# Talk:Path integral formulation

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## Evolution operator

I believe the evolution operator is:

$e^{-iHt\over{\hbar}}$

## Equivalence of formulations

I believe Dyson was the one that showed the approaches to be equivalent JeffBobFrank 01:21, 18 Feb 2004 (UTC)

## Last paragraph

The last paragraph says some contentious things. The sum-over-histories method is hardly "unpopular". The "sum-over-histories interpretation", however - that is, the attempt to elevate the sum-over-histories formalism into a physical ontology - is indeed little-known; I don't think I've ever seen it outside that paper coauthored by Sorkin. Let me quote the paper's last paragraph:

"... the sum-over-histories formulation goes a long way toward taking the 'mystery' out of quantum mechanics, or at least reducing it to the mystery inherent in the notion of probability itself. No doubt that mystery is enhanced somewhat by the presence of non-positive amplitudes and references to two-way paths, but the fundamental idea... remains the same..."

In my opinion this indicates the sophistical character of this sum-over-histories "interpretation". I'm reminded of a cartoon: a physicist stands at a blackboard, in front of a crowd of skeptical colleagues. In the middle step of his derivation, he has written, THEN A MIRACLE OCCURS. "See? It's all just probabilities. Of course, some of them are negative probabilities, a concept which makes no sense under either the frequentist or the subjectivist interpretation of the concept of probability; but that just shows that further research is required..."

There is something to the claim that "[this is] the only form of the theory which can explain [the EPR] paradox without breaking locality". The individual paths appearing in the formalism are indeed built purely from ontologically local entities (point particles, local field values), something which is not true in any formalism which countenances, say, entangled quantum states. Nonetheless, the paper by Sinha and Sorkin (in its concluding analysis) in fact expresses some doubt as to whether sum-over-histories is local after all, given the "global character" of how the final probabilities are calculated.

Wikipedia is hardly the place in which theoretical debates of this sort should be adjudicated, but I hope it's clear why I find that last paragraph somewhat problematic. I also want to emphasize again, for absolute clarity, that the sum-over-histories method is not being criticised here, because it is only an algorithm. It's the attempt to turn it into an ontology (an "interpretation") which is deeply problematic. I leave it to more experienced Wikipedians to decide what the just solution here is. Mporter 21 Feb 2004, 5.55pm AEST

As a sidelight, apropos your comments about negative probabilities, you may enjoy Feynman "Negative probability" in Quantum Implications, eds Hiley and Peat, where he makes a case for allowing them, as long as such an event is not measurable/verifiable. Like having negative dollars as you add up your bills, it may be calculationally allowed as long as certain restrictions on the state are true.GangofOne 07:04, 10 Jun 2005 (UTC)

## Merge with "Functional integral"

Should this article actually be merged with Functional integral (QFT)? While it is in principle the same subject, that article is both very specific in its application to quantum field theory (as opposed to, say, nonrelativistic single-particle QM), and is also very technical. This seems to be more the place for an introduction to the path-integral formulation. (If we do want to merge the articles, I say the other one should come here, and not the reverse, since this article has the more general title.) And I'd rather do it sooner than later. --Matt McIrvin 04:06, 27 Sep 2004 (UTC)

Well, I went ahead and did it... --Matt McIrvin 06:13, 27 Sep 2004 (UTC)
The material formerly in Functional integral (QFT) is now incorporated into a section here, and I've tried to write some introductory matter to make the symbols a little clearer, though the heavily mathematical part further down still needs a lot more explanatory text. I've put in an introduction and reorganized the whole page into sections and subsections; my new section on single-particle mechanics needs more development but is a start. Diagrams would be nice. I've kept the controversial section on QM interpretation at the very end; I'll let other people argue over that for now. --Matt McIrvin 07:15, 27 Sep 2004 (UTC)
Attempted to NPOVify the interpretation section. --Matt McIrvin 05:35, 2 Oct 2004 (UTC)

## Formulation

Is $\lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ e^{\frac{i}{\hbar}I(H(x_j, t))}$ realy correct?

