Majorana Condition

A Majorana condition selects the fixed points of an antilinear charge-conjugation map. After fixing that real structure, a discrete transformation must preserve it to act within the same real solution space. This requirement restricts phases that were arbitrary on an unconstrained complex Dirac space. In the conventions used here, parity needs an imaginary intrinsic phase, while the aligned time-reversal matrix already preserves the condition. Majorana Spinors owns the component construction, Lorentz covariance, and field-statistics qualifications; this page applies that construction to discrete transformations.

Required background. Majorana Spinors defines the real structure, and Symmetry Conventions, Parity, and Time Reversal fix the matrices and phases. Helpful background. Charge Conjugation distinguishes conjugate equations from unitary transformations of quantum states.

A fixed real structure on Dirac solutions

Let

$$J=BK,\qquad B=i\gamma^2,\qquad J^2=I.$$

The chosen Majorana condition is

$$J\psi=\psi.$$

The fixed-point set is a real vector space. If $\psi$ satisfies the condition, then $J(z\psi)=z^*\psi$, so multiplication by an arbitrary complex number does not preserve it. This is why phase compatibility is a nontrivial question.

For the free neutral equation with real mass, $J$ maps solutions to solutions. A transformation $S$ preserves the chosen Majorana space when $J(S\psi)=S\psi$ for every $J$-fixed solution. It is sufficient that the complete maps commute, $JS=SJ$. When either map is antilinear, scalar coefficients must be conjugated in checking that equation.

These statements do not replace the ordinary complex Hilbert space of physical one-particle states. The reality condition on classical solutions and the Majorana field mode expansion are related constructions, as explained on the canonical owner.

Why the parity phase is imaginary

Start with the ordinary Dirac parity $\Pi=\beta R$, where $R$ reverses the spatial argument. The charge-conjugation identity gives

$$B\beta^*B^{-1}=-\beta.$$

Since $J$ commutes with the coordinate reflection,

$$J\Pi=-\Pi J.$$

Thus $\Pi$ alone sends a $J$-fixed spinor to the opposite real-structure eigenspace. Allow an intrinsic phase and define

$$\Pi_{\rm M}=\eta_P\Pi,\qquad |\eta_P|=1.$$

For $J\psi=\psi$,

$$J\Pi_{\rm M}\psi =-\eta_P^*\Pi\psi.$$

To equal $\Pi_{\rm M}\psi=\eta_P\Pi\psi$, the phase must satisfy

$$\eta_P=-\eta_P^*, \qquad \eta_P=\pm i.$$

Either choice preserves the fixed Majorana condition. Its square on the spinor is

$$\Pi_{\rm M}^2=-I.$$

This is compatible with the ordinary complex-Dirac convention $\Pi^2=I$: the two conventions impose different requirements on the intrinsic phase. The coordinate inversion still squares to the identity. The additional minus sign concerns its spinor representation; bilinears are unchanged by $\psi\mapsto-\psi$.

An overall convention for $J$ or the spinor basis can also be changed, provided all maps are translated consistently. The invariant requirement is preservation of the specified real structure, not an isolated matrix entry being real or imaginary in every basis.

Time reversal in the aligned convention

The chapter uses

$$\Theta=U_TK,\qquad U_T=-\gamma^1\gamma^3.$$

The two antilinear maps commute because

$$BU_T^*=U_TB^*.$$

Therefore

$$J\Theta\psi=\Theta J\psi=\Theta\psi$$

for a Majorana solution. Including the reversed time argument does not change this algebraic conclusion. The operation preserves the fixed reality condition and still has $\Theta^2=-I$.

If $\Theta$ is multiplied by a phase $\eta_T$ while $J$ remains fixed, preservation requires $\eta_T=\eta_T^*$, hence $\eta_T=\pm1$ for unit modulus. The common alternative matrix $U_{T,0}=i\gamma^1\gamma^3$ differs from ours by the phase $U_T=iU_{T,0}$. Used without translating that relative phase, $U_{T,0}K$ sends the fixed Majorana space to the opposite eigenspace of $J$. Multiplying it by $i$ restores the aligned map.

On the unconstrained complex Dirac space, the isolated time-reversal maps are equivalent up to phase. After imposing a particular reality condition, their compatibility with that condition must also be checked. This is a useful example of why individually valid textbook formulas cannot always be combined unchanged.

Gauge charge and the meaning of neutrality

For an ordinary U(1) phase $U_\chi=e^{iq\chi/\hbar}$, antilinearity gives

$$J\,U_\chi\psi =e^{-iq\chi/\hbar}\psi$$

on a $J$-fixed spinor. For arbitrary local $\chi$, this equals $U_\chi\psi$ only for a trivial charge action. A single fixed Majorana real representation therefore cannot carry a nonzero ordinary U(1) charge.

The obstruction is present even if a particular background happens to vanish: a freely written self-conjugate solution does not make a nontrivial gauge phase preserve its real subspace. In nonzero backgrounds, Charge Conjugation also shows explicitly that $J$ relates the $q$ and $-q$ equations.

This statement concerns ordinary abelian charge, not the absence of every possible interaction. Neutral fields can have interactions consistent with their representation and statistics. For example, the absence of a diagonal Majorana Pauli moment follows from a fermionic tensor-bilinear identity, not merely from calling the field neutral; see Magnetic Moment. The detailed mass and component construction remains on Majorana Spinors.

A basis-independent compatibility test

Let $P=\beta$ denote the spinor matrix in the complete parity map $\Pi=\beta R$. For a constant unitary basis change $\psi'=V\psi$, transform

$$B'=VBV^{\mathsf T},\qquad P'=VPV^\dagger,\qquad U_T'=VU_TV^{\mathsf T}.$$

The corresponding complete maps obey the same commutation or anticommutation relations. For example, the parity test can be written entirely in matrices:

$$B'(\eta_PP')^* =(\eta_PP')B'.$$

It gives the same allowed relative phases as the original basis. A basis in which the reality condition looks like ordinary componentwise reality can simplify notation, but it cannot remove the need for this compatibility test. The matrix conversion rules are collected on Symmetry Conventions.

Exercises

The opposite eigenspace. If $J\psi=-\psi$, show that $i\psi$ is $J$-fixed. Why does this matter when comparing two phase conventions?

Solution

Antilinearity gives $J(i\psi)=-iJ\psi=i\psi$. The two real eigenspaces are related by multiplication by $i$. An unexplained phase difference can therefore exchange which reality condition appears to be preserved.

A failed real parity phase. Set $\eta_P=1$ and start with $J\psi=\psi$. Show exactly which condition the parity-transformed spinor satisfies.

Solution

Since $J\Pi=-\Pi J$, $J(\Pi\psi)=-\Pi\psi$. The result lies in the opposite eigenspace. The failure concerns preservation of this fixed real structure, not covariance of the unconstrained Dirac equation.

Rephase the time-reversal matrix. Starting from the common $U_{T,0}=i\gamma^1\gamma^3$, find a phase that makes the map preserve the chosen Majorana condition and verify its square.

Solution

Multiplication by $i$ gives $iU_{T,0}=-\gamma^1\gamma^3=U_T$. The resulting map commutes with $J$. Its square is $U_TU_T^*=-I$. The phase changes compatibility with $J$ but cannot change an antiunitary operator’s square.

References

• Beisert, Niklas. Quantum Field Theory I. ETH Zurich lecture notes, autumn semester 2012, chapter 5, section 5.3. Free Spinor Field.
• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002. Corrected manuscript.
• Thaller, Bernd. The Dirac Equation. Springer (1992). doi:10.1007/978-3-662-02753-0.