I What is the role of Relativistic QM?

[fxdung]

In non-relativity then there is QM, but in relativistic regime then there is QFT. Then what is useful role of Relativistics QM nowaday, or it is only has a historical meaning?Does non-existance wave function in relativistic regime make RQM meaningless?

 

[Demystifier]

The wave function exists, but it does not have the usual probabilistic interpretation in the position space. Its Fourier transform, however, has the usual probabilistic interpretation in the momentum space.

 

[A. Neumaier]

In non-relativity then there is QM, but in relativistic regime then there is QFT. Then what is useful role of Relativistics QM nowaday, or it is only has a historical meaning?

The latter, if you mean the Dirac equation.

But a more general approach to relativistic quantum mechanics is still well alive. See the article 'Is there a multiparticle relativistic quantum mechanics?' from my theoretical physics FAQ.

 

[meopemuk]

The wave function exists, but it does not have the usual probabilistic interpretation in the position space.

Why do you think that probabilistic interpretation does not apply to position measurements? There are Newton-Wigner position operators in each sector of the Fock space. So, I don't see much difference in the status of position in QFT and in non-relativistic QM.

Eugene.

 

[andresB]

Why do you think that probabilistic interpretation does not apply to position measurements? There are Newton-Wigner position operators in each sector of the Fock space. So, I don't see much difference in the status of position in QFT and in non-relativistic QM.

Eugene.

Take a look at Landau's words

 

[meopemuk]

I would like to disagree with Landau on these points.

1. One may have troubles with the "negative frequency" solutions, if one regards one-particle states as solutions of wave equations (such as the Dirac equation). But this kind of thinking and confusion was characteristic for early days of relativistic quantum theory. Now, I think, it is commonly accepted that the Dirac equation applies to quantum fields, rather than to particle wave functions. According to Wigner's theory of irreducible representations of the Poincare group (see Weinberg's vol. 1, chapter 2), one-particle wave functions are defined on the positive energy hyperboloid in the energy-momentum space, so the energy spectrum is strictly positive.

2. The argument about creating new particles when attempting to measure position does not look convincing. The Newton-Wigner operators of particle positions commute with particle number operators in the Fock space. So, while it is OK to apply the Heisenberg uncertainty principle to the non-commuting pairs position/momentum or position/energy, there is no good reason to apply this principle to the commuting pair of observables position/"number of particles". So, the formation of electron-positron pairs is not "inevitable".

Eugene.

 

[fxdung]

I do not understand why Dirac equation and Klein-Gordon equation still are taught in University nowaday,while the normal RQM is useless?

 

[weirdoguy]

Because they are still part of relativistic quantum field theory. They are just interpreted differently.

 

[throw]

...if one regards one-particle states as solutions of wave equations (such as the Dirac equation). But this kind of thinking and confusion was characteristic for early days of relativistic quantum theory. Now, I think, it is commonly accepted that the Dirac equation applies to quantum fields, rather than to particle wave function

This is a false dichotomy, one can set up a quantum field operator even for a single particle in non-relativistic quantum mechanics - they are usually introduced in a non-relativistic context for systems of identical particles by noting the matrix elements for transitions in such systems can be equivalently represented using things that turn out to be quantum field operators - so there is similarly no reason why QFT forces one to work with quantum field operators out of necessity.

This does not imply that one can interpret relativistic quantum mechanics as a multi-particle quantum mechanical theory of a finite number of particles (compare to the blog post on this issue linked above), because the number operator is not conserved for a relativistic Hamiltonian, thus one should treat it as a potentially infinite particle system - thus there is no reason why one cannot just use multi-particle stationary states built out of potentially infinite products, if one couldn't do such a thing in principle then quantum field theory wouldn't make any sense.

The argument about creating new particles when attempting to measure position does not look convincing.

Not only is the argument convincing (and argued for in a canonical reference), but immediately after the above-quoted passage they show the minimum uncertainty in any position measurement is always finite - a result which differs from the non-relativistic case due to the finite velocity of light - and thus explains e.g. why one finds non-local position-space photon wave functions when one tries to construct such things.

In other words, the naive interpretation of non-relativistic position-space wave functions is out the window from a relativistic perspective, and further (relating to the above paragraph I wrote) such systems are really multi-particle systems with non-conserved particle number operators, hence why field theory is such a useful tool.

