user:Waperkins

The idea that the photon is a composite particle composed of a neutrino-antineutrino pair was first suggested by de Broglie.

{{cite journal

|author=L. de Broglie

|year=1932

|title=

|journal=Compt. Rend.

|volume=195 |pages=536, 862

}}

{{cite book

|title=Une novelle conception de la lumiere

|author=L. de Broglie

|location=Paris (France)

|publisher=Hermann et. Cie.

|year=1934

}}

Historically, particles that were once thought to be elementary such as protons, neutrons, pions, and kaons have turned out to have to be composites. The idea that emission and absorption of a photon is the creation and annihilation of a particle-antiparticle pair is attractive. However, the neutrino theory of light has some serious problems, and according to the standard model the photon is an elementary particle and a gauge boson. Although there are some claims

{{cite arXiv

|author=V. V. Varlamov

|year=2001

|title=About Algebraic Foundation of Majorana-Oppenheimer Quantum Electrodynamics and de Brogie-Jordan Neutrino Theory of Light

|arxiv=math-ph/0109024

}}

{{cite journal

|author=W. A. Perkins

|year=2002

|title=Quasibosons

|journal=International Journal of Theoretical Physics

|volume=41 |pages=823-838

}}

of problems with the current photon model, the problems with the composite photon model are much worse.

Furthermore, there is no experiment evidence that the photon has a composite structure. The ideas and methods used in trying to form a composite photon are of historic importance and may be useful in forming some other composite particles. Some of the problems for the neutrino theory of light are the non-existence for massless neutrinos

{{cite journal

|author=Y. Fukuda et al. (Super-Kamilkande Collaboration)

|year=1998

|title=Evidence for oscillation of atmospheric neutrinos

|journal=Physical Review Letters

|volume=81 |pages=1562-1567

}}

with both spin parallel and antiparallel to their momentum and the fact that composite photons are not bosons. Attempts to solve some of these problems will be discussed, but the lack of massless neutrinos makes it impossible to form a massless photon with this theory.

De Broglie continued to work on the composite photon theory until 1950

{{cite journal

|author=L. de Broglie and M. A. Tonella

|year=1950

|title=

|journal=C. R. Acad. Sci. (Paris)

|volume=230 |pages=1329

}}

and there continues to be some interest

in recent years.

{{cite journal

|author=V. V. Dvoeglazov

|year=1999

|title=Speculations on the neutrino theory of light

|journal=Annales Fond. Broglie

|volume=24 |pages=111-127

}}

{{cite journal

|author=V. V. Dvoeglazov

|year=2001

|title=Again on the possible compositeness of the photon

|journal=Phys. Scripta

|volume=64 |pages=119-127

}}

{{cite book

|author=W. A. Perkins

|year=World Scientific, Singapore

|title=Interpreted History of Neutrino Theory of Light and Its Future

|pages=115-126

|title book=Lorentz Group, CPT and Neutrinos

|editors= A. E. Chubykalo, V. V. Dvoeglazov, D. J. Ernst, V. G. Kadyshevsky,

and Y. S. Kim

}}

De Broglie did not address

the problem of statistics for the composite photon. However,

"Jordan considered the essential part of the problem

was to construct Bose-Einstein amplitudes from Fermi-Dirac amplitudes", as Pryce

{{cite journal

|author=M. H. L. Pryce

|year=1938

|title=On the neutrino theory of light

|journal=Proc. Roy. Soc. (London)

|volume=A165 |pages=247-271

}}

noted. Jordan

{{cite journal

|author=P. Jordan

|year=1935

|title=Zur Neutrinotheorie des Lichtes

|journal=Z. Phys.

|volume=93 |pages=464-472

}}

"suggested that it is not the interaction

between neutrinos and antineutrinos that binds them

together into photons, but rather the manner in which

they interact with charged particles that leads

to the simplified description of light in terms of photons."

Jordan's hypothesis

eliminated the need for theorizing

an unknown interaction, but his hypothesis that the

neutrino and antineutrino are emitted in exactly

the same direction seems rather artificial as

noted by Fock.

{{cite journal

|author=Fock

|year=1937

|title=

|journal=Phys. Z. Sowjetunion

|volume=11 |pages=1

}}

His strong desire to obtain

exact Bose-Einstein commutation

relations for the composite photon led him to work

with a scalar or longitudinally polarized photon.

Actually, it is not difficult to obtain transversely polarized

photons from neutrinos.

{{cite journal

|author=R. de L. Kronig

|year=1936

|title=On a relativistically invariant formulation of the neutrino theory

of light

|journal=Physica

|volume=3 |pages=1120-1132

}}

{{cite journal

|author=W. A. Perkins

|year=1965

|title=Neutrino theory of photons

|journal=Physical Review B

|volume=137 |pages=1291-1301

}}

The neutrino field satisfies the Dirac equation with the mass set to zero,

::$$

\gamma^\mu p_\mu \Psi = 0.

The gamma matrices in the Weyl basis are:

::$$

\gamma^0 = \left( \begin{array}{cccc}

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0

\end{array} \right),

\; \; \; \; \gamma^1 = \left( \begin{array}{cccc}

0 & 0 & 0 & 1 \\

0 & 0 & 1 & 0 \\

0 & -1 & 0 & 0 \\

-1 & 0 & 0 & 0

\end{array} \right),

::$$

\gamma^2 = \left( \begin{array}{cccc}

0 & 0 & 0 & -i \\

0 & 0 & i & 0 \\

0 & i & 0 & 0 \\

-i & 0 & 0 & 0

\end{array} \right),

\; \; \; \; \gamma^3 = \left( \begin{array}{cccc}

0 & 0 & 1 & 0 \\

0 & 0 & 0 & -1 \\

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0

\end{array} \right).

