- Rahul,

There is no similar theory. However, the numerical value of the elementary charge can be derived from the "number" of irreducible two-particle states (=two-particle states of an irreducible representation of the Poincaré group) in relation to the "number" of product states within a volume element of the momentum space formed by the particles' momenta. This strongly indicates that the electromagnetic interaction is "caused" by the momentum entanglement of two "charged" particles, described by an irreducible two-particle representation. (Momentum entanglement is a general property of irreducible two-particle representations. In quantum electrodynamics, the interpretation is that the entanglement is caused by the "exchange of gauge particles.")

You will find details in:

W. Smilga, "Reverse Engineering Approach to Quantum Electrodynamics," Journal of Modern Physics, Vol. 4 No. 5, 2013, pp. 561-571. doi: 10.4236/jmp.2013.45079. - Rahul,

I think that the Higgs mechanism is not yet the last word. There must be some more fundamental reason that explains the spectrum of masses of elementary particles.

As to the electric charge, many years ago P.A.M. Dirac found that there is a connection between the electron charge 'e' and the quantum of magnetic pole which he considered disappointing, because it did not explain the numerical value of the pure number e^2/(hbar* c)=1/137.0359..... (called the fine structure constant and denoted by the greek letter 'alpha'). The alpha expresses the electron charge in terms of the Planck constant 'hbar=h/2pi' and the speed of light 'c' - having known this number we know the electric charge. Dirac came to the conclusion that the correct explanation would require some entirely new idea. I think that the correct way towards explaining this number was already given in 1989. The electric charge 'lives' at the spatial infinity (equivalent to the 3-dimensional de Sitter hyperboloid /spacetime/) -- total content of the Maxwell theory that concerns the field of an isolated charge can be reduced to a single 'massless' scalar field 'living' on the de Sitter hyperboloid. A quantization of that field leads to the Quantum Theory of the Electric Charge. In the framework of this theory one can study various quantum observables. These observables depend in a nontrivial way on the numerical value of the fine structure constant. In particular, some observables may exist or not, depending on the numerical value of alpha. Therefore, we can expect that this theory is capable to sharply predict the numerical value of alpha, provided one guesses correctly the appropriate observable. There is no other theory known with such a nontrivial behaviour (for example, nothing would be spoiled in the quantum electrodynamics /QED/ when alpha (or equivalently, e) was changed -- the electric charge is a free parameter of QED and could be arbitrary) . It has been shown that the quantum theory of the electric charge predicts some critical values for the fine structure constant. Moreover, it has been established that 0 < alpha/pi < 1/4. Up to now no one has found any observable or prescription of how this number could be determined precisely within this theory.

If you want read more -- see the following articles (the first introduces the theory and is more technical)

1) main article:

A. Staruszkiewicz (1989) Ann. Phys. (New York) 190, 354

QUANTUM MECHANICS OF PHASE AND CHARGE AND QUANTIZATION

OF THE COULOMB FIELD

http://www.sciencedirect.com/science/article/pii/0003491689900183

2) an argument against the existence of magnetic poles:

A. Staruszkiewicz (1997) Banach Center Publications, Vol. 41, 221

ON THE QUANTAL NATURE OF THE COULOMB FIELD

http://matwbn.icm.edu.pl/ksiazki/bcp/bcp41/bcp41221.pdf

3) on the paradox of quantization of the amplitude of the (classical) Coulomb field:

A. Staruszkiewicz (1998) QUANTUM MECHANICS OF THE ELECTRIC

CHARGE Acta Phys.Polon. B29, 4, 929

http://www.actaphys.uj.edu.pl/vol29/pdf/v29p0929.pdf

4) on the paradox between gauge invariance and stationarity of the Coulomb field

A. Staruszkiewicz (2002) Acta Phys.Polon. B33 2041

PHYSICS OF THE ELECTRIC CHARGE

http://www.actaphys.uj.edu.pl/vol33/abs/v33p2041,

Lukasz - Lukasz,

The sentence in your post

"There is no other theory known with such a nontrivial behavior (for example, nothing would be spoiled in the quantum electrodynamics /QED/ when alpha (or equivalently, e) was changed -- the electric charge is a free parameter of QED and could be arbitrary)"

may lead to misunderstandings.

In fact, there is a theory that predicts the numerical value of alpha: This theory is QED in connection with a basic rule of quantum mechanics.

This rule says: "Isolated quantum mechanical systems are described in space-time by irreducible representations of the Poincaré group."

If we respect this rule in setting up the perturbation algorithm of QED, then the coupling constant can be calculated. The result is a theoretical fine-structure constant that matches the empirical one.

Of course, the electric charge that enters QED as an empirical factor is by no means arbitrary. (I suppose you do not really mean that.) Feynman's rules do not say: put in an arbitrary number. His rules rather say: put in the empirical value of the fine-structure constant. Feynman did not realize that by this prescription he made his rules consistent with the basic rule just mentioned. - This is a very good question. In the first place it is a mystery why electrons and protons have the same numerical value of their charge, being, according to the standard model, in unrelated particle families.This is actually a strong argument for a "subquark" model.

