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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.
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."
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.
I think this should be with respect to . 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 .
Why did we drop the time-discretization?
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.
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?
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!
What Hilbert space does this operator act on? If we are groping around trying to find one, then please make this explicit.
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.
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 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)
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)
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)
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)