The neutrino theory of light is the proposal that the photon is a composite particle formed of a neutrino–antineutrino pair. It is based on the idea that emission and absorption of a photon corresponds to the creation and annihilation of a particle–antiparticle pair. The neutrino theory of light is not currently accepted as part of mainstream physics, as according to the Standard Model the photon is an elementary particle, a 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, Louis de Broglie 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 Pascual Jordan, Ralph Kronig, Max Born, and others worked on the theory. In 1938, Maurice Pryce 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. In 1966, V. S. Berezinskii reanalyzed Pryce's paper, giving a clearer picture of the problem that Pryce uncovered. Starting in the 1960s, work on the neutrino theory of light resumed, and there continues to be some interest in recent years. 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 with the current photon model. However, there is no experimental evidence that the photon has a composite structure. Experimental results show that only left-handed neutrinos and right-handed antineutrinos exist. Three sets of neutrinos have been observed, one that is connected with electrons, one with muons, and one with tau leptons. There is convincing evidence that some neutrinos have mass. In experiments at the SuperKamiokande researchers have discovered neutrino oscillations in which one flavor of neutrino changed into another. It has been shown that the smallest of the three neutrinos could have mass = 0. Also, a massless photon could be formed with non-zero mass neutrinos inside the Einstein-Cartan torsion. Here, it is assumed that the composite photon is formed of a massless electron neutrino, ν 2 {\displaystyle \nu _{2}} and a massless electron antineutrino ν ¯ 2 {\displaystyle {\overline {\nu }}_{2}} . 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.
The neutrino field The neutrino field satisfies the Dirac equation with the mass set to zero,
γ μ p μ Ψ = 0. {\displaystyle \gamma ^{\mu }p_{\mu }\Psi =0.}
The gamma matrices in the Weyl basis are:
γ 0 = ( 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ) , {\displaystyle \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),}
γ 2 = ( 0 0 0 − i 0 0 i 0 0 i 0 0 − i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 − 1 − 1 0 0 0 0 1 0 0 ) . {\displaystyle \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 γ 0 {\displaystyle \gamma ^{0}} is Hermitian while γ k {\displaystyle \gamma ^{k}} is antihermitian. They satisfy the anticommutation relation,
γ μ γ ν + γ ν γ μ = 2 η μ ν I {\displaystyle \gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }I}
where η μ ν {\displaystyle \eta ^{\mu \nu }} is the Minkowski metric with signature ( + − − − ) {\displaystyle (+---)} and I {\displaystyle I} is the unit matrix.
a 1 {\displaystyle a_{1}} and c 1 {\displaystyle c_{1}} are designated as the fermion annihilation operators for ν 1 {\displaystyle \nu _{1}} and ν ¯ 1 {\displaystyle {\overline {\nu }}_{1}} respectively, while a 2 {\displaystyle a_{2}} and c 2 {\displaystyle c_{2}} are designated as the annihilation operators for ν 2 {\displaystyle \nu _{2}} and ν ¯ 2 {\displaystyle {\overline {\nu }}_{2}} .
ν 1 {\displaystyle \nu _{1}} is a right-handed neutrino and ν 2 {\displaystyle \nu _{2}} is a left-handed neutrino. Since only ν 2 {\displaystyle \nu _{2}} and ν ¯ 2 {\displaystyle {\overline {\nu }}_{2}} have been observed, the neutrino field is given by, The neutrino field is given by,
Ψ ( x ) = 1 V ∑ k { [ a 2 ( k ) u − 1 + 1 ( k ) ] e i k x {\displaystyle \Psi (x)={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }\left\{\left[a_{2}(\mathbf {k} )u_{-1}^{+1}(\mathbf {k} )\right]e^{ikx}\right.}
+ [ c 2 † ( k ) u + 1 − 1 ( − k ) ] e − i k x } , {\displaystyle \left.+\left[c_{2}^{\dagger }(\mathbf {k} )u_{+1}^{-1}(\mathbf {-k} )\right]e^{-ikx}\right\},}
