I Problem of time in quantum field theory?

Summary
Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?
Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Everything I read about "the problem of time in quantum mechanics," i.e. absolute time in QM clashing with relativity's relative time coordinate and relativity of simultaneity, invokes non-relativistic QM to explain what the problem is. However, while I'm not that good at QFT I know that it is a Lorentz invariant theory, correct? (except for non-relativistic versions used in condensed matter systems) Thus QFT doesn't, or shouldn't, have an absolute time but rather an infinite number of Lorentz frames which are equivalent to each other as far as the laws of physics are concerned, but which have different time coordinates.

So what exactly is the problem? I'm assuming simultaneity has something to do with it but I can't quite put my finger on it. I'm not that up on QFT, does it in fact require an absolute time? Please help me.
 

vanhees71

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There is no problem of time in quantum mechanics, neither in non-relativistic nor in relativistic theory. Relativistic QT is most conveniently formulated in terms of quantum field theory, QFT, since at relativistic collision energies usually particles get created and destroyed.

One heuristic way to formulate QFTs is to use "canonical quantization" starting from classical field theories based on an analysis of the unitary representations of (the covering group of) the proper orthochronous Lorentz group. To guarantee the correct causality structure, i.e., the linked-cluster principle, you find that you can use massive ##p^2>0## and massless ##p^2=0## representations and then build local fields. Together with the demand that energy should be bounded from below you get the spin-statistics theorem (half-integer-spin fields have to be quantized as fermions integer-spin fields as bosons), and microcausality follows, i.e., operators for local observables (energy, momentum, angular-momentum density, several charge- and current-densities) commute at spacelike separations of its arguments. This leads also to a Hamiltonian which implies the validity of the linked-cluster principle.

For details see

Weinberg, S.: The Quantum Theory of Fields , volume 1, Cambridge University Press, 1995
Duncan, Anthony: The conceptual framework of quantum field theory , Oxford University Press, 2012
 

A. Neumaier

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Everything I read about "the problem of time in quantum mechanics,"
You should let us know where you have read this, to provide the context necessary to understand your query.
 
Well okay, but why do people say that there is a problem of time in QM if there isn't?
 
You should let us know where you have read this, to provide the context necessary to understand your query.
Well today on arXiv was posted this pre-print:


And that's what got me thinking, although it's a notion I've seen here and there before (often in the popular press, which of course doesn't explain it). The first ten references seem to be relevant to defining the "problem."
 

A. Neumaier

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Well today on arXiv was posted this pre-print:
This is about quantum gravity, not about quantum mechanics or quantum field theory! Quantum gravity is an area of physics where unsolved problems abound.

The ''problem of time'' is that in general relativity, time is not well-defined, as it depends on the curvilinear coordinate system used to coordinatize spacetime. If one considers spacetime itself as emergent (which is the case in some approaches to quantum gravity), there is the problem of how the notion of time emerges.
 

dextercioby

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Well today on arXiv was posted this pre-print:


And that's what got me thinking, although it's a notion I've seen here and there before (often in the popular press, which of course doesn't explain it). The first ten references seem to be relevant to defining the "problem."
Thank you for the reference. This is a very short note with strong bibliography, which is actually the asset of the paper.
 

arivero

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Is it the problem of not having an observable for time? This is turn is because the energy is a positive operator, on the other hand momentum has positive and negative values and then we have still a good operator for position.

I am not sure how the problem resolves itself in relativistic quantum mechanics. It could be said that the energy becomes negative too, or it could be argued that we also lose the position operator.
 

vanhees71

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I think it's again in the sense of thinking not "quantum theoretically" enough. There's this example often envoked using a Mach-Zehnder interferometer tuned such that with two beam splitters in place one exit has no signal due to destructive interference and the conclusion that then there'd be no "which-way information" on the photon. Taking out the beam splitter then leads to 50:50 chance for the photon to end up in either detector and then it's concluded that you have which-way information after the detection of the photon in one of the detectors, i.e., you apparently draw conclusions about the "past of the photon".

