Relativistic quantum mechanics and causality

[GDogg]

I was told in class that both the Dirac equation and the Klein-Gordon equation violate causality, even though they're relativistic invariants, and that this wasn't surprising because the 2 postulates of special relativity don't imply causality.

Is this true?

 

[jostpuur]

Hans and meopemuk didn't dare to answer this now?

GDogg, I don't know if that's true, but I know that we have had debates about precisely this topic here. So it very much seems a slightly unsettled question.

 

[meopemuk]

I was told in class that both the Dirac equation and the Klein-Gordon equation violate causality, even though they're relativistic invariants, and that this wasn't surprising because the 2 postulates of special relativity don't imply causality.

Is this true?

This is a loaded question and I don't have a short answer. As jostpuur said, we had a lot of discussions on this topic. See, for example, recent posts in

https://www.physicsforums.com/showthread.php?t=175155

and

https://www.physicsforums.com/showthread.php?t=185315

Eugene.

 

[OOO]

I was told in class that both the Dirac equation and the Klein-Gordon equation violate causality, even though they're relativistic invariants

If you mean the Dirac equation and Klein-Gordon equation as classical field equations then it is hard to see why they should violate causality. Simply put them on a finite difference lattice and observe how some movement here and now causes some movement over there some time later.

Analytically, if you look at the dispersion relation $$p_\mu p^\mu = m^2$$ i.e. $$\omega^2 =m^2+k^2$$, where I have assumed hbar=c=1, then you get for the group velocity

$$\frac{d\omega}{d k} = \frac{k}{\omega}$$, so $$\left(\frac{d\omega}{d k}\right)^2 = 1-\frac{m^2}{\omega^2}$$

which is clearly smaller than 1 (the speed of light in my system of units), at least if we are talking about positive m^2.

However the statement is most likely referring to quantum field theory, especially what Peskin-Schroeder writes in Chapter 2.4. He concludes that the propagation amplitude is nonzero (but exponentially small) for spacelike separation, but the commutator of the field operator vanishes for spacelike separation. Because the commutator $$[\phi(x),\phi(y)]$$ vanishes in this case, the field can be measured independently in both (spacelike separated) points and so there is no way for the first measurement to affect the second. Thus causality is indeed preserved, although it does not seem to at first sight.

 

[meopemuk]

However the statement is most likely referring to quantum field theory, especially what Peskin-Schroeder writes in Chapter 2.4. He concludes that the propagation amplitude is nonzero (but exponentially small) for spacelike separation, but the commutator of the field operator vanishes for spacelike separation. Because the commutator $$[\phi(x),\phi(y)]$$ vanishes in this case, the field can be measured independently in both (spacelike separated) points and so there is no way for the first measurement to affect the second. Thus causality is indeed preserved, although it does not seem to at first sight.

Unfortunately, Peskin & Schroeder didn't explain the connection between field commutators at different points and causality. It would be nice to have at least one example in which these commutators are related to the cause-effect connection between physical processes at different space-time points.

Eugene.

 

[OOO]

Unfortunately, Peskin & Schroeder didn't explain the connection between field commutators at different points and causality. It would be nice to have at least one example in which these commutators are related to the cause-effect connection between physical processes at different space-time points.

Eugene.

Maybe I am a bit too naive with respect to this. But isn't this simply quantum mechanics ?

commutative = simultaneously measurable = independent from each other = not causally related

 

[Hans de Vries]

If you mean the Dirac equation and Klein-Gordon equation as classical field equations then it is hard to see why they should violate causality. Simply put them on a finite difference lattice and observe how some movement here and now causes some movement over there some time later.

Analytically, if you look at the dispersion relation $$p_\mu p^\mu = m^2$$ i.e. $$\omega^2 =m^2+k^2$$, where I have assumed hbar=c=1, then you get for the group velocity

$$\frac{d\omega}{d k} = \frac{k}{\omega}$$, so $$\left(\frac{d\omega}{d k}\right)^2 = 1-\frac{m^2}{\omega^2}$$

which is clearly smaller than 1 (the speed of light in my system of units), at least if we are talking about positive m^2.

Exactly. The propagators never get outside the lightcone, neither if you simulate
the propagation on a lattice, nor with the exact analytical solutions.

However the statement is most likely referring to quantum field theory, especially what Peskin-Schroeder writes in Chapter 2.4. He concludes that the propagation amplitude is nonzero (but exponentially small) for spacelike separation, but the commutator of the field operator vanishes for spacelike separation. Because the commutator $$[\phi(x),\phi(y)]$$ vanishes in this case, the field can be measured independently in both (spacelike separated) points and so there is no way for the first measurement to affect the second. Thus causality is indeed preserved, although it does not seem to at first sight.

