The purpose of this paper is to show that the composite photon theory measures up well against the Standard Model’s elementary photon theory. This is done by comparing the two theories, area by area. Although the predictions of quantum electrodynamics are in excellent agreement with experiment (as in the anomalous magnetic moment of the electron), there are some problems, such as the difficulty in describing the electromagnetic field with the four-component vector potential because the photon has only two polarization states. In most areas the two theories give similar results, so it is impossible to rule out the composite photon theory. Pryce’s arguments in 1938 against a composite photon theory are shown to be invalid or irrelevant. Recently, it has been realized that in the composite theory the antiphoton does not interact with matter because it is formed of a neutrino and an antineutrino with the wrong helicity. This leads to experimental tests that can determine which theory is correct.
In the history of physics many particles, which were once believed to be elementary, later turned out to be composites. The idea that the photon is a composite particle dates back to 1932, when Louis de Broglie [
Jordan’s modifications made it easy for Pryce in 1938 to show that the theory was untenable. Pryce [
Neutrino oscillations in which one flavor of neutrino changes into another have been observed at the SuperKamiokande [
There has been some continuing work on the composite photon theory (see [
In the next section we will compare the elementary and composite theories, area by area. In Section 3 we re-examine Pryce’s arguments [
Intuitively, de Broglie’s idea makes reasonable the significant difference in characteristics exhibited by spin-1 photon and a spin-1/2 neutrino. When a photon is emitted, a neutrino-antineutrino pair arises from the vacuum. Later the neutrino and antineutrino annihilate when the photon is absorbed.
In the following sections we will examine the similarities and differences of the elementary and composite photon theories. Although the composite and elementary theories are similar, there are both subtle and major differences.
In noting the problem of quantizing the electromagnetic field, Bjorken and Drell [
This implies a vector potential,
For any electromagnetic field,
A satisfactory Lagrangian density is given by,
Using the standard method, we construct conjugate momenta from
We start with the neutrino field. Solving the Dirac equation for a massless particle,
where
We designate
where we have used only the two corresponding spinors, and
The fermion and antifermion are bound by this attractive local vector interaction of Equation (9) as discussed by Fermi and Yang [
with the annihilation operators for left-circularly and right-circularly polarized photons with momentum
where
Although many sets of gamma matrices satisfy the Dirac equation, one must use the Weyl representation of gamma matrices to obtain spinors appropriate for the composite photon. If a different set of gamma matrices is used, the photon field will NOT satisfy Maxwell equations. Kronig [
In classical Hamiltonian mechanics, the Poisson bracket is defined as,
where
In going over to quantum theory, it is hypothesized that the fundamental Poisson brackets become com- mutators with
The generalized coordinates and momenta for the classical electromagnetic field are,
Thus, the fundamental commutators become,
However, the third Equation of (16) is not consistent with Maxwell equations, so we must depart from the canonical path [
Expanding the
where
radiation gauge),
Also it is convenient to choose,
Inverting Equation (18) we obtain the amplitudes,
Following Bjorken and Drell [
Left-handed and right-handed circularly polarized annihilation operators are obtained from the combinations,
and they obey the commutation relations,
From this discussion it is evident that the elementary photon commutation relations were carried over from the classical canonical formalism and are not based on any fundamental principle. The photon distribution for Blackbody radiation can be calculated using the second quantization method [
Composite integral spin particles obey commutation relations [
In obtaining the commutation relations involving
and they obey the commutation relations,
One virtue of a good theory is simplicity. Although the composite photon commutations relations (26) and (28) appear more complex than the elementary commutations relations (22) and (24), they are really simpler because it is only necessary to postulate the fermion anticommutation relations and then derive boson commutation relations. A more detailed discussion is contained in Ref. [
The composite photon distribution for Blackbody radiation can be calculated using the second quantization method [
The
In the elementary theory the polarization vectors are chosen so that the electromagnetic field satisfies Maxwell equations. In composite theory there is no flexibility; the polarization vectors are given by the neutrino bis- pinors.
