View source for Physics:Neutrino theory of light

The '''neutrino theory of light''' is the proposal that the [[Physics:Photon|photon]] is a composite particle formed of a [[Physics:Neutrino|neutrino]]–antineutrino [[Physics:Pair production|pair]]. It is based on the idea that [[Physics:Emission (electromagnetic radiation)|emission]] and [[Physics:Absorption (electromagnetic radiation)|absorption]] of a photon corresponds to the creation and annihilation of a particle–antiparticle pair. The [[Physics:Neutrino|neutrino]] theory of light is not currently accepted as part of mainstream physics, as according to the [[Physics:Standard Model|Standard Model]] the photon is an [[Physics:Elementary particle|elementary particle]], a [[Physics:Gauge boson|gauge boson]].

== History ==
In the past, many particles that were once thought to be elementary such as protons, neutrons, pions, and kaons have turned out to be composite particles. In 1932, [[Biography:Louis de Broglie|Louis de Broglie]]<ref name="broglie1">
{{Cite journal |author=de Broglie |first=Louis |year=1932 |title=Sur une analogie entre l'équation de Dirac et l'onde électromagnétique |url=https://gallica.bnf.fr/ark:/12148/bpt6k73524/f536.image |journal=Compt. Rend. |language=fr |volume=195 |pages=536, 862}}</ref><ref name="broglie2">
{{Cite journal |author=de Broglie |first=Louis |year=1932 |title=Sur le champ électromagnétique de l'onde lumineuse |url=https://gallica.bnf.fr/ark:/12148/bpt6k73524/f862.image |journal=Compt. Rend. |language=fr |volume=195 |pages=862}}</ref><ref name="broglie3">
{{Cite book |author=de Broglie |first=Louis |title=Une novelle conception de la lumiere |publisher=Hermann et. Cie. |year=1934 |location=Paris, France |language=fr}}</ref> suggested that the photon might be the combination of a neutrino and an antineutrino. During the 1930s there was great interest in the neutrino theory of light and [[Biography:Pascual Jordan|Pascual Jordan]],<ref name="jordan2">
{{Cite journal |author=Jordan |first=Pascual |year=1935 |title=Zur Neutrinotheorie des Lichtes |journal=Z. Phys. |volume=93 |issue=7–8 |pages=464–472 |bibcode=1935ZPhy...93..464J |doi=10.1007/BF01330373 |s2cid=120050979}}</ref> [[Biography:Ralph Kronig|Ralph Kronig]], [[Biography:Max Born|Max Born]], and others worked on the theory.

In 1938, [[Biography:Maurice Pryce|Maurice Pryce]]<ref name="pryce">
{{Cite journal |author=Pryce |first=Maurice H. L. |year=1938 |title=On the neutrino theory of light |journal=Proceedings of the Royal Society |volume=A165 |issue=921 |pages=247–271 |bibcode=1938RSPSA.165..247P |doi=10.1098/rspa.1938.0058 |doi-access=free}}</ref> brought work on the composite photon theory to a halt. He showed that the conditions imposed by Bose–Einstein commutation relations for the composite photon and the connection between its spin and polarization were incompatible. Pryce also pointed out other possible problems,

“In so far as the failure of the theory can be traced to any one cause it is fair to say that it lies in the fact that light waves are polarized transversely while neutrino ‘waves’ are polarized longitudinally,”

and lack of [[Rotational invariance|rotational invariance]]. In 1966, V. S. Berezinskii reanalyzed Pryce's paper, giving a clearer picture of the problem that Pryce uncovered.<ref name="berez">
{{Cite journal |author=Berezinskii |first=V. S. |year=1966 |title=Pryce's theorem and the neutrino theory of photons |journal=Zh. Eksp. Teor. Fiz. |volume=51 |pages=1374–1384 |bibcode=1967JETP...24..927B}}

* translated in ''Soviet Physics JETP'', '''24''': 927 (1967)</ref>

Starting in the 1960s work on the neutrino theory of light resumed, and there continues to be some interest in recent years.<ref name="dvoeg1">
{{Cite journal |author=Dvoeglazov |first=V. V. |year=1999 |title=Speculations on the neutrino theory of light |journal=Annales de la Fondation Louis de Broglie |volume=24 |pages=111–127 |arxiv=physics/9807013 |bibcode=1998physics...7013D}}</ref><ref name="dvoeg2">
{{Cite journal |author=Dvoeglazov |first=V. V. |year=2001 |title=Again on the possible compositeness of the photon |journal=Phys. Scripta |volume=64 |issue=2 |pages=119–127 |arxiv=hep-th/9908057 |bibcode=2001PhyS...64..119D |doi=10.1238/Physica.Regular.064a00119 |s2cid=34939834}}</ref><ref name="perkins3">
{{Cite book |author=Perkins |first=W. A. |title=Lorentz Group, CPT and Neutrinos: Proceedings of the International Workshop, Zacatecas, Mexico, 23–26 June 1999 |publisher=World Scientific, Singapore |year=1999 |editor1=Chubykalo |editor-first=A. E. |location=Singapore |pages=115–126 |chapter=Interpreted History of Neutrino Theory of Light and Its Future |editor2=Dvoeglazov |editor-first2=V. V. |editor3=Ernst |editor-first3=D. J. |editor4=Kadyshevsky |editor-first4=V. G. |editor5=Kim |editor-first5=Y. S.}}</ref><ref name="sen2">
{{Cite journal |author=Sen |first=D. K. |year=2007 |title=Left- and right-handed neutrinos and baryon–lepton masses |journal=Journal of Mathematical Physics |volume=48 |issue=2 |page=022304 |bibcode=2007JMP....48b2304S |doi=10.1063/1.2436985}}</ref> Attempts have been made to solve the problem pointed out by Pryce, known as Pryce's theorem, and other problems with the composite photon theory. The incentive is seeing the natural way that many photon properties are generated from the theory and the knowledge that some problems exist<ref name="varlamov">
{{cite journal |author=Varlamov |first=V. V. |year=2002 |title=About Algebraic Foundation of Majorana–Oppenheimer Quantum Electrodynamics and de Brogie–Jordan Neutrino Theory of Light |journal=Annales Fond.broglie |volume=27 |pages=273–286 |arxiv=math-ph/0109024 |bibcode=2001math.ph...9024V}}</ref><ref name="perkins4">
{{Cite journal |author=Perkins |first=W. A. |year=2002 |title=Quasibosons |journal=International Journal of Theoretical Physics |volume=41 |issue=5 |pages=823–838 |arxiv=hep-th/0107003 |doi=10.1023/A:1015728722664|s2cid=696381 }}</ref> with the current photon model. However, there is no experimental evidence that the photon has a composite structure.