The Feynman rules uses the same propagators as what you referred to as the classical
field equations. The confusion starts when people start decomposing the propagator
(Green's function) rather then the field itself. For instance in positive and negative
energy components even though this is never done in the path integral approach.

The Green's function represents the response of the vacuum on a point event excitation.
People then associate the point field with the particle and the Green's function with the
wave function, subsequently they want to decompose or quantize the Green's function
and get odd results. The right thing to do however is to apply the full and unmodified
(forward) propagator to the quantized field to get the time evolution of the quantized field. Regards, Hans

 

[jostpuur]

Exactly. The propagators never get outside the lightcone, neither if you simulate
the propagation on a lattice, nor with the exact analytical solutions.

But I don't think that studying group velocities of some packets is sufficient to draw these conclusions.

 

[meopemuk]

Maybe I am a bit too naive with respect to this. But isn't this simply quantum mechanics ?

commutative = simultaneously measurable = independent from each other = not causally related

Yes, I also see these vague analogies. However, I think we cannot be satisfied with vague analogies in the matter of such a great importance. It is often said that the reason to introduce (quantum) fields is to guarantee the subluminal propagation of signals and causality, which is (supposedly) violated in theories of particles interacting via potentials. So, the issue of causality is in the foundation of the quantum field theory. It would be nice to devote a bit more attention and rigor to such a fundamental issue.

I would be entirely satisfied if Peskin & Schroeder presented at least one example of an interacting system with "cause" and "effect" events. It would be great if they could explain how quantum fields are used in the description of these events, and why the commutativity of the fields guarantees that the effect always precedes the cause in any reference frame. Since this explanation is absent, I remain unconvinced.

Eugene.

 

[OOO]

Yes, I also see these vague analogies.

Why do you think these are vague or even analogies ? This is what quantum mechanics wants us to believe about measurements.

I would be entirely satisfied if Peskin & Schroeder presented at least one example of an interacting system with "cause" and "effect" events.

I don't know what example you need. The fact that commuting observables have a common system of eigenvectors is purely mathematical. So if you believe in the statement that measurements result in eigenvectors you are forced to believe that commuting observables cannot affect each other's outcome, so neither one can be cause or effect for the other. Will you be happier when you see that this general principle holds for a special example ?

It would be great if they could explain how quantum fields are used in the description of these events, and why the commutativity of the fields guarantees that the effect always precedes the cause in any reference frame.

So the outside of the light cone is "causally empty", i.e. causal relationships are restricted to the inside of the light cone. The reference affairs shouldn't be hard since the Klein-Gordon field is scalar. Then you will not find a proper orthochronous Lorentz transform that turns the forward light cone into the backward light cone and vice versa. Thus there is no chance for reversing cause and effect.

 

[meopemuk]

I don't know what example you need. The fact that commuting observables have a common system of eigenvectors is purely mathematical. So if you believe in the statement that measurements result in eigenvectors you are forced to believe that commuting observables cannot affect each other's outcome, so neither one can be cause or effect for the other. Will you be happier when you see that this general principle holds for a special example ?

Yes I would be much happier if Peskin & Schroeder gave a concrete physical example and indicated which commuting observables they are talking about. As far as I know, scalar fields have not been directly measured in any physical experiment. So, I would hesitate to call them "observables".

Eugene.

 

[meopemuk]

The Feynman rules uses the same propagators as what you referred to as the classical
field equations.

I don't think that Feynman rules and propagators are relevant to the discussion of causality. Feynman diagrams and propagators are mathematical objects that are used to calculate the S-matrix of scattering events. In such events, the interaction between particles occurs almost instantaneously in a small region of space. So, in scattering experiments it is not possible to decide where is the cause, where is the effect and what is the distance and time separation between them.

Eugene.

 

[OOO]

Yes I would be much happier if Peskin & Schroeder gave a concrete physical example and indicated which commuting observables they are talking about. As far as I know, scalar fields have not been directly measured in any physical experiment. So, I would hesitate to call them "observables".

Eugene.

You're free to find an example for yourself. From what you say it appears to me you're quite sure that Peskin & Schroeder don't know what they are talking about. So are we still discussing the consequences of quantum field theory or is this about something new ?

 

[meopemuk]

You're free to find an example for yourself. From what you say it appears to me you're quite sure that Peskin & Schroeder don't know what they are talking about. So are we still discussing the consequences of quantum field theory or is this about something new ?