This is of course thinking in a classical-wave or classical-particle picture about the photon and then in terms of the idea of "complementarity" or "wave-particle dualism" of old quantum mechanics. Particularly for photons that's utterly misleading. It's not even possible to define a proper position observable for the photon. So the "which-way information" doesn't make sense to begin with.

In quantum theory, in this case particularly QED, the state of a photon is given by a statistical operator, which includes the information about the probabilities for detection in one or the other of the detectors in the Mach-Zehnder interferometer setups. You can also interpret it as measuring the energy density of the electromagnetic field, being prepared in a single-photon state (in general not a pure but a mixed state of course). The corresponding detection probabilities are all you can know about the photon in this setup, and you can calculate the time evolution of these probabilities using the standard dynamics of QED. Of course, it doesn't refer to the time evolution of some observable since the observable may not be determined due to the preparation of the state of the photon at the beginning of the experiment.
 
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The problem of time in QG is the problem that time in QM is itself a background dependent notion just like in Newtonian mechanics (i.e. you can evolve a wavefunction using the TDSE) in stark contrast to the notion of time in GR.

This conceptual inconsistency between these two different notions of time makes the direct quantization of GR mathematically intractable, if not impossible.
 
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Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Everything I read about "the problem of time in quantum mechanics," i.e. absolute time in QM clashing with relativity's relative time coordinate and relativity of simultaneity, invokes non-relativistic QM to explain what the problem is.
There's a very comprehensive book on the subject called The Problem of Time: Quantum Mechanics versus General Relativity by Dr.Edward Anderson. He's also got a number of preprints on the subject (which appear to form the basis of the book). There's one titled:
Problem of Time and Background Independence: the Individual Facets

In the book he suggests that "the Problem of Time is, in greater generality, a consequence of the mismatch between Background Dependent and Background Independent Paradigms of Physics".

He goes on to say that it "has nine facets--closely following [the review by] Isham and Kuchaˇr--resulting from nine corresponding aspects of Background Independence".

However, while I'm not that good at QFT I know that it is a Lorentz invariant theory, correct? (except for non-relativistic versions used in condensed matter systems) Thus QFT doesn't, or shouldn't, have an absolute time but rather an infinite number of Lorentz frames which are equivalent to each other as far as the laws of physics are concerned, but which have different time coordinates.

So what exactly is the problem? I'm assuming simultaneity has something to do with it but I can't quite put my finger on it. I'm not that up on QFT, does it in fact require an absolute time? Please help me.
In their paper, The problem with ‘The Problem of Time’, Bryan and Medved say that "the Minkowski metric of quantum field theory is generally regarded as a mathematical construct and not a real physical object." This would seem to imply that the absolute notion of time is retained in QFT.
 

Demystifier

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Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

So what exactly is the problem?
There are several problems of time in quantum theory.

1. Time in non-relativistic QM: Time can be observed, but unlike other quantum observables it is not described by a hermitian operator. Why is time different?

2. Time in relativistic QM: The Klein-Gordon equation does not define a probability density in space conserved in time. A common interpretation of that is that there is no relativistic position operator, but then where does the position operator (in 1. above) come from in the non-relativistic limit?

3. Time in relativistic QFT: In the Schrodinger picture the state ##|\psi(t)\rangle## does not satisfy a Lorentz-covariant equation, so how exactly is relativistic QFT Lorentz covariant in that picture?

4. Time in canonical quantum gravity: The time-reparametrization invariance of classical gravity implies the Hamiltonian constraint ##H=0##, which in the quantum case implies ##H|\psi\rangle=0##. Combining it with the Schrodinger equation ##H|\psi\rangle=i\partial_t|\psi\rangle## implies that the state ##|\psi\rangle## does not depend on time. So where does the change of Nature with time come from?

All those problems have some proposed solutions, but neither of those solutions seems completely satisfying. People are still working on it.
 

vanhees71

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Generally, time is not an observable in physics but a parameter that labels the causal order of events. You can infer about time only by measurements involving some observables (e.g., in classical mechanics the uniform rectilinear motion of a particle, by taking its position as a function of counting cycles of a pendulum or whatever more accurate clock you prefer).

