Majorana equation

The conventional starting point is to state that "the Dirac equation can be written in Hermitian form", when the gamma matrices are taken in the Majorana representation. The Dirac equation is then written as[6]

${\displaystyle \left(\,-i\,{\frac {\partial }{\partial t}}-i\,{\hat {\alpha }}\cdot \nabla +\beta \,m\,\right)\,\psi =0}$

with ${\displaystyle {\hat {\alpha }}}$ being purely real 4×4 symmetric matrices, and ${\displaystyle \beta }$ being purely imaginary skew-symmetric; as required to ensure that the operator (that part inside the parentheses) is Hermitian. In this case, purely real 4‑spinor solutions to the equation can be found; these are the Majorana spinors.

The Majorana equation is

${\displaystyle i\,{\partial \!\!\!{\big /}}\psi -m\,\psi _{c}=0~}$

with the derivative operator ${\displaystyle {\partial \!\!\!{\big /}}}$ written in Feynman slash notation to include the gamma matrices as well as a summation over the spinor components. The spinor ${\textstyle \,\psi _{c}\,}$ is the charge conjugate of ${\textstyle \,\psi \,.}$ By construction, charge conjugates are necessarily given by

${\displaystyle \psi _{c}=\eta _{c}\,C\,{\overline {\psi }}^{\mathsf {T}}~}$

where ${\displaystyle \,(\cdot )^{\mathsf {T}}\,}$ denotes the transpose, ${\displaystyle \,\eta _{c}\,}$ is an arbitrary phase factor ${\displaystyle \,|\eta _{c}|=1\,,}$ conventionally taken as ${\displaystyle \,\eta _{c}=1\,,}$ and ${\displaystyle \,C\,}$ is a 4×4 matrix, the charge conjugation matrix. The matrix representation of ${\displaystyle \,C\,}$ depends on the choice of the representation of the gamma matrices. By convention, the conjugate spinor is written as

${\displaystyle {\overline {\psi }}=\psi ^{\dagger }\,\gamma ^{0}~.}$

A number of algebraic identities follow from the charge conjugation matrix ${\displaystyle C.}$[lower-alpha 1] One states that in any representation of the gamma matrices, including the Dirac, Weyl, and Majorana representations, that ${\displaystyle \,C\,\gamma _{\mu }=-\gamma _{\mu }^{\mathsf {T}}\,C\,}$ and so one may write

${\displaystyle \psi _{c}=-\eta _{c}\,\gamma ^{0}\,C\,\psi ^{*}~}$

where ${\displaystyle \,\psi ^{*}\,}$ is the complex conjugate of ${\displaystyle \,\psi \,.}$ The charge conjugation matrix ${\displaystyle \,C\,}$ also has the property that

${\displaystyle C^{-1}=C^{\dagger }=C^{\mathsf {T}}=-C}$

in all representations (Dirac, chiral, Majorana). From this, and a fair bit of algebra, one may obtain the equivalent equation:

${\displaystyle i\,{\partial \!\!\!{\big /}}\psi _{c}-m\,\psi =0}$

A detailed discussion of the physical interpretation of matrix ${\displaystyle C}$ as charge conjugation can be found in the article on charge conjugation. In short, it is involved in mapping particles to their antiparticles, which includes, among other things, the reversal of the electric charge. Although ${\displaystyle \psi ^{c}}$ is defined as "the charge conjugate" of ${\displaystyle \psi ,}$ the charge conjugation operator has not one but two eigenvalues. This allows a second spinor, the ELKO spinor to be defined. This is discussed in greater detail below.

