Relativistic quantum mechanics and causality

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

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.
 
GDogg said:
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.
 
GDogg said:
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.
 
OOO said:
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.
 
meopemuk said:
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
 
OOO said:
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.

OOO said:
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
 
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Hans de Vries said:
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.
 
OOO said:
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.
 
  • #10
meopemuk said:
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.

meopemuk said:
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 ?

meopemuk said:
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.
 
  • #11
OOO said:
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.
 
  • #12
Hans de Vries said:
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.
 
  • #13
meopemuk said:
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 ?
 
  • #14
OOO said:
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.
 
  • #15
meopemuk said:
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.
 
  • #16
OOO said:
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.


OOO said:
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.
 
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  • #17
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?
 
  • #18
haelfix said:
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

meopemuk said:
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.
 
  • #19
jostpuur said:
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.

Haelfix said:
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.
 
  • #20
meopemuk said:
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.
 
  • #21
meopemuk said:
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.
 
  • #22
meopemuk said:
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
 
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  • #23
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?
 
  • #24
OOO said:
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.

OOO said:
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.

OOO said:
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.
 
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  • #25
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.
 
  • #26
Hans de Vries said:
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.
 
  • #27
GDogg said:
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
 
  • #28
GDogg said:
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.
 
  • #29
Haelfix said:
'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.
 
  • #30
meopemuk said:
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.
 
  • #31
genneth said:
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.

Actually, there is a promising line of research to build a quantum relativistic theory of particles that interact via instantaneous potentials. It is called the "dressed particle" theory, which I mentioned a few times already. It's predictive power is just as good or even better than in the traditional renormalized QFT. The "dressed particle" approach does not use Dirac or Klein-Gordon equations.

Eugene.
 
  • #32
meopemuk said:
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.

I have thought about this some time and I have to admit that it was a bit oversimplified from the point of view of QFT. I understand that it's the actual wave functional that determines the probability for a transition to a localized gauge field (absorption by silver atom). But on the other hand, classically, there can be no doubt that the energy density of the electromagnetic wave determines the number of silver ions to be reduced. That's what all photography is about.

Shouldn't the fact that the energy density of the electromagnetic wave determines the density of silver grains in the plate, correspond to QFT's wave functional view ?
 
  • #33
OOO said:
I have thought about this some time and I have to admit that it was a bit oversimplified from the point of view of QFT. I understand that it's the actual wave functional that determines the probability for a transition to a localized gauge field (absorption by silver atom). But on the other hand, classically, there can be no doubt that the energy density of the electromagnetic wave determines the number of silver ions to be reduced. That's what all photography is about.

Shouldn't the fact that the energy density of the electromagnetic wave determines the density of silver grains in the plate, correspond to QFT's wave functional view ?

The traditional way of handling properties of light seems totally inconsistent to me.

Everything is simple when we are dealing with one photon. We know that it is described by a complex-valued wave function, whose square exactly describes the probability of the photon hitting any given point on the photographic plate. For high intensity sources emitting many photons at once, individual particles do not interact with each other. So, each photon is described by the same wave function. And this is what we see in experiment: the interference picture does not depend on whether the photons are released one-by-one or in a high-intensity laser beam. So, it seems logical that in the high-intensity case the interference picture should be described by the same one-particle wave function that was used for a single photon.

However, this is not the way the modern description works. We are taught that the light wave is a classical field (actually, two fields \mathbf{E}(\mathbf{r},t) and \mathbf{B}(\mathbf{r},t)), and that the interference picture is defined by the "field energy". Moreover, we are told that in the quantum case the fields must be "quantized", i.e., changed to operators. So, basically, we have two conflicting explanations of the same effect (interference). One is usual quantum addition of complex amplitudes. Another is energy of the (quantum?) field. A consistent theory cannot have two descriptions of the same thing. Which one is the correct description? I prefer the one-particle quantum picture. I don't think that electromagnetic field (either classical or quantum) is the correct way to think about photons and their wave properties.

Eugene.
 