The matrix $\backslash gamma^0$ is Hermitian while $\backslash gamma^k$ is antihermitian. They satisfy the anticommutation relation,

::$$

\gamma^{\mu} \gamma^{\nu} + \gamma^{\nu} \gamma^{\mu} = 2 \eta^{\mu \nu}I

where $\backslash eta^\{\backslash mu\; \backslash nu\}$ is the Minkowski metric with signature $(+---)$ and $I$ is the unit matrix.

The neutrino field is given by,

::$$

\Psi(x) = {1 \over \sqrt{V}} \sum_\mathbf{k} \left\{

\left[ a_1(\mathbf{k}) u^{+1}_{+1}(\mathbf{k}) + a_2(\mathbf{k}) u^{+1}_{-1}(\mathbf{k})

\right] e^{i k x} \right.

::$\backslash left.\; +\; \backslash left[\; c\_1^\backslash dagger(\backslash mathbf\{k\})\; u^\{-1\}\_\{-1\}\; (\backslash mathbf\{-k\})$

+ c_2^\dagger(\mathbf{k}) u^{-1}_{+1}(\mathbf{-k}) \right]e^{-i k x} \right\},

where $k\; x$ stands for $\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; k\_0t$.

$a\_1$ and $c\_1$ are the fermion annihilation operators for $\backslash nu\_1$

and $\backslash overline\; \backslash nu\_1$ respectively, while $a\_2$ and $c\_2$ are

the annihilation operators for $\backslash nu\_2$ and $\backslash overline\; \backslash nu\_2$.

$\backslash nu\_1$ is a right-handed neutrino and $\backslash nu\_2$ is a left-handed neutrino.

The $u$'s are spinors with the superscripts and subscripts refering to the energy and helicity states respectively. Spinor solutions for the Dirac equation are,

::$$

u^{+1}_{+1}(\mathbf{p}) = \sqrt{ {E + p_3} \over 2 E}

\left( \begin{array}{c}

1 \\

{{p_1 + i p_2} \over {E + p_3}} \\

0 \\

0

\end{array} \right),

::$$

u^{-1}_{-1}(\mathbf{p}) = \sqrt{ {E + p_3} \over 2 E}

\left( \begin{array}{c}

{{-p_1 + i p_2} \over {E + p_3}} \\

1 \\

0 \\

0

\end{array} \right),

::$$

u^{-1}_{+1}(\mathbf{p}) = \sqrt{ {E + p_3} \over 2 E}

\left( \begin{array}{c}

0 \\

0 \\

1 \\

{{p_1 + i p_2} \over {E + p_3}}

\end{array} \right),

::$$

u^{+1}_{-1}(\mathbf{p}) = \sqrt{ {E + p_3} \over 2 E}

\left( \begin{array}{c}

0 \\

0 \\

{{-p_1 + i p_2} \over {E + p_3}} \\

1

\end{array} \right).

The neutrino spinors for negative momenta are related to those of positive momenta by,

::$$

u^{+1}_{+1}(\mathbf{-p}) = u^{-1}_{-1}(\mathbf{p}),

::$$

u^{-1}_{-1}(\mathbf{-p}) = u^{+1}_{+1}(\mathbf{p}),

::$$

u^{+1}_{-1}(\mathbf{-p}) = u^{-1}_{+1}(\mathbf{p}),

::$$

u^{-1}_{+1}(\mathbf{-p}) = u^{+1}_{-1}(\mathbf{p}).

De Broglie and Kronig suggested the use of a local interaction to bind the neutrino-antineutrino pair. (Rosen and Singer

{{cite journal

|author=N. Rosen and P. Singer

|year=1959

|title=The photon as a composite particle

|journal=Bulletin of the Research Council of Israel

|volume=8F |pages=51-62

}}

have used a delta-function interaction in forming a

composite photon.)

Fermi and Yang

{{cite journal

|author=E. Fermi and C. N. Yang

|year=1949

|title=Are mesons elementary particles

|journal=Physical Review

|volume=76 |pages=1739-1743

}}

used a local interaction to bind

a fermion-antiferminon pair in attempting to form a pion. A four-vector field can be created from a fermion-antifermion pair,

{{cite book

|title=Relativistic Quantum Fields

|author=J. D. Bjorken and S. D. Drell

|location=New York (NY)

|publisher=McGraw-Hill

|year=1965

}}

::$$

\Psi^\dagger \gamma_0 \gamma_{\mu} \Psi.

Forming the photon field can be done simply by,

::$$

A_\mu(x) = \sum_\mathbf{p} {-1 \over 2 \sqrt{V p_0}}\left\{

\left[Q_R(\mathbf{p}) u^{-1}_{-1}(\mathbf{p})^\dagger \gamma_0 \gamma_{\mu} u^{+1}_{+1}(\mathbf{p})

+ Q_L(\mathbf{p}) u^{+1}_{+1}(\mathbf{p})^\dagger \gamma_0 \gamma_{\mu}

u^{-1}_{-1}(\mathbf{p}) \right]e^{i p x}

\right.

::$\backslash left.\; +\; \backslash left[Q\_R^\backslash dagger(\backslash mathbf\{p\})\; u^\{+1\}\_\{+1\}(\backslash mathbf\{p\})^\backslash dagger\; \backslash gamma\_0\; \backslash gamma\_\{\backslash mu\}\; u^\{-1\}\_\{-1\}(\backslash mathbf\{p\})$

+ Q_L^\dagger(\mathbf{p}) u^{-1}_{-1}(\mathbf{p})^\dagger \gamma_0 \gamma_{\mu} u^{+1}_{+1}(\mathbf{p})

\right]e^{-i p x} \right\}, \quad\quad (1)

where $p\; x\; =\; \backslash mathbf\{p\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; p\_0t\; =\; \backslash mathbf\{p\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; Et$.