One should compare to the claims about momentum space in the surrounding paragraphs of the above quote.

So, the formation of electron-positron pairs is not "inevitable".

The Newton-Wigner operator argument and how it relates to electron-positron pairs is too vague to respond to, if you could provide some detail that would be interesting, but the point being argued for should be able to be made without recourse to this formalism.

However the quote about forming electron-positron pairs not being necessary in a measurement (interaction) process is completely unjustifiable.

I do not understand why Dirac equation and Klein-Gordon equation still are taught in University nowaday,while the normal RQM is useless?

It is not useless - one needs to use single-particle stationary states in order to construct identical multi-particle systems that can be described using single-particle quantum field operators satisfying the Dirac equation (etc...), these stationary states must satisfy the Dirac equation. The novel thing is why one must throw away the naive non-relativistic interpretation of wave functions, that many of the 'alternative' approaches to quantum mechanics get distracted by, while still salvaging a quantum theory rather than simply having nothing.

 

[Demystifier]

Why do you think that probabilistic interpretation does not apply to position measurements? There are Newton-Wigner position operators in each sector of the Fock space. So, I don't see much difference in the status of position in QFT and in non-relativistic QM.

Eugene.

Yes, but NW position operator is not relativistic covariant. For instance, if particle is localized in one Lorentz frame (with localization defined via NW position operator), then it is not localized in another Lorentz frame.

 

[A. Neumaier]

immediately after the above-quoted passage they show the minimum uncertainty in any position measurement is always finite - a result which differs from the non-relativistic case due to the finite velocity of light - and thus explains e.g. why one finds non-local position-space photon wave functions when one tries to construct such things.

This cannot be correct since it would also explain one finds non-local position-space Higgs wave functions when one tries to construct such things. But the latter can be constructed via the Newton-Wigner mechanism.

The Newton-Wigner operator argument and how it relates to electron-positron pairs is too vague to respond to, if you could provide some detail that would be interesting

One can do it in any representation of the extended Poincare group. See the articles

• Particle positions and the position operator
• Localization and position operators

from my Theoretical physics FAQ.

 

[A. Neumaier]

Yes, but NW position operator is not relativistic covariant. For instance, if particle is localized in one Lorentz frame (with localization defined via NW position operator), then it is not localized in another Lorentz frame.

Position measurements are also not relativistic covariant, since they are done in the rest frame of the detector.

 

[andresB]

What physical mechanism would be used to measure the position of an electron as precisely as wanted without putting too much energy into it that it triggers electron-positron pair creation (thus rendering the position of your original electron meaningless)?

 

[meopemuk]

Rules of quantum mechanics say: if position commutes with the number of electrons, then there should exist states with sharp values of position and a definite number of electrons (e.g., N=1). QM does not specify "physical mechanisms" of measurements.

Eugene.

 

[meopemuk]

Yes, but NW position operator is not relativistic covariant. For instance, if particle is localized in one Lorentz frame (with localization defined via NW position operator), then it is not localized in another Lorentz frame.

The principle of relativity does not say that all observers must see the same things. If the particle looks localized for one observer and delocalized for another observer, that's OK with the principle of relativity. This principle only insists that descriptions of different observers should be related by a unitary representation of the Poincare group. And the NW position operator does not contradict this requirement.

Eugene.

 

[PeterDonis]

if position commutes with the number of electrons

Yes, "if". So does it?

Physically, the fact that if I try to measure an electron's position with high enough accuracy, I will start creating electron-positron pairs, argues that it doesn't: measuring position first, then electron number gives different results from measuring electron number first, then position.

 

[meopemuk]

if I try to measure an electron's position with high enough accuracy, I will start creating electron-positron pairs,

How did your reach this conclusion?

Theoretically, Newton-Wigner position operators of particles are defined separately in each N-particle sector of the Fock space. So, by definition, NW position commutes with all particle number operators. They are measurable simultaneously. In theory.

Eugene.

 

[PeterDonis]

How did your reach this conclusion?

Isn't it obvious? To measure the position with higher accuracy, you need to use a higher energy probe. Eventually the probe energy is sufficient to create electron-positron pairs. We know we can create electron-positron pairs in the lab by applying sufficient energy, so it seems obvious that applying the energy to measure position more accurately would also do it.