  • #34
meopemuk said:
The traditional way of handling properties of light seems totally inconsistent to me.

Everything is simple when we are dealing with one photon. We know that it is described by a complex-valued wave function, whose square exactly describes the probability of the photon hitting any given point on the photographic plate. For high intensity sources emitting many photons at once, individual particles do not interact with each other. So, each photon is described by the same wave function. And this is what we see in experiment: the interference picture does not depend on whether the photons are released one-by-one or in a high-intensity laser beam. So, it seems logical that in the high-intensity case the interference picture should be described by the same one-particle wave function that was used for a single photon.

However, this is not the way the modern description works. We are taught that the light wave is a classical field (actually, two fields \mathbf{E}(\mathbf{r},t) and \mathbf{B}(\mathbf{r},t)), and that the interference picture is defined by the "field energy". Moreover, we are told that in the quantum case the fields must be "quantized", i.e., changed to operators. So, basically, we have two conflicting explanations of the same effect (interference). One is usual quantum addition of complex amplitudes. Another is energy of the (quantum?) field. A consistent theory cannot have two descriptions of the same thing. Which one is the correct description? I prefer the one-particle quantum picture. I don't think that electromagnetic field (either classical or quantum) is the correct way to think about photons and their wave properties.

Eugene.

But there are many classical phenomenon which would be difficult to account for with just a wavefunction for a photon. In particular, static fields generated by, say, a single electron. If you use an action at a distance model, then that's clearly violated by experimental evidence for the delay in feeling the motion of an electron by another electron. These are the considerations that led to field models classically. Moreover, these are extremely observed phenomenon. A quantum theory that violates them even slightly would be falsified. Personally, I'm satisfied by the need to quantise the classical fields, and for them to yield particles of spin 1 which interact with electrons in the correct way, as to account for particle accelerator experiments. However, I'm not yet convinced about the precise formalism used by QFT (and its application in QED and QCD). Like you, I'm dubious about the fact that a lack of sensible time evolution can be given. Furthermore, I'm disturbed by the background dependence. I guess I'm just a bit more conservative in my views? :wink:
 
  • #35
genneth said:
But there are many classical phenomenon which would be difficult to account for with just a wavefunction for a photon. In particular, static fields generated by, say, a single electron. If you use an action at a distance model, then that's clearly violated by experimental evidence for the delay in feeling the motion of an electron by another electron. These are the considerations that led to field models classically. Moreover, these are extremely observed phenomenon. A quantum theory that violates them even slightly would be falsified.

Yes, in classical electrodynamics the description of light and (retarded) electromagnetic interactions between charges are provided by the same set of fields \mathbf{E}, \mathbf{B}. In the "dressed particle" approach these two phenomena are described differently. Light is a flux of (quantum) particles - photons. Electric and magnetic interactions between charges are described by instantaneous distance- and velocity-dependent potentials. I am not sure if such an action-at-a-distance has been clearly rejected in experiments. There are some recent experiments (Chiao, Nimtz, etc.) which can be interpreted as evidence of faster-than-light interactions between charges. I don't want to claim that interpretation of these measurements is unambiguos. In my opinion, this issue is not closed yet.

Eugene.
 
  • #36
"I don't find anything "trivial" or "clear" in these statements."

Weinberg in his book links the original paper by Bohr and Rosenfeld
"translation by R Cohen and J Stachel" phys review. 78, 794 (1950).

I like reading papers from that time as the connection with experiment was much closer and transparent, so I'd recommend checking it out.

Absent a blackboard its hard to flush out what I mean mathematically in this setting, and its getting to the point where I can no longer english speak to explain what I mean, (which is fine).
 
  • #37
Haelfix said:
Weinberg in his book links the original paper by Bohr and Rosenfeld
"translation by R Cohen and J Stachel" phys review. 78, 794 (1950).

Haelfix,
Thank you for the reference. I'll check it out.

Eugene
 
  • #38
meopemuk said:
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.


..........

Then, what does?
Regards,
Reilly Atkinson
 
  • #39
reilly said:
meopemuk said:
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.
..........

