This posting is about the problems with the idea that you can simply formulate quantum mechanical systems by picking a configuration space, an action functional S on paths in this space, and evaluating path integrals of the form
$$\int_{\text{paths}}e^{iS[\text{path}]}$$
Necessity of imaginary time
This section has been changed to fix the original mistaken version.
If one tries to do this path integral for even the simplest possible quantum field theory case (a non-relativistic free particle in one space dimension), the answer for the propagator in energy-momentum space is
$$G(E,p)=\frac {1}{E-\frac{p^2}{2m}}$$
Fourier transforming to real-time is ill-defined (the integration goes through the location of the pole at $E=\frac{p^2}{2m}$). Taking $t$ complex and in the upper half plane, for imaginary $t$ the Fourier transform is a well-defined integral. One gets the real-time propagator then by analytic continuation as a boundary value. For a relativistic theory one has
$$G(E,p)=\frac{1}{E^2-(p^2+m^2)}$$
and two poles (at $E=\pm \sqrt{p^2+m^2}$) to deal with. Again Fourier-transforming to real-time is ill-defined, but one can Fourier transform to imaginary time, then use this to get a sensible real-time propagator by analytic continuation.
Trying to do the same thing for Yang-Mills theory, again one gets something ill-defined for real time, with the added disadvantage of no way to actually calculate it. Going to imaginary time and discretizing gives a version of lattice gauge theory, with well-defined integrals for fixed lattice spacing. This is conjectured to have a well-defined limit at the lattice spacing is taken to zero.
Not an integral and not needed for fermions
Actual fundamental matter particles are fermions, with an action functional that is quadratic in the fermion fields. For these there’s a “path integral”, but it’s in no sense an actual integral, rather an interesting algebraic gadget. Since the action functional is quadratic, you can explicitly evaluate it and just work with the answer the algebraic gadget gives you. You can formulate this story as an analog of an actual path integral, but it’s unclear what this analogy gets you.
Phase space path integrals don’t make sense in general
Another aspect of the fermion action is that it has only one time derivative. For actions of this kind, bosonic or fermionic, the variables are not configuration space variable but phase space variables. For a linear phase space and quadratic action you can figure out what to do, but for non-linear phase spaces or non-quadratic actions, in general it is not clear how to make any sense of the path integral, even in imaginary time.
In general this is a rather complicated story (see some background in the part I post). For an interesting recent take on the phase-space path integral, see Witten’s A New Look At The Path Integral Of Quantum Mechanics.
Update: A commenter pointed me to this very interesting talk by Neil Turok. The main motivation that Turok explains at the beginning of the talk (and also in the Q and A afterwards) is exactly one that I share. He argues that the lesson of the the last 40 years is that one should not try and solve problems by making the Standard Model more complicated. All one needs to do is look more closely at the Standard Model itself and its foundations. If you do that, one thing you find is that there’s a “trouble with path integrals”. In Turok’s words, the problems with the path integral indicate that “the field is without foundations” and “nobody knows what they are doing”.
I do though very much part company with him over the direction he takes to try and get better foundations. He argues that you shouldn’t Wick rotate (analytically continue in time), but should complexify paths, analytically continuing in path space. For some problems doing the latter may be a better idea than doing the former, and in his talk he works out a toy QM calculation of this kind. But the model he studies (anharmonic oscillator) doesn’t at all prove that going to the imaginary time theory is a bad idea, for some calculations that works very well. He’s motivated by defining the path integral for gravity, where Euclidean quantum gravity is a problematic subject, but the gravitational version of the toy model I think will also be problematic. The ideas I’ve been pursuing involving the way the symmetries of spinors behave in Euclidean signature I think give a promising new way to think about this, and you won’t get that from just trying to complexify the conventional variables used to describe geometries.
It’s completely off-topic, sorry, but I just want to bring to your attention this comment uploaded today on the ArXiv (ref: 2302.07897) that severely criticizes the paper by D. Jafferis et al. on traversable wormholes that generated so much hype recently. According to this new analysis, that looks very solid, several essential claims made in the paper are not justified.