Wouldn't it rather be like $\lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ \prod_{j=1}^{n-1}e^{\frac{i}{\hbar}I(H(x_j, t_j))}$

with $t_j = j \Delta t$

or is it

$\lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ e^{\frac{i}{\hbar}I(H(x_1, \ldots, x_{n-1}, t_j))}$

with different H for each n ?

The way I wrote it is perhaps not the best way of putting it; it needs to be more explicit. What I really wanted to get across is that in the integrand, $H$ is the function of time represented by a set of straight segments connecting the $x_j$ at times $t_j$, and $I$ is actually the integral of the Lagrangian $L(x, \dot x, t)$ over that path. I suppose in practice it would end up being the product of the exponential for each little segment, but that form is further from the spirit of the thing.
I probably should have abandoned the generic use of $H$ at that point... my mind's too fuzzy right now to make it better. --Matt McIrvin 00:27, 11 Oct 2004 (UTC)
Also each little segment would depend on $x_j$ and $x_{j+1}$... --Matt McIrvin 15:01, 11 Oct 2004 (UTC)
This is not a necessity, the limit inherent to integration would take care of this as $x_{j+1} = x_j +{\rm d}x$, see Riemann sums).

I have searched the net but didn't find anything better than stated here so I have tried some own thoughts.
Starting from the $\sum_{\rm all\ paths}e^{{\rm i} S}$ approach I came up with $\int_{\bar{\varphi} \in \{\bar\varphi | \bar\varphi(0) = \bar a; \bar\varphi(1) = \bar b\}}e^{{\rm i}\int_{\lambda = 0}^{1}\bar E \bar\varphi(\lambda){\rm d}\lambda}{\rm d}\mu(\bar\varphi)$

where $\bar\varphi$ varies over all paths in spacetime starting from $\bar\varphi(0)=\bar a$ and ending in $\bar\varphi(1)=\bar b$, $\bar E$ denoting the energy four-vector and $\mu$ is an aproptiate measure on the set of possible paths. With the paths approximated by segments of straight lines we are likely to end up with the official thing but with an additional benefit of a clearer understanding.

Alas, I am stuck on $\mu$ as well as on $\bar E$, especially in case where we have zero rest mass.
Can anyone do better please? 217.94.149.179 20:05, 20 Oct 2004 (UTC)

## Hidden time

Pavel V. Kurakin (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, me).

My idea is that many-paths are physically real, but in sub-quantum (not observed by us) world. Many-paths, amplified by transactional interpretation of quantum mechanics (TIQM) by John Cramer lead me to a 3rd new idea (after 1st: many-paths and 2nd: transactions). 3 together they constitute, I believe, an original theory, letting to explain quantum superposition of states, state vector reduction and non-local correlations like EPR (see quantum entenglement).

Shortly speaking, signals move in vacuum in so-called 'hidden time', which is not equivalent to our physical time. They move between all sources, which are to emit particles, and all (possible) detectors. In the simplest case we have one source and a set of possible detectors. How will a particle chose one of many detectors?

It explores the space and counts how much it likes different detectors, in full accordance with Feynman many-paths. While it explores (many copies of that particle travel and explore), phisical time does not tick. Finally the source prefers some definite detector. Copies of the particle (more strictly - signals) are killed all but one. This one ultimately comes to a detector we physically see our particle at.

How long can signals explore the space? Infinite time! :) -- In 'hidden' time. Physiacl time does tick (at detecting point) only when 'ultimate decision signal' comes to that detector.

More accurate arguments were published this year by Keldysh Institute of Applied Mathematics, Russian Academy of Sciences in my preprint.