Then, what does?
Regards,
Reilly Atkinson

Below I will briefly describe how I understand quantum field theory and its relationship to QM. I understand that my views do not look like the mainstream, but I believe they are in full accord with the way QFT presented in S. Weinberg, "The quantum theory of fields", vol. 1. So, here it goes...


Why quantum mechanics is no good enough? In quantum mechanics we normally deal with systems having fixed number of particles (N). The Hilbert space is built as a tensor product of N 1-particle Hilbert spaces where operators of observables, state vectors, dynamics, etc are defined. In relativistic physics energy can be converted to mass due to Einstein's E=mc^2, so we must also take into account the possibility of changing the number of particles (emission, absorption, annihilation, etc.). This can be achieved by switching from fixed-number-of-particles Hilbert spaces to the Fock space, which is simply a direct sum of N-particle Hilbert spaces, where N varies from 0 to infinity. Fundamentally, QFT is nothing else but quantum mechanics in the Fock space.

Operators of observables and wave functions in the Fock space One can define operators of particle observables in the Fock space in exactly the same way as in QM. For example, in each N-particle sector we have well-defined operators (of position, momentum, spin, etc.) for each of the N particles described there. These operators have usual common bases of eigenvectors, and an arbitrary N-particle state vector can be projected on these bases to obtain N-particle wave functions in different representations. The new feature is that one also has state vectors with non-zero projections on sectors with different N. Wave functions of states with such undefined particle numbers are just sets of N-particle wave functions, each with its own coefficient. The square of the coefficient is the probability to find N particles in this general state.

Unitary representation of the Poincare group. The most general way to construct a relativistic quantum theory is by defining an unitary representation of the Poincare group in the Hilbert space (in our case this is the Fock space) of the system. Basically, it is sufficient to construct interaction operators V and \mathbf{W} in the generators of time translations (the Hamiltonian) and boost

H = H_0 + V
\mathbf{K} = \mathbf{K}_0 + \mathbf{W}

so that the Poincare commutation relations between all 10 generators (including the total momentum \mathbf{P}_0 and the total angular momentum \mathbf{J}_0) remain preserved. This is a very non-trivial problem, and currently there is only one known solution which satisfies all requirements, such as the cluster separability, the possibility to describe particle-number-changing interactions, etc. [Please note that the fact that it is the only known solution does not imply that it is the only possible solution.] This is where quantum fields come into play:

1. Define particle creation and annihilation operators in the Fock space.
2. For each particle type build a certain linear combination (the free quantum field) \psi(\mathbf{r},t) of the creation and annihilation operators with two major properties
2a. (anti)commutativity at space-like separations
2b. manifestly covariant transformation laws with respect to the non-interacting representation of the Poincare group in the Fock space
3. Then operator V(t) can be constructed as an integral on \mathbf{r} of field products (or polynomials)

V(t) = \int d^3r \psi(\mathbf{r},t) \phi(\mathbf{r},t) A(\mathbf{r},t) \ldots

(for brevity I omit possible indices of operators \psi, \phi, A and summations over these indices). The "boost interaction" \mathbf{W}(t) is given by a similar formula.

It is important to note that we don't need to give any physical interpretation to quantum fields \psi, \phi, A, \ldots. They are just abstract mathematical quantities, whose role is to facilitate the construction of interaction operators V and \mathbf{W}. Once this construction is completed, we have a full interacting quantum theory in the Hilbert space with particle operators, wave functions, the time evolution operator

