Osterwalder-Schrader

I’ve been trying to write up some new ideas about Wick rotation for a long time now, keep getting stuck as it becomes clear at various points that I haven’t gotten to the bottom of what is going on. To take a little break from that I thought it might be useful to write some more informal things here on the blog, about parts of this story that I do understand.

One thing I want to write about are two important papers by Konrad Osterwalder and Robert Schrader. The first is their Axioms for Euclidean Green’s Functions, published in Communications in Mathematical Physics in 1973. I’ll refer to this as the OS reconstruction paper. The second is Euclidean Fermi fields and a Feynman-Kac formula for Boson-Fermion models, published in Helvetica Physica Acta, also in 1973, which I’ll refer to as the Euclidean Fermi fields paper.

Konrad Osterwalder was the instructor in my Math 55 class at Harvard, my first semester there in the fall of 1975. I just found out while looking for some information about him that he passed away quite recently (December 19 last year). Back in 1975 Osterwalder was an assistant professor in mathematical physics at Harvard, and the Math 55 class he taught followed quite closely chapters 0-4 of Loomis and Sternberg’s Advanced Calculus, which at the time was the standard textbook for the course.

During my last term at Harvard (spring 1979) I took an upper level graduate class from Arthur Jaffe on the foundations of QFT. As a requirement of the course, I had to pick a relevant paper and write about it. The paper I picked was the Osterwalder-Schrader Euclidean Fermi fields paper. I was pretty much mystified by it, and remained so for many, many years. I’m planning to write something about this paper in a later blog post, here will concentrate on the OS reconstruction paper.

For some amusing commentary on the story of the OS reconstruction paper, see Slava Rychkov’s talk “CFT Osterwalder Schrader Theorem” at this meeting in 2019, where he says:

these papers appeared in Communications in Mathematical Physics. If you start reading these papers you immediately get a headache. The first ten pages are just notation. You have to go through then another theorem, lemma, lemma theorem, Hille-Yosida theorem, things like that.

Very few people have read these papers and very few people know what has actually been done there. It’s almost irresistible, people love to cite these papers because it’s like a feeling of ancient magic books, the scriptures. Many normally very careful people misquote these papers and miscite them by attributing to them results which are not there.

The background for all of this is the story of Euclidean or imaginary time quantum field theory, which starts with Julian Schwinger’s 1958 paper On the Euclidean Structure of Relativistic Field Theory. For some relevant history, see here. One way of looking at quantum field theory is that it’s all about the vacuum expectation values of field operators, the Wightman functions. What Schwinger was suggesting was that one could define quantum field theories in terms of the analytic continuation of Wightman functions, evaluated at imaginary time (these are now known as Schwinger functions). This fits well with the modern point of view that QFTs should be defined by path integrals, since it is only imaginary time path integrals for which one can hope to have something one can make rigorous, not a purely formal object.

Schwinger was well aware that if you tried to define a QFT as a set of Schwinger functions, something missing was a way of recognizing when these corresponded to a physical theory in real time. In the discussion session of his presentation at the 1958 ICHEP, he said

The question of to what extent you can go backwards, remains unanswered, i.e. if one begins with an arbitrary Euclidean theory and one asks: when do you get a sensible Lorentz theory? This I do not know. The development has been in one direction only: the possibility of future progress comes from the examination of the reverse direction, and this is completely open.

The OS reconstruction paper was based on a crucial new idea for how to recognize a Schwinger function corresponding to a physical real-time theory, the condition of “reflection positivity”. Jaffe recounts here how this came about.

A crucial property for a quantum theory is that it has a Hermitian inner product on states, with states having positive norm in this inner product. The Hermitian nature of the inner product of two state vectors involves complex conjugation on one of them. On functions of time, this is just complex conjugation of the value of the function. When you work with complex time $z=t+i\tau$ instead of real time, the complex conjugation takes $z=t+i\tau$ to $\overline z=t-i\tau$. This is a reflection $\tau \rightarrow -\tau$ in the imaginary time axis, sometimes called the Osterwalder-Schrader reflection.

By the way, this seems to me a first indication of the possibility I’ve been trying to understand of spacetime transformations in Euclidean spacetime turning into internal symmetries in Minkowski spacetime (here reflection in time is turning into pointwise complex conjugation).

What Osterwalder and Schrader did in the OS reconstruction paper was provide a theorem stating when Schwinger functions came from Wightman functions. As Rychkov notes, this paper is very hard going. After spending a lot of time with it, I realized one reason why the whole thing is difficult, which I’ll try and explain here. This is something that has held up what I’ve been trying to do with Wick rotation. Quite possibly I’m missing something and maybe someone will explain to me what it is.

Analytically continuing from real time to imaginary time is relatively easy, because it’s an example of what mathematicians know as the Paley-Wiener theorem. If you have a function $f(t)$ with Fourier transform $\widetilde f(E)$ that is only supported at positive energy, you can do inverse Fourier transformation to complex values of time by
$$F(z)=\frac{1}{\sqrt{2\pi}}\int_0^\infty e^{-izE}\widetilde f(E)dE$$
Because
$$e^{-izE}=e^{-itE}e^{\tau E}$$
this integral will give a result holomorphic in $z$ for $\tau<0$, with boundary value at $\tau=0$ the original function $f(t)$. The “Wick rotation” to a function of $\tau$ is given by $F(-i\tau)$. If you change the conventions I’m using for $2\pi$ factors and the sign of the exponent, this is just the Laplace transform of $\widetilde f(E)$
$$\int_0^\infty e^{-\tau E}\widetilde f(E)dE$$

The problem is that going in the other direction is much trickier. Given a function $f_S(\tau)$ (S for “Schwinger”), if you try to analytically continue to get $f(t)$ by first inverting the Laplace transform to get $\widetilde f(E)$ (then inverse Fourier to get $f(t)$), there’s a problem. When you look up the formula for inverse Laplace transform it basically says “first analytically continue to $f(t)$, then Fourier transform to get $\widetilde f(E)$.”

