Talk:Path integral formulation

Bad and weak section on "Quantum action principle"

The section entitled "Quantum action principle" does not make much sense. It is undefined and unmotivated.

It could have been a model of how you open your mind, creatively re-interpret numbers as operators on the fly, and brilliantly construct and interpret Hilbert spaces for them to act on as you go. It could have been a gradual emergence of crisp ideas from a genial fog. But it's not. It's just someone mumbling to himself incomprehensibly.

Here is a rundown, until I run out of energy.

In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory.

Why do you need to know the trajectory to do the Legendre transform? When you go back and forth between L and H, it is just between two functions. There is no mention of trajectory. Perhaps the following is meant: "In quantum mechanics, it is hard to see what to do with the Lagrangian, because the motion is not over a definite trajectory."

In classical mechanics, with discretization in time,

$\epsilon H=p(t)(q(t+\epsilon )-q(t))-\epsilon L\,$

and

$p={\partial L \over \partial {\dot {q}}}\,$

What is the path p(t), q(t)? It hasn't been introduced. Are we solving for it? Are we varying it? Does it already satisfy Hamilton's equations? Are we going to do something "along" it?

Why is there a discretization in time? It looks like the author knows where he is headed, because he knows how it comes out, but he hasn't stopped to tell us the goal.

where the partial derivative with respect to q holds q(t + ε) fixed.

I think this should be with respect to ${\dot {q}}$. Or maybe with respect to q(t)? But how can it make sense to say that q(t + ε) be held fixed during the differentiation when L is not even a function of q(t + ε)? It is a function of q and ${\dot {q}}$.

The inverse Legendre transform is:

$\epsilon L=\epsilon p{\dot {q}}-\epsilon H\,$

Why did we drop the time-discretization?

In quantum mechanics, the state is a superposition of different states with different values of q, or different values of p, and the quantities p and q can be interpreted as noncommuting operators. The operator p is only definite on states that are indefinite with respect to q.

These commonplace remarks about QM are just a distraction at this point, except for the suggestion that we should interpret p and q as (possibly noncommuting) operators.

So consider two states separated in time and act with the operator corresponding to the Lagrangian:

$e^{i(p(q(t+\epsilon )-q(t))-\epsilon H(p,q))}\,$

This expression is entirely unmotivated at this point.

Is it supposed to be familiar to us from our study of classical mechanics? Then say this, and tell us what it is called in that field.

I notice that the same quantity, namely exp(iεL), appears the following section on work of Feynman. It is much easier to understand there.

I also have trouble with the fact that we are trying to interpret q(t) as an operator, where q(t) is the value of position at time t of a trajectory. Aren't the position and momentum operators universal entities, independent of t, in normal quantum mechanics? How can we use the value of position at a particular time to define a position operator? I would think it would end up being an eigenvalue of that operator.

Also p is written without t. Why?

two states separated in time

What kinds of things are these states? Are they complex numbers attached to each point in spacetime like the Feynman amplitudes in the following section? Or are they vectors in the usual Hilbert space H=L^2(R^3) that is used for Schroedinger's equation? From the text written here, I have no idea.

Is the state is given by a function f(t) taking values in H? Then say this, don't keep it a secret!

act with the operator corresponding to the Lagrangian

What Hilbert space does this operator act on? If we are groping around trying to find one, then please make this explicit.

If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean?

There is a problem here. Wikipedia writes reinterpreted. But the original interpretation has never been given!

What is the original, classical meaning of exp(iεL) in classical mechanics that we are trying to generalize to quantum mechanics? Does it have something to do with stationary phase, Huygens principle, or geometric optics? If so, this should be stated explicitly, not left for the astute reader to literally mind-read.

The first factor is

$e^{-ipq(t)}\,$

If this is interpreted as doing a matrix multiplication, the sum over all states integrates over all q(t), and so it takes the Fourier transform in q(t), to change basis to p(t). That is the action on the Hilbert space – change basis to p at time t.

Now that this clue has been given, we can guess that the Hilbert space might be L^2(R^n), born with "q" coordinates but also possessing "p" coordinates. But this raises more questions than it answers. First, why does q depend on t but p not?

Second, if the t is dropped and just $e^{-ipq}$ presented, then I can see that, if integrated with respect to q, it will do a Fourier transform.

But what gives us the right to perform an integration? This is pulled out of a hat here. If we were planning on doing an integration, this should have been announced in advance.

Third, putting the t back in, why can we perform an integration with respect to q(t)? This is no longer the free variable q. It is a function of the free variable t. Any integration would have to be with respect to t, but I don't think that's what Wikipedia wants us to do here.

Quite possibly, it is an integration with respect to all paths q(⋅). In this case, we actually do have to integrate with respect to the value q(t) assumed by the path q at the time t, and do this for every t. But this should have been explained in advance. Only the reader already familiar with the path integral formulation could be expected to guess this at this point.

In short, reading this section is an exercise in accident reconstruction. For a person who already knows the material, it might be possible to interpret the section accurately. For someone very sophisticated in mathematics and physics, but who does not already know this construction, it is very difficult. For someone who isn't as strong intellectually, but still really wants to know quantum, it's an invitation to have a mental breakdown and wake up as Deepak Chopra.

89.217.24.127 (talk) 01:37, 13 May 2015 (UTC)

"Strictly speaking the only question that can be asked in physics is: "What fraction of states satisfying condition A also satisfy condition B?""