Theoretically

If your theory can't account for an an actual physical process, your theory is wrong. And since we know, as above, that we can create electron-positron pairs in the lab, any theory that does not include this possibility, as the theory you describe obviously does not, must be wrong.

 

[meopemuk]

Isn't it obvious? To measure the position with higher accuracy, you need to use a higher energy probe. Eventually the probe energy is sufficient to create electron-positron pairs. We know we can create electron-positron pairs in the lab by applying sufficient energy, so it seems obvious that applying the energy to measure position more accurately would also do it.

Large energy does not necessarily mean pair creation. A single high-energy particle cannot create a pair.

Eugene.

 

[PeterDonis]

Large energy does not necessarily mean pair creation.

Unless you are disputing the experimental fact that pair creation can and has been done in the lab, your comment here is irrelevant to the actual point I was making. I notice that you did not respond at all to the last paragraph in my post #18, which makes the point more sharply.

 

[A. Neumaier]

If your theory can't account for an an actual physical process, your theory is wrong.

It is not his theory, but the theory of Newton and Wigner from 1952 that shows for any unitary representation of the full Poincare group (including parity and inversion) the existence of a position operator in each Lorentz frame.

This can be applied to subrepresentations of a quantum field theory. In the free case where a number operator exists, the eigenspaces of the number operator provide such representation spaces. Therefore the Newton-Wigner position operator defined on the full representation commutes with the number operator.

In the interacting case the situation is more complicated since nonperturbatively, no number operator exists. (This is related to Haag's theorem of the nonexistence of the interaction picture.) However, single-partice subrepresentations still exist (one for each elementary or bound state) and there are projection operators projecting to these subrepresentations, commuting with the Newton-Wigner position operator defined on the full representation. Thus single particle position is well-defined in each Lorentz frame.

 

[Demystifier]

Position measurements are also not relativistic covariant, since they are done in the rest frame of the detector.

The principle of relativity does not say that all observers must see the same things. If the particle looks localized for one observer and delocalized for another observer, that's OK with the principle of relativity. This principle only insists that descriptions of different observers should be related by a unitary representation of the Poincare group. And the NW position operator does not contradict this requirement.

Eugene.

Fine, you can measure position of a relativistic particle and it is possible to design the experimental setup such that the results of measurement are described by an appropriate NW position operator. But it is still true that there is a big difference between momentum and position, in the sense that one is Lorentz covariant and the other isn't. That's quite different from nonrelativistic QM, where position and momentum behave more symmetrically.

Above, of course, refers to the standard interpretation of quantum theory. In the Bohmian interpretation there are additional problems with a NW definition of particle position, but that's another story.

 

[meopemuk]

But it is still true that there is a big difference between momentum and position, in the sense that one is Lorentz covariant and the other isn't.

You are right: particle's momentum-energy forms a relativistic 4-vector, but the position transformation law is more complicated.

Eugene.

 

[A. Neumaier]

But it is still true that there is a big difference between momentum and position, in the sense that one is Lorentz covariant and the other isn't. That's quite different from nonrelativistic QM, where position and momentum behave more symmetrically.

This is a misleading comparison.

In nonrelativistic quantum mechanics, 3-position and 3-momentum behave symmetrically, whereas time and energy are treated very asymmetrically.

Exactly the same holds in relativistic quantum mechanics in each Lorentz frame. In the limit of low 3-momentum one recovers the nonrelativistic case in each Lorentz frame.

 

[PeterDonis]

the Newton-Wigner position operator defined on the full representation commutes with the number operator.

So how does all this account for the physical fact I mentioned before, that we can create electron-positron pairs in the lab? And that, if we are measuring an electron's position with sufficient accuracy, our measurement will involve sufficient energy to create an electron-positron pair, so physically we would not expect such a measurement to commute with a measurement of electron number?

 

[A. Neumaier]

So how does all this account for the physical fact I mentioned before, that we can create electron-positron pairs in the lab?

This is a fact derivable from relativistic quantum field theory, and hence is consistent with the theoretical consequences of the latter, including the Newton-Wigner theory. The latter is a theoretical result implying that one cannot argue based on noncommutativity and associated uncertainty relations.

Indeed, since time is not an operator, the uncertainty relation between time and energy which you allude to is quite different in spirit from uncertainty relations for noncommuting operators.