The Majorana operator, ${\displaystyle \,\mathrm {D} _{\text{L}}\,,}$ is defined as

${\displaystyle \mathrm {D} _{\text{L}}\equiv i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }+\eta \,m\,\omega \,K}$

where

${\displaystyle {\overline {\sigma }}^{\mu }={\begin{bmatrix}\sigma ^{0}&-\sigma ^{1}&-\sigma ^{2}&-\sigma ^{3}\end{bmatrix}}={\begin{bmatrix}I_{2}&-\sigma _{\text{x}}&-\sigma _{\text{y}}&-\sigma _{\text{z}}\end{bmatrix}}}$

is a vector whose components are the 2×2 identity matrix ${\displaystyle \,I_{2}\,}$ for ${\displaystyle \,\mu =0\,}$ and (minus) the Pauli matrices for ${\displaystyle \,\mu \in \{1,\,2,\,3\}\,.}$ The ${\displaystyle \,\eta \,}$ is an arbitrary phase factor, ${\displaystyle \,|\eta |=1\,,}$ typically taken to be one: ${\displaystyle \,\eta =1\,.}$ The ${\displaystyle \,\omega \,}$ is a 2×2 matrix that can be interpreted as the symplectic form for the symplectic group ${\displaystyle \,\operatorname {Sp} (2,\mathbb {C} )\,,}$ which is a double covering of the Lorentz group. It is

${\displaystyle \omega =i\,\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}~,}$

which happens to be isomorphic to the imaginary unit "i" (i.e. ${\displaystyle \omega ^{2}=-I\,}$ and ${\displaystyle \,a\,I+b\,\omega \cong a+b\,i\in \mathbb {C} \,}$ for ${\displaystyle \,a,b\in \mathbb {R} }$) with the matrix transpose being the analog of complex conjugation.

Finally, the ${\displaystyle \,K\,}$ is a short-hand reminder to take the complex conjugate. The Majorana equation for a left-handed complex-valued two-component spinor ${\displaystyle \,\psi _{\text{L}}\,}$ is then

${\displaystyle \mathrm {D} _{\text{L}}\psi _{\text{L}}=0}$

or, equivalently,

${\displaystyle i\,{\overline {\sigma }}^{\mu }\,\partial _{\mu }\psi _{\text{L}}(x)+\eta \,m\,\omega \,\psi _{\text{L}}^{*}(x)=0}$

with ${\displaystyle \,\psi _{\text{L}}^{*}(x)\,}$ the complex conjugate of ${\displaystyle \,\psi _{\text{L}}(x)\,.}$ The subscript L is used throughout this section to denote a left-handed chiral spinor; under a parity transformation, this can be taken to a right-handed spinor, and so one also has a right-handed form of the equation. This applies to the four-component equation as well; further details are presented below.

The Weyl equation describes the time evolution of a massless complex-valued two-component spinor. It is conventionally written as[8][9][10]

${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}$

Written out explicitly, it is

${\displaystyle I_{2}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}$

The Pauli four-vector is

${\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}$

that is, a vector whose components are the 2 × 2 identity matrix ${\displaystyle I_{2}}$ for μ = 0 and the Pauli matrices for μ = 1, 2, 3. Under the parity transformation ${\displaystyle {\vec {x}}\to {\vec {x}}^{\prime }=-{\vec {x}}}$ one obtains a dual equation

${\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}$

where ${\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}}$. These are two distinct forms of the Weyl equation; their solutions are distinct as well. It can be shown that the solutions have left-handed and right-handed helicity, and thus chirality. It is conventional to label these two distinct forms explicitly, thus:

${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0\qquad {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}$

The Weyl equation describes a massless particle; the Majorana equation adds a mass term. The mass must be introduced in a Lorentz invariant fashion. This is achieved by observing that the special linear group ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$ is isomorphic to the symplectic group ${\displaystyle \operatorname {Sp} (2,\mathbb {C} ).}$ Both of these groups are double covers of the Lorentz group ${\displaystyle \operatorname {SO} (1,3).}$ The Lorentz invariance of the derivative term (from the Weyl equation) is conventionally worded in terms of the action of the group ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$ on spinors, whereas the Lorentz invariance of the mass term requires invocation of the defining relation for the symplectic group.