I would be happy to know any criticism :)

• The article is very good, nice references and all that although i think you have "missed" the Semi-classical expansions for the Feynmann Path-integral as $\hbar \rightarrow 0$ could someone provide any reference to this?..thanks. —The preceding unsigned comment was added by 83.213.38.122 (talkcontribs) 21:49, 9 August 2006 (UTC2)
Um that's a funny idea, similar to a crazy idea of mine (which, probably, someone else had already too). However I do not like it. I am not a physicist however and I am referring to your layman summary not your paper so please forgive me if I misunderstood. What I don't like is 1. infinite zero time is essentially the same, or worse, as non-locality. Non-local theories exist [the easiest being "Everything is wave function and it's non-local"], and yours just requires a giant effort from the poor little particle. 2. You seem to have two kinds of mass in your theory, particles and detectors. Fault shared with Copenhagen ("why me worry, the measurement device is classical"). --88.74.163.241 (talk) 09:18, 18 May 2009 (UTC)

## Error

A lot of this stuff is way over my head, but the one thing I thought I understood looks wrong in this article... under the section "The path integral and the partition function", why does it say:

$|\alpha;t\rangle=e^{iHt\hbar}|\alpha;0\rangle$

shouldn't it be:

$|\alpha;t\rangle=e^{iHt\over{\hbar}}|\alpha;0\rangle$? At the very least to make the argument of the exponential unitless? Ed Sanville 16:52, 16 August 2005 (UTC)
Right you are. Fixed. GangofOne 04:59, 17 August 2005 (UTC)

NOt always... 'Edsanville' think user could be using natural units for Planck's constant or other --85.85.100.144 22:23, 16 February 2007 (UTC)

## What is the name of that interpretation?

Hey all, one particular section of the article is a death trap with no leads to further information. Does anybody know the name of the interpretation referenced in the path integral in quantum-mechanical interpretation section? Terms, phrases, some scientific history, anything would be helpful. The section links to another article on the interpretations of quantum mechanics, however there seems to be no segment there that seems a continuation. Thank you, -- kanzure 14:11, 28 July 2006 (UTC)

It may be: Sukanya Sinha and Rafael D. Sorkin, "A Sum-over-histories Account of an EPR(B) Experiment", Found. of Phys. Lett. 4:303-335 (1991). -- kanzure 14:56, 28 July 2006 (UTC)

## QFT?

The article links to QFT, which is a disambiguation page. However, I'm not knowledgeable enough to tell if it should be disambiguated to quantum field theory or quantum fourier transform. Could someone please disambiguate the link? –RHolton– 03:32, 11 November 2006 (UTC)

## Chapmann-Kolmogorov and Feynmann

It's a curious fact that hardly any book points a relationship between te so-called Chapmann-Kolmogorov equation for continous processes and Feynmann path integral formulation, in fact the C.Kolmogorov equation in differential form , is just the discretized SE or Difussion equation (imaginary time), the problem is given the Integral equation of C.K obtain the differential one and hence SE --85.85.100.144 22:21, 16 February 2007 (UTC)

## diffraction grating

I think we should add in this article the interpretation of diffraction grating from the view of path integral formulation. To me, it seems to be the best argument for the case of path integrals, as it effectively explains diffraction grating easily where non-path integral explanations leave much to be desired.

I'm no physicist, so I hesitate to do it myself, but if no one else rises to the challenge, I suppose I can add the section when I get my next holiday. — Eric Herboso 23:54, 23 September 2007 (UTC)

Yes, you are right. Presenting it as Feynman did it with rotating arrows helps to understand it quite intuitively. See also Wikiversity:Making Sense of QM. Arjen Dijksman (talk) 21:31, 27 November 2007 (UTC)

## Path of minimum action always dominates the integral?

In section The path integral and the partition function, it states: In the classical limit, $\hbar\to0$, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel. Wouldn't it be preferable to state: In the classical limit, $\mathcal{S}[x] >>\hbar$, ...? Arjen Dijksman (talk) 21:26, 27 November 2007 (UTC)

## Reality of Paths

The argument over whether the different paths are "real" or not is not really physical. In the Schrodinger equation, if you locate a particle at position x precisely and then a very short time later look for it at position y, you have an amplitude to find it anywhere in space. The first measurement localizes the particle, making its momentum infinitely uncertain. Does this mean that there is a "path" where the particle jumps from one point to another at very large speed? In the circumstances of this particular experiment it does. What if the particle is in a superposition of states at different positions which together have a small momentum? By linearity all the contributions to the enormous large jumps must wash out by superpositions. The phenomenon of wild paths contributing to the quantum mechanical amplitudes is independent of the formalism, it is a property of the theory. Whether the quantum amplitudes for each separate path should be thought of as "existing" is a hoary philosophical question, related to the interpretation debates which can go on with no end. I don't know if it's a good idea to bring them up here.Likebox (talk) 20:07, 14 May 2008 (UTC)