Some of the problems for the neutrino theory of light are the non-existence for massless neutrinos with both spin parallel and antiparallel to their momentum outside the [[Physics:Einstein–Cartan theory|Einstein-Cartan torsion]]<ref name="Neutrino oscillations in the presence of torsion">{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | doi=10.1007/BF02902051 | pages=479–500 }}</ref><ref name="Geometrical contribution to neutrino mass matrix">{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 }}</ref> and the fact that composite photons are not bosons.<ref name=fukuda>
{{Cite journal
|doi=10.1103/PhysRevLett.81.1562
|author=Y. Fukuda ''et al.'' (Super-Kamiokande Collaboration)
|year=1998
|title=Evidence for oscillation of atmospheric neutrinos
|journal=[[Physics:Physical Review Letters|Physical Review Letters]]
|volume=81
|issue=8 |pages=1562–1567
|bibcode=1998PhRvL..81.1562F
|arxiv = hep-ex/9807003 }}</ref> Attempts to solve some of these problems will be discussed, but the lack of massless neutrinos outside the [[Physics:Einstein–Cartan theory|Einstein-Cartan torsion]]<ref name="Neutrino oscillations in the presence of torsion">{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | doi=10.1007/BF02902051 | pages=479–500 }}</ref><ref name="Geometrical contribution to neutrino mass matrix">{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 }}</ref> makes it impossible outside the [[Physics:Einstein–Cartan theory|Einstein-Cartan torsion]]<ref name="Neutrino oscillations in the presence of torsion">{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | doi=10.1007/BF02902051 | pages=479–500 }}</ref><ref name="Geometrical contribution to neutrino mass matrix">{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 }}</ref> to form a massless photon with this theory. The neutrino theory of light is not considered to be part of mainstream physics.

==Forming photon from neutrinos==
It is possible to obtain transversely polarized photons from neutrinos.<ref name=kronig>
{{Cite journal
|doi=10.1016/S0031-8914(36)80340-1
|author=R. de L. Kronig
|year=1936
|title=On a relativistically invariant formulation of the neutrino theory of light
|journal=Physica
|volume=3
|issue=10 |pages=1120–1132
|bibcode = 1936Phy.....3.1120K }}</ref><ref name="perkins1">
{{Cite journal |author=Perkins |first=W. A. |year=1965 |title=Neutrino theory of photons |journal=[[Physics:Physical Review|Physical Review]] |volume=137 |issue=5B |pages=B1291–B1301 |bibcode=1965PhRv..137.1291P |doi=10.1103/PhysRev.137.B1291}}</ref>

===The neutrino field===
The neutrino field satisfies the [[Physics:Dirac equation|Dirac equation]] with the mass set to zero,

::<math>
\gamma^\mu p_\mu \Psi = 0.
</math>

The [[Gamma matrices|gamma matrices]] in the Weyl basis are:
::<math>
\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),
</math>

::<math>
\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).
</math>

The matrix <math>\gamma^0</math> is [[Hermitian matrix|Hermitian]] while <math>\gamma^k</math> is antihermitian. They satisfy the anticommutation relation,
::<math>
\gamma^{\mu} \gamma^{\nu} + \gamma^{\nu} \gamma^{\mu} = 2 \eta^{\mu \nu}I
</math>
where <math>\eta^{\mu \nu}</math> is the Minkowski metric with signature <math>(+---)</math> and <math>I</math> is the unit matrix.