Where does this quote come from? There's no citation. What is it even supposed to mean? Not only is this completely meaningless "generalized" semantics mumbo jumbo, it also doesn't even strike me as true. I'm not sure how something such as measuring the speed of light would be described by this question. Unless you define "states" as something dumb such as the readings on your instruments. --176.199.192.165 (talk) 18:19, 28 April 2016 (UTC)

Who is this article written for?

I view myself as a literate, relatively intelligent person. I'm interested in physics, and although I have no formal training in the field, as a hobby I have acquainted myself with a few of the basics of classical mechanics, general relativity, and quantum field theory. And still, I find this article mostly incomprehensible to me. I feel that the entire article is very technical and at no point in it is more widely accessible language used. I do not understand this particular subject well, so I can not solve this problem myself, but I'm hoping that one of the very knowledgeable people who wrote this article can! This information is so important, so fundamental to the workings of our universe. . .and yet I fear my lack of understanding will be a common experience other readers will also have. Wikipedia is an encyclopedia, and I think we need to make its information accessible to a general audience, to curious people who are not already specialists in a given field! Becca (talk) 00:26, 10 November 2017 (UTC)

To give you some concrete examples of my confusion, after trying to read this article I still don't really even know what the path integral formulation is. Is it an interpretation of quantum mechanics, comparable to the Copenhagen interpretation or the many-worlds interpretation? Or is it a specific equation like Schrodinger's equation? Or perhaps a set of equations like the Einstein field equations? Or something else entirely? After attempting to read this article, I still don't know. Also, I want to know whether the path integral formulation is supported by all particle physicists. . .or does it have some critics? What are their criticisms of it? Have experiments been conducted to try and test the formulation? Or is it the sort of thing that is not testable? As you can see, I have a lot of questions. I think the level of detail and math in this articleis excellent, but I think a more general and concise summary somewhere in the article would greatly improve it. Becca (talk) 01:09, 10 November 2017 (UTC)

Yes, the article is necessarily technical, and not a tutorial for anyone who has not mastered quantum mechanics first. It is virtually impossible to make a popularly accessible "story" about a mathematical formalism, and not merely purveying a Lie-to-children. The article states up front it is a formulation of QM, and not an interpretation. So any talk of interpretation here (beyond mathematical interpretation, a very different, technical concept) is misplaced--so, in my view, section 8 should be a footnote, and not a section, as it has potentially misled, e.g., you. It is a mathematical technique re-expressing standard QM, Schroedinger equations and all, in a language amenable to some types of calculation. But it is equivalent to the standard QM formulation, and, as such, it predicts identical experimental results as standard QM--so there could be no issue of "testing" it or "criticizing" it. I can't help feeling you are in the wrong article here. Popular science publications are aggressively awful at summarizing such things, and create torrents of misconceptions impossible to resolve in WP, so a simple physics book could serve as a sensible tutorial. Cuzkatzimhut (talk) 15:10, 10 November 2017 (UTC)

Thank you very much for your response, Cuzkatzimhut! And thanks to you, I have a much better idea of what the path integral formulation is. I was specifically helped by this section of your comment: "it is a mathematical technique re-expressing standard QM, Schroedinger equations and all, in a language amenable to some types of calculation. But it is 'equivalent' to the standard QM formulation, and, as such, it predicts identical experimental results as standard QM--so there 'could' be no issue of 'testing' it or 'criticizing' it." That's great! And honestly, I think some language like that would be an excellent addition to the opening of this article. With due respect, however, I do disagree with you on what function this article should serve. You say that this article is only written for people who have already mastered quantum mechanics. However, according to Wikipedia's policy WP:NOT, Wikipedia should be an encyclopedia for a general audience--NOT a scientific journal or research paper. The policy reads: "A Wikipedia article should not be presented on the assumption that the reader is well-versed in the topic's field. Introductory language in the lead (and also maybe the initial sections) of the article should be written in plain terms and concepts that can be understood by any literate reader of Wikipedia without any knowledge in the given field before advancing to more detailed explanations of the topic." This Wikipedia policy (like all Wikipedia policies) is binding, and it's what I'm advocating. I don't want any of the technical information removed. I actually commend your commitment to accuracy and precision and agree with you that popular science publications are often very misleading in their summaries! But just because a good summary is very difficult to write does not mean it shouldn't be attempted. What would be so bad about leaving all the technical information, but including a few sections of more accessible summary (especially in the opening)? I must say I am definitely not in the wrong article; ALL Wikipedia articles should be written so that a general, literate audience can understand them. That's Wikipedia's policy. Take a look at the article on quantum mechanics, for example. It's an article that many brillant scientists have contributed to over the years. In some sections, it has language as simple as this: "Many electronic devices operate under effect of quantum tunneling. It even exists in the simple light switch. The switch would not work if electrons could not quantum tunnel. . ." Now. . .is that inaccurate? Maybe I'm wrong, but I think if that article can contain passages that are that simple then so can this article (even if they are quite hard to write). Becca (talk) 20:47, 10 November 2017 (UTC)

Footnote of lead image.

It reads: "These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t). Paths which self-intersect or go backwards in time are not allowed."

Surely it's meant to say that paths which go backwards are not allowed, which could be a corollary of the non-intersection axiom. Otherwise it suggests something related to time travel.--TZubiri (talk) 04:03, 18 October 2020 (UTC)

I gather it is trying to exclude going backwards, e.g. S-curves. "In time" is a metaphor for antiparticles. Cuzkatzimhut (talk) 20:44, 18 October 2020 (UTC)