The annihilation operators for right-handed and left-handed photons formed of fermion-antifermion pairs are defined as

{{cite book

|title=Quantum Mechanics

|author=H. J. Lipkin

|location=Amsterdam (Holland)

|publisher=North-Holland

|year=1973

}}

{{cite journal

|author=H. L. Sahlin and J. L. Schwartz

|year=1965

|title=The many body problem for composite particles

|journal=Physical Review B

|volume=138 |pages=267-273

}}

{{cite book

|title=Quantum Mechanics II

|author=R. H. Landau

|location=New York (NY)

|publisher=Wiley

|year=1996

}}

{{cite journal

|author=W. A. Perkins

|year=1972

|title=Statistics of a composite photon formed of two fermions

|journal=Physical Review D

|volume=5 |pages=1375-1384

}},

::$$

Q_R(\mathbf{p}) = \sum_\mathbf{k} F^\dagger(\mathbf{k})

\left [ c_1(\mathbf{p}/2-\mathbf{k})a_1(\mathbf{p}/2+\mathbf{k})

+ c_2(\mathbf{p}/2+\mathbf{k})a_2(\mathbf{p}/2-\mathbf{k}) \right ]

::$$

Q_L(\mathbf{p}) = \sum_\mathbf{k} F^\dagger(\mathbf{k})

\left [ c_2(\mathbf{p}/2-\mathbf{k})a_2(\mathbf{p}/2+\mathbf{k})

+ c_1(\mathbf{p}/2+\mathbf{k})a_1(\mathbf{p}/2-\mathbf{k}), \right ].

$F(\backslash mathbf\{k\})$ is a spectral function, normalized by

$\backslash sum\_\backslash mathbf\{k\}\; \backslash left|\; F(\backslash mathbf\{k\})\; \backslash right|^2\; =\; 1.$

The polarization vectors corresponding to the combinations used

in Eq. (1) are,

::$$

\epsilon_\mu^1( p ) = {-1 \over \sqrt{2}} [u^{-1}_{-1}(\mathbf{p})]^\dagger

\gamma_0 \gamma_{\mu} u^{+1}_{+1}(\mathbf{p}),

::$$

\epsilon_\mu^2( p ) = {-1 \over \sqrt{2}} [u^{+1}_{+1}(\mathbf{p})]^\dagger

\gamma_0 \gamma_{\mu} u^{-1}_{-1}(\mathbf{p}).

Carrying out the matrix multiplications results in,

::$$

\epsilon_\mu^1(p) \!= \!{1 \over \sqrt{2}} \left(

{{-i p_1 p_2 \!+\!E^2 \!+\!p_3 E \!-\!p_1^2} \over {E(E + p_3)}},

{{- p_1 p_2 \! + \!iE^2 \! +\!ip_3 E \! - \!ip_2^2 }

\over {E(E + p_3)}},

{{\!-p_1 \!- \!i p_2} \over E}, 0 \right),

::$$

\epsilon_\mu^2(p) \!= \!{1 \over \sqrt{2}} \left(

{{i p_1 p_2 \!+\!E^2 \!+\!p_3 E \!-\!p_1^2} \over {E(E + p_3)}},

{{-p_1 p_2 \! - \!iE^2 \! -\!ip_3 E \! + \!ip_2^2 }

\over {E(E + p_3)}},

{{\!-p_1 \!+ \!i p_2} \over E}, 0 \right), \quad (2)

where $\backslash epsilon\_0^1(p)$ and $\backslash epsilon\_0^2(p)$ have been placed on the right.

For massless fermions the polarization vectors depend only upon the direction of

$\backslash mathbf\{p\}$. Let $\backslash mathbf\{n\}\; =\; \backslash mathbf\{p\}/\; |\backslash mathbf\{p\}|$.

::$$

\epsilon_\mu^1(n) \!= \!{1 \over \sqrt{2}} \left(

{{-i n_1 n_2 \!+\!1 \!+\!n_3 \!-\!n_1^2} \over {1 + n_3}},

{{- n_1 n_2 \!+ \!in_1^2 \!+ \!in_3^2 \!+ \!in_3}

\over {1 + n_3}},

\!-n_1 \!- \!i n_2, 0 \right),

::$$

\epsilon_\mu^2(n) \!= \!{1 \over \sqrt{2}} \left(

{{i n_1 n_2 \!+\!1 \!+\!n_3 \!-\!n_1^2} \over {1 + n_3}},

{{- n_1 n_2 \!- \!in_1^2 \!- \!in_3^2 \!- \!in_3}

\over {1 + n_3}},

\!-n_1 \!+ \!i n_2, 0 \right).

These polarization vectors satisfy the

normalization relation,

::$$

\epsilon_\mu^j(p) \cdot \epsilon_\mu^{j*}(p) = 1,

::$$

\epsilon_\mu^j(p) \cdot \epsilon_\mu^{k*}(p) = 0 \;\; \text{for} \;\; k \ne j.

The Lorentz-invariant dot

products of the internal four-momentum $p\_\backslash mu$

with the polarization vectors are,

::$$

p_\mu \epsilon_\mu^1(p) = 0,

::$$

p_\mu \epsilon_\mu^2(p) = 0. \quad\quad\quad\quad (3)

In three dimensions,

::$$

\mathbf{p} \cdot \mathbf{\epsilon^1}(\mathbf{p}) =

\mathbf{p} \cdot \mathbf{\epsilon^2}(\mathbf{p}) = 0,

::$$

\mathbf{\epsilon^1}(\mathbf{p}) \times

\mathbf{\epsilon^2}(\mathbf{p})= -i\mathbf{p} / p_0,

::$$

\mathbf{p} \times \mathbf{\epsilon^1}(\mathbf{p})=-i p_0

\mathbf{\epsilon^1}(\mathbf{p}),

::$$

\mathbf{p} \times \mathbf{\epsilon^2}(\mathbf{p})= i p_0

\mathbf{\epsilon^2}(\mathbf{p}). \quad\quad\quad\quad (4)

In terms of the polarization vectors, $A\_\backslash mu(x)$ becomes,

::$$

A_\mu(x) = \sum_\mathbf{p} {1 \over \sqrt{2 V p_0}}\left\{

\left[Q_R(\mathbf{p}) \epsilon_\mu^1(\mathbf{p}) + Q_L(\mathbf{p})

\epsilon_\mu^2(\mathbf{p})

\right]e^{i p x} \right.