The double-covering of the Lorentz group is given by

${\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}$

where ${\displaystyle \Lambda \in \operatorname {SO} (1,3)}$ and ${\displaystyle S\in \operatorname {SL} (2,\mathbb {C} )}$ and ${\displaystyle S^{\dagger }}$ is the Hermitian transpose. This is used to relate the transformation properties of the differentials under a Lorentz transformation ${\displaystyle x\mapsto x^{\prime }=\Lambda x}$ to the transformation properties of the spinors.

The symplectic group ${\displaystyle \operatorname {Sp} (2,\mathbb {C} )}$ is defined as the set of all complex 2×2 matrices ${\displaystyle S}$ that satisfy

${\displaystyle \omega ^{-1}S^{\textsf {T}}\omega =S^{-1}}$

where

${\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$

is a skew-symmetric matrix. It is used to define a symplectic bilinear form on ${\displaystyle \mathbb {C} ^{2}.}$ Writing a pair of arbitrary two-vectors ${\displaystyle u,v\in \mathbb {C} ^{2}}$ as

${\displaystyle u={\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}\qquad v={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}}$

the symplectic product is

${\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}$

where ${\displaystyle u^{\textsf {T}}}$ is the transpose of ${\displaystyle u~.}$ This form is invariant under Lorentz transformations, in that

${\displaystyle \langle u,v\rangle =\langle Su,Sv\rangle }$

The skew matrix takes the Pauli matrices to minus their transpose:

${\displaystyle \omega \sigma _{k}\omega ^{-1}=-\sigma _{k}^{\textsf {T}}}$

for ${\displaystyle k=1,2,3.}$ The skew matrix can be interpreted as the product of a parity transformation and a transposition acting on two-spinors. However, as will be emphasized in a later section, it can also be interpreted as one of the components of the charge conjugation operator, the other component being complex conjugation. Applying it to the Lorentz transformation yields

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}$

These two variants describe the covariance properties of the differentials acting on the left and right spinors, respectively.

Under the Lorentz transformation ${\displaystyle x\mapsto x^{\prime }=\Lambda x}$ the differential term transforms as

${\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}(x^{\prime })=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}$

provided that the right-handed field transforms as

${\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }(x^{\prime })=S\psi _{\rm {R}}(x)}$

Similarly, the left-handed differential transforms as

${\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}(x^{\prime })=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}$

provided that the left-handed spinor transforms as

${\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }(x^{\prime })=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)}$

The complex conjugate of the right handed spinor field transforms as

${\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}(x^{\prime })=S^{*}\psi _{\rm {R}}^{*}(x)}$

The defining relationship for ${\displaystyle \operatorname {Sp} (2,\mathbb {C} )}$ can be rewritten as ${\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega \,.}$ From this, one concludes that the skew-complex field transforms as

${\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}(x^{\prime })=\left(S^{\dagger }\right)^{-1}m\omega \psi _{\rm {R}}^{*}(x)}$

This is fully compatible with the covariance property of the differential. Taking ${\displaystyle \eta =e^{i\phi }}$ to be an arbitrary complex phase factor, the linear combination

${\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}$

transforms in a covariant fashion. Setting this to zero gives the complex two-component Majorana equation for the right-handed field. Similarly, the left-chiral Majorana equation (including an arbitrary phase factor ${\displaystyle \zeta }$) is

${\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}$

The left and right chiral versions are related by a parity transformation. As shown below, these square to the Klein–Gordon operator only if ${\displaystyle \eta =\zeta .}$ The skew complex conjugate ${\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }$ can be recognized as the charge conjugate form of ${\displaystyle \psi ~;}$ this is articulated in greater detail below. Thus, the Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form.

