Relationship between EM and matter fields in QED

[CSnowden]

TL;DR Summary

Matter and EM represented as independent, coupled fields in QED, yet EM emerges via local phase changes in the matter field to establish gauge symmetry - so is EM really an independent field?

It seems that QED treats the matter and EM fields as independent yet coupled fields. On the other hand the EM field equations emerge immediately under local change of the phase of the matter field, exactly as required to reestablish local (gauge) invariance. From that perspective it almost seems as if there is a single field (matter/EM), presenting in different ways as its phase may be locally changed - would appreciate any thoughts on the topic.

 

[PeterDonis]

the EM field equations emerge immediately under local change of the phase of the matter field, exactly as required to reestablish local (gauge) invariance

I don't think this is a correct description of gauge invariance. The EM field equations don't "emerge" under gauge invariance; you derive the field equations first, and then you find out that those field equations are gauge invariant. You can derive the EM field equations from the EM field Lagrangian alone, without any coupling to any matter fields.

 

[A. Neumaier]

Summary: Matter and EM represented as independent, coupled fields in QED, yet EM emerges via local phase changes in the matter field to establish gauge symmetry - so is EM really an independent field?

It seems that QED treats the matter and EM fields as independent yet coupled fields. On the other hand the EM field equations emerge immediately under local change of the phase of the matter field, exactly as required to reestablish local (gauge) invariance. From that perspective it almost seems as if there is a single field (matter/EM), presenting in different ways as its phase may be locally changed - would appreciate any thoughts on the topic.

If you look at the Lagrangian you can see that they are independent fields. Setting the coupling constant e to zero produces independent field equations. Only the coupling relates the two.

That one can use local phase changes to motivate gauge transformations doesn't change these facts.

 

[Heikki Tuuri]

A good question. I have been doing research on exactly this question in the past weeks.

We start from the Klein-Gordon equation for a massive particle.

Paul Dirac in 1928 realized that we can find a Lorentz covariant equation by factorizing the operator in the Klein-Gordon equation.

The factorization reveals the bispinor structure (4 components) of the electron equation. As Feynman put it, no one has found an intuitive "physical" explanation for why the factorization trick works.

We then notice that one can multiply the electron function by an arbitrary complex number whose absolute value is 1, and we get another solution for the Dirac equation. This is called the global U(1) symmetry of the Dirac lagrangian. U(1) refers to rotating the complex plane (for the values of the electron function).

The gauge theory principle claims that one can then implement an interaction of the particle with a "gauge field" by rotating the complex plane according to a function which is continuous in spacetime. We compensate the effect of the rotation on the lagrangian by adding a 4-vector gauge field A and replacing the partial derivative operators by the difference:

partial derivative - the corresponding
component of A.

The gauge field A turns out to be the electromagnetic field EM.

There is a flaw in the above derivation of the interaction lagrangian: it fails to take into account the change of the inertial mass of the electron in a static electric field.

The static field of a charge defines a preferred coordinate frame. Then we can (and must) remove the electric potential term from the covariant partial derivative with respect to t, and add the potential to the inertial mass m of the electron.

The flaw is the underlying reason for the Klein paradox:

https://en.wikipedia.org/wiki/Klein_paradox

Once we take into account the change in the inertial mass, the Klein paradox disappears.

One may ask if the EM field really is "independent" from the electron field? Does the gauge theory mechanism "derive" the EM field from the Dirac equation?

Looking at the Dirac equation, an obvious way to disturb the equation is to add perturbation terms to the partial derivatives. The EM field happens to be the simple interaction field which we obtain from this trick.

Could the EM field exist without the fermion fields? The only way to access the EM field is through charged particles. But electromagnetic waves carry energy which is detached from static electric charges. In that sense, the EM field is independent from fermion fields.

In the electroweak theory we add perturbation terms for each "degree of freedom" in U(1) × SU(2). We get a gauge field with 4 degrees of freedom, and 4 gauge bosons.

Philosophers have been studying the ontology of the Higgs mechanism and gauge fields in recent years.

 

[akhmeteli]

The following may be relevant.

Dirac (Proc. Roy. Soc., A, 209, 291 (1951)) showed that in classical theory of electromagnetic field one can choose the gauge in such a way that the resulting (modified) Maxwell equations describe interacting electromagnetic field and charged particles moving in accordance with the Lorentz equation.

In scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics) one can choose the gauge (following Schrödinger, Nature,169, 538 (1952)) in such a way that the scalar field is real. After that, the matter field can be algebraically eliminated from the equations of motion, and the resulting modified Maxwell equations describe independent evolution of electromagnetic field (Akhmeteli, Eur. Phys. J. C (2013) 73:2371 (https://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf). On the other hand, these equations describe the same physics as scalar electrodynamics, i.e., both electromagnetic field and scalar matter. One can also build a relevant Lagrangian depending on the electromagnetic field only.