And that, if we are measuring an electron's position with sufficient accuracy, our measurement will involve sufficient energy to create an electron-positron pair, so physically we would not expect such a measurement to commute with a measurement of electron number?

At least I wouldn't expect it since it is false. (Apart from being poorly formulated since measurements can be performed but cannot commute.)

 

[PeterDonis]

At least I wouldn't expect it since it is false.

In other words, it is false that measuring electron number, then position with enough accuracy to supply the energy to create an electron-positron pair, must always give the same result as measuring position with that accuracy, then measuring electron number?

If so, you are agreeing with the point I have been trying to make in response to @meopemuk. I agree I have been stating things in a highly non-rigorous manner, but it seems like you understand what I mean.

 

[throw]

Unfortunately it's a bit shocking to read some of this discussion and I will try to push back on the sense I'm getting from it e.g. this focus on the NW operator is completely unjustifiable, or the apparent distinction between RQM and QFT in any serious sense, or the discussion about time operators.

First of all: the majority of string theory books/notes 'first quantize' a relativistic point particle. In these notes they promote time (not proper time) to an operator and arrive e.g. at the Klein-Gordon equation which tells us point particle wave functions are independent of proper time. See for example section 4.1 of these notes.

If we trusted the discussion in this thread we would think every string theory book is incorrect on basic things like turning time into an operator in a relativistic context. Similarly every discussion of the quantized string promotes the time coordinate to an operator.

The deeper question is why in a relativistic context we can't impart the naive measurement interpretations to such operators - that of course is really related to the passage cited earlier, I would have come away thinking this is all incorrect if I took the earlier discussion of time operators at face value hence why I feel the need to push back.

The fact that Newton-Wigner does not treat time as an operator and is frame-dependent and breaks relativistic covariance is simply a complete embarrassment and is just one reason why this is a fringe topic nowadays.

The notion of a position operator for the position of a single particle at best depends on the interaction process being such that we can attach an interpretation to it that it did not create particle-anti-particle pairs and so completely change the system we were dealing with (if we were expecting a single particle remaining a single particle just interacting with some classical apparatus to hold at all times).

Why? Because of arguments such as that given in the quoted passage cited earlier, and the non-locality of position-space wave functions I mentioned earlier that people apparently also disagree with (despite this result being over 90 years old).

The claim here for example is that the NW operator is only meaningful in such a case up to a certain minimal localization and can be used to help establish the validity of a one-particle theory, which, while useful, is at most small potatoes just like modelling particles moving in external potentials or something, which is also very useful, although it's even less useful and frame-dependent, hence the fringe-topic-ness.

The fact that a position operator is to a large extent useless in QFT (apart from the sense in the notes mentioned earlier, which is usually always bypassed and was in the early days) was known decades before Newton-Wigner (results they never even mention in their original paper), and it's application to things such as helping to determine the domain of validity of a one-particle theory as in the link above might be useful but it's irrelevant to the bigger picture, and really just has no relevance when talking about RQM vs QFT.

The sense one would get from this thread is that Newton-Wigner is a very important idea in QFT when it simply isn't, again why people should push back against such claims.

It is thus a very basic misunderstanding of QFT to think there is any more meaning to position measurements, QFT just fundamentally changes things: creating particle-antiparticle pairs is inevitable so the idea of measuring e.g. a single particle using a measuring apparatus modeled as one or a fixed number of particles is simply the wrong way to think about things.

The main point is: relativistic quantum mechanics is quantum field theory, there is no essential difference between them except for some of the formal tools we use to derive results, in other words 1st vs 2nd quantization, just as there is no essential difference other than the tools we use when doing first vs second quantized non-relativistic quantum mechanics. One can go back and forth between these formalisms at will in principle, indeed nothing would make sense if one couldn't, it would be a huge flaw in quantum theory if this wasn't the case.

If this is shocking, one just needs to ask oneself: why is it that we can go back and forth between first and second quantization in NRQM but not in RQM? The answer is obvious that we can in both.

There are plenty of arguments justifying all of this for example in the textbook that the earlier quote was taken from if people are interested, none of this is really debatable stuff to be honest, none of it is thrown into question/doubt especially not due to the apparent 'importance' of the NW operator...

To address the reason people think RQM and QFT they are distinct.