@Peter
Have you read Padmanabhan’s Quantum field theory the why, what and how? His section 5.6.3 Propagator from a Fermionic Path Integral is fabulous. As someone who never really got how to extend my intuition of QM to QFT this book was a lifesaver 🙂
I am a little confused as to why you consider the discretization to be problematic. Is it because discretization is necessary for doing calculations, and if the theory doesn’t discretize properly, the calculations could give the wrong answer? Or is it because discretization is necessary for the fundamentals of physics, and if the theory doesn’t discretize properly, the theory is wrong?
More Anonymous,
That is a nice beginning book, one starting from the path integral. The “Fermionic Path Integral” though I think is a bit misleading, since it’s not an integral…
Robert A. Wilson,
Nothing to do with physics at short distances. Physicists are claiming to do certain integrals over infinite dimensional spaces. Mathematicians trying to make sense of this tend to get caught up in issues of the right formalism to define such integrals, which leads to a very complicated story.
I’m pointing out you can think about this more concretely: discretize the problem to make the integral finite-dimensional, then see if you can get a well-defined limit. For some theories like Yang-Mills, this procedure works, and is even necessary (there is no definition in infinite-dim, and the correct nature of the limit is very non-trivial). If you try to do this for other theories though (like the Chern-Simons or toy examples), trying to put the thing on a computer and calculate shows you clearly the serious problems that face attempts to get a well-defined path integral in those cases.
Peter,
“As written, this integral doesn’t exist.”
What do you mean by this? Isn’t that just the Fresnel integral?
Complete the square, use Euler’s formula to express exp in terms of sin and cos, and thus reduce the rhs to integrals of sin(k^2) and cos(k^2), i.e. Fresnel integrals. What doesn’t exist there?
Best, 🙂
Marko
Marko,
Yes, what I wrote isn’t right, was trying to refer to a problem with computing the propagator you see before you get to the formula I wrote. Will edit to fix.
My understanding is that the Osterwalder-Schrader theorem shows how this gets right to the heart of why it’s so hard to construct a QFT that really exists and satisfies the Wightman axioms. You can actually build a QFT iff you can actually define this measure (for imaginary time), and show reflection positivity (which allows you to Wick rotate back to real time). So isn’t the trickiness you describe exactly the right thing for mathematicians to get caught up in?
Blendletan,
I just fixed the first part of this post. The point I’m trying to make in that first section is that even for the simplest scalar free field theories the path integral as written is ill-defined. What is well-defined is the imaginary time path integral.
But even in imaginary time, making sense of the path integral for an interacting field theory is very hard. If you can do this, Osterwalder-Schrader give the conditions under which you can reconstruct a real-time theory.
I’ve been spending a lot of time looking at this story for spinor fields, where it is very non-trivial. If you’re a mathematician and want to write a QFT book using well-defined objects, you can do what Jaffe/Quinn did, sticking to scalar fields. Or, if you want to do spinor fields and QED like Folland/Talagrand, you need to avoid the path integral and define the theory in other ways.
What do you think about the “map” or “dictionary” from QFT to statistical mechanics using Wick rotation? The latter is such that the path integrals are well-defined Wiener integrals, so if you could justify the Wick rotation in the first place, perhaps the path-integrals of QFT would be trustworthy, no?
JT gravity,
This is what Osterwalder-Schrader do for you: they tell you the condition (reflection positivity) you need to justify Wick rotation to a real time theory.
What makes the subject difficult is not this, but that, especially in more than 1+1 dimension, the statistical mechanics systems corresponding to interesting QFTs are very challenging to say anything about. Euclidean lattice gauge theory is a 3+1d stat mech system, but if you can really understand what happens in the limit as lattice spacing goes to zero, there’s a million dollars waiting for you.
Peter,
Thanks for bringing forward and clearly these core mathematical problems in physic. BTW, it is amazing how physicists can usually make things look confused, so that any mathematician trying to help for the next step can be lost in the fog.
