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Is the Path Integral formulation of QM just a mathematical tool or does it offer deep physical insights on the nature of QM? Is it just an alternate way to describe Quantum Mechanics? Could someone say that the Schrodinger formulation is better/worse than the path integral formulation or are they just two different ways to describe the same thing? If the later is the correct answer, then why did the path integral formulation ever need to be developed? Does it offer something more that the original formulation does not offer?
And, well to sum it all up, is any of the two formulations more fundamental?

EDIT: I am not seeking opinions here. I am seeking facts. What i mean by that is for example: "did this formulation make more connections with classical mechanics, did this formulation predict more things or is this formulation regarded as just a mathematical tool(not being accepted as what actually goes on)?

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You can certainly use the path integral in the sense of an interpretation. It teaches you what a truly classical particle would have to do to reproduce the behavior of the quantum world. Curiously, since this is the season, an analogy to Santa's Sleigh is appropriate. Santa has to visit all homes in one night. The "particle" in a path integral has to take all possible paths while integrating over the exponential of the action. Whether you want to "believe" in either Santa or such a classical integrator particle is entirely up to you. I think you can guess my opinion.

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CuriousOne's response is one of the most strikingly beautiful explanations I've seen. But to answer your other questions:

"Is the Path Integral formulation of QM just a mathematical tool or does it offer deep physical insights on the nature of QM? Is it just an alternate way to describe Quantum Mechanics?"

The word "just" is key here. If you don't agree that it's possible to adopt an intuition via some sort of explanation congruent to that of CuriousOne, then you are still able to observe the connection to classical dynamics offered by the path integral formulation. The Lagrangian formulation of mechanics via the principle of least action has been around for a long, long time and represents the classical limit (or $\hbar \rightarrow 0$ limit. see ref [1]) of the path integral. So, it is not just a mathematical tool- it is the extension of a concept that underpins classical mechanics- that, in itself, does indeed give insight into the deeper nature of the quantum theory.

"Could someone say that the schrodinger formulation is better/worse than the path integral formulation or are they just two different ways to describe the same thing?"

We can show that the reproduce the same equations of motions in the non-relativistic case. So in that sense they describe the same things. However, things don't go so well with the Schrödinger equation when we approach the relativistic case. The path integral approach does better in this regard.

"If the later is the correct answer, then why did the path integral formulation ever need to be developed? Does it offer something more that the original formulation does not offer?"

Besides offering a deep connection to classical physics, the path integral approach turns out to make it much easier for us to do calculations in quantum field theory. It is, however, more difficult to make rigorous formulations for this approach (you can google the mathematical difficulties of "constructive quantum field theories" to find all the info you need on that). Despite this, the answer is "yes": it offers a lot more than just describing the same phenomena in different ways.


Reference above: Paul A. M. Dirac, "The Lagrangian in Quantum Mechanics", Physikalische Zeitschrift der Sowjetunion, 3 (1933) 64–72}}

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  • $\begingroup$ Since we are talking about beauty, Feynman's article "Space-Time Approach to Non-Relativistic Quantum Mechanics", R. P. Feynman Rev. Mod. Phys. 20, 367 is, in my opinion, one of the absolute highlights of scientific writing. I have a bound copy of it in my personal library. His book "QED: The Strange Theory of Light and Matter" is also one of my all time favorites of popular science writing. Having said that, I think you are right, the idea came from Dirac, didn't it? $\endgroup$
    – CuriousOne
    Dec 25, 2015 at 10:21
  • $\begingroup$ @CuriousOne your answer was great, but it was a close one between your answer and this answer $\endgroup$ Dec 25, 2015 at 14:38
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    $\begingroup$ Nice answer, but the "$\hbar\to 0$ thing... Is horrific. That is constant. We cannot make it to tend to zero! $\endgroup$
    – Les Adieux
    Dec 25, 2015 at 15:14
  • $\begingroup$ Kim Peek, horrific? No one said we are changing constants of nature. We simply see how quantum laws reproduce classic ones in certain analytical limits, and there are many articles that do this thought experiment with Planck's constant as well thought experiments that tinker with the fine structure constant- (and the FSC is dimensionless, so I'm guessing you might find those even more horrific) $\endgroup$ Dec 25, 2015 at 15:55
  • $\begingroup$ @CuriousOne yes, there's a famous funny story about how Feynman got the idea from Dirac, compared a propagator produced by this new principle to the established answer, and realized they agreed! $\endgroup$ Dec 25, 2015 at 16:00
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It's a consequence of quantum mechanical principles. I wouldn't consider it essential. To derive it one needs the essential mathematical framework of Dirac's quantum mechanics. In fact in Dirac's QM book he gives the formula of the relation of action from an initial to final state. The Feynman path integral is really just an analysis of further intermediary states, ad infinitum. Though I am not all that well versed in QFT, I believe sometimes the path integral approach is preferred because it starts out with a Lagrangian that can be made relativistic easier. But as far as I know, this isn't essential, and one can develop the theory relativistically using canonical methods. I definitely would not consider it more fundamental than the QM as described by Dirac. And you may want to study a bit more about QM, since Schrodinger is just one representation of QM, but its most general and abstract setting is given by Dirac. Feynman developed it I believe to better understand the interference effect of QM, in relationship to the double slit experiment. He thought what would happen if you made multiple slits, into an infinity of slits, and then deduced the particle must travel all paths. Reference: Principles of Quantum Mechanics by Dirac

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The classical conception of QM has two standard formalisms; the Schrodinger picture where states have a dependency on time; but observables do not; and the Heisenberg picture where the reverse is true.

The pictures are equivalent by 'basis change', which was first demonstrated by Dirac; but they are not quite equivalent in a physical sense; for example, the Heisenberg picture is better adapted for the move into relativistic QM.

The path integral is a third and different formalism; and is generally considered more useful in the development of QFT.

The physical motivation for its development, at least historically, was by Feynman, and along the lines suggested in the answer by Tenes; and which, in a way already demonstrates its physical relevance.

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