[x(t)] \mathcal{D}x(t).$$

After reading a few times what Peskin does, I came up with the following:

Definition: Let $[a,b]\subset \mathbb{R}$ and let a partition $P$ of $[a,b]$ be given: $$a=t_0<t_1<\cdots<t_{N-1}<t_N=b.$$ Let $x_0,\dots, x_{N}\in \mathbb{R}$. We call the piecewise smooth path $x : [a,b]\to \mathbb{R}$ given by $$x(t)=x_k+\frac{x_{k+1}-x_k}{t_{k+1}-t_k}(t-t_k),\quad t\in [t_{k},t_{k+1}],\quad k\in \{0,\dots, N\}$$ a linear interpolation of the points $x_0,\dots, x_N$.

Definition: Let $C_p^{\infty}([a,b];\mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let further $\mathfrak{F}: C_p^\infty([a,b];\mathbb{R})\to \mathbb{C}$ be a given functional acting on such paths.

Let further $P$ be a partition of $[a,b]$ into $N$ subintervals. We define the functional $\mathfrak{F}_P : \mathbb{R}^{N+1}\to \mathbb{C}$ with respect to this partition to be

$$\mathfrak{F}_P(x_0,\dots,x_N)=\mathfrak{F}[x_P(t)]$$

where $x_P(t)$ is the linear interpolation of the points $x_0,\dots, x_N$.

Definition: Let $C^\infty_p([a,b];\mathbb{R})$ be the space of piecewise smooth paths defined in $[a,b]$. Let a functional $\mathfrak{F}: C^\infty_p([a,b];\mathbb{R})\to \mathbb{C}$ be given.

We say that $\mathfrak{F}$ is functionally integrable if the limit:

$$\lim_{|P|\to 0}\int_{\mathbb{R}^{N+1}} \mathfrak{F}_P(x_0,\dots, x_N) dx_0\cdots dx_{N}$$

exists, where $|P|$ is the mesh of the partition, given by $$|P|=\max\{t_{k+1}-t_k | k\in \{0,\dots, N\}\}.$$ In that case we call such limit the functional integral of $\mathfrak{F}$ and denote it by

$$\int \mathcal{D}x(t) \mathfrak{F}[x(t)].$$

So is that the mathematically idea behind it? In the end for Phyics we would be interested in the case $\mathfrak{F}[x(t)]=e^{iS[x(t)]}$.

So in that process I think that the crucial step is:

1. The time-slice procedure together with the linear interpolation construction allows us to turn a functional $\mathfrak{F}$ into an ordinary function which obviously depend on the particular time-slicing (the partition).

2. The function can be integrated giving a result that depends on the partition. We then study the limit as the mesh goes to zero.

3. The thing with analytic continuation, Wick rotation and euclidean action, would then appear just for the specific functional $e^{iS[x(t)]}$ in order to get something that is functionally integrable.

Have I got it right? Is this the actual mathematically precise version of what is done in most QM/QFT texts?

If not, where did I got it wrong or what did I miss?

Answer 1

A mathematically rigorous and satisfactory definition of the path integral is related mostly to solving two problems:

1. To give a proper definition of measure on the space of paths (there is no Lebesgue measure - i.e. a $\sigma$-finite, translation invariant measure - and therefore another measure shall be used);

2. To give a proper definition of oscillatory integrals (an integral of an oscillating phase on a set of infinite measure is ambiguously defined).

In non-relativistic quantum mechanics, it is possible to define the path integral in imaginary time completely rigorously, making use essentially of stochastic Brownian integration (Wiener measure on paths). In fact, in imaginary time the oscillating phase becomes a damping exponential factor, thus simplifying a lot the definition of the integral, and part of the exponential is used to define the Wiener measure. This is very helpful to study semigroups of the type $e^{-\tau (-\Delta +V)}$, for suitable potentials $V$, whenever $\tau\geq 0$. Under suitable conditions this may also give some information on the unitary group $e^{-it (-\Delta +V)}$ (for example, it may be used to prove self-adjointness of $(-\Delta+V)$, thus guaranteeing the existence of the unitary group of evolution). The rigorous path integral formula for Schrödinger operators takes the name of Feynman-Kac formula. It can also be extended to some simple quantum field theories of particles interacting with a radiation field (either through minimal coupling or linearly).

In real time, there have been attempts to define the path integral as an oscillatory integral on the space of paths, using ideas from Hörmander. This is due to Albeverio, Høegh-Krohn, Mazzucchi, and others. However, there are serious complications in this case, and it is possible to give a coherent and consistent definition only in very few special simple cases (such as the harmonic oscillator, and some perturbations of it).

The second procedure is perhaps more adherent in its spirit to the one outlined by the OP, and it takes proper care of mathematical subtleties like the independence of the definition from the chosen approximation of the integral. Unfortunately, let me remark again that it is meaningful only for few specific systems.

Answer 2

Is this the actual mathematically precise version of what is done in most QM/QFT texts?

The description in the question looks correct. In QM, the basic idea is to first discretize time to make the number of integration variables finite (so the integrand reduces to an ordinary multi-variable function instead of a functional), then evaluate that ordinary multi-variable integral, and then take the limit of the result as the time step-size goes to zero — as proposed in the question.

The same idea is used to define a "path" integral in quantum field theory. In that case both time and space are discretized, because the integration "variables" are functions of both time and space. In other words, continuous spacetime is replaced with a discrete (and finite) lattice to define path integrals in QFT. But in most cases with one or more dimensions of space, such as QED and QCD, we don't actually know (rigorously) if a non-trivial continuum limit exists. It might not exist in QED, but it presumably does in QCD, and there's a Millenium Prize waiting for the first person to prove it.

An important caveat is that in QFT, when the lattice step-size is varied, the various parameters in the action (mass parameters and coupling coefficients) must also be varied in the right way in order to keep the model's low-resolution predictions fixed. "Low-resolution" here refers to scales much coarser than the step-size. Even if the continuum limit really does exist, in order for that limit to be non-trivial (meaning that it retains non-zero interactions among the various fields), the mass parameters and coupling coefficients must be regarded as appropriate functions of the step-size when taking the continuum limit. This is the foundation for the modern understanding of renormalization. Regarding renormalization in QM, see "Renormalization in quantum mechanics" (https://arxiv.org/abs/hep-th/9305052).

For nonabelian chiral gauge theories (such as the Standard Model), last time I checked, we still don't yet know how to define the path integral at all, not even in discrete spacetime.

But at least for the case of QM, as shown in the question, it works.