::$$

\left. + \left[Q_R^\dagger(\mathbf{p}) \epsilon_\mu^{1*}(\mathbf{p})

+ Q_L^\dagger(\mathbf{p}) \epsilon_\mu^{2*}(\mathbf{p})

\right]e^{-i p x} \right\}. \quad\quad\quad (5)

The electric field $\backslash mathbf\{E\}~$ and magnetic field

$\backslash mathbf\{H\}~$ are given by,

::$$

\mathbf{E}(x) = - { \partial \mathbf{A}(x) \over \partial t },

::$$

\mathbf{H}(x) = \nabla \times \mathbf{A}(x). \quad\quad\quad\quad (6)

Applying Eq. (6) to Eq. (5), results in,

::$$

E_\mu(x) = i \sum_\mathbf{p} {\sqrt{p_0} \over \sqrt{2 V }}\left\{

\left[Q_R(\mathbf{p}) \epsilon_\mu^1(\mathbf{p})

+ Q_L(\mathbf{p}) \epsilon_\mu^2(\mathbf{p}) \right]e^{i p x}

\right.

::$$

\left. - \left[Q_R^\dagger(\mathbf{p}) \epsilon_\mu^{1*}(\mathbf{p})

+ Q_L^\dagger(\mathbf{p}) \epsilon_\mu^{2*}(\mathbf{p}),

\right]e^{-i p x} \right\}.

::$$

H_\mu(x) = \sum_\mathbf{p} {\sqrt{p_0} \over \sqrt{2 V }}\left\{

\left[Q_R(\mathbf{p}) \epsilon_\mu^1(\mathbf{p})

- Q_L(\mathbf{p}) \epsilon_\mu^2(\mathbf{p}) \right]e^{i p x}

\right.

::$$

\left. + \left[Q_R^\dagger(\mathbf{p}) \epsilon_\mu^{1*}(\mathbf{p})

- Q_L^\dagger(\mathbf{p}) \epsilon_\mu^{2*}(\mathbf{p}),

\right]e^{-i p x} \right\}.

Maxwell's equations for free space are obtained as follows:

::$$

\partial E_1(x) / \partial x_1 =

i \sum_\mathbf{p} {\sqrt{p_0} \over \sqrt{2 V }}\left\{

\left[Q_R(\mathbf{p}) p_1 \epsilon_1^1(\mathbf{p})

+ Q_L(\mathbf{p}) p_1 \epsilon_1^2(\mathbf{p}) \right]e^{i p x}

\right.

::$$

\left. + \left[Q_R^\dagger(\mathbf{p}) p_1 \epsilon_1^{1*}(\mathbf{p})

+ Q_L^\dagger(\mathbf{p}) p_1 \epsilon_1^{2*}(\mathbf{p})

\right]e^{-i p x} \right\}.

Thus, $\backslash partial\; E\_1(x)\; /\; \backslash partial\; x\_1\; +\; \backslash partial$

E_2(x) / \partial x_2 +

\partial E_3(x) / \partial x_3 contains terms of the form

$p\_1\; \backslash epsilon\_1^1(\backslash mathbf\{p\})\; +\; p\_2\; \backslash epsilon\_2^1(\backslash mathbf\{p\})$

+ p_3 \epsilon_3^1(\mathbf{p}) which equate to zero by the first of Eq. (4).

This gives,

::$$

\nabla \cdot \mathbf{E}(x) = 0,

::$$

\nabla \cdot \mathbf{H}(x) = 0.

as $\backslash mathbf\{H\}$ contains similar terms.

The expression $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}(x)$ contains terms of the form

$\backslash mathbf\{p\}\; \backslash times\; \backslash mathbf\{\backslash epsilon^1\}(\backslash mathbf\{p\})$ while

$\backslash partial\; \backslash mathbf\{H\}(x)\; /\; \backslash partial\; t$

contains terms of form $i\; p\_0\; \backslash mathbf\{\backslash epsilon^1\}(\backslash mathbf\{p\})$. Thus, the last two equations of (4) can be used to show that,

::$$

\nabla \times \mathbf{E}(x) = - \partial \mathbf{H}(x) / \partial t,

::$$

\nabla \times \mathbf{H}(x) = \partial \mathbf{E}(x) / \partial t.

Although the neutrino field violates parity and charge

conjugation

{{cite journal

|author=T. D. Lee and C. N. Yang

|year=1957

|title=Parity nonconservation and two-component theory of the neutrino

|journal=Physical Review

|volume=105 |pages=1671-1675

}},

$\backslash mathbf\{E\}$ and

$\backslash mathbf\{H\}$ transform in the usual way

,

::$$

P \mathbf{E}(\mathbf{x},t) P^-1 = -\mathbf{E}(\mathbf{-x},t),

::$$

P \mathbf{H}(\mathbf{x},t) P^-1 = \mathbf{H}(\mathbf{-x},t),

::$$

C \mathbf{E}(\mathbf{x},t) C^-1 = -\mathbf{E}(\mathbf{x},t),

::$$

C \mathbf{H}(\mathbf{x},t) C^-1 = -\mathbf{H}(\mathbf{x},t).

$A\_\backslash mu$ satisfies the Lorentz condition,

::$$

\partial A_\mu / \partial x_\mu = 0

which follows from Eq. (3).

Although many choices for gamma matrices can satisfy the Dirac equation, it

is essential that one use the Weyl representation in order to get the correct photon polarization vectors and $\backslash mathbf\{E\}$ and $\backslash mathbf\{H\}$ that satisfy Maxwell's equations. Kronig

first realized this. In the Weyl representation,

the four-component spinors are describing two sets of two-component neutrinos.