In QT (at least for cases 1-3; I can't say anything about 4), because I've no clue about this branch of physics) the reason simply is that if time would be treated as an observable in the usual sense it should be the canonical conjugate to the Hamiltonian, which then wouldn't be bounded from below and we'd have no stable world we could live in and discuss about physics ;-)).

Ad 2. For massive particles you can construct position operators in standard QFT; you can't for massless particles with spin ##\geq 1## (i.e., in the real world for photons), but that's not a bug but a feature since it precisely describes how photons behave. Relativistic QM simply doesn't exist nor make it physical sense beyond the usual approximation in situations, where you are close to non-relativistic physics anyway.

Ad 3. That's not surprising since you break manifest Poincare covariance by just going to the Hamiltonian description. You don't, of course, really use Poincare covariance for observable quantities like S-matrix elements etc. Of course, it's often easier to keep everything manifestly covariant, e.g., by using the path-integral formalism and stick to the Lagrangian formulation (however only after carefully checking whether it's really applicable by first using the Hamltonian path-integral formalism!).
 

Demystifier

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For massive particles you can construct position operators in standard QFT; you can't for massless particles with spin ##\geq 1## (i.e., in the real world for photons)
So can you write down the explicit construction of the position operator for the massless spin-0 particle? And what exactly goes wrong when you attempt to generalize it to spin-1?
 

vanhees71

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See @A. Neumaier 's FAQ,

https://www.mat.univie.ac.at/~neum/physfaq/topics/position.html

and the references therein.

In which sense photons are "localized" though there's no position operator, see any good textbook on Quantum Optics. In lack of a position observable, not the photons themselves can be localized but there are only detection probabilities of a given detector which, as a massive object, can of course be localized. A good discussion can be found in

J. C. Garrison, R. Y. Chiao, Quantum Optics, Oxford Univ. Press (2008)
 

Demystifier

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Demystifier

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I get it now. All the problems start from the Lorentz invariant measure
$$\frac{d^3p}{p_0}$$
The position operator defined as being proportional to the momentum derivative in the post above then creates an additional term due to the momentum derivative acting on ##1/p_0##, making the position operator non-hermitian (that's nicely explained in the Schweber's QFT book). For that reason one must redefine the position operator by adding a compensating term that renders the position operator hermitian. It is this compensating term that depends on spin and mass and eventually makes it non-existent for zero mass and spin larger than ##1/2##.

But there is a way out of this problem. Instead of using the Lorentz invariant measure above, one can use the canonical measure ##d^3p## as in non-relativistic QM. Sure, the position operator will not be Lorentz covariant, but as Newton and Wigner have shown, the position operator is not Lorentz covariant even when the Lorentz invariant measure above is used.

So what is lost by using the canonical measure ##d^3p##? Newton and Wigner propose 4 axioms that a "reasonable" position operator should satisfy, called axioms (a), (b), (c) and (d). Their theorem (completed by others) says that there is no position operator satisfying all 4 axioms for zero mass and spin larger than ##1/2##. The axioms (a), (b) and (c) are axioms which are valid also in non-relativistic QM, so they are not violated by taking the canonical measure ##d^3p##. What is lost is the axiom (d), which is a certain regularity condition inspired by the requirement of Lorentz covariance. But since the resulting (Newton Wigner) position operator violates Lorentz covariance anyway, it does not seem that much more is lost when (d) is violated too. Hence it seems reasonable to give up the axiom (d) and construct the position operator basically as in non-relativistic QM. In this way the position operator becomes simple and does not depend on mass and spin. And the absence of Lorentz covariance makes perfect sense from the operational point of view, because the position operator is supposed to describe a situation in which the position of the particle is measured by an apparatus which defines a preferred Lorentz frame in which the apparatus is at rest.

To conclude, it is not correct to say that position operator does not exist for zero mass and spin larger than ##1/2##. The correct statement is that position operator satisfying Newton-Wigner axioms does not exist for zero mass and spin larger than ##1/2##. What I propose here is to give up one of those axioms (axiom (d)), in which case the position operator exists, does not depend on mass and spin, and is basically the same as in non-relativistic QM.
 
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