Two things recently made me think I should write something about path integrals: Quanta magazine has a new article out entitled How Our Reality May Be a Sum of All Possible Realities and Tony Zee has a new book out, Quantum Field Theory, as Simply as Possible (you may be affiliated with an institution that can get access here). Zee’s book is a worthy attempt to explain QFT intuitively without equations, but here I want to write about what it shares with the Quanta article (see chapter II.3): the idea that QM or QFT can best be defined and understood in term of the integral
$$\int_{\text{paths}}e^{iS[\text{path}]}$$
where S is the action functional. This is simple and intuitively appealing. It also seems to fit well with the idea that QM is a “many-worlds” theory involving considering all possible histories. Both the Quanta article and the Zee book do clarify that this fit is illusory, since the sum is over complex amplitudes, not a probability density for paths.
This posting will be split into two parts. The first will be an explanation of the context of what I’ve learned about path integrals over the years. If you’re not interested in that, you can skip to part II, which will list and give a technical explanation of some of the problems with path integrals.
I started out my career deeply in thrall to the idea that the path integral was the correct way to formulate quantum mechanics and quantum field theory. The first quantum field theory course I took was taught by Roy Glauber, and involved baffling calculations using annihilation and creation operators. At the same time I was trying to learn about gauge theory and finding that sources like the 1975 Les Houches Summer School volume or Coleman’s 1973 Erice lectures gave a conceptually much simpler formulation of QFT using path integrals. The next year I sat in on Coleman’s version of the QFT course, which did bring in the path integral formalism, although only part-way through the course. This left me with the conclusion that path integrals were the modern, powerful way of thinking, Glauber was just hopelessly out of touch, and Coleman didn’t start with them from the beginning because he was still partially attached to the out-of-date ways of thinking of his youth.
Over the next few years, my favorite QFT book was Pierre Ramond’s Field Theory: A Modern Primer. It was (and remains) a wonderfully concise and clear treatment of modern quantum field theory, starting with the path integral from the beginning. In graduate school, my thesis research was based on computer calculations of path integrals for Yang-Mills theory, with the integrals done by Monte-Carlo methods. Spending a lot of time with such numerical computations further entrenched my conviction that the path integral formulation of QM or QFT was completely essential. This stayed with me through my days as a postdoc in physics, as well as when I started spending more time in the math community.
My first indication there could be some trouble with path integrals I believe started in around 1988, when I learned of Witten’s revolutionary work on Chern-Simons theory. This theory was defined as a very simple path integral, a path integral over connections with action the Chern-Simons functional. What Witten was saying was that you could get revolutionary results in three-dimensional topology, simply by calculating the path integral
$$\int_{\mathcal A} e^{iCS[A]}$$
where the integration is over the space of connections A on a principal bundle over some 3-manifold. During my graduate student days and as a postdoc I had spent a lot of time thinking about the Chern-Simons functional (see unpublished paper here). If I could find a usable lattice gauge theory version of CS[A] (I never did…), that would give a way defining the local topological charge density in the four-dimensional Yang-Mills theory I was working with. Witten’s new quantum field theory immediately brought back to mind this problem. If you could solve it, you would have a well-defined discretized version of the theory, expressed as a finite-dimensional version of the path integral, and then all you had to do was evaluate the integral and take the continuum limit.
Of course this would actually be impractical. Even if you solved the problem of discretizing the CS functional, you’d have a high dimensional integral over phases to do, with the dimension going to infinity in the limit. Monte-Carlo methods depend on the integrand being positive, so won’t work for complex phases. It is easy though to come up with some much simpler toy-model analogs of the problem. Consider for example the following quantum mechanical path integral
$$\int_{\text {closed paths on}\ S^2} e^{i\frac{1}{2}\oint A}$$
Here $S^2$ is a sphere of radius 1, and A is locally a 1-form such that dA is the area 2-form on the sphere. You could think of A as the vector potential for a monopole field, where the monopole was inside the sphere.
If you think about this toy model, which looks like a nice simple version of a path integral, you realize that it’s very unclear how to make any sense of it. If you discretize, there’s nothing at all damping out contributions from paths for which position at time $t$ is nowhere near position at time $t+\delta t$. It turns out that since the “action” only has one time derivative, the paths are moving in phase space not configuration space. The sphere is a sort of phase space, and “phase space path integrals” have well-known pathologies. The Chern-Simons path integral is of a similar nature and should have similar problems.
I spent a lot of time thinking about this, one thing I wrote early on (1989) is available here. You get an interesting analog of the sphere toy model for any co-adjoint orbit of a Lie group G, with a path integral that should correspond to a quantum theory with state space the representation of G that the orbit philosophy associates to that orbit. Such a path integral that looks like it should make sense is the path integral for a supersymmetric quantum mechanics system that gives the index of a Dirac operator. Lots of people were studying such things during the 1980s-early 90s, not so much more recently. I’d guess that a sensible Chern-Simons path integral will need some fermionic variables and something like the Dirac operator story (in the closest analog of the toy model, you’re looking at paths moving in a moduli space of flat connections).