Polarization vectors for photons with spin parallel and antiparallel to their momentum (taken to be along the third axis) are given by,
In Section 2.2 we chose some properties of the polarization vectors in Equation (19) and (20). In four dimensions we have,
and the dot products with the internal four-momentum
Also in three dimensions,
To calculate the completeness relation, we use linear polarization vectors. Noting that the sum over polari- zation states only involves the two transverse polarizations and not the third direction
From Equation (10) we see that the polarization vectors are neutrino bispinors:
Carrying out the matrix multiplications results in,
(36)
Since the neutrino spinors and the polarization vectors only depend upon the direction of
As one can see these polarization vectors are good for any direction
and the dot products with the internal four-momentum
Also in three dimensions,
Using Equation (36) we calculate the completeness relation,
In the elementary theory, Maxwell equations are taken as an experimental result as discussed in Section 2.1.1. The vector potential,
In the composite theory, Maxwell equations are derived, as they must be if the composite theory is relevant. Substituting Equation (35) into Equation (10) gives
The electric and magnetic fields are obtained from
Using Equation (39) we obtain,
and with Equation (40) we obtain,
Using Equation (39) again, we see that
The numbers operator for an elementary photon is defined as,
When acting on a number state or Fock state, it returns the number of photons with momentum
for a state with
Acting on the one and zero particle states results in,
The number operators for right-handed and left-handed composite photons are defined as,
Perkins [
where
This result differs from that for the elementary photon because of the second term, which is small for large
where
applying
which is the same result as obtained with boson operators. The formulas in Equation (54) are similar to those in Equation (50) with correction factors that approach zero for large
The commutation relations for electric and magnetic fields in the elementary photon theory are [
and
With the composite photon theory, the commutation relations for
(59)
and
The antiphoton is identical to the photon. Thus the electromagnetic field can at most change by a factor of
the electromagnetic field must transform as,
in order to leave the product
Under the parity operator the vector potential transforms as,
This implies that the creation and annihilations operators change as,
Under the combined operation of CP,
In short-hand notation,
Under C (charge conjugation) and P (parity), the neutrino annihilation operator transform as follows:
We construct the composite antiphoton field in a manner similar to that of the composite photon field,
with the annihilation operators for left-circularly and right-circularly polarized antiphotons with momentum
Note that
where we have taken
since
Under the combined operation of CP,
In short-hand notation,
Since the internal structure of the composite photon is,
the antiphoton is,
Not only is
The photon and antiphoton are invariant only under the combined operation of charge conjugation and parity,
However, there can be photon states that are eigenstates of C and P. As is done with the neutral kaon, we create superpositions of the particle and antiparticle,
Under charge conjugation,
showing that
similar results under parity. In the composite photon theory the electromagnetic field transforms in the usual way only under the combined operation of CP.
Since the photon is its own antiparticle, all photons are identical. Thus, a state of two photons must be sym- metric under interchange. This result has been used to rule out certain reactions [
In the composite theory, four photon states exist, i.e.,
Here we examine Compton scattering, using Feynman diagrams. (The photo-electric effect is similar.)
Incoming electron:
Outgoing electron:
Propagator:
Incoming neutrino:
Incoming antineutrino:
Outgoing neutrino:
Outgoing antineutrino:
Incoming photon:
Outgoing photon:
Vertex:
The matrix element for Compton scattering as shown in
In the composite theory the matrix element for Compton scattering as shown in
The matrix element contains components,
and
Since the electron-neutrino interaction is V-A, we must insert the projection operator,
states with negative-helicity particles and positive-helicity antiparticles. With this insertion we have com- ponents,
and
Since
the insertion of
an electron, the terms contain components,
and
The
This indicates that antiphotons do NOT interact with elections in a matter world, because
In an antimatter world, the positron-neutrino interaction is V + A and
helicity particles and negative-helicity antiparticles. In a symmetric manner photons do not interact with positrons in an antimatter world [
Experiment [
Positrons interact with the electromagnetic field in a manner similar to that of electrons. Thus, the composite photon theory requires that the effect of virtual photons is the same in matter and antimatter worlds.
In comparing the elementary and composite photon theories, it is noted that in the elementary theory it is difficult to describe the electromagnetic field with the four-component vector potential. This is because the photon has only two polarization states. This problem does not exist with the composite photon theory. The commutation relations are more complex in the composite theory because of the composite photon’s internal fermion structure. However, this complexity is not unique to the composite photon; other composite particles with internal fermions have similar complexity. In the elementary theory the polarization vectors are chosen to give a transverse field, while in the composite theory they are determined by the fermion bispinors. The com- posite theory predicts Maxwell equations, while the elementary theory has been created to encompass it. Some differences are so slight that they are almost impossible to detect experimentally (i.e., Planck’s law). However, the composite theory predicts that the antiphoton is different than the photon.
Pryce [
Note that
(Jordan and Kronig were working on that assumption.) Pryce [
meant that
An important test of these ideas will occur when the photons from anti-Hydrogen are examined. The com- posite photon theory predicts that the antiphotons from anti-Hydrogen will have the wrong helicity for inter- action with electrons, and thus the antiphotons will not be detectable. Furthermore, ordinary photons have the wrong helicity for interaction with anti-hydrogen.
Helpful discussions with Prof. J. E. Kiskis are gratefully acknowledged.