How many years of math and physics training are usually needed to “understand” virtually the entire contents of the link regarding ‘Sum over Paths’?
By “understand”, I mean to have an intuitive sense what the symbols and equations mean, how they are derived, the assumptions underlying them, and the limitations of their application.
Is this undergraduate, albeit senior, physics? Strictly graduate? Post-graduate?
How many years of study are we looking at?
Related question: is this stuff pretty typical of what’s studied in mathematical physics? More advanced? Less?
I hope the mods will keep this in GQ since a) those best qualified to answer tend to be denizens of GQ, and b) there may be a factual answer (or very close).
Like many concepts in both mathematics and physics, you can study path integrals at different depths, and depending on that, you will need different levels of sophistication to deal with them. The general discipline, functional integration, I think to this day has some very important open questions (such as whether such integrals even exist in a rigorous sense) in the mathematics, so it’s an area of open research; on the other hand, if you’re fine with a certain amount of handwaving, you could easily introduce path integrals in an undergraduate course on quantum mechanics. The main mathematics you’ll need is functional analysis, plus a bit of linear algebra in order to make the connection to the usual formulation of QM via linear operators on Hilbert space.
The path integral is not in itself something especially sophisticated, it’s simply a way of doing quantum mechanics that’s equivalent to more widespread methods, such as using Schrödinger’s formulation, or Dirac/von Neumann’s, etc. Those formulations likewise can be studied at different levels of sophistication, from simple matrix multiplication up to the full operator-theoretic C*-algebra formalism. Path integral methods are particularly widely used in quantum field theory, which is a difficult subject in itself, but not because the path integral introduces any additional complexity (actually, I found the path integral method to be much more accessible than so-called canonic quantization).
In pursuing an undergraduate mechanical engineering degree, I had (IIRC) three semesters of calculus. That would have been enough to understand most of the mathematical symbols and concepts seen at your link.
However, in order to prep for the college calc classes, high school went like this:
I think there is a bit more than just multi-varaible calculus. Some of the path integrals look like they are over the complex plane so you will probably want some complex analysis, you will also need to pick up some understanding of Fourier transforms, but I saw that in a physics class before I saw it in a math class.
So assuming your starting point is at “just about ready to take calculus” you will need a couple of years worth of college level maths plus whatever math you learn in your physics courses on the way to understanding the physics required.
I’m a math guy more than a physics guy (although I did have 2 years of undergrad physics including some basic quantum mechanics) but to me this looks like early (first or second year) grad school level stuff.
Undergrad physics prof here. The path-integral formulation of quantum mechanics (sections 1–3 of the Wikipedia article) might be something that a undergraduate senior would see in a second-semester quantum mechanics course. It’s not really part of the standard curriculum at that level, though, and it might well be put on the back-burner in favor of a different topic that the professor prefers.
Sections 4–6, on the other hand, deal with quantum field theory. This is not something that is usually taught to undergraduates unless they’re incredibly advanced and motivated. Usually it’s something that would be taught in the first or second year of graduate school.
One of the things that struck me when I looked through the Wiki article was the degree of abstraction that (I think) it contained. Well, that, and the large number of distinct ‘pure’ math concepts it contained (i.e. path integrals of course, but also functionals, Fourier transforms, dual spaces, Green’s Function, Noether’s Theorem, and more). And that’s what motivated my question - it looked to me that you’d need a degree in math just to begin to approach the physics. So, second year graduate level makes a lot of sense.
Part of what it means to get a degree in a technical field is that, even if you’re not familiar with some concept you come across, you ought to be able to look into it on your own and come to some understanding of it without the hand-holding of a class. After all, the concepts that get taught in classes have to get developed, and learned by the first teachers, somehow.
Another thing to note is that some concepts are advanced enough that very few, if any, people in a given field can immediately come to terms with it without further study. It used to confuse me how even some decently highly regarded professors in a field would just outright say “I need time to digest this information” or “I don’t understand exactly what they’re doing here.”
Even without a class, I’ve been in meetings where a bunch of professors, grad students, and undergrad researchers sit around in a room and read a new paper and explain it to each other until everyone understands it. Usually the people who have been working in the field for 20 years come to terms with the info faster, but it’s not horribly uncommon for them to get just as lost as others at some of the tricker parts of the finding.
ETA: I’ve had some long-time professors in machine learning outright tell me that they “wing” some of the finer points of inferential statistics or (multi-variate) vector calc, rather than have an immediate intuitive understanding of it.
Most of the concepts you named, you’d encounter first in a course on theoretical mechanics, which is taught in the third semester here in Germany: functional analysis (at least some basics) you need to solve variational problems, finding the equations of motion of a system using the Euler-Lagrange equations, Fourier transforms are extensively used for all kinds of wave phenomena, and Noether’s theorem—the statement that any symmetry of the system leads to a conserved quantity—is used to motivate the various conservation laws (energy, momentum, angular momentum etc.). Green’s functions you’d maybe only learn in the next course, theoretical electrodynamics, while dual spaces are something you use implicitly for a long time before really picking up/being told what they are.
So these are things you would know in your second year of studying physics, however, as I said, they wouldn’t have been introduced at a mathematician’s level of rigour, or at the greatest possible sophistication. Often, you won’t learn a subject in detail in the first go-round, but revisit the subject, adding new layers to your understanding and solidifying the old ones, as you progress through your studies.
So I think it’s definitely feasible to introduce path integrals at the undergraduate level, though you’d probably be more likely to meet them in your first year of graduate studies.