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If anyone who has studied path integral of Feynman, you will know that it makes quantum mechanics more like classical mechanics. A student who has learned about the lagrangian will easily understand the concept of quantum mechanics through path integral (I think). Why do we still keep using Schroedinger's approach? It's history now.

Maybe to fully understand the Feynman's approach, you need to, erm, integrate so much? I mean, maybe it is based heavily on math?

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  • $\begingroup$ This is a good question. I agree. (Functional integration is a very powerful approach, in general.) I suggest however slightly rewording the question: "What historical facts contribute to the Feynman integral not being taught, today, more widely and earlier in the academic physics curriculum, Schroedinger's approach being taught instead in those cases?" Because Feynman's integration is well know, the question is really how early on it is taught and its use is recommended. $\endgroup$ – Guido Jorg Nov 24 '14 at 15:20
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    $\begingroup$ Short answer, based on some years of experience with the path integral formulation: 1) It is not mathematically well-founded. 2) The conceptual framework can be very hard to grasp for students 3) It does not make simple, intuitive calculations any easier. 4) Functional calculus is not basic, normal calculus is. In short, why would one teach it before QFT (where it does really improve matters)? $\endgroup$ – Danu Nov 24 '14 at 20:21
  • $\begingroup$ @Danu: Why does it not mathematically weel-found? $\endgroup$ – Ooker Nov 24 '14 at 22:02
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    $\begingroup$ @Ooker It is well-known that the path integral is ill-defined, at least in the critical QFT case. See for instance this reference. For some more discussion, see here $\endgroup$ – Danu Nov 24 '14 at 22:08
  • $\begingroup$ I fail to see how this is on topic. It's a question about modern pedagogy, not historical practice of science. Among practicing physicists, path integrals were adopted rather quickly. Neither of the answers is addressing the question from a historical context. I'm also questioning whether the premise of this question is even correct, considering that both the junior-level quantum mechanics class I'm now teaching and the course I took as a freshman undergraduate introduced path integrals to substantial effect. $\endgroup$ – Logan M Nov 25 '14 at 0:15
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I do not agree with the statement, that the lack of mathematical rigor is a major reason for not teaching the path integral formalism in quantum mechanics. The common physicist is normally not interested in complete mathematical rigor, as long as the concepts make sense from a physical point of view and produce the right results. A good example of this is the Dirac formalism - every physicist knows it, and everyone with an elementary education in functional analysis knows that the mathematical justification cannot rely on pure Hilbert space theory. I don't know any textbook which even does an attempt to formulate Dirac's formalism in a rigorous way (Apart from that usual gibberish about spectral decompositions in finite dimensions, which does of course not apply to most problems in QM), that is because the physicist doesn't care and the mathematical physicist does not use bras and kets. The mathematical formulation needs, apart from spectral theory, also some stuff about locally convex and nuclear spaces, which is also pretty specific.

The Feynman integral is a quite intuitive construct from a purely physical point of view, and everyone taking a course in quantum mechanics should have basic education in analytical mechanics, at least it's like this in Germany - usually Lagrange and Hamilton is covered, sometimes also a bit of Hamilton-Jacobi. I don't think this is too much to be asked from an undergrad student who had a course on mechanics. Furthermore, I also don't think that it is too hard for an undergrad to use it, therefore the skills required to introduce and work with the path integral should be given.

From my point of view, there are three major reasons, for the Schrödinger equation being the "standard approach" to quantum mechanics, and not the path integral formulation. Of course, these are correlated.

The first reason is of course, that it has always been done like this. For this reason, there exist a lot of good and established textbooks using this approach, whom I can base a lecture on. There is a wide range of good material and problems. Furthermore, depending on in which area you will work in, the Schrödinger approach is usually a good approach to work with, therefore it is by far not outdated - path integrals are primarily used by particle physicists, in other areas, Schrödinger is usually very useful.