The neutrino field is given by,
::<math>
\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. </math>
::<math>\left. + \left[ c_1^\dagger(\mathbf{k}) u^{-1}_{-1} (\mathbf{-k})
+ c_2^\dagger(\mathbf{k}) u^{-1}_{+1}(\mathbf{-k}) \right]e^{-i k x} \right\},
</math>

where <math>k x</math> stands for <math>\mathbf{k} \cdot \mathbf{x} - k_0t</math>.
<math>a_1</math> and <math>c_1</math> are the fermion annihilation operators for <math>\nu_1</math>
and <math>\overline \nu_1</math> respectively, while <math>a_2</math> and <math>c_2</math> are the annihilation operators for <math>\nu_2</math> and <math>\overline \nu_2</math>.
<math>\nu_1</math> is a right-handed neutrino and <math>\nu_2</math> is a left-handed neutrino.
The <math>u</math>'s are [[Physics:Spinors|spinors]] with the superscripts and subscripts referring to the energy and [[Physics:Helicity (particle physics)|helicity]] states respectively. [[Physics:Spinor|Spinor]] solutions for the [[Physics:Dirac equation|Dirac equation]] are,

::<math>
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),
</math>
::<math>
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),
</math>
::<math>
u^{-1}_{+1}(\mathbf{p}) = \sqrt{ {E + p_3} \over 2 E}
\left( \begin{array}{c}
0 \\
0 \\
1 \\
\end{array} \right),
</math>
::<math>
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).
</math>

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

::<math>
u^{+1}_{+1}(\mathbf{-p}) = u^{-1}_{-1}(\mathbf{p}),
</math>
::<math>
u^{-1}_{-1}(\mathbf{-p}) = u^{+1}_{+1}(\mathbf{p}),
</math>
::<math>
u^{+1}_{-1}(\mathbf{-p}) = u^{-1}_{+1}(\mathbf{p}),
</math>
::<math>
u^{-1}_{+1}(\mathbf{-p}) = u^{+1}_{-1}(\mathbf{p}).
</math>

===The composite photon field===
De Broglie<ref name=broglie1/> and Kronig<ref name=kronig/> suggested the use of a local interaction to bind the neutrino–antineutrino pair. (Rosen and Singer<ref name="rosen">
{{Cite journal |last1=Rosen |first1=N. |last2=Singer |first2=P. |name-list-style=amp |year=1959 |title=The photon as a composite particle |journal=Bulletin of the Research Council of Israel |volume=8F |issue=5 |pages=51–62 |osti=4200679}}</ref> have used a [[Physics:Delta potential|delta potential]] interaction in forming a composite photon.) Fermi and Yang<ref name="fermi-yang">
{{Cite journal |last1=Fermi |first1=Enrico |last2=Yang |first2=C. N. |name-list-style=amp |year=1949 |title=Are mesons elementary particles |journal=[[Physics:Physical Review|Physical Review]] |volume=76 |issue=12 |pages=1739–1743 |bibcode=1949PhRv...76.1739F |doi=10.1103/PhysRev.76.1739}}</ref> 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,<ref name="bjorken_qm">
{{Cite book |last1=Bjorken |first1=J. D. |url=https://archive.org/details/relativisticquan0000bjor_c5q0 |title=Relativistic Quantum Fields |last2=Drell |first2=S. D. |publisher=McGraw-Hill |year=1965 |location=New York, New York |language=en-us |url-access=registration |name-list-style=amp}}</ref>
::<math>
\Psi^\dagger \gamma_0 \gamma_{\mu} \Psi.
</math>

Forming the photon field can be done simply by,
::<math>
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. </math>
::<math>\left. + \left[Q_R^\dagger(\mathbf{p}) u^{+1}_{+1}(\mathbf{p})^\dagger \gamma_0 \gamma_{\mu} u^{-1}_{-1}(\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)
</math>
where <math>p x = \mathbf{p} \cdot \mathbf{x} - p_0t = \mathbf{p} \cdot \mathbf{x} - Et</math>.

The annihilation operators for right-handed and left-handed photons formed of fermion–antifermion pairs are defined as,<ref name="lipkin">
{{Cite book |author=Lipkin |first=H. J. |title=Quantum Mechanics |publisher=North-Holland |year=1973 |location=Amsterdam (Netherlands)}}</ref><ref name="sahlin">
{{Cite journal |last1=Sahlin |first1=H. L. |last2=Schwartz |first2=J. L. |name-list-style=amp |year=1965 |title=The many body problem for composite particles |journal=[[Physics:Physical Review|Physical Review]] |volume=138 |issue=1B |pages=B267–B273 |bibcode=1965PhRv..138..267S |doi=10.1103/PhysRev.138.B267}}</ref><ref name="landau1">
{{Cite book |author=Landau |first=R. H. |title=Quantum Mechanics II |publisher=Wiley |year=1996 |location=New York, New York |language=en-us}}</ref><ref name="perkins2">
{{Cite journal |author=Perkins |first=W. A. |year=1972 |title=Statistics of a composite photon formed of two fermions |journal=[[Physics:Physical Review D|Physical Review D]] |volume=5 |issue=6 |pages=1375–1384 |bibcode=1972PhRvD...5.1375P |doi=10.1103/PhysRevD.5.1375}}</ref>
::<math>
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 ] </math>
::<math>
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 ].
</math>

<math>F(\mathbf{k})</math> is a spectral function, normalized by
<math>\sum_\mathbf{k} \left| F(\mathbf{k}) \right|^2 = 1.</math>

==Photon polarization vectors==
The polarization vectors corresponding to the combinations used in Eq. (1) are,
::<math>
\epsilon_\mu^1( p ) = {-1 \over \sqrt{2}} [u^{-1}_{-1}(\mathbf{p})]^\dagger
 \gamma_0 \gamma_{\mu} u^{+1}_{+1}(\mathbf{p}),  </math>
::<math>
\epsilon_\mu^2( p ) = {-1 \over \sqrt{2}} [u^{+1}_{+1}(\mathbf{p})]^\dagger
\gamma_0 \gamma_{\mu} u^{-1}_{-1}(\mathbf{p}).
</math>

Carrying out the matrix multiplications results in,
::<math>
\epsilon_\mu^1(p) \!= \!{1 \over \sqrt{2}} \left(
{{- 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), </math>
::<math>
\epsilon_\mu^2(p) \!= \!{1 \over \sqrt{2}} \left(
{{-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)

</math>
where <math>\epsilon_0^1(p)</math> and <math>\epsilon_0^2(p)</math> have been placed on the right.

