# I QFT in Euclidean or Minkowski Spacetime

Tags:
1. Feb 23, 2017

### referframe

Forgetting for the moment about curved spacetime, does the relativistic QFT in use today by experimental physicists live in Euclidean spacetime or Minkowski spacetime.

2. Feb 23, 2017

### Staff: Mentor

Yes.

Seriously, both are used, and I don't know that either one could be said to be preferred. It depends on the specific case being modeled.

3. Feb 24, 2017

### Demystifier

It lives in Minkowski spacetime, but in some cases you can use a mathematical trick which allows you to make calculations in Euclidean spacetime.

4. Feb 24, 2017

### Demystifier

Do you think that causality of QFT can be formulated in Euclidean spacetime?

5. Feb 24, 2017

### Staff: Mentor

Aren't there some proposals, such as Hawking's "no boundary" proposal for the early universe, that are formulated in Euclidean spacetime?

AFAIK the transformation between Minkowski and Euclidean spacetime doesn't change the commutation relations between operators. So if I take a pair of operators at events that are spacelike separated in Minkowski spacetime, and transform to Euclidean spacetime, they will still commute, correct?

6. Feb 24, 2017

### Demystifier

I thought we are not talking about quantum gravity. But if we do, see the recent paper by Visser:
https://arxiv.org/abs/1702.05572

But what if we start from Euclidean spacetime? How do we distinguish betwen a spacelike and a timelike separation in a rotation invariant way?

7. Feb 24, 2017

### Staff: Mentor

Hm, interesting. I was not aware of the curved spacetime issues he discusses.

I don't know that you need to distinguish between spacelike and timelike in order to define causality. You just need to distinguish causally connected from non-causally-connected events. Non-commutation vs. commutation of operators at those events would be one way to do it. But I admit I am not very familiar with the details of how theories on Euclidean spacetime are formulated, and the Visser paper you linked to has made me less confident of what I thought I knew.

8. Feb 25, 2017

You get from Lorentzian to Euclidean spacetime by analytic continuation. So causality is encoded in the branch cut and pole structure of correlation functions in the complex plane.

In Euclidean gravity you find that the time variable needs to be periodic in 1/T where T is the Hawking temperature. You can compute this easily for an AdS black hole and also "thermal AdS" (which just means choosing time to be periodic with no black hole. The Hawking page transition happens when the temperature is low enough that thermal AdS is more energetically favorable than a black hole.

9. Feb 26, 2017

### A. Neumaier

Only if you distinguish a timelike direction. This works for free fields in Euclidean space and for thermal fields where the background specifies a distinguished time direction, but not in general for dynamics in curved space.

In curved space, local Lorentz transformations and hence a Minkowski structure of the fields in the tangent planes are essential. Even in flat space. dynamics and causality are expressed in terms of the Minkowksi space, and the Euclidean space formulation is secondary.

10. Feb 26, 2017

### referframe

Great discussion. I can see there is a lot to learn. QUESTION: I assume that, in the above discussion, the QFT-related spacetime "events" (both spacelike and timelike) being referred to are quantum measurement events. Is that correct?

Last edited: Feb 26, 2017
11. Feb 27, 2017

### A. Neumaier

No. In relativity, event is synonymous to space-time point. Measurement is not involved.

12. Feb 27, 2017

### referframe

Yes, but in a couple of the posts, the word "events" was used in the context of commuting or non-commuting observables.

13. Feb 27, 2017

### atyy

Yes, in QFT in the Copenhagen interpretation, events are quantum measurement events - ie. measurement events within quantum theory; in quantum theory a measurement is a classical event that produces a classical outcome.

14. Feb 27, 2017

### A. Neumaier

Yes, commuting or non-commuting field operators at space-time arguments = events.
But in this thread, the posts before yours didn't use the term in that sense.

15. Feb 27, 2017

### atyy

Events in relativistic QFT are the same as in classical relativity: things that really happen. They are invariant under transformation.