Similar (but less satisfactory) results were obtained for the spinor electrodynamics (Dirac-Maxwell electrodynamics (see my article quoted above).

 

[ftr]

The following may be relevant.

...

I have some questions if you don't mind

1) What was the motivation for eliminating the matter field?
2) Was the matter field eliminated or "replaced with", also you can you describe what is mass or charge after the "elimination/replacement"?
3) Can you use your system to be applied to two interacting electrons(Dirac two particle)?
4) In your THIRD interpretation wouldn't explaining the electron by more electron/positron is a bit circular.
5) Would you agree to the that your theory hints at the old idea that all matter are of electromagnetic origin.

BTW, I like this work and I think although you and A. Neumaier have heated discussions but I think both TI and your idea are connected.

 

[akhmeteli]

Thank you very much for your interest and kind words. Let me try to answer your questions.

1) What was the motivation for eliminating the matter field?

That was not the initial motivation. Initially, I just looked for some attractive deterministic interpretation of quantum theory, but at an early stage of this work I got interested in Majorana solutions of the Dirac equation, as they looked promising for my purpose. In this context, a gauge condition $A^\mu A_\mu=0$ arose pretty naturally, so I looked for earlier work using similar gauge conditions and found the (all but forgotten) articles by Dirac and Schrödinger quoted in my post. The Dirac's work described elimination of matter in a classical theory, and the Schrödinger's work set stage for eventual elimination of the matter field from scalar electrodynamics. Then it was natural to look for similar results for spinor electrodynamics.

2) Was the matter field eliminated or "replaced with"

I am not sure I understand this question. On the one hand, the mathematical term "elimination" is certainly applicable (the matter field is eliminated from the equations), on the other hand, the resulting equations differ from those for free electromagnetic field, so one can argue that matter is still present, at least its functions are now fulfilled by electromagnetic field.

can you describe what is mass or charge after the "elimination/replacement"?

Charge and mass still enter the resulting equations for electromagnetic field, and properties of the solutions strongly depend on the specific values $m$ and $e$. I am not sure I can give a more physical answer.

3) Can you use your system to be applied to two interacting electrons(Dirac two particle)?

I do not have a clear idea, but transition to many-particle theories is discussed in my EPJC article (where I follow nightlight).

4) In your THIRD interpretation wouldn't explaining the electron by more electron/positron is a bit circular.

I don't quite see why it is circular. I consider an interpretation where a single (quantum) electron is modeled by (classical) $N+1$ electrons and $N$ positrons.

5) Would you agree to the that your theory hints at the old idea that all matter are of electromagnetic origin.

To some extent, yes, but please note that, until recently, I only considered electrodynamics, so, for example, strong interaction was not considered at all. However, I also have interesting results for Yang-Mills fields (https://arxiv.org/abs/1811.02441). So maybe all matter is of gauge field origin (not just electromagnetic origin)?:-)

BTW, I like this work and I think although you and A. Neumaier have heated discussions but I think both TI and your idea are connected.

I did have "heated discussions" with A. Neumaier, but most of them were not directly related to our respective interpretations. For example, I objected to his specific critique of the Bohm interpretation and of the Born rule. As for his thermal interpretation, unfortunately, I am not familiar with it (as far as I know, there are no peer-reviewed publications on TI so far), so I cannot praise or criticize it. He did criticize my interpretation, stating, for example, that it cannot describe Helium atom. Maybe he is right, or maybe it will be possible to do that in the future. I believe though that some of my results are valuable no matter what interpretation one prefers, for example: 1. the Dirac equation in electromagnetic or Yang-Mills field is equivalent to an equation for one real function; 2. matter field can be eliminated from scalar electrodynamics, and the resulting equations describe independent evolution of electromagnetic field.

 

[ftr]

Thanks for the detailed reply. I have to absorb what you have said and maybe I can sharpen my questions.

 

[Paul Colby]

The gauge field A turns out to be the electromagnetic field EM.

As I recall, one introduces independent gauge fields along with their required transformation under gauge transformation in order that the combine system is then gauge invariant. The gauge fields are introduced as new degrees of freedom in both the classical and as quantized theories.

 

[ftr]

I am not sure I understand this question. On the one hand, the mathematical term "elimination" is certainly applicable (the matter field is eliminated from the equations), on the other hand, the resulting equations differ from those for free electromagnetic field, so one can argue that matter is still present, at least its functions are now fulfilled by electromagnetic field.

I guess I am interested in solving your equations for free electron or under some potential and compare to the original and to see if any insight can be obtained.

 

[akhmeteli]

I guess I am interested in solving your equations for free electron or under some potential and compare to the original and to see if any insight can be obtained.

Which equations? The fourth order Dirac equation? Or equations of scalar electrodynamics? Let me note that results of my work cannot be used for free electron, but this does not seem to be a problem as charged field always creates some electromagnetic field.

