Majorana Spinors

A Majorana spinor is a Dirac spinor constrained to equal its charge conjugate. In four-dimensional Minkowski spacetime, this is an antilinear reality condition that leaves four real component functions rather than four independent complex ones. It is compatible with a free real mass, but it does not mean “a positive-energy electron with real components,” nor can it coexist with a nonzero ordinary conserved electric charge.

Required background. Dirac Spinors constructs the chiral representation and mass coupling; Gamma-Matrix Conventions fixes the basis used below.

Charge conjugation as an antilinear map

Use natural units and the chiral basis. Fix the conventional phase by

$$C=i\gamma^2\gamma^0,\qquad \Psi^c=C\bar\Psi^{\,T}=B\Psi^*,\qquad B=i\gamma^2.$$

The matrices satisfy

$$C(\gamma^\mu)^TC^{-1}=-\gamma^\mu,\qquad B(\gamma^\mu)^*B^{-1}=-\gamma^\mu,\qquad BB^*=I.$$

The star denotes componentwise complex conjugation. It is part of the transformation: multiplying by $C$ alone is not charge conjugation. The last identity gives $(\Psi^c)^c=\Psi$. For proper orthochronous spinor transformations, $BS(A)^*=S(A)B$, so the reality condition is Lorentz covariant.

To check dynamics, complex conjugate the free equation $(i\gamma^\mu\partial_\mu-m)\Psi=0$ with real $m$. Multiplication by $B$ returns the same equation for $\Psi^c$. In a real electromagnetic background, the analogous calculation reverses the charge parameter $q$. Self-conjugacy is therefore compatible with the free neutral equation, while charged equations must be treated with their conjugate internal representation.

Reconstructing the spinor from one Weyl column

Let $\epsilon=i\sigma^2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. For $\Psi=(\chi_L,\chi_R)^T$, the conjugate is

$$\Psi^c= \begin{pmatrix} \epsilon\chi_R^*\\ -\epsilon\chi_L^* \end{pmatrix}.$$

The condition $\Psi=\Psi^c$ is solved by

$$\Psi_M= \begin{pmatrix} \chi\\ -\epsilon\chi^* \end{pmatrix}.$$

Indeed $\epsilon^2=-I$, so substituting the lower block into the upper condition returns $\chi$. Two arbitrary complex component functions of $\chi$ determine all four components of $\Psi_M$. The free equation for the independent column becomes

$$i\bar\sigma^\mu\partial_\mu\chi=-m\epsilon\chi^*.$$

This equation is real-linear, not complex-linear, for nonzero $m$. The complex conjugate column supplies the other chirality. Calling $\chi$ a two-component variable does not make the massive equation a single decoupled massless Weyl equation.

Before imposing the equation of motion, the Majorana condition reduces eight real component functions to four. After quantization, a massive Majorana species has two spin states and no independent antiparticle species; a massive Dirac field has two particle and two antiparticle spin states. These are different counts at different stages of the construction.

Reality depends on the basis

Under a unitary change of spinor basis $\Psi'=U\Psi$, the antilinear map is represented by

$$B'=UBU^T,\qquad \Psi'^c=B'\Psi'^*.$$

It is a congruence, not the similarity transformation used for gamma matrices. A Majorana basis makes $B'=I$, so the condition becomes ordinary componentwise reality there. Such a basis exists for this four-dimensional Minkowski real structure. In the chiral or Dirac basis, demanding that every component be real would impose a different, generally incorrect constraint.

A change in the phase convention for charge conjugation changes the written relation between the blocks. Physical statements must be compared after translating that convention.

Charge conjugation exchanges chiralities: $B(\gamma^5)^*B^{-1}=-\gamma^5$. Therefore a spinor that is both self-conjugate and purely left-chiral must vanish, and likewise for a purely right-chiral spinor. There is no nonzero Majorana–Weyl spinor in $3+1$-dimensional Minkowski spacetime. This assertion depends on dimension and signature.