Spacetime is classical in both relativistic QFT and classical relativity. The spacetime location is part of the data of an event, but it is not the full data. Measurement settings and their outcomes are also data specifying events in QFT, and they are classical.

In contrast, the wave function is not invariant under transformation, so it is not real unless we agree that an invisible preferred frame exists.

16. Feb 28, 2017

### A. Neumaier

The well-known book by Misner, Thorne, and Wheeler, the bible of relativity, defines on p.5,
This is the authoritative definition, formally specifying the technical meaning of the notion of ''event'' in relativity. Like many other formal notions of physics it is only loosely related to the informal natural language notion with the same label.
The informal meaning (including your interpretation of it) belongs to the interpretation of relativity, not to its formal (shut-up-and-calculate) part.

Given the state of the QFT, the event (according to the definition given by Misner-Thorne-Wheeler) completely specifies what happens there (i.e., in a small neighborhood of the event) - namely through the n-point correlation functions with arguments in this neighborhood. There is nothing else in QFT. What we can observe there is contained in the least oscillating contributions to this correlations. (The spatial and temporal high frequency part is unobservable due to the limited resolution of our instruments.)

There is no classical apparatus for measuring our planetary system when the latter is modeled by quantum fields. Thus the Copenhagen interpretation cannot apply to finite time QFT of large systems, only to the results of few-particle scattering calculations derived from it.

Last edited: Feb 28, 2017
17. Mar 1, 2017

### referframe

All wave functions in general are not Lorentz invariant, but interestingly enough, as Feynman pointed out, the function EXP[ i(px - Et)] is invariant because it contains the Minkowski inner product of the energy-momentum four-vector and the spacetime four-vector and all such inner products are invariant.

18. Mar 1, 2017

### atyy

When I say "invariant", I don't mean just Lorentz invariant - I mean invariant under a general coordinate transformation.

Events are real happenings, eg. lightning striking a tree at a certain place and time. So they cannot disappear by renaming.

19. Mar 2, 2017

### A. Neumaier

These are conspicuously large field expectation values in an extended region of space-time. They have nothing to do with the technical notion of an event in relativity theory.

20. Mar 2, 2017

### DarMM

The full correspondence, the Feynman-Kac relation, is that there exists a stochastic process in Euclidean space whose moments (with a specific ordering) can be analytically continued to the correlation functions of a QFT in Minkowski space.
However there are aspects of the Euclidean theory, e.g. certain moments which are not correctly ordered in the first coordinate (the one which will be continued to time), that are not related to the QFT in Minkowski space.

21. Mar 2, 2017

### DarMM

Wavefunctions in QFT are Lorentz invariant, but they are not functions of the coordinates, but functionals of the classical field configurations:
$\Psi[\phi(\textbf{x})] \hspace{10mm} \textbf{x} \in \mathbb{R}^{3}$

22. Sep 4, 2017

### Thinkor

You stated "Given the state of the QFT, the event (according to the definition given by Misner-Thorne-Wheeler) completely specifies what happens there (i.e., in a small neighborhood of the event)"

Didn't you mean "the quantum field" rather than "the QFT"? I suspect I am just nit-picking, but I want to be sure I understand what you mean.

23. Sep 4, 2017

### Staff: Mentor

Gee I always thought is was Wald

Seriously while a good book I lost my copy but never replaced it because I found myself returning to Wald.

Of relevance to this thread if anyone is interested in how one gets curved space-time from a flat one then Ohanians book is the one to get:
https://www.amazon.com/Gravitation-Spacetime-Second-Hans-Ohanian/dp/0393965015

It was the first book I learned GR from and it's non-geometrical approach is different to any I have read since.

Thanks
Bill

24. Sep 5, 2017

### A. Neumaier

It wouldn't change the meaning if there is only one quantum field. But in general there may be multiple fields and there is only one state. hence it is the state of the theory and not of the field.