U(t) = \exp(\frac{i}{\hbar} Ht) [/itex]......(1)<br /> <br /> etc, i.e., everything one would need to solve any kind of physical problem.<br /> <br /> <b>Renormalization and dressing</b> Unfortunately, this nice theory has a serious problem. It appears that for all realistic interaction operators V the scattering (S-) matrix cannot be calculated, because its matrix elements come out infinite. To solve this problem one can add (as Tomonaga, Schwinger, and Feynman did) infinite renormalization counterterms to the Hamiltonian H (and to the boost operator \mathbf{K} as well). Then one can get a very accurate S-matrix, but the Hamiltonian H becomes infinite and useless for calculating the time evolution (1) and for finding bound states via diagonalization. This problem can be solved by applying an &quot;unitary dressing transformation&quot; to the Hamiltonian and all other generators of the Poincare group. Then the theory assumes the form very similar to ordinary quantum mechanics: We have a full description for the system of particles interacting with each other via instantaneous potentials, which, in addition, can change the number of particles in the system. The Poincare commutators, cluster separability and other important physical requirements are exactly satisfied. Quantum fields are not needed for the physical interpretation of this theory..<br /> <br /> Eugene.
 
  • #40
meopemuk said:
Actually, there is a promising line of research to build a quantum relativistic theory of particles that interact via instantaneous potentials. It is called the "dressed particle" theory, which I mentioned a few times already. It's predictive power is just as good or even better than in the traditional renormalized QFT. The "dressed particle" approach does not use Dirac or Klein-Gordon equations.

Eugene.

Whilst that may be for a QFT (or equivalents there of), I was thinking more elementarily, of a theory for just a simple free relativistic particle, without any complications of pair creation and the such.
 
  • #41
I guess what meopemuk wants is a unambiguous Alice-Bob thought experiment, hopefully in which we can prove if [A,B]=0 no superluminal signal can be sent, and if [A, B]!=0, a superluminal can be sent.
And we can indeed accomplish this "proof" in QM, for example, in this link:http://everything2.com/title/Quantum+entanglement+and+faster+than+light+communication
under the "general case" the author proved Cooper can never know whether Alice has done the measurement or not, with a premise that Alice and Cooper has a simultaneous set of eigenbasis (namely [A,B]=0). Vice versa, if we [A,B]!=0, we can construct such a state so that Cooper can know whether Alice has done the measurement .
 
  • #42
meopemuk said:
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.


..........

Then, what does?
Regards,
Reilly Atkinson

Meopemuk -- what do you mean by a photon wave function? When you say it is not related to A_{\mu}(x) do you refer to a representation based on EM field strength, such as the Riemann-Silberstein representation (as described in [1-3])? As far as I can see, this representation does not admit a unitary representation of the Poincare group and therefore can not be interpreted as a probability density, so it doesn't get much traction in the community. Nevertheless I like the idea.

[1-3]
Bialynicki-Birula, I. Exponential Localization of Photons, Phys. Rev. Lett., 1998, 80, 5247-
Smith, B. J. & Raymer, M. G. Two-photon wave mechanics, Phys. Rev. A, 2006, 74, 062104
Hawton, M. Photon wave functions in a localized coordinate space basis, Phys. Rev. A, 1999, 59, 3223-
 
  • #43
meopemuk said:
The traditional way of handling properties of light seems totally inconsistent to me.

Everything is simple when we are dealing with one photon. We know that it is described by a complex-valued wave function, whose square exactly describes the probability of the photon hitting any given point on the photographic plate.

Who is the ''we'' that knows that?

There is no such wave function. The photon (i.e., the massless spin 1 representation of the Poincare group) does not admit a canonical position representation in which |psi(x)|^2 could be interpreted as a probability density. Attempted constructions of such a position representation depend on additional ingredients that change under gauge transforms. But a position probability would have to be gauge independent.
 
  • #44
A. Neumaier said:
Who is the ''we'' that knows that?

There is no such wave function. The photon (i.e., the massless spin 1 representation of the Poincare group) does not admit a canonical position representation in which |psi(x)|^2 could be interpreted as a probability density. Attempted constructions of such a position representation depend on additional ingredients that change under gauge transforms. But a position probability would have to be gauge independent.

Thank you for reminding me about that. You are right, it is not possible to define a position operator with commuting components (X,Y,Z) in the 1-photon Hilbert space. So, strictly speaking, there are no position-space wave functions associated with photons.