Possibly, we have different opinions on what constitutes a "proof" in theoretical physics. I don't buy the space-like (anti)commutativity of quantum fields as a proof of causality. The causality in QFT is a rather subtle issue, and there are dozens of papers discussing it without reaching any satisfactory conclusion, in my opinion.

Eugene.

 

[OOO]

Possibly, we have different opinions on what constitutes a "proof" in theoretical physics. I don't buy the space-like (anti)commutativity of quantum fields as a proof of causality. The causality in QFT is a rather subtle issue, and there are dozens of papers discussing it without reaching any satisfactory conclusion, in my opinion.

Eugene.

Have I used the word proof ? No, I haven't. I'm not a mathematician. If physicists always did proofs by mathematical standards, they'd get stuck in the mud of irrelevant discussions, like mathematician do all too often. All I can say is that it sounds plausible to me and obviously I am not the only one. Since you seem to have no alternative to offer but your skepticism, I tend to be skeptical about your skepticism as well.

 

[meopemuk]

Have I used the word proof ? No, I haven't. I'm not a mathematician. If physicists always did proofs by mathematical standards, they'd get stuck in the mud of irrelevant discussions, like mathematician do all too often. All I can say is that it sounds plausible to me and obviously I am not the only one.

I actually disagree. I don't think that mathematicians engage in "irrelevant discussions" more than theoretical physicists do. Mathematicians rigorously prove theorems based on a well-defined set of axioms. Once a theorem is proven, everybody agree about that and move on. I wish theoretical physics was designed by the same rules: a few axioms everybody can agree upon, and the rest is proven by rules of logic. Unfortunately, we are far from that, and such unscientific factors as personal preferences, celebrity status, etc. play excessively prominent role.

Since you seem to have no alternative to offer but your skepticism, I tend to be skeptical about your skepticism as well.

Actually I do have an alternative positive message for this discussion. You can find it in

E. V. Stefanovich, "Is Minkowski space-time compatible with quantum mechanics?", Found. Phys. 32 (2002), 673.

E. V. Stefanovich, "Relativistic quantum dynamics", http://www.arxiv.org/abs/physics/0504062 (see esp. chapter 10)

and in a few more papers available from my website http://www.geocities.com/meopemuk

Eugene.

 

[Haelfix]

Eugene, if you agree that commutators of gauge invariant operators vanish outside the lightcone, I really don't see what the problem is for causality.

If you don't agree with that statement, then I can probably find various different proofs in different formalisms with varying lvls of rigor (for regular QFT, some old bootstrap papers by Cho and others contain proofs, for slightly more rigorous Aqft, and wightman commutators, the Streeter-Wightman book should suffice).

However, I do think you agree with the statement, so I am a little puzzled. Can you give an example maybe of what you have in mind?

 

[jostpuur]

Eugene, if you agree that commutators of gauge invariant operators vanish outside the lightcone, I really don't see what the problem is for causality.

I think that this

As far as I know, scalar fields have not been directly measured in any physical experiment. So, I would hesitate to call them "observables".

sums it.

The operators don't describe physical quantities here, but are used to create and annihilate particles. So how could it make sense to say "causality is preserved because measurements don't affect each others"?

To me this looks like that physicists have a well defined mathematical statement here, but don't really know what it means physically.

 

[meopemuk]

The operators don't describe physical quantities here, but are used to create and annihilate particles. So how could it make sense to say "causality is preserved because measurements don't affect each others"?

To me this looks like that physicists have a well defined mathematical statement here, but don't really know what it means physically.

I agree with jostpuur completely.

Eugene, if you agree that commutators of gauge invariant operators vanish outside the lightcone, I really don't see what the problem is for causality.

There is actually no need to prove that scalar quantum fields commute at space-like intervals. In Weinberg's "The quantum theory of fields" vol. 1 (a book which I respect very much) this requirement is a part of definition of the field. So, the commutativity is not an issue. The issue is what physical conclusions can be made from this mathematical fact?

Experimentally, nobody measures scalar of spinor quantum fields or their commutators. We are measuring observables of particles (positions, momenta, spins, etc.). So, commutators of fields do not tell much about what happens in experiment.

If we really want to study the question of causality in QFT we should build a model of an interacting system described in terms of constituent particles. We should define which configuration of particles we are going to call "the cause" and which configuration of particles (at a later time) is "the effect". We should make sure that these two events are related to each other through interacting time evolution. Then we should transform the entire description to the moving reference frame and check that the temporal order of these events (the effect is later than the cause) is frame-independent. This would be a satisfactory proof, in my opinion. Nobody has done this so far, and handwavings about quantum field commutativity do not convince me at all.