The second reason is, that simple problems one can imagine are usually easier to solve using the Schrödinger equation rather than the path integral formulation. For example, try to compute the harmonic oscillator using path integrals. It's complicated, not elegant and you do not get a big physical insight in what happens. The hydrogen atom is even worse, and I don't want to start with Delta potentials. Another major disadvantage, which makes it quite complicated to solve certain problems, is that it is difficult to change representation in order to compute something. The Feynman formalism deals well with certain problems which primarily appear in QFT, but no one in his right mind would try to solve the hydrogen atom using path integrals. Of course, the choice of what we call "simple and standard problems" is affected by the way we teach quantum mechanics, so this point is correlated to the first one.

The third reason is, that the Schrödinger approrach (in my opinion) has more educational value. Firstly, because it is an approach we in principle already know from mechanics and electrodynamics: Here you have a differential equation, go solve it. In mechanics, it's Newton/Lagrange/Hamilton, in electrodynamics its Maxwell, and in quantum mechanics its the Schrödinger equation. Furthermore, while solving the SE, you usually solve the eigenvalue problem for the Hamiltonian, which of course has a very direct and experimentally observable interpretation as the energy states of the respective system. The kernel of time evolution is a rather abstract construct, which does not have a direct interpretation like this.

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  • $\begingroup$ Hi Daniel, nice to see you around here! I respectfully disagree with your first point (in teaching, I think there is a big difference between knowing there is a rigorous formulation and not bothering to completely explain it vs. knowing there isn't any), but +1 nonetheless for a lot of interesting info on the side. $\endgroup$ – Danu Nov 27 '14 at 19:52
  • $\begingroup$ Very interesting. I like the idea that Schrödinger's equation is more intuitive than the kernel. "Furthermore, while solving the SE, you usually solve the eigenvalue problem for the Hamiltonian, which of course has a very direct and experimentally observable interpretation as the energy states of the respective system", +1 for this. Also, I'm so surprise that mathematicians don't use kets and bras. $\endgroup$ – Ooker Nov 28 '14 at 7:09
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I would like to add a slightly more technical answer to supplement the other (although it is a bit late). It may be helpful, I think, and it touches a broader issue which interesting. It's a conflict of history of teaching versus history of the mathematical prerequisites for the material.

First of all, supposing the topic material is taught early on, how would that work? I like the following books:---they benefit from the long experience of the authors in teaching this material.

Cartier P, DeWitt-Morette C, 2006, Functional Integration, Cambridge University Press. (B)

DeWitt B, 2003, Global Approach to Quantum Field Theory, I, II, Oxford University Press. (C)

Prior this, the following textbook would introduce all basics, for example to student starting with basic knowledge of calculus.

Choquet-Bruhat Y, DeWitt-Morette C, 1982, Analysis, Manifolds, and Physics, I, Elsevier. (A)

This would be a selfcontained series of courses: A, B, C, which would also teach a good chunk of other material, so a pair of introductory mathematical courses, whose content is covered in (A) would be dropped. So, only one more course, in total, would be required. Why isn't this happening?

(X) The answer is indeed historical, due to historically the prerequisite mathematics for the subject, when first developed, gaining the reputation of being "very advanced material" with few introductory texts, and this reputation stuck. Even when no longer justified, it influences how departments construct their curriculum. Mathematics is synergistic. What was once difficult becomes easier when other concepts are discovered and developed. Advances in method over time made the calculations easier and no longer really something difficult in the usual instances, and plenty of introductory books to the material now exist. However, reputations change more slowly than the environment that generated them.

For example, A.E. above brought up the point that analytic continuation is not taught to undergraduates in many places, and complex variables aren't mandatory. That's my experience too. I suggest the same circumstances being the reason.

For rigorous work with functional integration, distribution methods must be used. (Now I prefer sheaf cohomology methods leading to generalized functions. But the justification and application is the same in this case.) The historical issue is: category theory, sheaves, and generalized functions became widely known and fully developed in the 1970's. (After Feynman's integral was in use for over twenty years!) To know how to do calculations in the arbitrary case, beyond the intuitive setting up of the problems (which was developed 1920's-1950's), these later methods are required.