The argument in the OS reconstruction paper is tricky, partly because they can’t directly do this inverse Laplace transform. Instead, given $f_S(\tau)$, they define a function of $E$ by Laplace transform, but this function is not $\widetilde f(E)$ (does anyone know of a nice relation between them?), although it has properties they can use to prove their reconstruction theorem.

It turned out that the proof in the OS reconstruction paper was flawed. Their lemma 8.8 claimed to show that the way they were dealing with this problem for a single variable would continue to work for multiple variables, but this was wrong, with a counterexample soon found. They later wrote a second paper, which fixes the problem, but at the cost of a very difficult argument, and assuming a particular property of the Schwinger functions. The Rychkov talk linked to above explains that when he tried to understand the exact relationship between Euclidean and Minkowski in conformal field theory, he was shocked to realize that the OS reconstruction theorem did not apply, because there was no viable way of knowing if the Euclidean Schwinger functions had the necessary property. At this point, the best way to try and understand the OS reconstruction paper is not by reading it, but by looking at explanations from Rychkov (videos here and here, or section 9 of a paper with collaborators).

The OS reconstruction argument is an impressive and important piece of mathematical physics, but its impenetrability has had the unfortunate effect of convincing most people (myself included for many years…) that the relation between Minkowski and Euclidean quantum field theories is something straightforward and well-understood. This matches up with an equally unfortunate conviction that the problems of defining QFTs by path integrals are not serious, with Minkowski vs. Euclidean nothing but a different sprinkling of factors of i in an integral.

This story is just one aspect of fundamental problems about understanding QFTs which go much deeper than that of not being able to provide rigorous proofs. Already by the time I was a student it was clear there was a mismatch between the scalar QFTs studied by mathematical physicists using Euclidean methods and the ones relevant to the real world. In addition, the bottom line about such scalar QFTs turns out to be that they exist and are non-trivial only in two and three spacetime dimensions, must be trivial in four or more spacetime dimensions.

The Standard Model QFT is mainly built on spinor Fermi fields and Yang-Mills gauge fields. I’m sure that’s why back in 1979 I was interested in the Osterwalder-Schrader Fermi fields paper (much more about this in another blog posting). Attempts to fully understand Yang-Mills gauge fields soon moved to the discretized lattice gauge theory. During my graduate student years I was seduced by the simplicity of Euclidean spacetime pure Yang-Mills lattice gauge theory, which is basically a geometrically beautiful statistical mechanics system that can be studied with statistical mechanics methods, including straightforward Monte-Carlo calculations. That experience, coupled with not understanding the subtlety of the OS reconstruction theorem, left me convinced that the way to understand QFT was using path integrals in a Euclidean spacetime theory, with the question of the relation to physics just one of how to do the analytic continuation to real time after the theory was solved.

More and more I’ve become convinced that this was a misguided point of view. A better starting point may be the following. A fundamental aspect of quantum theory is the existence of the Hamiltonian H and a unitary operator $U(t)=e^{-itH}$ which represents translation in time and provides the dynamics of the theory. The significance of Wick rotation is that it is telling you that if you think of time as complex variable, positivity of the energy implies that $U(t)$ is only part of the story, a boundary value of a holomorphic representation $U(z)=e^{-izH}$ of a holomorphic semigroup (complex time translations with imaginary time one sign only). The fundamental quantum field theory of the real world likely should not be thought of as a statistical system, but as having a holomorphic aspect, involving much deeper mathematics.

Update: Glad to see a relevant comment from Yoh Tanimoto. In order to make clear exactly what is bothering me about the OS reconstruction argument and explain the comments about Laplace and inverse Laplace in the posting, here are some more details (I’m simplifying by ignoring spatial variables).

OS are getting Wightman distributions W from Schwinger functions S in 4.1. There in 4.12 they define the Fourier transform $\widetilde W$ as the unique thing whose Laplace transform is S. They do this by invoking the lemma 8.8 that gets them in trouble. If they had a formula for the inverse Laplace transform, explicitly giving $\widetilde W$ in terms of S, they wouldn’t have trouble.

What’s really bothering me is what they do in section 4.3 (which, besides being a complicated argument, is atrociously written, hard to decode). There they are trying to show that the following constructions of the physical state space are the same:

• Euclidean construction $\mathcal K$ as test functions on the positive imaginary time line modulo those null in the inner product given by $S$ with the OS reflection.
• Usual Wightman real time reconstruction of the state space $\mathcal H$ as test functions of real time modulo those null in the inner product given by W.

Here the problematic Laplace transform is that of equation 4.20. They Laplace transform (NOT inverse Laplace transform) the first kind of test function to get the Fourier transform of the second kind of test function. I guess they are able to get an identification of the two Hilbert spaces this way, but I’m wondering why you don’t instead use the inverse Laplace transform and be matching analytic continuations of the two kinds of test functions. Presumably because without a formula for the inverse Laplace transform you can’t match the norm using S with the norm using W.

To say a bit more about my motivation, it’s that I’d like to know why there’s not a more straightforward version of the Wick rotation between two different definitions of the physical state space. I’d like a holomorphic construction that specializes to the two different cases (have been trying to do this using hyperfunctions).