This spring I’ve been teaching a course aimed at math graduate students, starting with quantum mechanics and trying to get to an explanation of the Standard Model by the end of the semester. Course notes for the first half of the course are available here, but still quite preliminary, in particular I need to do a lot of work on section 9.4 and add material to chapter 10. Hoping to get to this tomorrow.

There won’t be much progress on the notes for the next couple weeks. This coming week I’m hoping to spend some time trying to understand Peter Scholze’s IAS lectures, will go down to the IAS on Tuesday. On Thursday I’m heading out on a spring break vacation to the Arizona-Utah desert.

Perhaps a good way to think about these notes is that they’re both aimed more at mathematicians than physicists (although I hope accessible to many physicists) and also designed more to supplement than to replace the discussions in the standard physics texts. So, a lot of the standard material is not there, since it’s well-covered elsewhere, but there are a lot of topics covered that usually aren’t.

One unusual aspect of the notes is that I spend a lot of time trying to explain non-relativistic quantum field theory, since that seems to me to be a better starting point than immediately diving into the relativistic case. I’d be curious to know if anyone can point me to a good discussion of the path integral formalism for non-relativistic quantum field theory, which is something I haven’t found. This is one reason it’s taking a while to finish writing up my own version.

Also original here I think is a careful discussion of the real forms of spinors and twistors. This in some sense is background for the new ideas about “spacetime is right-handed” which I’ve been working on. Nothing in the notes now about the new ideas, but I hope the explanation of the conventional story in these notes is useful.

Hey Peter,

From the Physics side of things, there are some recommended texts that discuss the path-integral for non-relativistic QFT. In my experience, they are usually associated with advanced statistical mechanics and condensed matter. Not sure if this is what you are looking for, but a couple of standard choices are:

– QFT in Condensed Matter Physics, Naoto Nagaosa

– Quantum Many-Particle Systems, John W. Negele and Henri Orland

All the best

The path integral for non-relativistic quantum fields (the so-called coherent state path integral) is everywhere in condensed matter theory. Two few textbooks, just off the top of my mind:

Altland and Simons, “Condensed Matter Field Theory”, Chapter 4

Dupuis, “Field Theory of Condensed Matter and Ultracold Gases”, Chapter 1 (that chapter is accessible for free here https://www.worldscientific.com/doi/epdf/10.1142/9781800613911_0001 )

This kind of path integrals can be generalized to many Lie algebras (su(2), su(1,1), etc), see e.g. Perelomov “Generalized coherent states and their applications”.

Rosenfelder, Roland. “Path integrals in quantum physics.” arXiv preprint arXiv:1209.1315 (2012).

https://arxiv.org/abs/1209.1315

Rosenfelder, R., 1994. Particles and shadows: a generalized path-integral approach to non-relativistic field theories. Journal of Physics A: Mathematical and General, 27 (10), p.3523.

https://iopscience.iop.org/article/10.1088/0305-4470/27/10/027/meta

Thanks for the references!

Closest to what I’ve been looking for are section 2.2 of Nagaosa and section 1.4 of Dupuis. These give an imaginary time path integral of the right sort, but do it for finite imaginary time, to do a finite temperature calculation. Not seeing a discussion of zero temperature and the relation via analytic continuation to the real time Wightman function.

Yes, one can think of these as “coherent state path integrals”, as well as “phase space path integrals”. I wrote something very long ago about the problems with these, see

https://www.math.columbia.edu/~woit/tqtandrepthy.pdf

At the time I got interested in this when thinking about the Chern-Simons path integral. If you look at something very simple with the same structure (path integral over paths on a sphere as phase space) you see that there are very serious problems making any sense of such things. A fundamental aspect of quantization is that you can’t just go directly from a symplectic manifold to a quantum system, preserving the symplectic invariance (the infinite-dimensional group of symplectomorphisms). There’s something fundamentally conceptually wrong with the phase space/coherent state path integral, because it appears to do this. For the thing to make sense you need somewhere to smuggle in a polarization or somehow break the symplectic invariance.

When your integral is just a Gaussian, then you can formally do the exact integral and get something sensible, at least in imaginary time. There’s something interesting going on here which I don’t fully understand. The formal “derivations” of a coherent state path integral though aren’t really helpful, since they have obvious problems (you’re “integrating” over “paths” with no continuity property, with formulas involving their derivatives).

Peter,

It is possible to do some calculations (in imaginary time) even if the action is not Gaussian. In the case of a quartic term, one can use a Hubbard-Stratonoich (HS) transformation to render the integral gaussian, integrate out the field, and then HS field. Because, as you remark, neither the original field nor the HS field are continuous, one needs to be extra careful. However, using insight from Ito calculus (the HS field is effectively a weight noise), one can recover the exact result, see Sec. II of https://arxiv.org/abs/1906.02571.

I’ve become interested in this problem when trying to understand how to (properly) perform non-linear change of variables in such integrals. Contrary to standard Feynman path integrals, here it is much more subtle because the field is not continuous (in the Feynman case, it is continuous but not differentiable, which allows for controlling the derivatives of new variables, however with the introduction of new “quantum potential” terms). No such luck here however. As far as I know, it is still an open problem.

Adam,

Thanks for the comment and reference. These seem to be consistent with my understanding, that for configuration space path integrals you have continuity and if careful can do transformations on the the fields with results consistent with usual quantization. But for phase space or coherent state path integrals, no continuity, and (beyond linear transformations), the “integral” doesn’t at all have sensible properties under change of variables.