Define a pair of operators, the Majorana operators,

${\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}&=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K&\mathrm {D} _{\rm {R}}&=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\end{aligned}}}$

where ${\displaystyle K}$ is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as

${\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\mapsto \mathrm {D} _{\rm {L}}^{\prime }&=S\mathrm {D} _{\rm {L}}S^{\dagger }&\mathrm {D} _{\rm {R}}\mapsto \mathrm {D} _{\rm {R}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\mathrm {D} _{\rm {R}}S^{-1}\end{aligned}}}$

whereas the Weyl spinors transform as

${\displaystyle {\begin{aligned}\psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }&=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}&\psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }&=S\psi _{\rm {R}}\end{aligned}}}$

just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take

${\displaystyle {\begin{aligned}\mathrm {D} _{\rm {L}}\psi _{\rm {L}}&=0&\mathrm {D} _{\rm {R}}\psi _{\rm {R}}&=0\end{aligned}}}$

as a pair of complex 2-spinor Majorana equations.

The products ${\displaystyle \mathrm {D} _{\rm {L}}\mathrm {D} _{\rm {R}}}$ and ${\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}}$ are both Lorentz covariant. The product is explicitly

${\displaystyle \mathrm {D} _{\rm {R}}\mathrm {D} _{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)}$

Verifying this requires keeping in mind that ${\displaystyle \omega ^{2}=-1}$ and that ${\displaystyle Ki=-iK~.}$ The RHS reduces to the Klein–Gordon operator provided that ${\displaystyle \eta \zeta ^{*}=1}$, that is, ${\displaystyle \eta =\zeta ~.}$ These two Majorana operators are thus "square roots" of the Klein–Gordon operator.

Under parity, the left-handed spinors transform to right-handed spinors. The two eigenvectors of the charge conjugation operator, again in the Weyl basis, are

${\displaystyle \psi _{\rm {{R},{\text{Weyl}}}}^{(\pm )}={\begin{pmatrix}\pm \eta \omega \psi _{\rm {R}}^{*}\\\psi _{\rm {R}}\end{pmatrix}}}$

As before, both solve the four-component Majorana equation, but only one also solves the Dirac equation. This can be shown by constructing the parity-dual four-component equation. This takes the form

${\displaystyle \beta \left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)=i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}}$

where

${\displaystyle \delta _{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }-\eta m\omega K}$

Given the two-component spinor ${\displaystyle \psi _{\rm {R}}}$ define its conjugate as ${\displaystyle \chi _{\rm {L}}=-\eta \omega \psi _{\rm {R}}^{*}.}$ It is not hard to show that ${\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=-\eta \omega (\delta _{\rm {L}}\chi _{\rm {L}})}$ and that therefore, if ${\displaystyle \mathrm {D} _{\rm {R}}\psi _{\rm {R}}=0}$ then also ${\displaystyle \delta _{\rm {L}}\chi _{\rm {L}}=0}$ and therefore that

${\displaystyle 0=\left(\delta _{\rm {L}}\oplus \mathrm {D} _{\rm {R}}\right)\left(\chi _{\rm {L}}\oplus \psi _{\rm {R}}\right)}$

or equivalently

${\displaystyle 0=(i\gamma ^{\mu }\partial _{\mu }+m{\mathsf {C}}){\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}$

This works, because ${\displaystyle {\mathsf {C}}(\chi _{\rm {L}}\oplus \psi _{\rm {R}})=-(\chi _{\rm {L}}\oplus \psi _{\rm {R}})}$ and so this reduces to the Dirac equation for

${\displaystyle \psi _{{\rm {R}},{\text{Weyl}}}^{(-)}=\chi _{\rm {L}}\oplus \psi _{\rm {R}}={\begin{pmatrix}\chi _{\rm {L}}\\\psi _{\rm {R}}\end{pmatrix}}}$

To conclude, and reiterate, the Majorana equation is

${\displaystyle 0=\left(i\gamma ^{\mu }\partial _{\mu }-m{\mathsf {C}}\right)\psi =i\gamma ^{\mu }\partial _{\mu }\psi -m\psi _{c}}$

It has four inequivalent, linearly independent solutions, ${\displaystyle \psi _{\rm {L,R}}^{(\pm )}.}$ Of these, only two are also solutions to the Dirac equation: namely ${\displaystyle \psi _{\rm {L}}^{(+)}}$ and ${\displaystyle \psi _{\rm {R}}^{(-)}~.}$