The real problem/issue here is that people apparently think that when we do relativistic quantum mechanics it somehow implies we are forced to work with one or a finite number of particles which would mean viewing RQM as an almost exact copy of elementary NRQM problems up to choices of potentials etc... (and so we can talk about position operators for such systems, despite the critical flaw of ignoring that interactions create particle-antiparticle pairs thus changing the number of particles one needs to include to accurately describe the system, simply a fatal flaw to ignore...).

But it is simply a completely unjustifiable assumption to expect the number of particles to remain fixed in a relativistic theory. This unavoidable fact has to be accounted for when doing RQM in a non-2nd quantized form. Furthermore the finite speed of light (RQM) vs an infinite speed of light (NRQM) has to be accounted for in the uncertainty principle which leads to such serious implications that it even changes our interpretation of the wave function. Even worse, time and space have to be treated on an equal footing, which is why the string theory notes cited above do exactly this.

The fact that we don't attribute the naive meaning to these time-space operators, and don't demand such things reproduce NW operators or other things, is to do with the quoted L&L passage that people are disagreeing with - one would be completely lost when seeing quantized time operators in a QFT context otherwise so one should just trust good textbooks more than discussion forums.

The reason QFT is so useful is that in one fell swoop we can deal with any number of identical particles as part of one big 'field' - it's not mandatory that we take this 2nd quantized view but everything becomes clear/easy (e.g. a 'number operator' and it not being conserved is obvious) when one does. It especially clears up the interpretation of the 'negative energy' eigenfunctions for example.

 

[vanhees71]

How did your reach this conclusion?

Theoretically, Newton-Wigner position operators of particles are defined separately in each N-particle sector of the Fock space. So, by definition, NW position commutes with all particle number operators. They are measurable simultaneously. In theory.

Eugene.

For non-interacting particles that may be right, but those you can't detect nor localize.

 

[dextercioby]

@throw Well, there are very well founded theories of a quantization of time in non-relativistic QM, but for some unknown reason they are not reaching (advanced) textbook-level, hence most people are unaware of them and keep dwelling on the same arguments for ages. This is also because of the immense success of QFT to which no obvious generalization (such as a time operator on equal footing with the Hamiltonian) of QM was felt needed (well, up to known theoretical/mathematical problems).

 

[meopemuk]

For non-interacting particles that may be right, but those you can't detect nor localize.

Are you saying that interaction changes particle observables/operators?

Then our definitions of "interaction" are quite different. In my view, relativistic interaction should be introduced in a theory by adding operators V and W to generators of the non-interacting representation of the Poincare group. It seems that S. Weinberg would agree with me. See eqs. (3.3.18) and (3.3.20) in his "The quantum theory of fields" vol. 1.

So, relativistic interaction of the Dirac-Weinberg type modifies only 4 total observables of the physical system. The remaining theory (particle observables, particle number operators, Fock space structure) remains the same as in the non-interacting case.

But I can also agree that Weinberg's approach to QFT is very different from usual textbooks, where interacting quantum fields are placed in the center, and it is not possible to derive particles, their positions, Fock space etc. from the fields in the interacting regime.

Eugene.

 

[throw]

I would like to see Weinberg's perspective directly applied to the non-relativistic case as much as possible, the closest general reference seems to be Ballentine.

 

[andresB]

I would like to see Weinberg's perspective directly applied to the non-relativistic case as much as possible, the closest general reference seems to be Ballentine.

Ballentine takes from older sources, like Jordan's "linear operators in QM" or Jauch's "foundation of QM".

Btw, the same treatment can be given to classical systems, where it also happens that in the relativistic case the position operator behaves like a Newton-Wigner operator.
https://arxiv.org/abs/2105.13882
https://arxiv.org/abs/2004.08661

 

[vanhees71]

Are you saying that interaction changes particle observables/operators?

Then our definitions of "interaction" are quite different. In my view, relativistic interaction should be introduced in a theory by adding operators V and W to generators of the non-interacting representation of the Poincare group. It seems that S. Weinberg would agree with me. See eqs. (3.3.18) and (3.3.20) in his "The quantum theory of fields" vol. 1.

So, relativistic interaction of the Dirac-Weinberg type modifies only 4 total observables of the physical system. The remaining theory (particle observables, particle number operators, Fock space structure) remains the same as in the non-interacting case.

But I can also agree that Weinberg's approach to QFT is very different from usual textbooks, where interacting quantum fields are placed in the center, and it is not possible to derive particles, their positions, Fock space etc. from the fields in the interacting regime.

