The neutrino field

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

${\displaystyle \gamma ^{\mu }p_{\mu }\Psi =0.}$

The gamma matrices in the Weyl basis are:

${\displaystyle \gamma ^{0}=\left({\begin{array}{cccc}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{array}}\right),\;\;\;\;\gamma ^{1}=\left({\begin{array}{cccc}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{array}}\right),}$

${\displaystyle \gamma ^{2}=\left({\begin{array}{cccc}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{array}}\right),\;\;\;\;\gamma ^{3}=\left({\begin{array}{cccc}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{array}}\right).}$

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

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

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

The neutrino field is given by,

${\displaystyle \Psi (x)={1 \over {\sqrt {V}}}\sum _{\mathbf {k} }\left\{\left[a_{1}(\mathbf {k} )u_{+1}^{+1}(\mathbf {k} )+a_{2}(\mathbf {k} )u_{-1}^{+1}(\mathbf {k} )\right]e^{ikx}\right.}$

${\displaystyle \left.+\left[c_{1}^{\dagger }(\mathbf {k} )u_{-1}^{-1}(\mathbf {-k} )+c_{2}^{\dagger }(\mathbf {k} )u_{+1}^{-1}(\mathbf {-k} )\right]e^{-ikx}\right\},}$

where ${\displaystyle kx}$ stands for ${\displaystyle \mathbf {k} \cdot \mathbf {x} -k_{0}t}$. ${\displaystyle a_{1}}$ and ${\displaystyle c_{1}}$ are the fermion annihilation operators for ${\displaystyle \nu _{1}}$ and ${\displaystyle {\overline {\nu }}_{1}}$ respectively, while ${\displaystyle a_{2}}$ and ${\displaystyle c_{2}}$ are the annihilation operators for ${\displaystyle \nu _{2}}$ and ${\displaystyle {\overline {\nu }}_{2}}$. ${\displaystyle \nu _{1}}$ is a right-handed neutrino and ${\displaystyle \nu _{2}}$ is a left-handed neutrino. The ${\displaystyle u}$'s are spinors with the superscripts and subscripts referring to the energy and helicity states respectively. Spinor solutions for the Dirac equation are,

${\displaystyle u_{+1}^{+1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}1\\{{p_{1}+ip_{2}} \over {E+p_{3}}}\\0\\0\end{array}}\right),}$

${\displaystyle u_{-1}^{-1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}{{-p_{1}+ip_{2}} \over {E+p_{3}}}\\1\\0\\0\end{array}}\right),}$

${\displaystyle u_{+1}^{-1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}0\\0\\1\\{{p_{1}+ip_{2}} \over {E+p_{3}}}\end{array}}\right),}$

${\displaystyle u_{-1}^{+1}(\mathbf {p} )={\sqrt {{E+p_{3}} \over 2E}}\left({\begin{array}{c}0\\0\\{{-p_{1}+ip_{2}} \over {E+p_{3}}}\\1\end{array}}\right).}$

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

${\displaystyle u_{+1}^{+1}(\mathbf {-p} )=u_{-1}^{-1}(\mathbf {p} ),}$

${\displaystyle u_{-1}^{-1}(\mathbf {-p} )=u_{+1}^{+1}(\mathbf {p} ),}$

${\displaystyle u_{-1}^{+1}(\mathbf {-p} )=u_{+1}^{-1}(\mathbf {p} ),}$

${\displaystyle u_{+1}^{-1}(\mathbf {-p} )=u_{-1}^{+1}(\mathbf {p} ).}$

The composite photon field

De Broglie^[1] and Kronig^[16] suggested the use of a local interaction to bind the neutrino–antineutrino pair. (Rosen and Singer^[18] have used a delta potential interaction in forming a composite photon.) Fermi and Yang^[19] 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,^[20]