## Did Candlin come up with Grassmann integration?

Many people reference Brezin's textbook, but it's a textbook. I found a reference to this article by Candlin in Nuovo Cimento 1956, but I do not have access to this journal, and I don't know if this is the primary source. If anyone knows, please say.Likebox (talk) 02:28, 16 May 2008 (UTC)

With regards to this, Mandelstam references Candlin, as do a couple of other people, so I think it is provisionally safe to credit him, but it would be nice to do a full literature review regarding this matter, especially since Candlin seems to have fallen silent. Schwinger has a faux Grassman integration in the 50s, which comes up whenever he uses his action principle with anticommuting fields, but he doesn't give a general rule for path integration in the anticommuting case, he just piddled around until he found a consistent set of formal rules for differenting the action. Feynman has a path-oriented path integral which reproduces the statistics and is in principle equivalent to Grassmann integrals, but it's diagram/particle-path based. Brezin's account does do the whole deal, fermi coherent states and all, but he is writing it as if it is already well accepted folklore.Likebox (talk) 20:47, 5 June 2008 (UTC)
I finally read his paper--- it is a beautiful, complete treatment of Grassman integration. It is strange that this person invented a classic tool and then vanished. He has no other papers that I could find, I wonder if anybody knows what happened to him?Likebox (talk) 20:48, 27 August 2008 (UTC)
David John Candlin has a page now, but I don't know any more than the sketchy details provided by the Princeton University catalog of members. Hopefully someone out there does.Likebox (talk) 21:39, 27 August 2008 (UTC)
Ah--- the theory/experiment disconnect. There is an active D.J. Candlin in experimental physics, who wrote 177 papers according to Spires, as part of large collaborations. Perhaps its the same person.Likebox (talk) 04:19, 28 August 2008 (UTC)

## Dirac Fretting about Path Integral

A comment on this page was deleted which asked, if Dirac understood a heuristic version of the path-integral before Feynman then:

"Can someone explain why it was that Dirac fretted about the uncertainty principle when Feynman presented his results..."

This is confusing two frets. It was Bohr who fretted about the uncertainty principle when Feynman presented the diagrams somewhere or other. Dirac fretted about Unitarity.

Bohr's complaint was specious, as Bohr later came to understand, but Dirac's complaint was substantive. Feynman had shown how to pass from the Canonical formalism to the Lagrangian path integral formalism only in certain special cases, that is when the Hamiltonian is quadratic in the momentum. Dirac knew that in other cases, and for a general Hamiltonian, it is difficult to define the proper generalization of the Legendre transformation which will give the right Lagrangian. His worry was that Unitarity is not obvious for a given Lagrangian which is not appropriately related to a unitary Hamiltonian, and this complaint might be the reason that Dirac did not formulate a full path integral formalism. Feynman went ahead probably because he at the time didn't appreciate the severity of the problem, or else because he had a strong physical intuition about the specific cases of quantum field theories, which are always quadratic in the field momentum.Likebox (talk) 06:29, 11 June 2008 (UTC)

## Pseudohistory

The pseudohistory is that Freeman Dyson showed that Feynman's path integral was equivalent to older methods. This is not accurate, Feynman showed this long before Dyson. Dyson showed that it is possible to derive Feynman diagrams from an operator expansion, which when Feynman's path integral was unfamiliar was the easiest way for a physicist to learn some of the new methods. But Dyson's methods are inferior and have been replaced by the path integral.Likebox (talk) 20:49, 19 August 2008 (UTC)

## Normalization

I removed this comment from the article:

However, if the time-sliced path integral is formulated in the phase space of the variables $x(t)$ and $p(t)$, the measure of integration $\prod_j\int dx(t_j)\int d p(t_j)/(2\pi \hbar)$ yields the properly normalized amplitude. The integral over all $p(t_j)$ produces the correct normalization factors for the Feynman integral over all $x(t_j)$.