One convenient starting point for writing the solutions is to work in the rest frame way of the spinors. Writing the quantum Hamiltonian with the conventional sign convention ${\displaystyle H=i\partial _{t}}$ leads to the Majorana equation taking the form

${\displaystyle i\partial _{t}\psi =-i{\vec {\alpha }}\cdot \nabla \psi +m\beta \psi _{c}}$

In the chiral (Weyl) basis, one has that

${\displaystyle \gamma ^{0}=\beta ={\begin{pmatrix}0&I\\I&0\end{pmatrix}},\quad {\vec {\alpha }}={\begin{pmatrix}{\vec {\sigma }}&0\\0&-{\vec {\sigma }}\end{pmatrix}}}$

with ${\displaystyle {\vec {\sigma }}}$ the Pauli vector. The sign convention here is consistent with the article gamma matrices. Plugging in the positive charge conjugation eigenstate ${\displaystyle \psi _{\text{Weyl}}^{(+)}}$ given above, one obtains an equation for the two-component spinor

${\displaystyle i\partial _{t}\psi _{\rm {L}}=-i{\vec {\sigma }}\cdot \nabla \psi _{\rm {L}}+m(i\sigma _{2}\psi _{\rm {L}}^{*})}$

and likewise

${\displaystyle i\partial _{t}(i\sigma _{2}\psi _{\rm {L}}^{*})=+i{\vec {\sigma }}\cdot \nabla (i\sigma _{2}\psi _{\rm {L}}^{*})+m\psi _{\rm {L}}}$

These two are in fact the same equation, which can be verified by noting that ${\displaystyle \sigma _{2}}$ yields the complex conjugate of the Pauli matrices:

${\displaystyle \sigma _{2}\left({\vec {k}}\cdot {\vec {\sigma }}\right)\sigma _{2}=-{\vec {k}}\cdot {\vec {\sigma }}^{*}.}$

The plane wave solutions can be developed for the energy-momentum ${\displaystyle \left(k_{0},{\vec {k}}\right)}$ and are most easily stated in the rest frame. The spin-up rest-frame solution is

${\displaystyle \psi _{\rm {L}}^{(u)}={\begin{pmatrix}e^{-imt}\\e^{imt}\end{pmatrix}}}$

while the spin-down solution is

${\displaystyle \psi _{\rm {L}}^{(d)}={\begin{pmatrix}e^{imt}\\-e^{-imt}\end{pmatrix}}}$

That these are being correctly interpreted can be seen by re-expressing them in the Dirac basis, as Dirac spinors. In this case, they take the form

${\displaystyle \psi _{\text{Dirac}}^{(u)}={\begin{bmatrix}e^{-imt}\\0\\0\\-e^{imt}\end{bmatrix}}}$

and

${\displaystyle \psi _{\text{Dirac}}^{(d)}={\begin{bmatrix}0\\e^{-imt}\\-e^{imt}\\0\end{bmatrix}}}$

These are the rest-frame spinors. They can be seen as a linear combination of both the positive and the negative-energy solutions to the Dirac equation. These are the only two solutions; the Majorana equation has only two linearly independent solutions, unlike the Dirac equation, which has four. The doubling of the degrees of freedom of the Dirac equation can be ascribed to the Dirac spinors carrying charge.

In a general momentum frame, the Majorana spinor can be written as

Particles corresponding to Majorana spinors are known as Majorana particles, due to the above self-conjugacy constraint. All the fermions included in the Standard Model have been excluded as Majorana fermions (since they have non-zero electric charge they cannot be antiparticles of themselves) with the exception of the neutrino (which is neutral).

Theoretically, the neutrino is a possible exception to this pattern. If so, neutrinoless double-beta decay, as well as a range of lepton-number violating meson and charged lepton decays, are possible. A number of experiments probing whether the neutrino is a Majorana particle are currently underway.[14]