${\displaystyle \Psi ^{\dagger }\gamma _{0}\gamma _{\mu }\Psi .}$

Forming the photon field can be done simply by,

${\displaystyle A_{\mu }(x)=\sum _{\mathbf {p} }{-1 \over 2{\sqrt {Vp_{0}}}}\left\{\left[Q_{R}(\mathbf {p} )u_{-1}^{-1}(\mathbf {p} )^{\dagger }\gamma _{0}\gamma _{\mu }u_{+1}^{+1}(\mathbf {p} )+Q_{L}(\mathbf {p} )u_{+1}^{+1}(\mathbf {p} )^{\dagger }\gamma _{0}\gamma _{\mu }u_{-1}^{-1}(\mathbf {p} )\right]e^{ipx}\right.}$

${\displaystyle \left.+\left[Q_{R}^{\dagger }(\mathbf {p} )u_{+1}^{+1}(\mathbf {p} )^{\dagger }\gamma _{0}\gamma _{\mu }u_{-1}^{-1}(\mathbf {p} )+Q_{L}^{\dagger }(\mathbf {p} )u_{-1}^{-1}(\mathbf {p} )^{\dagger }\gamma _{0}\gamma _{\mu }u_{+1}^{+1}(\mathbf {p} )\right]e^{-ipx}\right\},\quad \quad (1)}$

where ${\displaystyle px=\mathbf {p} \cdot \mathbf {x} -p_{0}t=\mathbf {p} \cdot \mathbf {x} -Et}$.

The annihilation operators for right-handed and left-handed photons formed of fermion–antifermion pairs are defined as,^{[21][22][23][24]}

${\displaystyle Q_{R}(\mathbf {p} )=\sum _{\mathbf {k} }F^{\dagger }(\mathbf {k} )\left[c_{1}(\mathbf {p} /2-\mathbf {k} )a_{1}(\mathbf {p} /2+\mathbf {k} )+c_{2}(\mathbf {p} /2+\mathbf {k} )a_{2}(\mathbf {p} /2-\mathbf {k} )\right]}$

${\displaystyle Q_{L}(\mathbf {p} )=\sum _{\mathbf {k} }F^{\dagger }(\mathbf {k} )\left[c_{2}(\mathbf {p} /2-\mathbf {k} )a_{2}(\mathbf {p} /2+\mathbf {k} )+c_{1}(\mathbf {p} /2+\mathbf {k} )a_{1}(\mathbf {p} /2-\mathbf {k} ),\right].}$

${\displaystyle F(\mathbf {k} )}$ is a spectral function, normalized by ${\displaystyle \sum _{\mathbf {k} }\left|F(\mathbf {k} )\right|^{2}=1.}$

Bose–Einstein commutation relations

It is known that a photon is a boson.^[27] 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

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

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

${\displaystyle \{a(\mathbf {k} ),a^{\dagger }(\mathbf {l} )\}=\delta (\mathbf {k} -\mathbf {l} ),}$

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

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

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

${\displaystyle \left[b(\mathbf {k} ),b^{\dagger }(\mathbf {l} )\right]=\delta (\mathbf {k} -\mathbf {l} ).\quad \quad (7)}$

The creation and annihilation operators of composite particles formed of fermion pairs adhere to the commutation relations of the form^{[21][22][23][24]}

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

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

${\displaystyle \left[Q(\mathbf {k} ),Q^{\dagger }(\mathbf {l} )\right]=\delta (\mathbf {k} -\mathbf {l} )-\Delta (\mathbf {k} ,\mathbf {l} ).\quad \quad (8)}$

with

${\displaystyle \Delta (\mathbf {p} ^{\prime },\mathbf {p} )=\sum _{\mathbf {k} }F^{\dagger }(\mathbf {k} )\left[F(\mathbf {p} ^{\prime }/2-\mathbf {p} /2+\mathbf {k} )a^{\dagger }(\mathbf {p} -\mathbf {p} ^{\prime }/2-\mathbf {k} )a(\mathbf {p} ^{\prime }/2-\mathbf {k} )\right.}$

${\displaystyle \left.+F(\mathbf {p} /2-\mathbf {p} ^{\prime }/2+\mathbf {k} )c^{\dagger }(\mathbf {p} -\mathbf {p} ^{\prime }/2+\mathbf {k} )c(\mathbf {p} ^{\prime }/2+\mathbf {k} )\right].\quad \quad (9)}$