where k x {\displaystyle kx} stands for k ⋅ x − k 0 t {\displaystyle \mathbf {k} \cdot \mathbf {x} -k_{0}t} . The u {\displaystyle u} 's are spinors with the superscripts and subscripts referring to the energy and helicity states respectively. Spinor solutions for the Dirac equation are,
u + 1 + 1 ( p ) = E + p 3 2 E ( 1 p 1 + i p 2 E + p 3 0 0 ) , {\displaystyle u_{+1}^{+1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}1\\{{p_{1}+ip_{2}} \over {E+p_{3}}}\\0\\0\end{array}}\right),}
u − 1 − 1 ( p ) = E + p 3 2 E ( − p 1 + i p 2 E + p 3 1 0 0 ) , {\displaystyle u_{-1}^{-1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}{{-p_{1}+ip_{2}} \over {E+p_{3}}}\\1\\0\\0\end{array}}\right),}
u + 1 − 1 ( p ) = E + p 3 2 E ( 0 0 1 p 1 + i p 2 E + p 3 ) , {\displaystyle u_{+1}^{-1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}0\\0\\1\\{{p_{1}+ip_{2}} \over {E+p_{3}}}\end{array}}\right),}
u − 1 + 1 ( p ) = E + p 3 2 E ( 0 0 − p 1 + i p 2 E + p 3 1 ) . {\displaystyle u_{-1}^{+1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}0\\0\\{{-p_{1}+ip_{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 ( − p ) = u − 1 − 1 ( p ) , {\displaystyle u_{+1}^{+1}(\mathbf {-p} )=u_{-1}^{-1}(\mathbf {p} ),}
u − 1 − 1 ( − p ) = u + 1 + 1 ( p ) , {\displaystyle u_{-1}^{-1}(\mathbf {-p} )=u_{+1}^{+1}(\mathbf {p} ),}
u − 1 + 1 ( − p ) = u + 1 − 1 ( p ) , {\displaystyle u_{-1}^{+1}(\mathbf {-p} )=u_{+1}^{-1}(\mathbf {p} ),}
u + 1 − 1 ( − p ) = u − 1 + 1 ( p ) . {\displaystyle u_{+1}^{-1}(\mathbf {-p} )=u_{-1}^{+1}(\mathbf {p} ).}
The composite photon field De Broglie and Kronig suggested the use of a local interaction to bind the neutrino–antineutrino pair. (Rosen and Singer have used a delta potential interaction in forming a composite photon.) Fermi and Yang used a local interaction to bind a fermion–antifermion pair in attempting to form a pion. A four-vector field can be created from a fermion–antifermion pair,
Ψ † γ 0 γ μ Ψ . {\displaystyle \Psi ^{\dagger }\gamma _{0}\gamma _{\mu }\Psi .}
Forming the photon field can be done simply by,
A μ ( x ) = ∑ p − 1 2 V p 0 { [ Q R ( p ) u − 1 + 1 ( p ) † γ 0 γ μ u + 1 − 1 ( p ) + Q L ( p ) u + 1 − 1 ( p ) † γ 0 γ μ u − 1 + 1 ( p ) ] e i p x {\displaystyle A_{\mu }(x)=\sum _{\mathbf {p} }{-1 \over 2{\sqrt {Vp_{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^{ipx}\right.}
+ [ Q R † ( p ) u + 1 − 1 ( p ) † γ 0 γ μ u − 1 + 1 ( p ) + Q L † ( p ) u − 1 + 1 ( p ) † γ 0 γ μ u + 1 − 1 ( p ) ] e − i p x } , ( 1 ) {\displaystyle \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^{-ipx}\right\},\quad \quad (1)}
where p x = p ⋅ x − p 0 t = p ⋅ x − E t {\displaystyle px=\mathbf {p} \cdot \mathbf {x} -p_{0}t=\mathbf {p} \cdot \mathbf {x} -Et} . The annihilation operators for right-handed and left-handed photons formed of fermion–antifermion pairs are defined as,
Q R ( p ) = ∑ k F † ( k ) [ c 2 ( p / 2 + k ) a 2 ( p / 2 − k ) ] {\displaystyle Q_{R}(\mathbf {p} )=\sum _{\mathbf {k} }F^{\dagger }(\mathbf {k} )\left[c_{2}(\mathbf {p} /2+\mathbf {k} )a_{2}(\mathbf {p} /2-\mathbf {k} )\right]}
Q L ( p ) = ∑ k F † ( k ) [ c 2 ( p / 2 − k ) a 2 ( p / 2 + k ) ] . {\displaystyle 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} )\right].}
F ( k ) {\displaystyle F(\mathbf {k} )} is a spectral function, normalized by ∑ k | F ( k ) | 2 = 1. {\displaystyle \sum _{\mathbf {k} }\left|F(\mathbf {k} )\right|^{2}=1.}
Photon polarization vectors The polarization vectors corresponding to the combinations used in Eq. (1) are,
ϵ μ 1 ( p ) = − 1 2 [ u − 1 + 1 ( p ) ] † γ 0 γ μ u + 1 − 1 ( p ) , {\displaystyle \epsilon _{\mu }^{1}(p)={-1 \over {\sqrt {2}}}[u_{-1}^{+1}(\mathbf {p} )]^{\dagger }\gamma _{0}\gamma _{\mu }u_{+1}^{-1}(\mathbf {p} ),}
ϵ μ 2 ( p ) = − 1 2 [ u + 1 − 1 ( p ) ] † γ 0 γ μ u − 1 + 1 ( p ) . {\displaystyle \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,
ϵ μ 1 ( p ) = 1 2 ( − i p 1 p 2 + E 2 + p 3 E − p 1 2 E ( E + p 3 ) , − p 1 p 2 +