The connection between the photon antisymmetric tensor and the two-component Weyl equation was also noted by Sen.

{{cite journal

|author=D. K. Sen

|year=1964

|title=A theoretical basis for two neutrinos

|journal=Il Nuovo Cimento

|volume=31 |pages=660-669

}}

One can also produce the above results using a two-component neutrino theory.

To compute the commutation relations for the photon field,

one needs the equation,

::$$

\sum_{j=1}^2 \epsilon_{\mu}^j(\mathbf{p}) \epsilon_{\nu}^{j*}(\mathbf{p})

= \sum_{j=1}^2 \epsilon_{\mu}^{j*}(\mathbf{p}) \epsilon_{\nu}^j(\mathbf{p})

= \delta_{\mu \nu} - {p_{\mu} p_{\nu} \over E^2}.

To obtain this equation, Kronig

wrote a relation between the neutrino spinors that was not

rotationally invariant as pointed out by Pryce.

However, as Perkins showed, this equation

follows directly from summing over the polarization vectors,

Eq. (2), that were obtained by

explicitly solving for the neutrino spinors.

If the momentum is along the third axis, $\backslash epsilon\_\backslash mu^1(n)$

and $\backslash epsilon\_\backslash mu^2(n)$ reduce to the usual polarization vectors

for right and left circularly polarized photons respectively.

::$$

\epsilon_\mu^1(n) = {1 \over \sqrt{2}}(1,i,0,0),

::$$

\epsilon_\mu^2(n) = {1 \over \sqrt{2}}(1,-i,0,0).

Although composite photons satisfy many properties of real photons,

there are major problems with this theory.

It is well-know that the photon is a boson.

{{cite journal

|author=C. Amsler et al. (Particle Data Group)

|year=2008

|title=The review of particle physics

|journal=Physics Letters B

|volume=667 |pages=1-1340

}}

Does the composite photon

satisfy Bose-Einstein commutation relations?

Fermions are defined as the particles whose creation and

annihilation operators adhere to the anticommutation relations

::$$

\{a(\mathbf{k}),a(\mathbf{l})\} = 0,

::$$

\{a^\dagger(\mathbf{k}),a^\dagger(\mathbf{l})\} = 0,

::$$

\{a(\mathbf{k}),a^\dagger(\mathbf{l})\}

= \delta(\mathbf{k}-\mathbf{l}),

while bosons are defined as the particles

that adhere to the commutation relations,

::$$

\left[b(\mathbf{k}),b(\mathbf{l})\right] = 0,

::$$

\left[b^\dagger(\mathbf{k}),b^\dagger(\mathbf{l})\right] = 0,

::$$

\left[b(\mathbf{k}),b^\dagger(\mathbf{l})\right]

= \delta(\mathbf{k}-\mathbf{l}). \quad\quad (7)

The creation and

annihilation operators of

composite particles formed of fermion pairs

adhere to the commutation relations of the form,

::$$

\left[Q(\mathbf{k}),Q(\mathbf{l})\right] = 0,

::$$

\left[Q^\dagger(\mathbf{k}),Q^\dagger(\mathbf{l})\right] = 0,

::$$

\left[Q(\mathbf{k}),Q^\dagger(\mathbf{l})\right]

= \delta(\mathbf{k}-\mathbf{l})- \Delta(\mathbf{k},\mathbf{l}). \quad\quad (8)

with

::$$

\Delta(\mathbf{p}^{\prime},\mathbf{p}) =

\sum_\mathbf{k} F^\dagger(\mathbf{k}) \left[

F(\mathbf{p}^{\prime}/2-\mathbf{p}/2+\mathbf{k})

a^\dagger(\mathbf{p}-\mathbf{p}^{\prime}/2-\mathbf{k})

a(\mathbf{p}^{\prime}/2-\mathbf{k}) \right.

::$$

\left.

+ F(\mathbf{p}/2-\mathbf{p}^{\prime}/2+\mathbf{k})

c^\dagger(\mathbf{p}-\mathbf{p}^{\prime}/2+\mathbf{k})

c(\mathbf{p}^{\prime}/2+\mathbf{k}) \right]. \quad\quad (9)

For Cooper electron pairs

,

"a" and "c" represent different spin directions. For

nucleon pairs (the deuteron)

,

"a" and "c" represent proton and neutron. For

neutrino-antineutrino pairs

,

"a" and "c" represent neutrino and antineutrino.

The size of the deviations from pure Bose behavior,

$\backslash Delta(\backslash mathbf\{p\}^\{\backslash prime\},\backslash mathbf\{p\}),$

depends on the degree

of overlap of the fermion wave functions

and the constraints of the Pauli exclusion principle.

If the state has the form,

::$$

|\Phi \rangle = a^\dagger(\mathbf{k_1})

a^\dagger(\mathbf{k_2})...a^\dagger(\mathbf{k_n})

c^\dagger(\mathbf{q_1})c^\dagger(\mathbf{q_2})...c^\dagger(\mathbf{q_m})|0 \rangle

then the expectation value of Eq. (9) vanishes for

$\backslash mathbf\{p\}^\{\backslash prime\}\; \backslash ne\; \backslash mathbf\{p\}$, and the expression for

$\backslash Delta(\backslash mathbf\{p\}^\{\backslash prime\},\backslash mathbf\{p\})$ can be approximated by,

::$$

\Delta(\mathbf{p}^{\prime},\mathbf{p}) =

\delta(\mathbf{p}^{\prime}-\mathbf{p})

\sum_\mathbf{k} \left| F(\mathbf{k}) \right|^2

\left[ a^\dagger(\mathbf{p}/2-\mathbf{k})

a(\mathbf{p}/2-\mathbf{k}) \right.

::$$

\left.

+ c^\dagger(\mathbf{p}/2+\mathbf{k})

c(\mathbf{p}/2+\mathbf{k}) \right].