BTW: It is difficult to understand how e can be derived from a space-time related symmetry. Walter, can you please provide a reference? - David,

Please find the reference in my answer to Rahul on top of this thread

or in http://arxiv.org/pdf/1004.0820.pdf - Walter,

I am not sure I understand your paper in detail, please correct me, if I did not after just browsing through it. I understand that you (and/or Wyler) suggest the following: The "entangled pair" is created independently of the coupling of the fermion field to the gauge field - it is so to speak some kinematics seazoned with the Pauli exclusion principle. The normalization of the corresponding S matrix element miraculously reproduces the value of the fine structure constant, and is only interpreted as a charge in (Feynman) perturbation theory.

Now my question: Provided, you are right and I understood you properly: Why do quarks have a different charge then? They also obey the Dirac equation and should be able to build entangled two particle states. And how about neutrinos? - David,

In my paper I (not Wyler) suggest that there is no coupling to a gauge field at all. Instead, I claim that the entangled two-particle states within the S matrix elements "look" as if they were generated by the exchange of gauge quanta. (Entanglement is a natural property of irreducible two-particle representations.) The essential point is that the interpretation by gauge quanta does not fix the coupling constant, whereas states of an irreducible two-particle representation require a normalization that is different from the normalization of the incoming and outgoing product states. I have calculated this factor. It matches the empirical value of alpha.

Now to your questions: how do you know that quarks are able to build the same entangled two particle states as electrons? They are certainly not described by the same representation as electrons, because they show a different kinematical behavior: There are no "free" quarks.

Neutrinos are massless. Their representation is again different from the electrons.

My paper is about the mathematical structure of quantum electrodynamics, not about QCD and not about weak interaction. So I have to ask for some patience. I will try to answer also other questions of particle physics, but please understand that I am not able to answer them all in one paper. - Walter,

I do not think my reply was confusing.

In QED the electric charge is indeed a free parameter. It is present as a parameter in all QED amplitudes (it enters them through the fine structure constant alpha) , within QED (when corrections from weak or hadronic interactions are neglected) we can calculate the Lamb shift, the anomalous magnetic moment of the electron, the probability that an electron will absorb a photon, etc, they all depend on the numerical value of the elementary charge, but we do not know why the experimental value of e^2 is approx. 1/137 and no other. There is nothing in QED that would predict this number.

In all QED calculations we substitute the experimental value approx. 1/137 (which we must do unless we calculate it precisely). True, Feynman's rules say that we should put in the empirical value of the fine-structure constant. What I said in my previous reply was that QED does not say why this experimental value is approx. 1/137 and no other.

If You state that QED predicts this number, give a reference to the literature - surely you will find no single reference. The electric charge can be determined experimentally, for example, with the help of the Schwinger formula (a QED calculation) that expresses the anomalous magnetic moment of the electron in terms of a series expansion in the fine structure constant. Having measured the anomalous moment of the electron (eg. in the Penning trap) one can derive the value of the fine structure constant by equating the Schwinger formula with the experimental value of the electron moment, and then solving the power series for alpha. However, there is no formula for the fine structure constant alone!

I think that the most convincing argument that the numerical value of the pure number alpha cannot be derived from QED, are Feynman's words from his book "The Strange Theory of Light and Matter". Feynman undoubtedly perfectly understood QED and knew what he was saying about alpha in the following words:

///// "There is a most profound and beautiful question associated with the observed coupling constant, e, the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!" ///// - Lukasz,

I thought the word "arbitrary" might confuse some readers of this blog. Of course, I knew what you intended to say. Thank you for the clarification.

When you carefully read my post, you will not find any statement like "QED predicts this number". My statement is rather (here in more detail): A well-known rule of quantum mechanics says that isolated systems are described by irreducible representations of the Poincaré group. If you respect this rule when setting up the perturbation algorithm of QED, then a normalization factor (normalizing the two-particle states in the algorithm) is required. This factor enters the S matrix formalism in the same position as the empirical value of alpha. The calculation shows that the numerical value of this factor matches the empirical value of alpha. From this matching I conclude that a) the rule mentioned above is in fact taken care of in QED (without being aware of it) by inserting the empirical value of alpha, b) alpha has to be understood as the normalization factor, required for mathematical consistency with this rule.

You will find details in my paper under the references given in my previous posts. (I must admit that I am the first who identified alpha with this normalization factor (as far as I know), therefore, you will not find anything about this in standard text books.)

Feynman is one of my personal heroes, but that does not mean that I understand his well-known words as an invitation to stop thinking about QED and leave alpha a mystery until all eternity. I am sure this would not be in his spirit. - Walter,

Surely, this would not be in Feynman's spirit as he said: "all good theoretical physicists put this number up on their wall and worry about it." saying this he encouraged everyone to think about how this mysterious number could be constructed.

Thank you for drawing your paper to my attention, I will read it with interest -- I'm quite intrigued by the ratio of the three volumes you gave in your paper which is close to the experimental value of alpha (however, I have the impression [ having NOT read your paper YET ] that there must be some misleading coincidence, not a real solution, but who knows?). - Lukasz,

Thank you for your readiness to read my paper. I hope I can convince you and, if not, I will be interested in hearing your arguments.

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