For massless fermions the polarization vectors depend only upon the direction of
<math>\mathbf{p}</math>. Let <math>\mathbf{n} = \mathbf{p}/ |\mathbf{p}|</math>.
::<math>
\epsilon_\mu^1(n) \!= \!{1 \over \sqrt{2}} \left(
{{- n_1 n_2 \!+ \!in_1^2 \!+ \!in_3^2 \!+ \!in_3}
\over {1 + n_3}},
\!-n_1 \!- \!i n_2, 0 \right), </math>
::<math>
\epsilon_\mu^2(n) \!= \!{1 \over \sqrt{2}} \left(
{{- n_1 n_2 \!- \!in_1^2 \!- \!in_3^2 \!- \!in_3}
\over {1 + n_3}},
\!-n_1 \!+ \!i n_2, 0 \right).
</math>
These polarization vectors satisfy the normalization relation,
::<math>
\epsilon_\mu^j(p) \cdot \epsilon_\mu^{j*}(p) = 1, </math>
::<math>
\epsilon_\mu^j(p) \cdot \epsilon_\mu^{k*}(p) = 0 \;\; \text{for} \;\; k \ne j.
</math>

The Lorentz-invariant dot
products of the internal four-momentum <math>p_\mu</math>
with the polarization vectors are,
::<math>
p_\mu \epsilon_\mu^1(p) = 0, </math>
::<math>
p_\mu \epsilon_\mu^2(p) = 0. \quad\quad\quad\quad (3)
</math>

In three dimensions,
::<math>
\mathbf{p} \cdot  \mathbf{\epsilon^1}(\mathbf{p}) =
\mathbf{p} \cdot  \mathbf{\epsilon^2}(\mathbf{p}) = 0, </math>
::<math>
\mathbf{\epsilon^1}(\mathbf{p}) \times
\mathbf{\epsilon^2}(\mathbf{p})= -i\mathbf{p} / p_0, </math>
::<math>
\mathbf{p} \times \mathbf{\epsilon^1}(\mathbf{p})=-i p_0
\mathbf{\epsilon^1}(\mathbf{p}), </math>
::<math>
\mathbf{p} \times \mathbf{\epsilon^2}(\mathbf{p})= i p_0
\mathbf{\epsilon^2}(\mathbf{p}). \quad\quad\quad\quad (4)
</math>

==Composite photon satisfies Maxwell’s equations==
In terms of the polarization vectors, <math>A_\mu(x)</math> becomes,
::<math>
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.
</math>
::<math>
\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)
</math>

The electric field <math>\mathbf{E}~</math>and magnetic field <math>\mathbf{H}~</math>are given by,
::<math>
\mathbf{E}(x) = - { \partial \mathbf{A}(x) \over \partial t }, </math>
::<math>
\mathbf{H}(x) = \nabla \times \mathbf{A}(x).  \quad\quad\quad\quad (6)

</math>

Applying Eq. (6) to Eq. (5), results in,
::<math>
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.
</math>
::<math>
\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\}.
</math>
::<math>
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.
</math>
::<math>
\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\}.
</math>

[[Physics:Maxwell's equations|Maxwell's equations]] for free space are obtained as follows:
::<math>
\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.
</math>
::<math>
\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\}.
</math>
Thus, <math>\partial E_1(x) / \partial x_1 + \partial
E_2(x) / \partial x_2 +
\partial E_3(x) / \partial x_3</math> contains terms of the form
<math>p_1 \epsilon_1^1(\mathbf{p}) + p_2 \epsilon_2^1(\mathbf{p})
+ p_3 \epsilon_3^1(\mathbf{p})</math> which equate to zero by the first of Eq. (4). This gives,
::<math>
\nabla \cdot \mathbf{E}(x) = 0, </math>
::<math>
\nabla \cdot \mathbf{H}(x) = 0.
</math>
as <math>\mathbf{H}</math> contains similar terms.

The expression <math>\nabla \times \mathbf{E}(x)</math> contains terms of the form
<math>\mathbf{p} \times \mathbf{\epsilon^1}(\mathbf{p})</math> while
<math>\partial \mathbf{H}(x) / \partial t</math>
contains terms of form <math>i p_0 \mathbf{\epsilon^1}(\mathbf{p})</math>. Thus, the last two equations of (4) can be used to show that,
::<math>
\nabla \times \mathbf{E}(x) = - \partial \mathbf{H}(x) / \partial t, </math>
::<math>
\nabla \times \mathbf{H}(x) = \partial \mathbf{E}(x) / \partial t.
</math>