The most troublesome point is that calculations outlined above cannot be performed within standard QFT, even in principle. Renormalized QFT does not have a well-defined finite Hamiltonian, so it is impossible to talk about the time evolution there. Yes, in QFT one can calculate the S-matrix to the exceptional level of precision. This is guaranteed by cancelation of infinities in each perturbation order. However, these cancelations do not occur when the time evolution in a finite time interval is considered.

Eugene.

 

[OOO]

Experimentally, nobody measures scalar of spinor quantum fields or their commutators. We are measuring observables of particles (positions, momenta, spins, etc.). So, commutators of fields do not tell much about what happens in experiment.

I am wondering why you doubt that a scalar field can be an observable. Whether there is a scalar particle in nature is irrelevant. It's just a mnemonic for the general approach, a toy model. If you are willing you may generalize it to the electromagnetic spin-1 field or the su(2), su(3) fields, whatever. Don't you accept the physical reality of the electromagnetic field ?

1) Accept: Well, then why should't it be an observable ? If you detect a grain of silver on a photographic plate, have you got any doubt that the triggering event could be called a "photon" and that the probability of it was determined by the electromagnetic field ?

2) Reject: What you say sounds a bit like Wheeler-Feynman theory - I admit that I don't know much about it. But, as I remember, Feynman himself said that this was a dead end.

I am still trying to understand what drives your skepticism.

 

[OOO]

Yes, in QFT one can calculate the S-matrix to the exceptional level of precision. This is guaranteed by cancelation of infinities in each perturbation order.

So what you really mean is QED/Electroweak theory not QFT. QCD and perturbation theory don't fit together well.

 

[Hans de Vries]

I don't think that Feynman rules and propagators are relevant to the discussion of causality. Feynman diagrams and propagators are mathematical objects that are used to calculate the S-matrix of scattering events. In such events, the interaction between particles occurs almost instantaneously in a small region of space. So, in scattering experiments it is not possible to decide where is the cause, where is the effect and what is the distance and time separation between them.

No, on the contrary, please forget your human dimensions and timescale and zoom in a bit

If you would travel through a 1 TeV scattering zone then you encounter ~1 billion
phase changes for each nanometer you travel. That's why Feynman's plane wave
approximation works so well. If each phase change takes about one second in your
zoomed in state then it takes you 30 years to travel through one nanometer of
scattering zone.

Regards, Hans

 

[GDogg]

This looks like a very interesting discussion but I'm still an undergrad and haven't studied QFT yet, so I'm having problems following it. Let me ask a couple of questions:

Is there agreement on the fact that non-QFTs, i.e the Dirac and Klein-Gordon equations for particles interacting via potentials, violate causality? And since these are derived from special relativity, does this mean that causality is not embedded in special relativity?

 

[meopemuk]

Don't you accept the physical reality of the electromagnetic field ?

1) Accept: Well, then why should't it be an observable ? If you detect a grain of silver on a photographic plate, have you got any doubt that the triggering event could be called a "photon" and that the probability of it was determined by the electromagnetic field ?

When a photon leaves its mark on the photographic plate, we measure an observable "photon's position". The probability for such a measurement is determined by the square of photon's wave function in the position space. I'm afraid that photon's quantum field $A_{\mu}(x)$ has nothing to do with this probability.

My attitude toward quantum fields of various particle types is similar to that explained in S. Weinberg "The quantum theory of fields" vol. 1: Quantum fields are purely mathematical objects whose role is to provide "building blocks" for interaction Hamiltonians in the Fock space and to guarantee that the obtained interaction (and the S-matrix) is relativistically invariant, i.e., that the commutation relations of the Poincare group remain valid in the case of interacting particles.

2) Reject: What you say sounds a bit like Wheeler-Feynman theory - I admit that I don't know much about it. But, as I remember, Feynman himself said that this was a dead end.

As I explained in another thread https://www.physicsforums.com/showthread.php?t=186364, I think that the correct description of dynamics of interacting particles is given by the "dressed particle" theory. It is similar to the Wheeler-Feynman approach in the sense that both are action-at-a-distance theories. However, here the similarities end. The "dressed particle" formalism doesn't use retarded and advanced potentials. It predicts the instantaneous character of interactions between particles.

So what you really mean is QED/Electroweak theory not QFT. QCD and perturbation theory don't fit together well.

Yes, my comments are made mostly with QED (which, I believe, I understand pretty well) in mind. However, I think that the same ideas are applicable to other quantum field theories.

Eugene.

 

[Haelfix]

Good, so you agree that microcausality is true. So i guess you're question is

'Why does field micro causality guarantee particle observable causality'?