Using the fermion number operators $n\_a(\backslash mathbf\{k\})$ and

$n\_c(\backslash mathbf\{k\})$, this can be written,

::$$

\Delta(\mathbf{p}^{\prime},\mathbf{p}) =

\delta(\mathbf{p}^{\prime}-\mathbf{p})

\sum_\mathbf{k} \left| F(\mathbf{k}) \right|^2

\left[ n_a( \mathbf{p}/2-\mathbf{k})

+ n_c(\mathbf{p}/2+\mathbf{k})

\right]

::$$

= \delta(\mathbf{p}^{\prime}-\mathbf{p})

\sum_\mathbf{k} \left[ \left| F(\mathbf{p}/2-\mathbf{k}) \right|^2

n_a(\mathbf{k}) + \left| F(\mathbf{k}- \mathbf{p}/2) \right|^2

n_c(\mathbf{k}) \right]

::$$

= \delta(\mathbf{p}^{\prime}-\mathbf{p})

\overline {\Delta} (\mathbf{p},\mathbf{p})

showing that it is the average number

of fermions in a particular state $\backslash mathbf\{k\}$ averaged

over all states with weighting factors

$F(\; \backslash mathbf\{p\}/2-\backslash mathbf\{k\})$ and

$F(\backslash mathbf\{k\}-\backslash mathbf\{p\}/2)$.

In 1928, Jordan noticed that commutation relations for

pairs of fermions were similar to those for bosons.

{{cite journal

|author=P. Jordan

|year=1928

|title=Die Lichtquantenhypothese: Entwicklung und gegenwärtiger Stand

|journal=Ergebnisse der exakten Naturwissenschaften

|volume=7 |pages=158-208

}}

Compare Eq. (7) with Eq. (8).

From 1935 till 1937, Jordan, Kronig, and others

{{cite journal

|author=M. Born and N. S. Nagendra Nath

|year=1936

|title=

|journal=Proc. Indian Acad. Sci.

|volume=A3 |pages=318

}}

tried to obtain exact Bose-Einstein commutation

relations for the composite photon. Terms were added to the

commutation relations to cancel out the delta term in Eq. (8).

These terms corresponded to "simulated photons."

For example, the absorption of a photon of momentum $\backslash mathbf\{p\}$ could

be simulated by a Raman effect in which a neutrino with momentum

$\backslash mathbf\{p+k\}$ is absorbed while another of another with opposite spin and

momentum $\backslash mathbf\{k\}$ is emitted. (It is now known that single neutrinos or antineutrinos interact so weakly that they cannot simulate photons.)

This led Jordan to work

with a scalar or longitudinally polarized photons

instead of transversely polarized ones like real photons.

In 1938, Pryce showed that one cannot obtain

both Bose-Einstein statistics and transversely-polarized photons from

neutrino-antineutrino pairs. Construction of transversely-polarized

photons is not the problem.

{{cite journal

|author=K. M. Case

|year=1957

|title=Composite particles of zero mass

|journal=Physical Review

|volume=106 |pages=1316-1320

}}

As Berezinski

{{cite journal

|author=V. S. Berezinskii

|year=1966

|title=Pryce's theorem and the neutrino theory of light

|journal=Zh. Eksperim. i Teor. Fiz.

|volume=51 |pages=1374-1384}} translated in {{cite journal

|title=Pryce's theorem and the neutrino theory of light

|year=1967

|journal=Soviet Physics JETP

|volume=24 |pages=927-933

}}

noted, "The only actual difficulty is that the construction of a transverse

four-vector is incompatible with the requirement of statistics."

In some ways Berezinski gives a clearer picture of the

problem. A simple version of the proof is as follows:

The expectation values of the commutation relations for composite

right and left-handed photons are:

::$$

\left[ Q_R(\mathbf{p}^{\prime}),

Q_R(\mathbf{p}) \right] = 0, \;

\left[ Q_L(\mathbf{p}^{\prime}),

Q_L(\mathbf{p}) \right] = 0,

::$$

\left[ Q_R(\mathbf{p}^{\prime}),

Q_R^\dagger(\mathbf{p}) \right]

= \delta( \mathbf{p}^{\prime} - \mathbf{p})

(1 -{\overline \Delta_{12}}(\mathbf{p},\mathbf{p})),

::$$

\left[ Q_L(\mathbf{p}^{\prime}),

Q_L^\dagger(\mathbf{p}) \right]

= \delta( \mathbf{p}^{\prime} - \mathbf{p})

(1 -{\overline \Delta_{21}}(\mathbf{p},\mathbf{p})),

::$$

\left[ Q_R(\mathbf{p}^{\prime}),

Q_L(\mathbf{p}) \right] = 0, \;

\left[ Q_R(\mathbf{p}^{\prime}),

Q_L^\dagger(\mathbf{p}) \right] = 0, \quad\quad\quad\quad (10)

where

::$$

{\overline \Delta_{12}}(\mathbf{p},\mathbf{p}) =

\sum_\mathbf{k} \left[

\left| F(\mathbf{k}-\mathbf{p}/2) \right|^2 (n_{a1}(\mathbf{k}) + n_{c2}(\mathbf{k}) )

\right.

::$$

\left.

+ \left| F(\mathbf{p}/2-\mathbf{k}) \right|^2 ( n_{c1}(\mathbf{k}) + n_{a2}(\mathbf{k}) )

\right]. \quad\quad\quad\quad (11)

The deviation from Bose-Einstein statistics is caused

by $\backslash overline\; \backslash Delta\_\{12\}(\backslash mathbf\{p\},\backslash mathbf\{p\})$ and

$\backslash overline\; \backslash Delta\_\{21\}(\backslash mathbf\{p\},\backslash mathbf\{p\})$,

which are functions of the neutrino numbers operators.