Over the years my attention has moved on to other things, with the point of view that representation theory is central to quantum mechanics. To truly play a role as a fundamental formulation of quantum mechanics, the path integral needs to find its place in this context. There’s a lot more going on than just picking an action functional and writing down
$$\int_{paths}e^{iS[\text{path}]}$$
Interesting. I mentioned in an earlier thread that I was currently working through Blundell and Lancaster’s QFT For The Gifted Amateur. Before taking up that book I looked at Zee’s Nutshell. It’s notable that Zee starts right off introducing path integrals in chapter 1 while in Blundell/Lancaster they don’t show up until halfway through the book. I was wondering whether that was purely a personal choice or whether it represented a broader difference in schools of thought about how the subject should be taught.
Academic Lurker,
and Michel Talagrand in What is a Quantum Field Theory? A First Introduction for Mathematicians (Peter, thanks for mentioning this book!) completely ignores path integrals: “I have also decided to stay entitely away from path integrals […]. There is no doubt that from the point of view of physics, path integrals are a correct approach, but they are not well defined mathematically, and I do not see what yet another heuristic discussion of them would bring.”
In fact, I am working through all three (Talagrand, Zee’s Nutshell and Blundell/Lancaster) and I have just finished Zee’s QFTaSaP which I have to return to aSaP ;).
AcademicLurker,
Different people have different points of view, depends a lot on what your experience with qft is and what sorts of calculations you’ve worked on. For a long time I would have argued that the path integral is the place to start (and I think I did this when I taught a QFT course in the math department back in 2003). More recently, in the QM course I’ve been teaching and the book I wrote, I avoided path integrals completely. One reason for this though was that the course doesn’t cover interacting QFTs.
At this point it seems to me that it’s a good idea to start off with free field theories as an infinite collection of harmonic oscillators and emphasize the propagator, without getting into path integrals at the beginning. By the way, the path integral is a horrible way to calculate things in the QM simple harmonic oscillator). The path integral formalism is a very nice way of organizing the perturbation expansion, and it comes into its own in Yang-Mills theory. But it should be taught with caveats about the necessity of analytically continuing in time, difference between configuration space path integral and (much more problematic) phase space path integral. I think the whole field is suffering from a lack of awareness of the problems with the path integral formalism (which likely point to ways in which the fundamentals of the subject are unsatisfactory).
Andrzej Daszkiewicz mentions correctly that the Talagrand book (also Folland’s) are good examples of where the authors saw no way to give the kind of careful discussion they wanted in terms of path integrals. There are other books by mathematicians (see Glimm/Jaffe) that are based on path integrals, but then they stick to a certain class of theories where the path integral works well (and stay in Euclidean space-time).
A belated remark on many worlds and complex amplitudes (as opposed to probability densities): this situation is not specific to QFT at all. Think of the more elementary setting where unitary transformations are applied in discrete time steps, as is often the case in quantum information and computation. Then we get a finite sum instead of a path integral, and it is a sum of complex amplitudes (not probabilities). And that doesn’t prevent many worlds from being a viable interpretation.
Pascal,
No intent to start a discussion of many-worlds. My point was just that readers of the Quanta article hearing that a sum over all possible paths is “how the world really is”, might think this sounds like the many worlds multiverse, but it’s really something quite different. This is especially clear in the Euclidean time picture, where QM is precisely a probabilistic measure on possible histories of worlds, but this is a very different thing than the Born rule probability.
Hi Peter
The sphere integral is the “spin coherent state path integral” for a single spin, i.e. the first-order term is a Wess-Zumino-Novikov-Witten term. At least for 2D WZNW CFTs (Witten’s nonabelian bosonization), there is no problem reproducing results from the formal affine Lie algebra rep theory (e.g. scaling dims for local operators) via perturbation theory in the corresponding matrix sigma model. The perturbation theory works at large levels. Maybe the restriction to finite KM primary fields is a nonperturbative effect that would require a more serious formulation of the path integral?
Matthew Foster,
Yes, my toy example is the “spin coherent state path integral” and I’m claiming it’s problematic to think of it as an actual integral. You can certainly use it for a semi-classical approximation, but that’s different.
I wrote up all this way back in in 1989, published in an obscure location. A copy of the paper is here https://www.math.columbia.edu/~woit/tqtandrepthy.pdf
This is about 0+1 theories, for 1+1 QFTs, there’s a more complicated story. The WZW path integral is a configuration space path integral (analogous to taking the group manifold as configuration space in 0+1d)
There’s lots and lots more to say about these 1+1 d QFTs related to representations of loop groups or affine Lie algebras. From what I remember it was always my intention to write a follow up to the above paper that would discuss this, but I don’t think I ever finished that project, should go back to it.
Hi Peter,
In condensed matter, the main use of WZNW terms of this form is in the context of topological boundary modes, and the connection to associated topological theta terms in one higher dimension. For example, a 1+0-D path integral that encodes chiral edge-state wave functions of a quantum Hall droplet, in the presence of disorder, realizes a WZNW action that is a generalization of the spin coherent-state path integral. This allows one to connect directly to the topological character of the quantum Hall state in one higher spatial dimension, since WZNW is the boundary term accrued from the integration of the topological theta term in a finite system. More sophisticated “anomaly matching” arguments (e.g. with a Chern-Simons “response” action for the edge or bulk) give the same conclusions.
Usually we have an alternative nonperturbative framework for the boundary WZNW theory, so one doesn’t have to evaluate directly the likely-ill-defined path integral. Where we run into problems is with the higher-dimensional bulk, with a topological theta term that is hard to deal with and even hard to sensibly derive. This is because the coefficient of the theta term accrues contributions from the ultraviolet, and a satisfactory UV completion (in terms of a concrete lattice model in our case) is often too unwiedly to do much with.