Well I should point out its trivial in the case of the electromagnetic field. Performing a measurement of the field components at any spacetime point yields a *measurable* quantity. Clearly no superluminal signal can be sent between them under this condition. Of course, its less clear what one means by *measurable* components when we are dealing with say a dirac electron.

I guess people looked at this later from a different point of view by analyzing dispersion relations, as well as analyzing the behaviour of partial wave expansions and found that the construction is at least consistent.

In constructive field theory, things are much clearer, b/c they have very rigorous requirements and theorems on localization, observables and causality.

 

[meopemuk]

If you would travel through a 1 TeV scattering zone then you encounter ~1 billion
phase changes for each nanometer you travel. That's why Feynman's plane wave
approximation works so well. If each phase change takes about one second in your
zoomed in state then it takes you 30 years to travel through one nanometer of
scattering zone.

Yes, I agree with you that lots of things (a non-trivial time evolution of particle wave functions) are happening in the scattering zone. However, I think the following two statements are also true:

1. Experimentaly we don't have any access to this non-trivial time evolution. We can only measure states of free particles either entering the collision zone or leaving the collision zone.

2. Theoretically, we currently do not have tools to calculate this interacting time evolution in the collision zone. All we can do is to calculate the S-matrix, i.e., the mapping from free states before the collision to free states after the collision.

So, we have a "happy" state of affairs when both experiment and theory are ignorant about interaction dynamics.

Eugene.

 

[OOO]

Is there agreement on the fact that non-QFTs, i.e the Dirac and Klein-Gordon equations for particles interacting via potentials, violate causality? And since these are derived from special relativity, does this mean that causality is not embedded in special relativity?

The mainstream on QFT argues that you get into trouble when you try to interpret Dirac- and Klein-Gordon as equations for particles. But if you exclude this phrase about particles then it's obvious that (classically) Dirac- and Klein-Gordon equations do not violate causality. Information propagates with speeds smaller than c (inside the light cone), see my arguments on group velocity. But, according to special relativity, a change of reference frame could only reverse the temporal order of events outside the light cone, but these are not causally related.

Your final question is: is this always so in special relativistic classical theories ? My answer would be: no. A simple counter example is the phi^4 theory (a modification of the Klein-Gordon equation) which you might have heard about. There the classical field amplitude is subject to a potential the graph of which looks like a mexican hat. I.e. the potential has a local maximum where the field becomes unstable. If you do a taylor expansion around this maximum you get negative m^2 (or imaginary mass) and if you insert this into the expression for the group velocity, you see that it becomes larger than c ! Of course it makes limited sense to form wave packets from these waves since they are unstable, similar to the inflection point of the van-der-Waals curve in thermodynamics. Likewise they can't be considered tachyon particles.

But nevertheless this shows that special relativity does not necessarily imply that signal propagation is always smaller than c (i.e. causality). It can only guarantee the following:
1) if v<c then v<c in every reference frame
2) if v>c then v>c in every reference frame
3) if v=c then v=c in every reference frame

 

[meopemuk]

This looks like a very interesting discussion but I'm still an undergrad and haven't studied QFT yet, so I'm having problems following it. Let me ask a couple of questions:

Is there agreement on the fact that non-QFTs, i.e the Dirac and Klein-Gordon equations for particles interacting via potentials, violate causality? And since these are derived from special relativity, does this mean that causality is not embedded in special relativity?

I don't think that there exists a consistent relativistic quantum theory that you call "non-QFTs, i.e the Dirac and Klein-Gordon equations for particles interacting via potentials". In order to get a consistent theory you need to go all the way to QFT, and that's where things get really messy.

One thing you are definitely right about is that "causality is not embedded in special relativity". Einstein's postulates tell nothing about causality. The causality (i.e., the fact that the effect occurs after the cause in all frames of reference) should be proven from these postulates.

Eugene.

 

[meopemuk]

'Why does field micro causality guarantee particle observable causality'?

Well I should point out its trivial in the case of the electromagnetic field. Performing a measurement of the field components at any spacetime point yields a *measurable* quantity. Clearly no superluminal signal can be sent between them under this condition.

I don't find anything "trivial" or "clear" in these statements. They sound very abstract to me. I would better have a concrete example of a physical system with two experimentally identifiable events connected by the cause-effect relationship.

Eugene.

 

[genneth]

I don't think that there exists a consistent relativistic quantum theory that you call "non-QFTs, i.e the Dirac and Klein-Gordon equations for particles interacting via potentials". In order to get a consistent theory you need to go all the way to QFT, and that's where things get really messy.

I think that's a great shame, and actually a good direction to research. After all, it doesn't feel like there's any a priori reason for a relativistic quantum particle to not make sense. There are lines of attack that suggest a consistent theory can be found.