Linear polarization

photon operators are defined by,

::$$

\xi( \mathbf{p}) = {1 \over \sqrt{2}} \left[ Q_L(\mathbf{p})

+ Q_R(\mathbf{p}) \right],

::$$

\eta( \mathbf{p}) = {i \over \sqrt{2}} \left[ Q_L(\mathbf{p})

- Q_R(\mathbf{p}) \right]. \quad\quad\quad\quad (12)

A particularly interesting commutation relation is,

::$$

[\xi( \mathbf{p}^{\prime}),\eta^\dagger( \mathbf{p})]

= {i \over 2} \delta( \mathbf{p}^{\prime} - \mathbf{p})

[\overline \Delta_{21}(\mathbf{p},\mathbf{p})

-\overline \Delta_{12}(\mathbf{p},\mathbf{p})], \quad\quad (13)

which follows from (10) and (12).

For the composite photon to obey Bose-Einstein commutation relations, at

the very least,

::$$

[\xi( \mathbf{p}^{\prime}),\eta^\dagger( \mathbf{p})]

= 0 \quad\quad\quad\quad (14)

Pryce noted .

From Eq. (11) and Eq. (13) the

requirement is that

::$$

\sum_\mathbf{k} \left[

\left| F(\mathbf{k}-\mathbf{p}/2) \right|^2

(n_{a1}(\mathbf{k}) + n_{c2}(\mathbf{k}) - n_{a2}(\mathbf{k}) - n_{c1}(\mathbf{k}) )

\right.

::$$

\left. + \left| F(\mathbf{p}/2-\mathbf{k}) \right|^2

( n_{c1}(\mathbf{k}) + n_{a2}(\mathbf{k}) - n_{c2}(\mathbf{k}) - n_{a1}(\mathbf{k}) )

\right]

gives zero when applied to any

state vector. Thus, all the coefficients of

$n\_\{a1\}(\backslash mathbf\{k\})$ and $n\_\{c1\}(\backslash mathbf\{k\})$,

etc. must vanish separately. This means $F(\backslash mathbf\{k\})\; =\; 0$,

and the composite photon does not exist

,

completing the proof.

Perkins

reasoned that the photon does

not have to obey Bose-Einstein commutation relations, because the non-Bose

terms are small and they may not cause any detectable effects.

Perkins

noted, "As presented in many quantum mechanics

texts it may appear that Bose statistics follow from basic principles, but it is really from the classical canonical formalism. This is not a reliable procedure as evidenced by the fact that it gives the completely wrong result for spin-1/2 particles." Furthermore,

"most integral spin particles (light mesons, strange mesons, etc.) are composite particles formed of quarks. Because of their underlying fermion structure, these integral spin particles are not fundamental bosons, but composite quasibosons. However, in the asymptotic limit, which generally applies, they are essentially bosons. For these particles, Bose commutation relations are just an approximation, albeit a very good one. There are some differences; bringing two of these composite particles close together will force their identical fermions to jump to excited states because of the Pauli exclusion principle."

Berezinskii in reaffirming Pryce's theorem argues

that commutation relation (14) is necessary for the

photon to be truly neutral. However, Perkins

has shown that a neutral photon in the usual sense can be

obtained without Bose-Einstein commutation relations.

The number operator for a composite photon is defined as,

::$$

N( \mathbf{p}) = Q^\dagger(\mathbf{p}) Q(\mathbf{p}).

Lipkin

suggested for a rough estimate to assume

that $F(\backslash mathbf\{k\})=\; 1\; /\; \backslash Omega$

where $\backslash Omega$ is a constant equal

to the number of states used to construct the wave packet.

Perkins

showed that the effect

of the composite photon’s

number operator acting on a state of $m$ composite photons is,

::$$

N(\mathbf{p}) (Q^\dagger(\mathbf{p}))^m|0\rangle \;

= \left( m - {m(m-1) \over \Omega }

\right) (Q^\dagger(\mathbf{p}))^m|0\rangle,

using $N(\backslash mathbf\{p\})|0\backslash rangle\; =\; 0$.

This result differs from the usual

one because of the second term which is small for large $\backslash Omega$.

Normalizing in the

usual manner

{{cite book

|title=Quantum Mechanics of Many Degrees of Freedom

|author=D. S. Koltun and J. M. Eisenberg

|location=New York (NY)

|publisher=Wiley

|year=1988

}},

::$$

Q^\dagger(\mathbf{p})|n_\mathbf{p} \rangle \;

= \sqrt{ (n_\mathbf{p} +1)

\left( 1- {n_\mathbf{p} \over \Omega} \right) }

|n_\mathbf{p} +1\rangle,

::$$

Q(\mathbf{p})|n_\mathbf{p} \rangle \;

= \sqrt{ n_\mathbf{p}

\left( 1- {(n_\mathbf{p}-1) \over \Omega} \right) }

|n_\mathbf{p} -1\rangle, \quad\quad\quad\quad (15)

where $|n\_\backslash mathbf\{p\}\backslash rangle$ is the state of $n\_\backslash mathbf\{p\}$

composite photons having momentum $\backslash mathbf\{p\}$ which is created

by applying $Q^\backslash dagger(\backslash mathbf\{p\})$ on the vacuum $n\_\backslash mathbf\{p\}$ times.

Note that,

::$$

Q^\dagger(\mathbf{p})|0 \rangle = | 1_\mathbf{p}\rangle,

::$$

Q(\mathbf{p})|1_\mathbf{p}\rangle = |0\rangle,

which is the same result as obtained

with boson operators. The formulas in Eq. (15)

are similar to the usual ones with correction factors

that approach zero for large $\backslash Omega$.

The main evidence indicating that photons

are bosons comes from the Blackbody radiation experiments which are

in agreement with Planck's distribution.

Perkins calculated the photon distribution

for Blackbody radiation using the

second quantization method ,

but with a composite photon.