Eugene.

No, because "particle observables" refer to asymptotic free states. For transient states, there is no viable particle interpretation.

What I meant is that Landau&Lifshitz is right (BTW I don't like vol. IV of LL too much, because there are much more up-to-date books for QED, but this introductory chapter is well worth studying carefully). If you want to localize particles you need some interactions to do so. E.g., you can use a trap (Penning or Paul traps are examples), but if you try to squeeze a particle on a spot smaller than its Compton wavelength, $h/(m c)$, you rather create new particles than to localize the original particle better. The extreme are massless "particles" like photons, which cannot be localized at all, because there's no energy necessary to create new ones when trying to localize them. Despite this argument, even free massless particles with spins $\geq 1$ do not even admit a position operator.

Weinberg's approach to QFT is just the standard approach to its presentation for the 21st century. It gets rid of all the dust of the difficult (and still not finished) development of the theory since 1925 and gives a clear exposition of the foundations as far as they are understood today. Particularly there is no viable first-quantization approach beyond approximations for situations close to the non-relativistic limit. The clear analysis of Wigner's ground-breaking analysis of the representation theory of the proper orthochronous Poincare group (of 1939!) is the key for a real understanding of the theory. Together with the microcausality/locality concept it delivers the fundamental properties of the theory:

-necessity of antiparticles
-spin-statistics relation (half-integer spin fields must be quantized as fermions; integer-spin fields must be quantized as bosons)
-non-conservation of particle numbers (but rather charges) -> Creation and annihilation processes need a many-body description, and the most convenient one is QFT
-particle interpretation only for (asymptotic) free states

 

[vanhees71]

@throw Well, there are very well founded theories of a quantization of time in non-relativistic QM, but for some unknown reason they are not reaching (advanced) textbook-level, hence most people are unaware of them and keep dwelling on the same arguments for ages. This is also because of the immense success of QFT to which no obvious generalization (such as a time operator on equal footing with the Hamiltonian) of QM was felt needed (well, up to known theoretical/mathematical problems).

Can you give an example for a (review/introductory) paper about how "time is quantized"? The old argument by Pauli is very convincing, and I don't see what's wrong with it, no matter whether you consider non-relativistic or relativistic QT.

 

[throw]

In the non-relativistic case, time-energy uncertainty involves energy measurements at two different times, it's hard to believe this is from some operator relation since the whole reason for NRQM in the first place is the inability of simultaneously (i.e. at the same time) prescribing values of complementary variables; successive measurements of the same quantity (thus at different times, note we're also talking about exact values for the energy at these times as L&L set it up) is just completely different.

The RQM case means either generalizing this to include time-energy as part of complementary space-time momentum-energy measurements... It simply starts to look like nothing makes any sense until we give up something else - in this case it is the very idea of space-time energy-momentum complementarity that is challenged in the RQM context (see the position vs momentum space wave function comment earlier, or the discussion around the earlier-cited passage, for example).

This has interesting comments on 'Pauli's theorem' even in a non-relativistic context, it seems very hard to believe something can be an operator some of the time but not all of the time in a NRQM context depending on the spectrum which seems to be the argument, and this seems to be the road one has to go down to try to force a time operator.

But regarding the relativistic case - the particle action is gauge invariant, so the Hamiltonian is not even fixed until one fixes a gauge (parametrization), and it possesses primary constraints so it should be treated as a constrained system which also incorporates the massless case. The Schrodinger equation associated to this properly formulated version (in which the gauge choice does not ruin space-time covariance) is not even the naive 3D Schrodinger equation either, for example section 3 here: this appears to be the proper context in which Pauli's theorem should be applied regarding the choice of Hamiltonian and a potentially 'proper time' operator being associated to it, note it regards not 'time' but now a 'proper time' operator.

Thus, in this covariant case, (in which it should again be stressed that the Hamiltonian is determined by the choice of gauge usually taken to keep things covariant) operator commutation relations like $[X^{\mu},P_{\nu}] = i \delta^{\mu}_{\nu}$ are simply unaffected by Pauli's argument about discrete spectra of the Hamiltonian, which would really relate to whether it is possible to establish a proper time operator associated to a covariant-gauge-chosen Hamiltonian.

