# Relativistic quantum mechanics and causality

1. Sep 21, 2007

### 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?

2. Sep 21, 2007

### 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.

3. Sep 21, 2007

### meopemuk

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

and

Eugene.

4. Sep 22, 2007

### OOO

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.

5. Sep 22, 2007

### meopemuk

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.

6. Sep 22, 2007

### OOO

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

7. Sep 22, 2007

### Hans de Vries

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

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

Last edited: Sep 22, 2007
8. Sep 22, 2007

### jostpuur

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

9. Sep 22, 2007

### meopemuk

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.

10. Sep 22, 2007

### OOO

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

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 ?

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.

11. Sep 22, 2007

### meopemuk

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.

12. Sep 22, 2007

### meopemuk

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.

13. Sep 22, 2007

### OOO

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 ?

14. Sep 22, 2007

### meopemuk

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.

15. Sep 22, 2007

### OOO

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.

16. Sep 22, 2007

### meopemuk

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.

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.

Last edited: Sep 22, 2007
17. Sep 23, 2007

### 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 im a little puzzled. Can you give an example maybe of what you have in mind?

18. Sep 23, 2007

### jostpuur

I think that this

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.

19. Sep 23, 2007

### meopemuk

I agree with jostpuur completely.

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.

20. Sep 23, 2007

### OOO

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.