The atoms in the

walls of the cavity are taken to be

a two-level system with photons emitted from

the upper level $\backslash beta$ and absorbed

at the lower level $\backslash alpha$.

The transition probability for

emission of a photon is enhanced when $n\_\backslash mathbf\{p\}$

photons are present,

::$$

w_{\alpha \beta}( n_\mathbf{p} + 1 \leftarrow n_\mathbf{p} )

= (n_\mathbf{p} + 1) \left( 1 - {n_\mathbf{p} \over \Omega}

\right) w_{\alpha \beta}( 1_\mathbf{p} \leftarrow 0 ), \quad (16)

where the first of (15) has been used.

The absorption is enhanced less

since the second of (15) is used,

::$$

w_{ \beta \alpha}( n_\mathbf{p} - 1 \leftarrow n_\mathbf{p} )

= n_\mathbf{p} \left( 1 - {n_\mathbf{p}-1 \over \Omega}

\right) w_{ \beta \alpha}( 0 \leftarrow 1_\mathbf{p} ). \quad (17)

Using the equality,

::$$

w_{ \beta \alpha}( 0 \leftarrow 1_\mathbf{p} )

= w_{ \alpha \beta}( 1_\mathbf{p} \leftarrow 0 ),

of the transition rates,

Eqs. (16) and (17) are

combined to give,

::$$

{w_{\alpha \beta}( n_\mathbf{p}+1 \leftarrow n_\mathbf{p} )

\over

w_{ \beta \alpha}( n_\mathbf{p} - 1 \leftarrow n_\mathbf{p} )}

= {(n_\mathbf{p}+1) \left( 1 - {n_\mathbf{p} \over \Omega}

\right) \over

n_\mathbf{p} \left( 1 - {(n_\mathbf{p}-1) \over \Omega}

\right) }.

The probability of finding the system

with energy E is proportional to $e^\{-E/kT\}$

according to Boltzmann's distribution law.

Thus, the equilibrium between emission

and absorption requires that,

::$$

w_{\alpha \beta}( n_\mathbf{p}+1 \leftarrow n_\mathbf{p} )

e^{-E_{\beta} /kT} =

w_{ \beta \alpha}( n_\mathbf{p} - 1 \leftarrow n_\mathbf{p} )

e^{-E_{\alpha} /kT},

with the photon energy $\backslash omega\_p\; =\; E\_\{\backslash beta\}\; -\; E\_\{\backslash alpha\}$.

Combining the last two equations results in,

::$$

n_\mathbf{p} = {2 \over u+(u+2)/ \Omega +

\sqrt{u^2(1+2/\Omega) + (u+2)^2 /\Omega^2}},

with $u\; =\; e^\{\backslash omega\_p/kT\}\; -\; 1$.

For $\backslash Omega(\backslash omega\_p\; /kT)\; >>\; 1$, this reduces to

::$$

n_\mathbf{p} = {1 \over e^{\omega_p /kT}

\left( 1 + {1 \over \Omega} \right) - 1}.

This equation differs from Planck’s law because of the

the $1\; /\; \backslash Omega$ term.

For the conditions

used in the Blackbody radiation

experiments of Coblentz

{{cite journal

|author=W. W. Coblentz

|year=1916

|title=

|journal=Natl. Bur. Std. (U.S.) Bull.

|volume=13 |pages=459

}}, Perkins estimates that

,

and the maximum deviation from Planck's law is less than

one part in $10^\{-8\}$, which is too small to be detected.

Experimental results show that only left-handed neutrinos

and right-handed antineutrinos exist. Three sets of neutrinos

have been observed

{{cite journal

|author=G. Danby, J-M Gaillard, K. Goulianos, L. M. Lederman,

N. Mistry, M. Schwartz, and J. Steinberger,

|year=1962

|title=Observation of high-energy neutrino interactions and the existence

of two kinds of neutrinos

|journal=Physical Review Letters

|volume=9 |pages=36-44

}}

{{cite journal

|author=K. Kodama et al. (DONUT collaboration)

|year=2001

|title=Observation of tau neutrino interactions

|journal=Physics Letters B

|volume=504 |pages=218-224

}}, one

that is connected with electrons, one

with muons, and one with tau leptons.

{{cite journal

|author=M. L. Perl et al.

|year=1975

|title=Evidence for anomalous lepton production in e+ - e- annihilation

|journal=Physical Review Letters

|volume=35 |pages=1489-1492

}}

In the standard model the pion and muon decay modes are:

::$$

\pi^+ \rightarrow \mu^+ + \nu_\mu,

::$$

\mu^+ \rightarrow e^+ + \nu_e + \overline\nu_\mu.

To form a photon, which satisfies parity and charge

conjugation, two sets of two-component neutrinos

(i.e., right-handed and left-handed neutrinos) are needed.

Perkins (see Sec. VI of Ref. )

attempted to solve this problem by noting that the needed

two sets of two-component neutrinos would exist if the

positive muon is identified as

the particle and the negative muon as the

antiparticle. The reasoning is as follows: let $\backslash nu\_1$ be

the right-handed neutrino and $\backslash nu\_2$ the left-handed neutrino

with their corresponding antineutrinos (with opposite helicity).

The neutrinos involved in beta decay are $\backslash nu\_2$

and $\backslash overline\backslash nu\_2$, while those for $\backslash pi-\backslash mu$ decay are

$\backslash nu\_1$ and $\backslash overline\backslash nu\_1$.

With this scheme the pion and muon decay modes are:

::$$

\pi^+ \rightarrow \mu^+ + \overline\nu_1,

::$$

\mu^+ \rightarrow e^+ + \nu_2 + \nu_1.

There is convincing evidence that neutrinos have mass.

In experiments at the SuperKamiokande researchers

have discovered neutrino oscillations in which one flavor of

neutrino changed into another. This means that neutrinos have

non-zero mass.

Since massless neutrinos are needed to form a massless photon,

a composite photon is not possible.

==References==