In a specific gauge in which proper time is taken to be the usual time (section two of the above notes for example), non-locality immediately arises - we now get a non-local Schrodinger equation, the square of which as usual leads to 'negative energy' eigenvalues which then require a direct argument - at best one has to artificially throw away the negative energy eigenvalues and replace them with positive energy eigenvalues for a separate species of plane wave associated to what we call anti-particles, or else talk about going backward in time, to keep those energy eigenvalues positive, and so the meaning of position space is just destroyed as usual making time operators even less relevant, it's unavoidable.

 

[vanhees71]

The counterexample is of course invalid, because the there discussed Hamiltonian
$$\hat{H}=\frac{1}{2m} \hat{P}^2 -q E \hat{Q},$$
where I made deliberately the dimension of the potential right ($q$ electric charge of the particle, $E=\text{const}$ electric field). Of course here $[\hat{H},\hat{P}]=\text{const}$, but this Hamiltonian is indeed not bounded from below, i.e., there's no stable ground state. Of course a homogeneous electric field everywhere is fictitious and thus this is a idealized approximate Hamiltonian not describing a real-world situation.

The energy-time uncertainty relation is of a different kind than the uncertainty relations between observables. To measure time you have to measure an observable, e.g., the hyperfine transition of Cs which defines the SI unit of time, the second. To get this transition, i.e., the difference of the two energy levels, accurately you need to measure for a long time.

Another argument is that for a measurement of time you have to measure some observable $\hat{A}$. Then the uncertainty relation of this observable with the Hamiltonian/energy is
$$\Delta A \Delta E \geq \frac{1}{2} |\langle [\hat{A},\hat{H}] \rangle|=\frac{\hbar}{2}|\langle \dot{A} \rangle|,$$
where $\dot{A}$ is the operator representing the time derivative of the observable $A$. Then
$$\Delta E \geq \hbar \frac{|\langle \dot{A} \rangle|}{2 \Delta A} = \frac{\hbar}{2 \Delta t},$$
where $\Delta t$ is an estimator for the time it takes for the observable $A$ to change by an amount of its standard deviation $\Delta A$, i.e., where a change can be expected to be significantly detected. Again it tells you that to determine the energy of a system you need a sufficiently long time for measuring it.

 

[Demystifier]

In these notes they promote time (not proper time) to an operator and arrive e.g. at the Klein-Gordon equation which tells us point particle wave functions are independent of proper time.

There is no problem of making time an operator in that sense, in a Lorentz covariant way. But this operator is not an operator on a Hilbert space. It is an operator on the space of functions of spacetime position $x=(x^0,x^1,x^2,x^3)$, but those functions are not square integrable. That's because solutions of the Klein-Gordon equation cannot be localized in spacetime; they can be localized in 3-space, but not in time.

 

[throw]

Are you implying that the fact that continuous spectrum eigenfunction solutions of a Schrodinger equation are not square integrable, and so technically not in a Hilbert space (an issue that does not stop books like L&L), somehow let's us rationalize the existence of a time operator in a relativistic context, but not in a non-relativistic context? I'm not sure I could agree with anything like that.

 

[Demystifier]

Are you implying that the fact that continuous spectrum eigenfunction solutions of a Schrodinger equation are not square integrable, and so technically not in a Hilbert space (an issue that does not stop books like L&L), somehow let's us rationalize the existence of a time operator in a relativistic context, but not in a non-relativistic context? I'm not sure I could agree with anything like that.

No. I am saying that the scalar product (both relativistic and non-relativistic) between two wave functions involves a 3-dimensional integration over space, and not a 4-dimensional integral over spacetime. This means that the Hilbert space is a Hilbert space of functions of space only, not of space and time. For reasonable wave packets, this 3-dimensional integral is finite. But there are no wave packets for which the 4-dimensional integral would be finite. Just the opposite, the scalar product, namely the 3-dimensional integral, is conserved in time, it is a constant. If one would then integrate the constant over the infinite time interval, one would get infinity.

 

[vanhees71]

One should, however, also pose the caveat that a "wave function" (or the so-called "first-quantization formalism") has a very different and pretty limited sense in the relativistic case compared to the non-relativistic case. The only successful fully consistent description of relativistic QT for (interacting) particles is local relativistic QFT, providing the only consistent framework to formulate the Standard Model of high-energy particles physics, which still as one of the most successful theories ever discovered.