Although the neutrino field violates [[Physics:Parity|parity]] and charge
conjugation,<ref name="lee-yang">
{{Cite journal |last1=Lee |first1=T. D. |last2=Yang |first2=C. N. |name-list-style=amp |year=1957 |title=Parity nonconservation and two-component theory of the neutrino |journal=[[Physics:Physical Review|Physical Review]] |volume=105 |issue=5 |pages=1671–1675 |bibcode=1957PhRv..105.1671L |doi=10.1103/PhysRev.105.1671 |s2cid=13280252}}</ref> <math>\mathbf{E}</math> and <math>\mathbf{H}</math> transform in the usual way,<ref name=perkins1/><ref name=perkins2/>
::<math>
P \mathbf{E}(\mathbf{x},t) P^-1 = -\mathbf{E}(\mathbf{-x},t), </math>
::<math>
P \mathbf{H}(\mathbf{x},t) P^-1 = \mathbf{H}(\mathbf{-x},t), </math>
::<math>
C \mathbf{E}(\mathbf{x},t) C^-1 = -\mathbf{E}(\mathbf{x},t), </math>
::<math>
C \mathbf{H}(\mathbf{x},t) C^-1 = -\mathbf{H}(\mathbf{x},t).
</math>

<math>A_\mu</math> satisfies the Lorenz condition,
::<math>
\partial A_\mu / \partial x_\mu = 0
</math>
which follows from Eq. (3).

Although many choices for [[Gamma matrices|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 <math>\mathbf{E}</math> and <math>\mathbf{H}</math> that satisfy [[Physics:Maxwell's equations|Maxwell's equations]]. Kronig<ref name=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.<ref name="sen">
{{Cite journal |author=Sen |first=D. K. |year=1964 |title=A theoretical basis for two neutrinos |journal=Il Nuovo Cimento |volume=31 |issue=3 |pages=660–669 |bibcode=1964NCim...31..660S |doi=10.1007/BF02733763 |s2cid=122038895}}</ref> One can also produce the above results using a two-component neutrino theory.<ref name=perkins3/>

To compute the commutation relations for the photon field, one needs the equation,

::<math>
\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}.
</math>
To obtain this equation, Kronig<ref name=kronig/> wrote a relation between the neutrino spinors that was not rotationally invariant as pointed out by Pryce.<ref name=pryce/> However, as Perkins<ref name =perkins1/> 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, <math>\epsilon_\mu^1(n)</math>
and <math>\epsilon_\mu^2(n)</math> reduce to the usual polarization vectors for right and left circularly polarized photons respectively.
::<math>
\epsilon_\mu^1(n) = {1 \over \sqrt{2}}(1,i,0,0), </math>
::<math>
\epsilon_\mu^2(n) = {1 \over \sqrt{2}}(1,-i,0,0).
</math>

==Inconsistencies==
Although composite photons satisfy many properties of real photons, there are major problems with this theory.

===Bose–Einstein commutation relations===
It is known that a photon is a boson.<ref name=amsler>{{Cite journal
|doi=10.1016/j.physletb.2008.07.018
|author=C. Amsler ''et al.'' (Particle Data Group)
|year=2008
|title=The review of particle physics
|journal=[[Physics:Physics Letters B|Physics Letters B]]
|volume=667
|issue=1–5
|pages=1–1340
|bibcode=2008PhLB..667....1A
|url=https://www.zora.uzh.ch/id/eprint/11249/2/scalarsV.pdf
|hdl=1854/LU-685594
|s2cid=227119789
|hdl-access=free
|access-date=2019-12-14
|archive-date=2020-09-07
|archive-url=https://web.archive.org/web/20200907211544/https://www.zora.uzh.ch/id/eprint/11249/2/scalarsV.pdf
|url-status=dead
}}</ref> 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

::<math>
\{a(\mathbf{k}),a(\mathbf{l})\} = 0,  </math>
::<math>
\{a^\dagger(\mathbf{k}),a^\dagger(\mathbf{l})\} = 0,  </math>
::<math>
\{a(\mathbf{k}),a^\dagger(\mathbf{l})\}
= \delta(\mathbf{k}-\mathbf{l}),
</math>

while bosons are defined as the particles that adhere to the commutation relations

::<math>
\left[b(\mathbf{k}),b(\mathbf{l})\right] = 0,  </math>
::<math>
\left[b^\dagger(\mathbf{k}),b^\dagger(\mathbf{l})\right] = 0,  </math>
::<math>
\left[b(\mathbf{k}),b^\dagger(\mathbf{l})\right]
= \delta(\mathbf{k}-\mathbf{l}). \quad\quad (7)
</math>

The creation and annihilation operators of composite particles formed of fermion pairs adhere to the commutation relations of the form<ref name=lipkin/><ref name=sahlin/><ref name=landau1/><ref name=perkins2/>

::<math>
\left[Q(\mathbf{k}),Q(\mathbf{l})\right] = 0, </math>
::<math>
\left[Q^\dagger(\mathbf{k}),Q^\dagger(\mathbf{l})\right] = 0, </math>
::<math>
\left[Q(\mathbf{k}),Q^\dagger(\mathbf{l})\right]
= \delta(\mathbf{k}-\mathbf{l})- \Delta(\mathbf{k},\mathbf{l}). \quad\quad (8)
</math>

with

::<math>
\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.  </math>
::<math>
\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)
</math>

For [[Physics:Cooper pair|Cooper electron pairs]],<ref name=landau1/> "''a''" and "''c''" represent different spin directions. For nucleon pairs (the [[Physics:Deuterium|deuteron]]),<ref name=lipkin/><ref name=sahlin/> "''a''" and "''c''" represent proton and neutron. For neutrino–antineutrino pairs,<ref name=perkins2/> "a" and "c" represent neutrino and antineutrino. The size of the deviations from pure Bose behavior,

:<math>\Delta(\mathbf{p}^{\prime},\mathbf{p}), </math>

depends on the degree of overlap of the fermion wave functions and the constraints of the [[Physics:Pauli exclusion principle|Pauli exclusion principle]].

