Anti-symmetric operators for the Dirac or Majorana spinors

Question

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 (?).

Answer 1

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 $H_{nm}=-H_{mn}$.}