There are plenty of basic introductory books, published 1990's-present, which are not too advanced for undergraduates. E.g., Lawvere's book, MacLane's several books, etc., assume the reader begins with no knowledge whatsoever about even the most elementary mathematical concepts! And the introductory books on generalized functions and distributions begin easily: right hand limits of appropriate series of sums and products applied to 19th century problems, to solve them more easily than is traditional.

The issue is that they are, in fact, books. Buying a book is not sufficient to learn anything. It must be read. There often (for various institutional reasons) simply isn't enough time in the major curriculum to present all the material without having the undergraduate read something in some book that is not presented in class. But in most universities, at this level, use books, if at all, mostly for homework problems... :_( Historically, the education system has been drifting away from required reading the STEM fields.

On the other hand, the other methods can be presented without required reading. And while the typical undergraduate, coming from a typical high school can read books, they usually don't. If they buy them, it's because they contain required homework problems. Many undergraduate courses are not structured to require reading at all. The textbooks are selected only for their changing homework sets.

As you can imagine, and as is indeed the case, whatever cannot be taught this way, or has the reputation of "advanced" isn't mandatory or taught at all until well into graduate coursework. This is merely one case, it seems to me.

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  • $\begingroup$ Thanks for your answer. So in brief, your point is that old method has enough reputation to continue to be taught, and because of the limitation of time, professors don't have enough time to teach and students don't have enough time (or they just don't want) to read new methods. Am I missing your point? $\endgroup$ – Ooker Nov 26 '14 at 22:57
  • $\begingroup$ It's the new method (FI) that has a old reputation, of being "too advanced" for early presentation, despite changed circumstances (availability of rigorous introductory texts). And there isn't sufficient time to teach prerequisite classes without giving some sections of introductory texts as readings, which goes against the historical trend in how to teach ("books are for homework sets", etc., etc.). Now, if GEC courses far outside the major were made not mandatory for graduation, there'd be time to teach everything early on in the usual way, but that isn't happening any time soon. $\endgroup$ – Guido Jorg Nov 27 '14 at 7:56
  • $\begingroup$ @GuidoJorg Thanks for the nice answer. Could you supply some more precise references to the books you refer to (MacLane, for instance?) I'm quite interested, and would like to obtain them. $\endgroup$ – Danu Nov 27 '14 at 20:15
  • $\begingroup$ Sure, I'll update my answer with a categorized list tomorrow. I'll include too papers and monographs that are impressively accessible (I think) but show how to significantly simplify calculations in the most frequently occurring special cases. $\endgroup$ – Guido Jorg Nov 28 '14 at 1:59
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Let me try to summarize what was said in the comments and add something. (I am a teacher of mathematics, not physics, but this is close). First of all not all students study Lagrangian mechanics at an early stage. It requires much more mathematical sophistication than most undergraduate students have. (I am talking of US here).

Second, Feynman's integral itself cannot be called fully mathematically justified. There were many attempts to formulate it rigorously, I am not prepared to analyse these attempts and how successful they are, but one thing is clear: this justification requires by far more advanced mathematics than most physicists know (let alone undergraduates). In their daily work most physicists just do not care about rigorous mathematical justification. But teaching is another matter:-)

Third. It is not true that Schrodinger's formulation has only historic value. It is simple and convenient, and there is a rich and rigorous mathematical theory behind it. It is widely used by physicists and mathematicians. (Heisenberg's approach (matrix mechanics) is less popular, but even that has much more to it than "historical significance").

Fourth. Education is EXTREMELY conservative. Let me give another example: multivariate calculus (Stokes theorem and its relatives). The modern exposition is based on the formalism of differential forms of H. Cartan, invented in the beginning of 20-th century. It is much SIMPLER and more powerful than what we teach to undergraduates. Still undergraduate calculus is taught with 19-th century formalism of "vector analysis" almost everywhere (except France). The reason for this is that teachers tend to teach the way they were taught themselves:-)

But in the case of Feynman integral, I think that the main reasons are the second and third.

