In noting the problem of quantizing the electromagnetic field, Bjorken and Drell [17] declared, “It is ironic that of the fields we shall consider it is the most difficult to quantize.” Srednicki [18] commented, “Since real spin-1 particles transform in the representation of the Lorentz group, they are more naturally described as bispinors than as 4-vectors.” Varlamov [19] also noted that, “the electromagnetic four-potential is transformed within representation of the homogeneous Lorentz group...” Usually a canonical procedure for quantization is used although it is not manifestly covariant. We can describe the electromagnetic field with the four-component vector potential, but the photon only has two polarization states. One method of handling the problem is to introduce two non-physical photons along with the real ones, the Gupta-Bleuler procedure [20] . Another method is to give the photon a very, very small mass [21] . Following Bjorken and Drell [17] we will take only the transverse components and “abandon manifest covariance.” We start with Maxwell equations (in the absence of source charges and currents),

This implies a vector potential, , that satisfies,

For any electromagnetic field, and, there are many’s that differ by a gauge transformation.

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, , with , results in the spinors,

where, and the superscripts and subscripts on refer to the energy and helicity states re- spectively. The gamma matrices in the Weyl basis were used in solving the Dirac equation:

We designate as the annihilation operator for, the right-handed neutrino, and as the annihilation operator for, the left-handed antineutrino. We assign as the annihilation operator for, the left- handed neutrino, and as the annihilation operator for, the right-handed antineutrino. Since only and have been observed, we take the neutrino field to be,

where we have used only the two corresponding spinors, and stands for. A four-vector field can be created from a fermion-antifermion pair,

The fermion and antifermion are bound by this attractive local vector interaction of Equation (9) as discussed by Fermi and Yang [22] . We postulate that this local interaction between the neutrino and antineutrino is re- sponsible for their interaction with the electromagnetic coupling constant “” while a single neutrino interacts with the weak coupling constant “”. Both Kronig [23] and de Broglie [1] suggested local interactions in their work on the composite photon theory. Since the neutrino and antineutrino momenta are in opposite directions, we take the photon field to be [12] ,

with the annihilation operators for left-circularly and right-circularly polarized photons with momentum given by,

where is a spectral function.

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 [23] was the first to realize this, but he did not mention the deeper significance, i.e., two-component neutrinos are required for a composite photon. At that time a two-component neutrino theory would have been rejected because it violated parity. The connection between the photon antisymmetric tensor and the two-component Weyl equation was also noted by Sen [24] . Although we are working at the four-component level, one can form a composite photon at the two-component level [12] .

In classical Hamiltonian mechanics, the Poisson bracket is defined as,

where are the generalized coordinate and are the generalized momenta. If we use and in place and, we obtain the fundamental Poisson brackets,

In going over to quantum theory, it is hypothesized that the fundamental Poisson brackets become com- mutators with, and becoming operators,

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 [17] and replace it with,

Expanding the and into plane waves,

where, and and are identified as annihilation and creation operators for polarization

. We take the two unit polarization vectors to be perpendicular to in order to satisfy (i.e.,

radiation gauge),

Also it is convenient to choose,

Inverting Equation (18) we obtain the amplitudes, and,

Following Bjorken and Drell [17] , we use Equations (16) and (17) to obtain commutation relations for the annihilation and creation operators,

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 [25] , including commutation relations of Equation (22), resulting in Planck’s law,

Composite integral spin particles obey commutation relations [26] - [28] that are derived from the fermion anti- commutation relations of their constituents. For composite photons we have,

In obtaining the commutation relations involving and, we have taken the expectation values. Here the linearly-polarized photon annihilation operators are defined as,

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. [7] .

The composite photon distribution for Blackbody radiation can be calculated using the second quantization method [25] as above, but with the composite photon commutation relations. This results [7] in,

The component is less than 10⁻⁹, so the difference between Equation (25) and (29) is too small to measure.

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, we can set.

As one can see these polarization vectors are good for any direction, while the elementary polarization vectors, Equation (30), are only given along the third axis. These polarization vectors satisfy the normalization relation,

and the dot products with the internal four-momentum give,

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, , is then created to satisfy Maxwell equations.

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 in terms of the polarization vectors,

The electric and magnetic fields are obtained from and as usual,

Using Equation (39) we obtain,

and with Equation (40) we obtain,

Using Equation (39) again, we see that satisfies the Lorentz condition,

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 and polarization.

for a state with photons. Normalizing in the usual manner [25] ,

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 [7] showed that the effect of the composite photon’s number operator acting on a state of right- handed composite photons is,

where is a constant equal to the number of states used to construct the wave packet, and.

This result differs from that for the elementary photon because of the second term, which is small for large. Normalizing,

where is the state of right-handed composite photons having momentum which is created by

applying on the vacuum times. Note that,

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 [29] ,

and

With the composite photon theory, the commutation relations for and are similar to the ones for the elementary photon theory. However, the extra terms in composite commutation relations (26) result in extra terms for the and commutation relations [7] . With the extra terms the commutation relations do not vanish for space-like intervals, indicating that composite particles have a finite extent in space [7] .

(59)

and

The antiphoton is identical to the photon. Thus the electromagnetic field can at most change by a factor of under charge conjugation. Since the electromagnetic current, , changes sign under the operation of charge conjugation,

the electromagnetic field must transform as,

in order to leave the product in the Lagrangian invariant. For in the plane-wave re- presentation, Equation (18), this means,

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 given by,

Note that contains the other two spinors from Equation (5). Appying the charge conjugation and parity operators on the composite photon annihilation operators gives,

where we have taken to be symmetric in. Applying the charge conjugation and parity operators on the composite photon field gives,

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 different than, but its neutrinos types have never been observed. Under C and P,

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 is an eigenstate of C with value, while is an eigenstate of C with value with

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 [30] [31] .

In the composite theory, four photon states exist, i.e., , , , and. If the photons are not identical, a state of two photons can be antisymmetric (as well as symmetric) under interchange. Therefore, a vector particle can decay into two photons [13] .

The matrix element for Compton scattering as shown in Figure 1(a) is,

In the composite theory the matrix element for Compton scattering as shown in Figure 1(b) is,

The matrix element contains components,

and

Since the electron-neutrino interaction is V-A, we must insert the projection operator, to select

states with negative-helicity particles and positive-helicity antiparticles. With this insertion we have com- ponents,

and

Since designates a positive-helicity antiparticle and designates a negative-helicity particle

the insertion of does not change the result [13] . However, for the interaction of an antiphoton with

an electron, the terms contain components,

and

The and terms equate to zero as,

This indicates that antiphotons do NOT interact with elections in a matter world, because and have the wrong helicity.

In an antimatter world, the positron-neutrino interaction is V + A and selects states with positive-

helicity particles and negative-helicity antiparticles. In a symmetric manner photons do not interact with positrons in an antimatter world [13] .

Experiment [33] shows that all the photons in positronium are detected. Therefore, the photons involved must be and, the superposition of and.

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.