Frequency sectors and the quantum interpretation

Complex conjugation reverses the Fourier phase. A positive-frequency term $u(p)e^{-ip\cdot x}$ is mapped into a negative-frequency term. Consequently a nonzero massive c-number solution satisfying the Majorana condition cannot contain only positive frequencies. The condition relates the two frequency sectors; it does not select the positive spectral subspace of a one-particle Hamiltonian.

In a quantized Majorana field, those related coefficients multiply annihilation and creation operators of the same species. A positive-energy one-particle state remains a valid excitation of that field. There is no requirement that its external positive-frequency spinor coefficient, considered alone, be a real classical solution.

Self-conjugacy also removes a continuous overall phase symmetry of the constrained field. If $\Psi_M$ satisfies the fixed condition, then $(e^{i\alpha}\Psi_M)^c=e^{-i\alpha}\Psi_M$. Preservation for arbitrary $\alpha$ is impossible unless the field vanishes; the signs $\pm1$ remain. A single self-conjugate field therefore cannot carry a nonzero ordinary $U(1)$ charge. This does not rule out all interactions or every real nonabelian representation.

Commuting amplitudes and fermionic fields obey different identities

The statistics of the components matters when two spinors are interchanged. For a commuting complex column $\chi=(a,b)^T$,

$$\chi^T\epsilon\chi=ab-ba=0.$$

For Grassmann components the same expression is $2ab$, generally nonzero. This is why a two-component Majorana mass bilinear cannot be interpreted by replacing its anticommuting field components with ordinary commuting numbers and keeping every field identity unchanged.

A particularly useful distinction concerns the vector current. For an identical anticommuting Majorana field, the vector bilinear vanishes as a fermionic bilinear identity; quantum coincident-point products require an appropriate definition, such as the normal-ordered free current. For a commuting Majorana amplitude, $\Psi_M^\dagger\Psi_M=2\chi^\dagger\chi$ is positive and generally nonzero.

At one spacetime point, for example, $\chi=(1,0)^T$ gives $\Psi_M=(1,0,0,1)^T$ and the c-number current

$$\bar\Psi_M\gamma^\mu\Psi_M=(2,0,0,-2).$$

It is a nonzero future-null probability current, not an electric charge current of a charged Majorana particle. Applying the fermionic vanishing identity to this commuting column would be a category error.

The construction makes Majorana mass terms a possible description of neutral fermions. It does not establish whether a particular observed neutral particle has Majorana mass; that is a question for its interactions and experimental evidence.

Exercises

1. Verify $(\Psi^c)^c=\Psi$ using the two chiral blocks.

Solution

Conjugating twice gives upper block $\epsilon(-\epsilon\chi_L^*)^*=-\epsilon^2\chi_L=\chi_L$ and lower block $-\epsilon(\epsilon\chi_R^*)^*=-\epsilon^2\chi_R=\chi_R$. Here $\epsilon$ is real and $\epsilon^2=-I$.

2. If $\Psi_1$ and $\Psi_2$ satisfy the fixed Majorana condition, which linear combinations certainly do so?

Solution

All real linear combinations do. Antilinearity sends $a\Psi_1+b\Psi_2$ to $a^*\Psi_1+b^*\Psi_2$; generic complex coefficients violate the condition. The fixed-point space is a real vector space, even though it is embedded in a complex Dirac component space.

3. Derive the basis transformation $B'=UBU^T$.

Solution

$\Psi'^c=UB\Psi^*=UB(U^\dagger\Psi')^* =UBU^T\Psi'^*$ for unitary $U$. Using $UBU^\dagger$ would omit the conjugation acting on the basis change.

References

• H. K. Dreiner, H. E. Haber, and S. 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 arXiv:0812.1594v6, 2022, especially appendix G — conjugation, reality conditions, and component statistics.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — Majorana fields and mass terms.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes, 2006, chapter 4 — Majorana reality and its field interpretation.