Hi Peter,
Maybe I should not interject here, but there is much heat and little light on many discussions of path integrals. Although there are problems with the real-time Dirac -Feynman path integral for Chern-Simons and first-time derivative actions, there is a way that such things make sense.
If 1. the Lie-Trotter formula is valid and 2. there is a sensible completeness or overcompleteness relation (resolution of the identity), then the real-time path integral can be sensibly interpreted rigorously! This sort of thing is discussed in an old books by Barry Simon and by Larry Schulman. This is not to say there are not problems with certain physical systems, like those you mention, but for, say standard Hamiltonian mechanics, everything is fine. Other mathematicians (like T. Kato) and physicists also worked on this. Some of the subtleties, like the mid-point prescription are automatically explained by this method.
This method goes over to real-time bosonic field theories, provided they are put on lattices. There is still the technical issue of taking the continuum limit, but that is not unique to the path-integral formulation.
I always just think of the (non-Wick rotated) path integral as an application of 1. and 2. above. You usually can’t go wrong this way, although Chern-Simons models still have the problems you mention.
Hi Peter,
The problem is that, yes, you can start with a quantum theory, break up the exponential of the Hamiltonian using Lie-Trotter, and if you do this in a certain way, you can claim the infinite sum is a phase space path integral. The problem is that it doesn’t have lots of the properties you would expect of a path integral, since your paths are discontinuous and you crucially had to do very specific things in getting the infinite sum (eg. alternating specifying q and p).
The “path integral” you get doesn’t in particular have the invariances you expect. Since it is written in terms of dpdp in the measure and pdq in the exponent, you expect it to be invariant under symplectic transformations, but it isn’t (if it were, you could choose coordinates in which it was a harmonic oscillator, semiclassical approximation exact, see Schulman chapter 31). You also get into very confusing questions when you try to decide what the state space of the theory is starting with the path integral.
There’s a fundamental issue behind this: there is no way to start with a classical phase space (e.g. symplectic manifold) and quantize, getting a corresponding quantum system, unless you introduce extra structure (e.g. a polarization). Without smuggling in somewhere this extra structure, your phase space path integral is doing something impossible.
… but you don’t need phase space, only q space in Lie-Trotter. This is very close to the Schr{\”o}dinger representation. We don’t write wave functions or wave functionals as functions of phase space.
If you want to introduce p as well as q, one way is to write every other factor (exponential of the kinetic term) as a Gaussian integral, with p as the integration variable.
Dirac and Feynman did emphasize phase space (see Dirac’s 1933 paper, Feynman and Hibbs or Feynman’s papers from the 40’s), but I never saw the point of this.
Anyway, plenty of applications don’t use phase space. The book of Simons I mentioned, from the 70’s, actually proves nontrivial results from this (and some yes, with real, not imaginary time).
Hi Peter,
Sorry I thought your comment was referring to the third point in the posting, about path integrals over phase space. These do get used, and it’s not often recognized that path integrals for Chern-Simons theory and usual fermion theories are phase space path integrals, not configuration space path integras (since the action is first order in the time derivative).
To the extent we’re discussing instead the first point, and path integrals in configuration space, what I first wrote was too strong (since fixed), but there still are problems with real-time path integrals. I was just looking at the late 70s Simons book “Functional Integration and Quantum Physics”. The way he puts things in the introduction is:
“By extending the notion of measure, various attempts at defining [the real time path integral} have been made; see e.g [gives references]. These methods, while useful computationally and for formal heuristics, have not thus far turned out to be analytically powerful; we will not discuss them further.”
In the preface he explains “I decided to limit myself to the well-defined Wiener integral rather than the often ill-defined Feynman integral.”
My copy of this book is out of reach, in my office, but I will take your word for it that Simon confines the bulk of the discussion to the Wiener integral. But I do know that he ALSO discusses the proof of Lie-Trotter formula for REAL time at the beginning of the book (which is useful, even he does not use it further). I should check, but I remember the “non-analytically powerful” methods Simon refers to are some formal definitions of C. DeWitt, among others. There is also discussion of the real-time case in the second volume of Reed and Simon.