However, I still believe that ordinary 1-particle quantum mechanics should be applicable to photon states. In this quantum description there should exist some approximate (or non-commutative) representation of the photon's position. It must be so, because experimentally we can measure photon's position, e.g., as dark spots of the photographic plate. This quantum theory should also provide an explanation for the Young's double-slit experiment.

The point of my comment was that there exists another (non-quantum) theory that pretends to provide a different explanation for the wave properties of light. This is Maxwell's electrodynamics, in which the light is described as an electromagnetic wave. In my opinion, it is totally intolerable when the same physical effect (e.g., the light interference) has two completely different explanations. Only one explanation must survive. I think that the quantum-mechanical explanation is the correct one.

Happy New Year!
Eugene.
 
  • #45
meopemuk said:
I still believe that ordinary 1-particle quantum mechanics should be applicable to photon states. In this quantum description there should exist some approximate (or non-commutative) representation of the photon's position. It must be so, because experimentally we can measure photon's position, e.g., as dark spots of the photographic plate.

The situation is actually quite complex. See
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf

meopemuk said:
Maxwell's electrodynamics, in which the light is described as an electromagnetic wave. In my opinion, it is totally intolerable when the same physical effect (e.g., the light interference) has two completely different explanations. Only one explanation must survive. I think that the quantum-mechanical explanation is the correct one.

The two views are closely related, and are part of the common picture called quantum electrodynamics (QED). It has both observables for the electromagnetic field and for photon number. Photons appear as the limit of geometric optics, while the Maxwell equations appear in the mean field limit.

Happy New Year!
 
  • #46
OOO said:
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


or
Parameter Independence and Outcome Independence.
 
  • #47
An interesting physical way to understand how [O_1(t), O_2(t&#039;) ] relates to causality is through the linear response (Kubo) formalism. Suppose the system begins in the vacuum state, and you perturb the Hamiltonian with an external source V(t) = J(t) O_1. This could, for instance, be a background EM field, or an injection of particles. We wish to measure the observable O_2(t&#039;) at a later point in time, to lowest order in J. Standard time dependent perturbation theory, carried out in the interaction or Heisenberg picture, shows that the change in the observable O_2(t&#039;) depends on J(t) through the stated commutator ( ie, the retarded Green's function). If it vanishes, jostling O_1(t) leaves future measurements of O_2(t&#039;) unaffected.
 
  • #48
meopemuk said:
I still believe that ordinary 1-particle quantum mechanics should be applicable to photon states. In this quantum description there should exist some approximate (or non-commutative) representation of the photon's position.

Ordinary 1-particle quantum mechanics is applicable to photon states as long as you consistently work in the momentum representation, and restrict your states to be transversal to 4-momentum.

There are non-commutative position operators in the literature, but the formulas work both for photons and electrons, and in the latter case do not reduce to the Newton-Wigner position operator with commuting components that is well-known to correspond to the position representation of electrons. Thus the non-commutative position operators do not seem to have physical relevance.

meopemuk said:
It must be so, because experimentally we can measure photon's position, e.g., as dark spots of the photographic plate.

This only gives a very approximate position, consistent with the nonexistence of a position operator (that would produce sharp positions).

Note that most real measurement are not related to observables in the textbook sense but to so-called positive-operator-valued-measures (POVMs). The measurable photon position is of the latter kind.
 
  • #49
If a photon wave function is spread out over a very large area then how can it collapse over that large area instantly. Remember that it is observed at one location only. So how would an identical observation at a far distant location on the same wave function area 'know' that it had already been observed.

maybe we have to break -Bohm fashion- weak causality and admit that the photon knew its destination at its outset. The instant transmission of 'knowledge' is allowed in this context (but not information).
 
  • #50
wawenspop said:
If a photon wave function is spread out over a very large area then how can it collapse over that large area instantly. Remember that it is observed at one location only. So how would an identical observation at a far distant location on the same wave function area 'know' that it had already been observed.

You forget that measurements take time and are only approximately described by the highly idealized, instantaneous and complete measurements that figure in typical discussions of the foundations of quantum mechanics. The collapse is an idealization, too.
 
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