If the state has the form

::<math>
|\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
</math>

then the expectation value of Eq. (9) vanishes for <math>\mathbf{p}^{\prime} \ne \mathbf{p}</math>, and the expression for <math>\Delta(\mathbf{p}^{\prime},\mathbf{p})</math> can be approximated by

::<math>
\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.  </math>
::<math>
\left.
+ c^\dagger(\mathbf{p}/2+\mathbf{k})
c(\mathbf{p}/2+\mathbf{k}) \right].
</math>

Using the fermion number operators <math>n_a(\mathbf{k}) </math> and <math>n_c(\mathbf{k}) </math>, this can be written,

::<math>
\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] </math>
::<math>
= \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] </math>
::<math>
= \delta(\mathbf{p}^{\prime}-\mathbf{p})
\overline {\Delta} (\mathbf{p},\mathbf{p})
</math>
showing that it is the average number of fermions in a particular state <math>\mathbf{k}</math> averaged
over all states with weighting factors <math> F( \mathbf{p}/2-\mathbf{k}) </math> and <math> F(\mathbf{k}-\mathbf{p}/2) </math>.

====Jordan’s attempt to solve problem====

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<ref name=pryce/> noted. Jordan<ref name=jordan2/> "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.<ref name=fock>
{{Cite journal
|author=Fock
|year=1937
|title=
|journal=Phys. Z. Sowjetunion
|volume=11 |page=1
}}</ref> 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. Greenberg and Wightman<ref name="greenberg">
{{Cite journal |last1=Greenberg |first1=O. W. |last2=Wightman |first2=A. S. |name-list-style=amp |year=1955 |title=Re-examination of the neutrino theory of light |journal=[[Physics:Physical Review|Physical Review]] |volume=99 |issue=2 |pages=675 A |bibcode=1955PhRv...99..605. |doi=10.1103/PhysRev.99.605}}</ref> have pointed out why the one-dimensional case works, but the three-dimensional case does not.

In 1928, Jordan noticed that commutation relations for pairs of fermions were similar to those for bosons.<ref name="jordan1">
{{Cite journal |author=Jordan |first=P. |year=1928 |title=Die Lichtquantenhypothese: Entwicklung und gegenwärtiger Stand |journal=Ergebnisse der Exakten Naturwissenschaften |volume=7 |pages=158–208 |bibcode=1928ErNW....7..158J |doi=10.1007/BFb0111850}}</ref> Compare Eq. (7) with Eq. (8). From 1935 until 1937, Jordan, Kronig, and others<ref name="born">
{{Cite journal |author1=Born |first=Max |author2=N. S. Nagendra Nath |name-list-style=amp |year=1936 |title=The neutrino theory of light |journal=Proc. Indian Acad. Sci. |volume=A3 |page=318 |doi=10.1007/BF03046268 |s2cid=198143285}}</ref> 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 <math>\mathbf{p}</math> could be simulated by a Raman effect in which a neutrino with momentum <math>\mathbf{p+k}</math> is absorbed while another of another with opposite spin and momentum <math>\mathbf{k}</math> is emitted. (It is now known that single neutrinos or antineutrinos interact so weakly that they cannot simulate photons.)

====Pryce’s theorem====
In 1938, Pryce<ref name=pryce/> showed that one cannot obtain both [[Physics:Bose–Einstein statistics|Bose–Einstein statistics]] and transversely polarized photons from neutrino–antineutrino pairs. Construction of transversely polarized photons is not the problem.<ref name="case">
{{Cite journal |author=Case |first=K. M. |year=1957 |title=Composite particles of zero mass |journal=[[Physics:Physical Review|Physical Review]] |volume=106 |issue=6 |pages=1316–1320 |bibcode=1957PhRv..106.1316C |doi=10.1103/PhysRev.106.1316}}</ref> As Berezinski<ref name=berez/> 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:

::<math>
\left[ Q_R(\mathbf{p}^{\prime}),
Q_R(\mathbf{p}) \right] = 0, \;
\left[ Q_L(\mathbf{p}^{\prime}),
Q_L(\mathbf{p}) \right] = 0, </math>
::<math>
\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})),  </math>
::<math>
\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})), </math>
::<math>
\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)
</math>

where

::<math>
{\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.  </math>
::<math>
\left.
+ \left| F(\mathbf{p}/2-\mathbf{k}) \right|^2 ( n_{c1}(\mathbf{k}) + n_{a2}(\mathbf{k}) )
\right].  \quad\quad\quad\quad (11)
</math>

The deviation from [[Physics:Bose–Einstein statistics|Bose–Einstein statistics]] is caused by <math>\overline \Delta_{12}(\mathbf{p},\mathbf{p})</math> and
<math>\overline \Delta_{21}(\mathbf{p},\mathbf{p})</math>, which are functions of the neutrino numbers operators.