 

[vanhees71]

Ehm, if a theory is not applicable to a free electron, I'd rather rethink the theory...

 

[akhmeteli]

Ehm, if a theory is not applicable to a free electron, I'd rather rethink the theory...

I can understand that this may sound ridiculous to you, but theories need to describe Nature, not a free electron. Let us consider the case of the Dirac equation in electromagnetic field. If the fourth-order equation of my work gives exactly the same results as the Dirac equation in arbitrarily weak field, do I really need to worry about a free electron? Who has ever seen a free electron? A free electron that does not even have its own Coulomb field? If you have a single charged particle in the entire Galaxy, you don't have a free electron anywhere, even if the Coulomb field of the particle is well-screened.

Why does the equation of my work look so sensitive to an arbitrarily weak field? I don't know. In my work http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (J. Math. Phys., 52, 082303 (2011)) I wrote: "While this elimination could not be performed for zero electromagnetic fields, this does not look like a serious limitation, as in reality there always exist electromagnetic fields in the presence of charged fields, although they may be very small. However, it should be noted that free spinor field presents a special case and is not considered in this work, as it does not satisfy the equations of spinor electrodynamics. It is not clear how free field being a special case is related to the divergencies in quantum electrodynamics."

 

[ftr]

Which equations? The fourth order Dirac equation? Or equations of scalar electrodynamics? Let me note that results of my work cannot be used for free electron, but this does not seem to be a problem as charged field always creates some electromagnetic field.

Let us say "The fourth order Dirac equation". I am not interested in exact solution , even a qualitative one is fine. You can choose any potential you like.

 

[Paul Colby]

Why does the equation of my work look so sensitive to an arbitrarily weak field? I don't know.

Interesting paper. I assume your work doesn't claim to eliminate charge so wouldn't one always expect some field in the neighborhood of an electron?

Also, is it possible that the $F^\mu$ are not unique? Give $\psi_1$ could one find ${F'}^\mu\ne F^\mu$ which also satisfy (21)?

 

[akhmeteli]

Let us say "The fourth order Dirac equation". I am not interested in exact solution , even a qualitative one is fine. You can choose any potential you like.

So you can choose the Coulomb potential. You will get the same results as for the exact solutions of the standard Dirac equation.

 

[akhmeteli]

Interesting paper. I assume your work doesn't claim to eliminate charge so wouldn't one always expect some field in the neighborhood of an electron?

Again, which equations are we talking about? If it is the Dirac equation, then one usually treats it as a linear equation and does not include the electron's own Coulomb field. If it is spinor electrodynamics, then yes, the equations are nonlinear, and one does take into consideration the electron's Coulomb field. Let me note that the equations I derive are generally equivalent to the standard Dirac equation and the equations of spinor electrodynamics.

Also, is it possible that the $F^\mu$ are not unique? Give $\psi_1$ could one find ${F'}^\mu\ne F^\mu$ which also satisfy (21)?

I am not sure I understand this. $F^i$ is just the electromagnetic field. For example, $F^1=E^1+iH^1$.

 

[Paul Colby]

Again, which equations are we talking about?

I think that's what the funny numbers next to the equations are. If this is indeed correct then (21) as stated.

I am not sure I understand this.

Is there more than one choice of EM field in equation (21) which will yields the very same $\psi^1$? It's completely unclear to me that these are unique since the EM is an independent degree of freedom even within a classical theory. Equation (21) may be a valid constraint but not complete.

 

[akhmeteli]

I think that's what the funny numbers next to the equations are. If this is indeed correct then (21) as stated.

I understand that, but still equation (21) can be interpreted in two different ways.

If it is treated as an isolated equation, then it is implied that the electromagnetic field is an external field, and we consider motion of the Dirac particle in the external field.

If equation (21) is considered as part of the equations of spinor electrodynamics, then we have mutually consistent electromagnetic and spinor fields.

Is there more than one choice of EM field in equation (21) which will yields the very same $\psi^1$? It's completely unclear to me that these are unique since the EM is an independent degree of freedom even within a classical theory. Equation (21) may be a valid constraint but not complete.

If electromagnetic field is an external field, your question seems moot, as one cannot change the external field. However, for the equation to make physical sense, one needs to define the current (see below).

If we have mutually consistent electromagnetic and spinor fields, then equation (21) is indeed incomplete, as you need to define the current in the right-hand side of the Maxwell equations (however, eq. (21) does not look more incomplete than the Dirac equation in this respect). To do this, one starts with $\psi_1$, restores the Dirac spinor, and builds the current. This procedure (and the definition of equivalency of the fourth-order equation and the Dirac equation) is presented in my article https://arxiv.org/abs/1502.02351 (peer-reviewed version: in Quantum Foundations, Probability and Information, A. Khrennikov, T. Bourama (Eds.) (2018), p. 1) after Eq. 50.

 

[CSnowden]

Many thanks to everyone for some very insightful replies on the topic!