For Cooper electron pairs,^[23] "a" and "c" represent different spin directions. For nucleon pairs (the deuteron),^[21][22] "a" and "c" represent proton and neutron. For neutrino–antineutrino pairs,^[24] "a" and "c" represent neutrino and antineutrino. The size of the deviations from pure Bose behavior,

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

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

${\displaystyle |\Phi \rangle =a^{\dagger }(\mathbf {k_{1}} )a^{\dagger }(\mathbf {k_{2}} )...a^{\dagger }(\mathbf {k_{n}} )c^{\dagger }(\mathbf {q_{1}} )c^{\dagger }(\mathbf {q_{2}} )...c^{\dagger }(\mathbf {q_{m}} )|0\rangle }$

then the expectation value of Eq. (9) vanishes for ${\displaystyle \mathbf {p} ^{\prime }\neq \mathbf {p} }$, and the expression for ${\displaystyle \Delta (\mathbf {p} ^{\prime },\mathbf {p} )}$ can be approximated by

${\displaystyle \Delta (\mathbf {p} ^{\prime },\mathbf {p} )=\delta (\mathbf {p} ^{\prime }-\mathbf {p} )\sum _{\mathbf {k} }\left|F(\mathbf {k} )\right|^{2}\left[a^{\dagger }(\mathbf {p} /2-\mathbf {k} )a(\mathbf {p} /2-\mathbf {k} )\right.}$

${\displaystyle \left.+c^{\dagger }(\mathbf {p} /2+\mathbf {k} )c(\mathbf {p} /2+\mathbf {k} )\right].}$

Using the fermion number operators ${\displaystyle n_{a}(\mathbf {k} )}$ and ${\displaystyle n_{c}(\mathbf {k} )}$, this can be written,

${\displaystyle \Delta (\mathbf {p} ^{\prime },\mathbf {p} )=\delta (\mathbf {p} ^{\prime }-\mathbf {p} )\sum _{\mathbf {k} }\left|F(\mathbf {k} )\right|^{2}\left[n_{a}(\mathbf {p} /2-\mathbf {k} )+n_{c}(\mathbf {p} /2+\mathbf {k} )\right]}$

${\displaystyle =\delta (\mathbf {p} ^{\prime }-\mathbf {p} )\sum _{\mathbf {k} }\left[\left|F(\mathbf {p} /2-\mathbf {k} )\right|^{2}n_{a}(\mathbf {k} )+\left|F(\mathbf {k} -\mathbf {p} /2)\right|^{2}n_{c}(\mathbf {k} )\right]}$

${\displaystyle =\delta (\mathbf {p} ^{\prime }-\mathbf {p} ){\overline {\Delta }}(\mathbf {p} ,\mathbf {p} )}$

showing that it is the average number of fermions in a particular state ${\displaystyle \mathbf {k} }$ averaged over all states with weighting factors ${\displaystyle F(\mathbf {p} /2-\mathbf {k} )}$ and ${\displaystyle F(\mathbf {k} -\mathbf {p} /2)}$.

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,^[35][36] one that is connected with electrons, one with muons, and one with tau leptons.^[37]

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

π+	→	μ+	+	ν μ
μ+	→	e+	+	ν e	+	ν μ

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.^[17]) 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
ν
₁ be the right-handed neutrino and
ν
₂ the left-handed neutrino with their corresponding antineutrinos (with opposite helicity). The neutrinos involved in beta decay are
ν
₂ and
ν
₂, while those for π–μ decay are
ν
₁ and
ν
₁. With this scheme the pion and muon decay modes are:

π+	→	μ+	+	ν 1
μ+	→	e+	+	ν 2	+	ν 1

Absence of massless neutrinos

There is convincing evidence that neutrinos have mass. In experiments at the SuperKamiokande researchers^[15] have discovered neutrino oscillations in which one flavor of neutrino changed into another. This means that neutrinos have non-zero mass outside the Einstein-Cartan torsion.^[13][14] Since massless neutrinos are needed to form a massless photon, a composite photon is not possible outside the Einstein-Cartan torsion.^[13][14]