Linear polarization photon operators are defined by

::<math>
\xi( \mathbf{p}) = {1 \over \sqrt{2}} \left[ Q_L(\mathbf{p})
+ Q_R(\mathbf{p}) \right], </math>
::<math>
\eta( \mathbf{p}) = {i \over \sqrt{2}} \left[ Q_L(\mathbf{p})
- Q_R(\mathbf{p}) \right]. \quad\quad\quad\quad (12)

</math>
A particularly interesting commutation relation is,
::<math>
[\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)

</math>
which follows from (10) and (12).

For the composite photon to obey Bose–Einstein commutation relations, at the very least,

::<math>
[\xi( \mathbf{p}^{\prime}),\eta^\dagger( \mathbf{p})]
= 0   \quad\quad\quad\quad (14)
</math>
Pryce noted.<ref name=pryce/> From Eq. (11) and Eq. (13) the requirement is that
::<math>
\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. </math>
::<math>
\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]
</math>

gives zero when applied to any state vector. Thus, all the coefficients of
<math>n_{a1}(\mathbf{k})</math> and <math>n_{c1}(\mathbf{k})</math>,
etc. must vanish separately. This means <math>F(\mathbf{k}) = 0</math>, and the composite photon does not exist,<ref name=pryce/><ref name=berez/> completing the proof.

====Perkins’ attempt to solve problem====
Perkins<ref name=perkins1/><ref name=perkins2/> 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<ref name=perkins4/> 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 [[Physics:Pauli exclusion principle|Pauli exclusion principle]]."

Brzezinski in reaffirming Pryce's theorem argues that commutation relation (14) is necessary for the photon to be truly neutral. However, Perkins<ref name=perkins2/> 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

::<math>
N(\mathbf{p}) = Q^\dagger(\mathbf{p}) Q(\mathbf{p}).
</math>

Lipkin<ref name=lipkin/> suggested for a rough estimate to assume
that <math>F(\mathbf{k})= 1 / \Omega</math> where <math>\Omega</math> is a constant equal to the number of states used to construct the [[Physics:Wave packet|wave packet]].

Perkins<ref name=perkins4/> showed that the effect of the composite photon's
number operator acting on a state of <math>m</math> composite photons is,

::<math>
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,
</math>
using <math>N(\mathbf{p})|0\rangle = 0</math>. This result differs from the usual
one because of the second term which is small for large <math>\Omega</math>. Normalizing in the
usual manner,<ref name="koltun">
{{Cite book |last1=Koltun |first1=D. S. |title=Quantum Mechanics of Many Degrees of Freedom |last2=Eisenberg |first2=J. M. |publisher=Wiley |year=1988 |location=New York, New York |language=en-us |name-list-style=amp}}</ref>

::<math>
 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,  </math>
::<math>
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)

</math>

where <math>|n_\mathbf{p}\rangle</math> is the state of <math>n_\mathbf{p}</math>
composite photons having momentum <math>\mathbf{p}</math> which is created by applying <math>Q^\dagger(\mathbf{p})</math> on the vacuum <math>n_\mathbf{p}</math> times. Note that,
::<math>
Q^\dagger(\mathbf{p})|0 \rangle =  | 1_\mathbf{p}\rangle, </math>
::<math>
Q(\mathbf{p})|1_\mathbf{p}\rangle  =  |0\rangle,
</math>
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 <math>\Omega</math>.

=====Blackbody radiation=====
The main evidence indicating that photons are bosons comes from the Blackbody radiation experiments which are in agreement with Planck's distribution. Perkins<ref name=perkins4/> calculated the photon distribution for Blackbody radiation using the [[Physics:Second quantization|second quantization]] method,<ref name=koltun/> 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 β and absorbed at the lower level α. The transition probability for emission of a photon is enhanced when ''n''<sub>'''p'''</sub> photons are present,
::<math>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)</math>
where the first of (15) has been used. The absorption is enhanced less since the second of (15) is used,
::<math>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)</math>
Using the equality,
::<math>w_{ \beta \alpha}( 0 \leftarrow  1_\mathbf{p}) = w_{ \alpha \beta}(  1_\mathbf{p} \leftarrow 0 ),</math>
of the transition rates, Eqs. (16) and (17) are combined to give,
::<math>{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)}.</math>

The probability of finding the system with energy E is proportional to e<sup>&minus;E/kT</sup> according to Boltzmann's distribution law. Thus, the equilibrium between emission and absorption requires that,
::<math>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}, </math>
with the photon energy <math> \omega_p = E_{\beta} -  E_{\alpha} </math>. Combining the last two equations results in,
::<math>n_\mathbf{p} = {2 \over u+(u+2)/ \Omega + \sqrt{u^2(1+2/\Omega) + (u+2)^2 /\Omega^2}}, </math>
with <math>u =  e^{\omega_p/kT} - 1</math>. For <math>\Omega(\omega_p /kT) \gg  1</math>, this reduces to
::<math>n_\mathbf{p} = {1 \over e^{\omega_p /kT} \left( 1 + {1 \over \Omega} \right) - 1}.</math>

This equation differs from Planck's law because of the <math>1 / \Omega</math> term. For the conditions used in the blackbody radiation experiments of W. W. Coblentz,<ref name="coblentz">
{{Cite journal |author=Coblentz |first=W. W. |year=1916 |title=Constants of spectral radiation of a uniformly heated inclosure or so-called black body. II |journal=Bulletin of the Bureau of Standards |volume=13 |issue=3 |page=459 |doi=10.6028/bulletin.310 |doi-access=free}}</ref> Perkins estimates that {{nowrap|1 / Ω < {{val|e=-9}}}}, and the maximum deviation from [[Physics:Planck's law|Planck's law]] is less than one part in {{val|e=-8}}, which is too small to be detected.