Peter,

Are you saying that such non-linear changes of variables would be completely impossible? Would you have a reference that would substantiate that?

If it is in the reference you gave in a comment above, could you explain the argument for someone not really knowledgeable in CS TFT?

Adam,

I haven’t thought much about this in a very long time, and even then, I was focused on a different problem: how do you make sense of path integrals over paths in G/T (G compact, T maximal torus) and recover the usual quantization story there, where it’s the representation theory of G?

There is a no-go theorem, the Groenewold-van Hove theorem. Because of that theorem, something must go wrong if you do non-linear symplectic transformations on the variables, in the sense that such transformations can’t correspond to unitary transformations on the states. A different aspect of this is that, in addition to a phase space you need a polarization on that phase space to get a quantization, and the quantization depends subtly on the polarization. What I’m wondering about phase space path integrals is where the polarization choice comes in.

Again, a problem for me here is that I haven’t thought seriously about this in a long time.

For the path integral, i would suggest

https://link.springer.com/book/10.1007/978-3-540-76956-9

Mathematical Theory of Feynman Path Integrals

An Introduction

Authors: Sergio A. Albeverio , Raphael J. Høegh-Krohn , Sonia Mazzucchi

Part of the book series: Lecture Notes in Mathematics (LNM, volume 523)

That is what a math student would go for. It starts with non-relativistic path integrals and then goes over to quantum field theory, but all in theorem proof style. Some of the authors work at Bonn university.

Hi Peter,

there are some path integral books by Kleinert. I’m not so sure how close it is to what you are looking for. Here some version from his homepage.

https://hagenkleinert.de/documents/pi/HagenKleinert_PathIntegrals.pdf

Peter,

you wrote, ”There’s something fundamentally conceptually wrong with the phase space/coherent state path integral, because it appears to do this. For the thing to make sense you need somewhere to smuggle in a polarization or somehow break the symplectic invariance.”

But you dind’t notice that this is already smuggled in in the coherent state integral, since (as especially John Klauder emphasized repreatedly) a coherent state provides not only a symplectic structure but also a compatible metric structure, which is just a way to define the required polartization.

Arnold Neumaier,

When you naively write down the coherent state path integral, looking at the measure and action, only difference with the phase space integral is the boundary conditions. Like in the phase space path integral, choice of polarization appears just in the boundary conditions.

It’s been a long time since I looked at Klauder’s attempts to define coherent state path integrals. I guess you can try and use the Kahler structure polarization to do this. But after spending a lot of time looking at this kind of thing years ago, I never found anything that seemed convincing.

I’m just a student, but I like the following books on path integrals:

1)G.Roepstorff- Path Integral Approach to Quantum Physics ( Grad level mathematical physics book);

2)L.S. Schulman- Techniques and Applications of Path Integration (Undergrad-Grad level theoretical physics book)

Anon,

Nothing in the Roepstorff book about phase space/coherent state path integrals, but it has an excellent detailed treatment of Euclidean quantum fields. The Schulman book near the end has a chapter on the phase space path integral, carefully explaining the problems with it.

Hi Peter,

you might find a series of papers by Hashimoto et al useful, for example “Borel-Weil theory and Feynman path integrals on flag manifolds” (https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-23/issue-2/Borel-Weil-theory-and-Feynman-path-integrals-on-flag-manifolds/10.32917/hmj/1206128252.full) or “The Borel-Weil theorem and the Feynman path integral” (https://mathweb.tifr.res.in/sites/default/files/publications/studies/SM_13-Geometry-and-Analysis.pdf#page=325). Building on earlier work of Alekseev, Fadeev, and Shatashvili, they describe how to perform path integral quantization of coadjoint orbits the form G/T and, for the linear Hamiltonians corresponding to Lie group generators, how to regularize the path integral to obtain the usual representation theory of G. (I will admit I haven’t fully gone through these papers, so I’m not sure how correct they are.)

PS,

I’ve looked a lot at this kind of thing, but I don’t think they’re really “path integrals” in any useful sense. They depend crucially on exactly how you’ve chosen to discretize the “path” (which is an infinite set of points in a space, with no continuity properties). If you can’t change variables, is this really an integral in any coordinate-invariant sense?

I see. So your issue is with the phase space path integral in general. I guess from geometric quantization, it’s not a surprise that one can only give a simple definition of the phase space path integral for classical observables whose flow preserves the polarization, and perhaps has to resort to a BKS-type construction for more general observables. The fact that the coordinate space path integral is invariant under general coordinate transformations would correspond to the fact that the cotangent lift of such transformations preserve the vertical polarization. But I have to admit, I’ve never given it much thought (I don’t use path integrals much).

I’ve found one paper that discusses this: “Symplectic structures, BKS kernels and path integrals of BRST bosonic strings” (https://link.springer.com/article/10.1007/BF01550943).

Peter,

”It’s been a long time since I looked at Klauder’s attempts to define coherent state path integrals”

See:

J. R. Klauder and I. Daubechies,Quantum mechanical path integrals with Wiener measures for all polynomial Hamiltonians, Phys. Rev. Lett. 52 (1984), 1161.

I. Daubechies and J. R. Klauder, Quantum mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. J. Math. Phys. 26 (1985), 2239.