This statement is true, but (for the case at hand of quadratic kinetic energy) it is just as true for the x-version of the path integral, using $\prod_j \int dx(t_j) \sqrt{2\pi\hbar m}$, the momenta can be integrated out. So the sentence is really just stating what the normalization choice for the path integral should be, but without motivating the choice.

The choice of normalization can be motivated by formal considerations, like computing a propagator and unitarizing, but this is not very illuminating conceptually, but there is a more conceptual way. The factor of sqrt 2pi can be naturally understood as coming from the imaginary time stochastic evolution. The overall normalization of the path integral has a factor of $exp(iE_0t)$, the ground state energy, and the overall scale of the integral in imaginary time shrinks or grows according to the amount of ground state energy, which can be adjusted by adding a constant to the Hamiltonian. The best way to state the condition that fixes the normalization is to demand that the ground state energy is zero, so that the ground state is invariant under path integral time evolution.

When the ground state wavefunction has energy zero, the inner product of any wavefunction with the ground state is invariant in time. This inner product is the integral of psi, so that the total integral of psi is constant in time. This allows psi to be thought of as an imaginary time probability (when the imaginary time action is real), and the evolution is a stochastic process. With this point of view, the factors of sqrt 2pi are obvious--- they give the spreading gaussian normalization for a random walk. This connection to stochastic processes is stated in Feynman and Kac, but is often obscured in modern treatments.Likebox (talk) 20:36, 10 September 2008 (UTC)

Okay--- I think I see the point of the comment--- it is pointing out that the integration measure in the x-p version is simple and universal, while in the x-version it depends on parameters in the action. This is an important point.Likebox (talk) 20:42, 10 September 2008 (UTC)

## Needed improvements in derivation of path integral

This article would benefit greatly from an actual derivation of the path integral, which is not difficult.

1) Start with the matrix element of the time-ordered exponential, which is the time evolution operator.

2) Discretize time and rewrite the time-ordered exponential integral as a product of simple exponentials, exp(-iH(p_i,x_i) Delta t/ hbar).

3) Insert sums of complete states |x_i><x_i| integrated over each x_i in between the exp(-iH Delta t/ hbar) factors.

4) Evaluate <x_i| exp(-iH Delta t/ hbar) | x_{i-1}> by expanding the exponential to first order in Delta t, and going to the momentum representation to express <x_i|x_{i-1}> = int dp/(2pi) exp(i p(x_i-x_{i-1}))/hbar ). Similarly <x_i|p^2|x_{i-1> = int dp/(2pi) exp(i p(x_i-x_{i-1}))/hbar) p^2. Approximate x_i-x_{i-1} = \dot\x times Delta t. Re-exponentiate before doing the integral over p. By completing the square, the p integral is gaussian and gives an irrelevant normalization constant times exp(i Delta t L(x,\dot x)/hbar), where L is the Lagrangian.

5) Putting together all the factors at different x_i's and taking the limit Delta t -> 0, we get the path integral of exp( i \int dt L/hbar) = exp( i S/hbar).

Sorry, I don't have time to actually make these changes to the article.

The reference to "Path Integrals in Quantum Theories: A Pedagogic 1st Step" is useless; this is just a lot of hand-waving.

Jcline1 (talk) 21:04, 1 January 2011 (UTC)

## a better picture of a path?

i noticed this picture on the wiener process article and i think it would be a much more accurate and pedagogical representation of a path in the path integral formulation. In truth, the vast majority of paths look a lot more like this than like the current picture in the article. Kevin Baastalk 19:09, 16 March 2011 (UTC)

## First section terms

This article commits a cardinal sin in the first section of not defining any of the terms in the equations. Not too bad for physics undergrads, but useless for others looking into the topic (which is surely what Wikipedia is catering for). Some knowledgeable PhD want to sort it? 87.194.228.87 (talk) 09:22, 5 October 2011 (UTC)

