QFT in Euclidean or Minkowski Spacetime

[LarryS]

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

Thanks in advance.

 

[PeterDonis]

does the relativistic QFT in use today by experimental physicists live in Euclidean spacetime or Minkowski spacetime.

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.

 

[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.

 

[Demystifier]

I don't know that either one could be said to be preferred.

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

 

[PeterDonis]

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

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

Do you think that causality of QFT can be 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?

 

[Demystifier]

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

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

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?

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

 

[PeterDonis]

I thought we are not talking about quantum gravity. But if we do, see the recent paper by Visser

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

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

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.

 

[radium]

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.

 

[A. Neumaier]

You get from Lorentzian to Euclidean spacetime by analytic continuation.

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.

 

[LarryS]

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?

 

[A. Neumaier]

the QFT-related spacetime "events" (both spacelike and timelike) being referred to are quantum measurement events. Is that correct?

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

 

[LarryS]

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

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

 

[atyy]

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?

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.

 

[A. Neumaier]

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

Yes, commuting or non-commuting field operators at space-time arguments = events.

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.

But in this thread, the posts before yours didn't use the term in that sense.

 

[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.

 

[A. Neumaier]

The well-known book by Misner, Thorne, and Wheeler, the bible of relativity, defines on p.5,

Give a point in spacetime the name ''event''

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 spacetime location is part of the data of an event, but it is not the full data.

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.

 

[LarryS]

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

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.

 

[atyy]

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.

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.

 

[A. Neumaier]

lightning striking a tree at a certain place and time.

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.

 

[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.

 

[DarMM]

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.

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

 

[Thinkor]

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.

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.

 

[bhobba]

The well-known book by Misner, Thorne, and Wheeler, the bible of relativity,

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/dp/0393965015/?tag=pfamazon01-20

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

 

[A. Neumaier]

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.

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.