Following on my notes about Euler’s formula, I’ve finally finished some work on another piece of elementary exposition, a discussion of the free quantum particle, which can be found as chapters 10, 11 and 12 of the book I’m working on.
These chapters are a complete rewrite and major expansion of what used to be there, a rather slap-dash single chapter on the subject. The excuse for this in my mind had been that it’s a topic treated in detail in every quantum mechanics textbook, so best if I passed over it quickly and moved on to things that weren’t so well treated elsewhere. Another reason for this was that my understanding of analysis has never been what it should be, and it seemed best if I not make that too obvious by how I handled the mathematics of this subject.
This summer I started rewriting the book from the beginning, and once I hit the chapter on the free particle it became very clear that it needed improvement, both for its own sake and for how the material was needed in later chapters. I spent some time doing some remedial study in analysis, and after a while got to a point such that I felt capable of writing something that captured more of the relevant mathematics. Finally, today I got to the point where these three chapters are in decent shape, and soon I’ll move on to later ones.
One thing that I’d never thought much about before, but that struck me while rewriting these chapters, is the quite peculiar nature of a position eigenstate in quantum mechanics. Normally one only thinks about this in relativistic quantum field theory, where the problems associated with localizing a relativistic particle motivate the move to a quantum field theory. Of course a position eigenstate is just a delta-function, but what is peculiar is the dynamics, what happens if you take that as an initial condition. See the end of chapter 12 for what I’m talking about (be sure you have the latest version, today’s date on the front), I won’t try and reproduce that here. Part of this story is the tricky nature of the free-particle propagator in real time, as opposed to its much better behavior in imaginary time. The issue of analytic continuation in time continues to fascinate me, including the quite non-trivial nature of what happens even for the supposedly trivial case of a free particle in one spatial dimension.
Hi,
a small typo on the pg.146:
“At any time T >0, no matter now small, this will be a phase factor with
constant amplitude extending out to infnity in position space”
in that quote: n->h
as someone with a physics background, but with interest in more mathematical topics I think I’ll start with your book when I try to refresh my memory of the nonrelativistic QM.
thanks for making the drafts public!
Peter,
Page 145, equation for U(T,q_0,q_T). You say “to make sense of such an integral, on can…” [define it using Wick rotation, basically]. This is not really correct, and is a common misconception. The integral is perfectly well-defined as it stands, and can be evaluated with no additional analytic continuation, Wick rotation or otherwise. It’s called the Fresnel integral (there is a Wikipedia article about it), and it can be solved by using the solution of the Gauss integral on the real line, the Cauchy residuum theorem, and a creative choice of the integration contour.
The Wick rotation is not necessary to define that integral. Rather, it has to do with the fact that the Dalambertian operator is hyperbolic, and does not have an inverse. Then one uses the Wick rotation to transform the Dalambertian to a Laplacian (which is elliptic and therefore has an inverse), and then define the inverse of the Dalambertian by analytic continuation of the inverse of the Laplacian.
HTH, 🙂
Marko
vmarko,
“creative choice of integration contour”
I should look more closely, but I’m still not convinced that’s not a definition that uses analytic continuation.
Agreeing with vmarko about page 145. For another treatment, for example see Bogoliubov&Shirkov, page 147ff.
vskrin,
Thanks. Fixed.
Peter Morgan,
That’s the relativistic propagator, which is trickier. As far as I can tell Bogoliubov and Shirkov use the +i\epsilon method to deal with the issue I’m discussing. Doing this defines the integral as the boundary value of something well-defined in a complex half-plane, away from the boundary. I don’t think it’s inaccurate to characterize this as analytic continuation definition.
Seeing your response to vmarko, perhaps all that’s necessary is to replace “To make sense of such an integral …” by something like “To derive an analytic expression for such an integral …”.
The propagator is more complicated, precisely because of the distinction between hyperbolic and elliptic operators, i.e. Minkowski vs. Euclidean spacetime. For that story you really need the Wick rotation. But for the integral discussed in Peter’s book, this is unnecessary, because the integral can be evaluated without any analytic continuation.
The integration contour is drawn in the Wikipedia article on the evaluation of the Fresnel integral, and the rough steps of the evaluation are explained briefly. It’s a simple exercise in complex analysis. Moreover, the function under the integral is the exponential, and since it doesn’t have any poles the calculation does not contain any undefined quantities or such, so no analytic continuation involved.
HTH, 🙂
Marko
I took a look again at Brian Hall’s “Quantum Theory for Mathematicians”, and it has an excellent treatment of the free particle (see chapter 4), as well as much more about the subtleties of the Heisenberg uncertainty relation (chapter 12).
In exercise 2 of chapter 4 he derives the integral discussed here in a manner that doesn’t initially seem to be the usual boundary of a holomorphic function derivation. I’ll have to look more carefully at that…
Dear Peter,
to show the convergence of the Fresnel integrals (without complex analysis) one can use the Dirichlet test. Evaluation is more painful though, but is in most classical analysis books. For a newer exposition, you can check this book by S.Popescu:
http://civile.utcb.ro/cmat/cursrt/ma2e.pdf
The convergence is in Example 50, p.122 (of the text), the evaluation is Example 60, p. 164.