By the way, Feynman himself, in his Lectures on Physics explains quantum mechanics without his integral. He has one special lecture (a kind of appendix) on the Principle of least action, but warns that this is a much more advanced stuff than the rest of the book:-)

P.S. I do not want to register for the Physics site to answer the question whether "Lagrangian" must be capitalized. But my opinion is this: there are rules of the language. They may be logical or not, we may like them or not. But there are codified rules, and it is better to obey them. In Russian all these words are written with lower case, in German they write riemannsche Flache, but in English: Riemann surface. So in different languages different words in the same expression are capitalized.

P.P.S. For many years I was looking in bookstores the book of Feynman and Hibbs, Quantum mechanics and path integrals. The shelves are always full of "Are you joking Mr. Feynman?" and such, but this book is long out of print... Finally I purchased a Russian translation in a used book store in Ukraine.

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  • $\begingroup$ Small note: It is, in fact, standard to take at least one course using Lagrangian mechanics in a European undergraduate education in physics. However, the treatment is not rigorous (i.e. no real understanding of variational methods etc.) $\endgroup$ – Danu Nov 24 '14 at 22:13
  • $\begingroup$ +1 For mentioning the value of the Schrödinger formulation, by the way. Mathematical physicists are able to prove many theorems in functional analysis using methods that rely heavily on the theory of Schrödinger operators etc. to rigorously justify the handwaving of quantum mechanics as most physicists know it. $\endgroup$ – Danu Nov 24 '14 at 22:17
  • $\begingroup$ I'm so surprise that path integral is not mathematically rigorous. $\endgroup$ – Ooker Nov 25 '14 at 1:00
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    $\begingroup$ @Ooker you're in for a lot of surprises if you assume physics is always rigorous... $\endgroup$ – Danu Nov 25 '14 at 10:16
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    $\begingroup$ @Ooker that assumption is quite naive ;) $\endgroup$ – Danu Nov 25 '14 at 11:58
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For quantum mechanics path integral can be done rigorously using Feynman-Kac formula and analytic continuation, but the approach is by no means elementary. A single look at the Feynman-Kac formula shows why it is not a great place for students to start at. Moreover, understanding where it comes from is itself a chore. While path integral heuristics are intuitively attractive, implementing them in practice - with actual computations - is a different matter.

While path integral is called "integral" even in the context of quantum mechanics it is something more complicated, a pairing between a distribution and a test function. Analytically continued Feynman-Kac formula is impractical for actual calculations, but what it amounts to is solving an initial-boundary problem for the Schrödinger equation. There are many more methods for doing that, separation of variables and Laplace transform for example, and they are more elementary than the theory of measures and distributions on infinite dimensional spaces.

On the other hand, teaching Feynman diagrams early was argued convincingly by John Baez and others, not in the context of QFT, but in the context of computing finite-dimensional integrals perturbatively, see here for an elementary exposition.

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  • $\begingroup$ Sorry, I did not see your answer when I was writing my own, but now I see that it appeared earlier. I am not sure that "analytic continuation" that you mention was rigorously justified in all cases. Can you give a reference on a complete rigorous treatment ? $\endgroup$ – Alexandre Eremenko Nov 24 '14 at 22:27
  • $\begingroup$ In a US university where I teach undergraduates do not know what "analytic continuation" is:-( A course in complex variables is not mandatory even for Math undergraduates, and we do not teach analytic continuation in the undergraduate course anyway. $\endgroup$ – Alexandre Eremenko Nov 24 '14 at 22:29
  • $\begingroup$ There is a book about it by Albeverio et al. that uses Gaussian measures on Hilbert spaces, "cliff notes" are here scholarpedia.org/article/…. One needs restrictions on the potential to make it rigorous of course. $\endgroup$ – Conifold Nov 24 '14 at 23:46
  • $\begingroup$ Yes, there is a book of Albeverio. Not exactly an undergraduate stuff:-) $\endgroup$ – Alexandre Eremenko Nov 25 '14 at 15:12
  • $\begingroup$ See my answer, which continues this line of thought in a different way ;) $\endgroup$ – Guido Jorg Nov 26 '14 at 9:58

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