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 (?).

  • $\begingroup$ 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. $\endgroup$ May 28, 2020 at 3:46

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

  • $\begingroup$ 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? $\endgroup$ May 28, 2020 at 13:08
  • $\begingroup$ I think the answer is very nice and seems perfectly fine -- I will accept it in a week if no other better answers. (Voted up) $\endgroup$ May 28, 2020 at 23:09
  • $\begingroup$ thank you again - accepted $\endgroup$ Jun 14, 2020 at 22:13
  • $\begingroup$ Sorry, may I confirm: What do we mean by the matrix is real or imaginary? $\endgroup$ Jun 15, 2020 at 13:24
  • $\begingroup$ (1) when we say matrix $P$ is Hermitian, it mens $P^\dagger =P$,. when we say matrix $P$ is anti Hermitian, it mens $P^\dagger = -P$,. $\endgroup$ Jun 15, 2020 at 13:26

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