Peter,
The “tricky nature of the free-particle propagator in real time” is part of the Feynman path-integral approach as presented in Feynman and Hibbs, if I recall correctly. And, of course, if you convolve the kernel against any reasonable function, it works as it should.
Back in the mid-’70s, I had a conversation with Weinberg about how this is different in the QFT case due to the pesky 1/E factor (1/sqrt(E) in terms of the definition of the quantum field) required by relativistic invariance of the volume element in momentum space. Steve said he had never thought of the fact that that this messed up localizing the particle at a point.
Of course, there is another way of looking at it — you can view the vacuum as having non-trivial correlations at short distances (revealed in the non-zero VEVs), but Steve likes looking at particles as primary and the fields as just a way of describing interactions among the particles.
I’m still bothered as to how this works out in the fermion case — I know the math, but I am not sure how to view it physically, especially if one tries to think in terms of fields rather than the infinite number of particles of the Dirac sea. Alice Rogers has had some things to say relevant to this, but I’m not sure I understand her.
Dave
It is not hard to see that he Fresnel integral exists as an improper Riemann integral without any need for analytic continuation or deformation of integration contour away from the real line. As you take the integration limits to infinity, the function oscillates more and more rapidly (cancelling itself out more and more) so the contribution of any interval of given size at larger and larger distances becomes less and less. You can show the limit exists.
This is not true, however, for the moments of the Fresnel integral we often need to calculate, e.g., \int dx x^n e^{- alpha x^2}.
These can be calculated, though, without analytic continuation also. A nice source on how to do this is
Albeverio et al., J. Funct. Anal. 113, 177
Basically they do this by the generalization of the e^{-epsilon x^2} convergence factor we often add in Physics, do the integral, and take the limit epsilon -> 0. They generalize this to a general class of test functions and show that the limit is unique and independent of the test function.
We do something very similar to calculate Fourier transforms of tempered distributions
Hi Peter,
At the end of chapter 12, you’ve given the expression for a time-evolved position eigenstate as a Gaussian that broadens over time. What do you mean when you describe it as “a phase factor with constant amplitude extending out to infinity”?
Can you tell us more about the process of writing a book? How did it come about? Was it commissioned by a publisher? Or did you begin a draft and later contact a publisher?
By the way, while the Fresnel integral straightforwardly converges as a improper Riemann integral, I believe it doesn’t exist as a Lebesgue integral. To define it in the sense of Lebesgue, you do need the kind of convergence factor trick as in the Albeverio paper I mentioned above. Since you need that trick anyway as soon as you want to calculate moments, that might as well be your starting point. Still, no analytic continuation is ever necessary in either case.
Many thanks to all who provided references about the question of convergence of this integral. I wasn’t aware of ones not using some sort of i\epsilon argument (or equivalently a -\epsilon^2 convergence factor. I still think though the highly singular nature of what you get when you try and localize a non-relativistic particle (as opposed to the good behavior in imaginary time) is remarkable.
Bob,
It’s only a Gaussian in imaginary time, in real time it’s a phase factor.
Andrew,
The book came about because I started teaching a quantum mechanics course for mathematicians. There was no appropriate book covering a lot of what I wanted to do (e.g. the representation theory point of view, for other topics there are good books, for instance the Brian Hall book is great on the analysis side of the story). So I started writing notes for the class, at some point it became clear these were turning into a book. This project has now though gone on much longer than I wanted, I hope to finish soon.
I heard from several different publishers interested in what I was writing, earlier this year signed a contract with Springer. No, in case you’re wondering, there’s no money to be made writing this kind of thing, the motivation is purely the feeling that someone should do it, and that one learns a lot by working on something like this.
Hi Peter
So it is. You’re right, it is strange.
If you take a normalisable but very sharply peaked wavefunction at t=0, you can presumably convolute it with that object and get the corresponding wavefunction at t=T. The further you go spatially from the original peak, the more rapidly that phase changes with distance, so beyond some distance the contribution it makes to the wavefunction at time T will be arbitrarily small, provided the original function has a finite gradient everywhere.
I see that it works, but the way that it works is certainly strange.
Anonymous wrote:
That’s right, functions like sin(x^2) or cos(x^2) or exp(ix^2) are not Lebesgue integrable, because Lebesgue integration can’t handle the “infinite cancellation” required to get a finite answer. But you can use any reasonable way of first imposing a cutoff or forcing the function to decay for large x and then removing this; they all give the same answer. Analytic continuation is essentially one of these methods.
This goes to show that the Lebesgue theory is not the last word on integration, and not nearly good enough to do path integrals in real time.
The Henstock-Kurzweil integral is able to handle this particular integral.
Thanks John,
The lack of Lebesgue integrability makes precise the problem with the definition I was trying to put my finger on.
Bob, these propagators are certainly strange objects. It may help sometimes to remind oneself that propagators are not generally functions. Rather, they are distributions (generalized functions), which generally don’t even need to have any representation as pointwise functions. Even when they do have such a representation on part of their range (as happens to be the case with the t>0 free particle propagator), you can do all kinds of things with them (e.g. Fourier transforms, moments, etc.) involving “integrals” that are divergent in the Riemann or Lebesgue sense. I promise that for the most part all this makes intuitive sense once one familiarizes oneself sufficiently with generalized functions.
The canonical source for generalized functions is the brilliant multi-volume:
I.M Gel’fand and G.E. Shilov, Generalized functions