Anyway, the real Lie-Trotter formula is completely adequate for most of the applications you can find in, e.g., Feynman and Hibbs.
But phase-space PI are another matter, and I apologize if took things away on a tangent.
Thanks Peter,
Just took a look at Reed-Simon, vol. 2. In section X.11, they do use the free particle propagator and the Trotter product formula to write down an expression for the time evolution operator as the limit of an expression that one can interpret as a path integral using polygonal paths for a fairly general potential V.
But then they do very quickly say that while this is beautiful and heuristically an integral over path space, an actual measure on path space like Wiener measure is something quite different, and they go to imaginary time and use Wiener measure to get Feynman-Kac.
Besides my experience with lattice gauge theory and Yang-Mills, of course my current fascination with the way the symmetries of spinors work in Euclidean signature is another motiviation for thinking of imaginary time as not just a trick, but something fundamental. While it’s true that for QM path integrals you can make sense of the free particle propagator without going to imaginary time (thanks Marko…) and then use this to define a limit as a path integral, for QFT, I don’t think this is true. The argument now in the first section shows that then to even define the free particle propagator you have to do something, which has a natural interpretation as thinking of the propagator as something holomorphic in the upper half complex time plane.
Just a short note – regarding the propagator and Wick rotation, I believe the problem is a bit deeper, and the issues with path integrals are just a symptom.
Namely, there is a fundamental difference between elliptic and hyperbolic PDE’s, where the former have a well-defined Green function, while the latter really don’t. Since the Klein-Gordon equation is a hyperbolic PDE, the only way to introduce its Green function is to Wick-rotate to imaginary time, find the Green function in Euclidean theory, and then perform analytic continuation back into real-time. It’s essentially due to the difference in signature between the d’Alembertian and a 4D Laplacian.
This is a problem for all relativistic wave equations (and a consequence of Lorentz-invariance), so whenever you discuss free field theory (or perturbative one), you are bound to run into this issue. It has nothing specifically to do with path integrals, any approach to quantizing field theory will suffer from this (though it might be hidden away or appear in some other form).
Best, 🙂
Marko
Marko,
Where can I read in more technical details what you are saying? Is this related to the problem of choosing a vacuum for a QFT defined on a curved spacetime background? I guess this is so because, on the one hand, the choice of vacuum defines unambiguously the notion of positive/negative frequencies for the one-particle states (e.g., a positive-frequency particle state w.t.r. to Rindler is not so w.t.r. to a Minkowski observer), while on the other hand, the notion of positive/negative frequencies depends on the time-orientability of Lorentzian structure, which is avoided in Euclidean signature.
JT gravity,
An important thing to notice is that, just looking at the formula for the time/energy Fourier transform, positive energy means holomorphic in one complex time half plane, not the other.
Another version of the problem with the real time path integral that I was trying to point out is that it obscures a fundamental piece of structure of a qft: what is positive vs. negative energy or, by Fourier transform, what is positive vs. negative imaginary time? In the imaginary time path integral, the asymmetry is fundamentally built in and explicit
Peter,
I went through a similar period of liking and then understanding the limits of the Path Integral.
It can actually be difficult however to find a proper canonical treatment of issues normally discussed with the Path Integral, e.g. Yang-Mills theory and the Higgs Mechanism. For this reason I recommend the books of Strocchi “An Introduction to Non-Perturbative Foundations of Quantum Field Theory” and Nakanishi and Ojima “Covariant Operator Formalism Of Gauge Theories And Quantum Gravity”.
Darran,
I very much agree. There are very tricky issues with how gauge invariance works in these cases, highly unclear that the path integral solution (in lattice gauge theory, just integrate over all degrees of freedom, gauge or otherwise) really works as advertised. One example of this is the no-go theorem for BRST in lattice gauge theory. I agree with the recommendation of the Strocchi and Nakanishi/Ojima books.
