A Osterwalder-Schrader theorem and non-relativistic QM

1. Dec 3, 2017

king vitamin

I have recently learned a bit about the Osterwalder-Schrader theorem. From my understanding, this tells you when a Euclidean path integral can be analytically continued to a valid relativistic Hermitian quantum field theory (one needs reflection positivity etc.).

I am curious about corresponding results for non-relativistic theories. Let's say I have a Euclidean field theory where there isn't full rotational symmetry; if a definite example is needed, let's say I want it to Wick-rotate to a Galilean-invariant theory. Is there a corresponding theorem for what properties are needed in the Euclidean theory for the Wick rotation (which is used all the time) to guarantees a valid Hermitian quantum field theory?

I don't have an extremely strong background in the mathematics behind AQFT. I did find this statement of the OS theorem by Urs Schreiber (who I know posts here), which gives rotational invariance as an assumption. So my question boils down to whether this assumption can be relaxed.

2. Dec 4, 2017

A. Neumaier

By a Wick rotation one obtains from the symmetry group $ISO(4)$ of the Euclidean path integral one with symmetry group $ISO(1,3)$, which is the Poincare group and not the Galilei group. To get the latter you'd need to perform a contraction instead of a Wick rotation.

3. Dec 4, 2017

rubi

In order to reconstruct a quantum theory from a path integral, you just need reflection positivity (in order to get the Hilbert space) and (imaginary) time-translation invariance, in order to reconstruct the (imaginary) time-evolution operators. If time is continuous and the time evolution is strongly continuous, you can reconstruct the Hamiltonian, which generates ordinary unitary time-evolution. All the other baggage in the OS axioms just ensures that the corresponding quantum theory is a Wightman quantum field theory, but if you don't care about that, then the subset of the axioms that I stated, is enough.

4. Dec 4, 2017

king vitamin

I don't follow you here. I am asking about Euclidean-signature theories which do not have $ISO(4)$ symmetry and their relation to quantum theories which do not have $ISO(1,3)$ symmetry. There is certainly a Wick rotation involved, as seen in any textbook. I assume by "contraction" you're referring to the Inönü-Wigner method for getting the Galilei group from the Poincaré? Since I'm not interested in Poincaré invariance I do not understand this comment either.

This is really helpful, thanks. To repeat what I think I have understood of your post: if I want to apply the path integral to a quantum system with a finite Hilbert space (e.g. it is not a continuum field theory), I can formulate it in terms of a Euclidean path integral where the Euclidean theory requires reflection positivity and imaginary time-translation invariance. The other assumptions of the statement of OS are only needed if I want to guarantee that the corresponding theory is also a "Wightman QFT."

Unfortunately I'm not really familiar with what a Wightman QFT is; I'm looking at the Scholarpedia article now. It seems that it is specifically a Poincaré invariant QFT. I am largely interested in field theories within condensed matter/stat mech where we do not worry much in the passage from a finite lattice to the continuum limit, so I'm not too worried these extra issues provided the finite-dimensional case holds. My question mostly stemmed from thinking of mappings between theories of classical and quantum critical phenomena.

5. Dec 5, 2017

A. Neumaier

One can get the Galilei group both from ISO(4) and from ISO(1,3) by Inönü-Wigner contraction.

If the field theory you start with has no big symmetry group such as ISO(4) then Wick rotation also produces a theory with little symmetry only - not with the Galilei group.

In this case you cannot get the Galilei-invariant theory you had asked for. It needs fields defined on 3 space variables.

6. Dec 5, 2017

king vitamin

But there are certainly Euclidean field theories which Wick rotate to Galilean invariant quantum field theories. I still don't see the point of starting with $ISO(4)$ symmetry. In any case, as I said in my first post, the Galilean symmetry was only mentioned if a definite example is needed.

Let's forget the Galilei group. Assume I start with a continuum Euclidean field theory with full translation invariance along one of the dimensions, and no assumptions about the symmetry along/between the others. I now choose the translation-invariant direction as my "imaginary time" direction which will be Wick rotated. I'm curious about what assumptions are needed for the Wick rotated theory to be a valid QFT.

My understanding from rubi's post is that there is some extra baggage in the OS theorem needed for us to get a "Wightman QFT", and while I don't think I really care about this for the applications I'm currently considering, I am curious how important the assumption of spacetime symmetry is.

Yes, that much is obvious (perhaps you missed the context of the sentence you've quoted?).

7. Dec 6, 2017

A. Neumaier

You get a valid quantum theory but it has no symmetries apart from time translation symmetry. The relevent piece of theory is the Feynman-Kac formula. Whether the result is a valid QFT depends on your definition of the latter. (There is no standard definition in the nonrelativistic case.)