I agree, all variables and fields should be defined. — Preceding unsigned comment added by 194.171.38.2 (talk) 21:32, 17 September 2014 (UTC)

## Question about the first formula

In what is currently the first formula in the article, is

\epsilon H

just the product, or does it mean the amount by which H changes when time changes by \epsilon? Actually on second thought it is pretty clear that this just is multiplication by \epsilon, and in fact dividing that first equation by epsilon gives H = p. (q(t+e)-q(t))/e + L where the term (q(t+e)-q(t))/e approximates dq/dt. Assuming q' =dq/dt this is pq'+L as usual.Createangelos (talk) 13:14, 20 February 2012 (UTC)

## Rating as petty "mid-importance"??...

Izno: you would disagree. I set the importance to "high" this time, and really couldn't care less if its "quite how importance works" - to hell with "rules" becuase I care more that Feynman rewrote quantum mechanics with path integral formulation,

and a breakthrough in theoretical physics

The article itself says:

"This formulation has proved crucial to the subsequent development of theoretical physics, because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system.
The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks. For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics."

So yes IT IS a very deep and important topic reaching into other parts of physics (even if it doesn't apply in other context it has relevance): special relativity, QM, classical mechanics, and optics. 11:25, 8 June 2012 (UTC)

Regardless, you mistake the use of importance. Importance is not about how important it is to a given person or the world, it is about how important it is to have a good quality topic about it in the encyclopedia. Which can be informed by how important it is in the world, but is not what directly determines it. See WP:1.0.
High may be the appropriate place for it. It's always good to ask the WikiProject, though I would suggest that you have a look at WP:WikiProject Physics/Quality Control#Importance scale... --Izno (talk) 12:16, 8 June 2012 (UTC)
As you may see from the edit history I did look there and used that to back my points up, but then reverted becuase it was excessive point-making on my part. 12:36, 8 June 2012 (UTC)

## Mathematical foundation for the path integral is absent

One issue that this article does not seem to address, but definitely should, is the matter of whether these path integrals are even well-defined to begin with! From a strict mathematical perspective, it is of no value whatsoever that you can carry out reasonable-looking manipulations on a formula to derive useful conclusions, if you cannot even establish that the original formula is well-defined. Until it has been established that a formula is a well-defined expression for some mathematical object, one cannot even begin to prove that it also has the properties that justify the reasonable-looking manipulations, and only when that has been done can the original work rigorously amount to anything.

In the case of the Feynman path integrals, there are at least two points which are problematic:

• The space of all paths is too large for a measure to be straightforwardly definable on it.
• The integrand, and even a simplified version such as $\int_\mathbb{R} e^{ipx}\,dx$, couldn't be Lebesgue integrable even if a measure was given, since both the positive and negative parts of the integral are infinite ($\infty-\infty$ is undefined).

I'm not sure about the measure part (this might even be an open problem in mathematical physics), but the integrand issue requires that the value of the integral is considered to be a distribution (mathematics), does it not? Since a quantity being a distribution carries with it certain caveats regarding what one may do to it, this is an issue that the reader needs to be explictly warned about. 81.170.129.141 (talk) 09:26, 5 March 2013 (UTC)

The mathematical foundations are still unknown, in particular there is no translation invariant Borel measure on infinite dimensional spaces, i.e. $\mathcal{D}q$ does not exist. There are several approaches to this problem. The Wiener measure does exist and the Feynman "integral" can in some circumstances interpreted as some kind of analytic continuation to imaginary times. Another way is to use Feynmans approach and interpret it as a limit of oscillating integrals (this involves distribution theory as you suggested). A third way is defining it as a Fourier transform of measures, sometimes referred to as Fresnel integrals. I think the last two are identical on $\mathbb{R}^n$ but the latter one will not work on manifolds. I think none of these methods work pretty well put heavy restriction on the potential energy.
In the case of field theory there are the Osterwalder Schrader axioms. There was some limited success in constructing lower dimensional models but I think most people gave up. DvHansen (talk) 03:06, 12 August 2014 (UTC)