# Anti-symmetric operators for the Dirac or Majorana spinors

In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) has an anti-symmetric Dirac operator.

1. Say, if the $$\psi$$ is a Dirac spinor, he wrote down an action $$\int d^2x \sqrt{g} \bar{\psi} (i \gamma^\mu D_\mu) \psi$$ and (I think) he claims that the operator $$(i \gamma^\mu D_\mu)$$ is an anti-symmetric matrix.

2. Say, if the $$\chi$$ is a Majorana spinor, he wrote down an action $$\int d^2x \sqrt{g} {\chi} (i \gamma^\mu D_\mu) \chi$$ and (I think) he claims that the operator $$(i \gamma^\mu D_\mu)$$ is also an anti-symmetric matrix.

Is this true that the anti-symmetric matrix has something to do with these fermions (spinors)? or fermion statistics? Why?

p.s. Maybe the first case the $$D$$ operator is complex, and the second case that the $$D$$ operator is real (?).

• This seems difficult to parse. Anti-symmetric in what indices? The usual requirement is that $\gamma^{0}$ be Hermitean and the $\gamma^{i}$ anti-Hermitean. One can find Hermitean matrices that are symmetric, and ones that are antisymmetric ... perhaps a definite representation of the Dirac matrices is intended? Certainly, $iD_{\mu }$ is Hermitean. And all this shouldn't depend crucially on this being in 1+1 dimensions. May 28, 2020 at 3:46

In terms of the Hamiltonian $$H$$ the antisymmetry follows simply: Majorana fields $$\Psi(x,t)$$ are real, and they satisfy a real wave equation $$\partial\Psi/\partial t =-iH\Psi,$$ where $$iH$$ is real, hence $$H$$ is imaginary. Since $$H$$ must also be Hermitian, it means that $$H^T=-H$$ (antisymmetric).
More explicitly, $$H=\gamma_{\rm M}^\mu \partial_\mu$$, with $$\gamma_{\rm M}^\mu$$ the Dirac matrices in the Majorana representation, for which $$\gamma_{\rm M}$$ is a purely imaginary $$4\times 4$$ matrix. The antisymmetry of $$H$$ becomes manifest if we discretize the derivative operator so that $$H$$ becomes a matrix and $$H_{nm}=-H_{mn}$$.
• The plot thickens ... now, $H$ is constructed from the spatial components only - furthermore, since $H$ is related to $i\gamma_{M}^{i} D_i$ through a factor $\gamma_{M}^{0}$, which is antisymmetric, this would actually imply that $i\gamma_{M}^{i} D_i$ is symmetric, not antisymmetric. Could it be that the spoken word in a Zoom lecture, remembered from memory, is not quite as reliable as the written word? May 28, 2020 at 13:08
• (1) when we say matrix $P$ is Hermitian, it mens $P^\dagger =P$,. when we say matrix $P$ is anti Hermitian, it mens $P^\dagger = -P$,. Jun 15, 2020 at 13:26