===Only left-handed neutrinos exist===
Experimental results show that only left-handed neutrinos
and right-handed antineutrinos exist. Three sets of neutrinos
have been observed,<ref name="danby">
{{Cite journal |author=Danby |first1=G. |last2=Gaillard, K. Goulianos, L. M. Lederman, N. Mistry, M. Schwartz, and J. Steinberger |first2=J-M. |last3=Goulianos |first3=K. |last4=Lederman |first4=L. M. |last5=Mistry |first5=N. |last6=Schwartz |first6=M. |last7=Steinberger |first7=J. |year=1962 |title=Observation of high-energy neutrino interactions and the existence of two kinds of neutrinos |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=9 |issue=1 |pages=36–44 |bibcode=1962PhRvL...9...36D |doi=10.1103/PhysRevLett.9.36 |s2cid=120314867}}</ref><ref name=kodama>
{{Cite journal
|doi=10.1016/S0370-2693(01)00307-0
|author=K. Kodama ''et al.'' (DONUT collaboration)
|year=2001
|title=Observation of tau neutrino interactions
|journal=[[Physics:Physics Letters B|Physics Letters B]]
|volume=504
|issue=3 |pages=218–224
|arxiv = hep-ex/0012035 |bibcode = 2001PhLB..504..218D |s2cid=28649613
}}</ref> one that is connected with electrons, one with muons, and one with tau leptons.<ref name="perl">
{{Cite journal |author=Perl |first=M. L. |display-authors=0 |year=1975 |title=Evidence for anomalous lepton production in e+ – e− annihilation |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=35 |issue=22 |pages=1489–1492 |bibcode=1975PhRvL..35.1489P |doi=10.1103/PhysRevLett.35.1489 |s2cid=122060301}}</ref>

In the standard model the pion and muon decay modes are:
::{|
|{{Subatomic particle|Pion+}} || → || {{Subatomic particle|Muon+}} || + || {{Subatomic particle|Muon neutrino}}
|----
|{{Subatomic particle|Muon+}} || → || {{Subatomic particle|Positron}} || + || {{Subatomic particle|Electron neutrino}} || + || {{Subatomic particle|Muon antineutrino}}
|}

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.<ref name=perkins1/>) 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 {{Subatomic particle|Neutrino}}<sub>1</sub> be the right-handed neutrino and {{Subatomic particle|Neutrino}}<sub>2</sub> the left-handed neutrino with their corresponding antineutrinos (with opposite helicity). The neutrinos involved in beta decay are {{Subatomic particle|Neutrino}}<sub>2</sub> and {{Subatomic particle|Antineutrino}}<sub>2</sub>, while those for π–μ decay are {{Subatomic particle|Neutrino}}<sub>1</sub> and {{Subatomic particle|Antineutrino}}<sub>1</sub>. With this scheme the pion and muon decay modes are:
::{|
|{{Subatomic particle|Pion+}} || → || {{Subatomic particle|Muon+}} || + || {{Subatomic particle|Antineutrino}}<sub>1</sub>
|----
|{{Subatomic particle|Muon+}} || → || {{Subatomic particle|Positron}} || + || {{Subatomic particle|Neutrino}}<sub>2</sub> || + || {{Subatomic particle|Neutrino}}<sub>1</sub>
|}

===Absence of massless neutrinos===
There is convincing evidence that neutrinos have mass. In experiments at the SuperKamiokande researchers<ref name=fukuda/> have discovered neutrino oscillations in which one flavor of neutrino changed into another. This means that neutrinos have non-zero mass outside the [[Physics:Einstein–Cartan theory|Einstein-Cartan torsion]].<ref name="Neutrino oscillations in the presence of torsion">{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | doi=10.1007/BF02902051 | pages=479–500 }}</ref><ref name="Geometrical contribution to neutrino mass matrix">{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 }}</ref> Since massless neutrinos are needed to form a massless photon, a composite photon is not possible outside the [[Physics:Einstein–Cartan theory|Einstein-Cartan torsion]].<ref name="Neutrino oscillations in the presence of torsion">{{cite journal | author1=V. De Sabbata | author2=M. Gasperini  | title=Neutrino oscillations in the presence of torsion | journal=Il Nuovo Cimento A (1971-1996) | volume=65 | year=1981 | doi=10.1007/BF02902051 | pages=479–500 }}</ref><ref name="Geometrical contribution to neutrino mass matrix">{{cite journal | author1=Subhasish Chakrabarty | author2=Amitabha Lahiri | title=Geometrical contribution to neutrino mass matrix | journal=The European Physical Journal C | year=2019 | volume=79 | pages=697 | doi=10.1140/epjc/s10052-019-7209-2 | arxiv=1904.06036 }}</ref>

==References==
<!--- See Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist}}

==External links==
*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/de_broglie_-_new_conception_of_light.pdf L. de Broglie "A new conception of light"] (English translation)

{{DEFAULTSORT:Neutrino Theory Of Light}}
[[Category:Light]]
[[Category:Obsolete theories in physics]]

{{Sourceattribution|Neutrino theory of light}}