I should also point out that a very specific problem with the culture of “string theory” is that it promotes the ideology that there’s nothing to see here and the only people who think about or work on these issues are losers, too incompetent to realize that these subjects are a closed book and serious theorists are those who move on to schemes for getting emergent space-time out of string-theory grounded ideas.
I regard the BV-BVF complex cohomology to be the most powerful and deep quantisation scheme for QFT, and I also consider that the true meaning of “supermathematics” is not the supersymmetry of the target-space manifold relating fermions to bosons but in the mathematical structure of graded manifolds in the sense of Kostant modelling the phase-space of a quantum dynamical systems.
You raised an important sociological question regarding the general dogmatic attitude of string theories on their faith in the path integral technique. You should remember of Bert Schroer’s views on functional integrals and string theories.
Finally, returning to the question of postive/negative frequencies, it’s intersting that the Fourier decomposition is not conformally invariant while the decomposition into positive/negative frequencies itself is conformally invariant. Thus, it looks like there is a deep geometric structure hidden here which becomes transparent if you holomorphically continue the green function into the Riemann sphere, so that the choice of a positive frequency is equivalent to the choice of a hemisphere. And this is one of the key motivations for twistor theory!
I’m intrigued when Peter writes that his research is, “…another motivation for thinking of imaginary time as not just a trick, but something fundamental.” In 1953, Feynman used the path integral method with Weiner functionals and the Wick rotation between imaginary time and inverse thermal energy to write the partition function for the helium superfluid system. His imaginary time trajectories are now often referred to as “ring polymers” in modern quantum simulations (see Ceperley’s 1995 review of Path Integral Monte Carlo methods). Although the use of the imaginary time/inverse thermal energy dimension is always stressed as being a trick, Feynman’s partition function is mathematically the same as that of classical, coarse-grained representations of polymer systems under the Gaussian thread model, which in turn are equivalent to quantum density functional theory. The theorems of DFT therefore allow a one-to-one mapping between a classical polymer systems with the fictitious imaginary time dimension and regular, non-relativistic quantum mechanics, without it. It is very tempting therefore to look on the imaginary time as playing some fundamental role. Even Feynman and Hibbs mused about related matters in their 1965 quantum book (see section 10.5, Remarks on Methods of Derivation).
Russell Thompson,
It is truly remarkable that if you take your system defined in imaginary time to be periodic in imaginary time, you get statistical mechanics with temperature inverse to the period. In the limit as you get rid of periodicity (periodicity length infinite), you get the zero temperature limit, which is where you do conventional particle physics. “Just a trick?” I suspect not.
Hi,
Indeed there are several mathematical issues for defining the functional integrals in Lorentz signature, and it seems also a correct statement that for fermionic fields, actually this has little to do with integrals at all in terms of measure theory. But one can translate the formal Feynman functional integrals to a differential reformulation (the master Dyson-Schwinger equation), which is an equation for the partition function, and is formally equivalent to the functional integral formulation. Then, one can further translate this equation for the partition function to the language of (Feynman type) field correlators, and there the corresponding operators and function spaces are well defined.
That is, one could say that Feynman functional integrals are just shorthands for the master Dyson-Schwinger equation, expressed on the (Feynman type) field correlators. This kind of treatment also removes this puzzling issue concerning the fermion fields mentioned by Peter.
Best regards,
Andras
Some of you may be interested in a recent (2022) work by Feldbrugge and Turok that approaches path integrals without using Wick rotation.
The paper: https://arxiv.org/abs/2207.12798
Related paper: https://arxiv.org/abs/1909.04632
A talk by Turok: https://www.youtube.com/watch?v=L17Cx-iD8uU
Amateur Mathematician,
Thanks a lot: I just watched the Turok talk, found it very interesting. His motivation is very close to mine, although he takes a very different view of imaginary time, which I strongly disagree with. I’ll add something about this to the posting, with more comments.