Are there any definitions about the chirality of (2+1)D Dirac equation? For the (3+1)D Dirac equation, the Dirac field can be written as the sum of left- and right-hand Weyl field. Can this be reduced to the lower dimension, thus lead to the definition of chirality for the (2+1)D or even (1+1)D?


2 Answers 2


There isn't a good definition of chirality in (2+1)D or any other odd dimension. This is because the $\gamma_5$ matrix can't be defined usefully in a Clifford algebra with an odd number of generators.

For instance try to define $\gamma_5 = \gamma^0\gamma^1\gamma^2$. This commutes (not anti-commutes) with $\gamma^0,\gamma^1,\gamma^2$ and thus commutes with the whole Clifford algebra, including anything like a parity operator. In an irreducible representation it will just be a multiple of the identity.

In (1+1) there is no problem defining chirality. A common representation of the Clifford algebra in terms of the Pauli matrices is $$\gamma^0=\sigma_2$$ $$\gamma^1=-i\sigma_1,$$ This is a nice representation because $\gamma_5 = \gamma^0\gamma^1$ is diagonal and also the gamma matrices are completely imaginary. So it is like both the chiral (Weyl) representation and the Majorana representation.

The Dirac equation takes the same form, just with fewer spacetime dimensions. $$(i\gamma^\mu\partial_\mu-m)\psi =0.$$ Label the two components of the spinor $\psi = (\psi_L,\psi_R)^T,$ then if you write out the components of the Dirac equation for a massless spinor $$\partial_0 \psi_R = -\partial_1 \psi_R$$ $$\partial_0 \psi_L = +\partial_1 \psi_L,$$ which tells you $\psi_R$ is a right-moving wave and $\psi_L$ is a left-moving wave. But of course if you keep a mass term the two chiralities get mixed up (just like in 3+1).

  • $\begingroup$ Would you help me with some references about it, please? I want to know more about that. $\endgroup$
    – Shane
    Sep 4, 2015 at 6:59
  • $\begingroup$ If you read 'Advanced Topics in QFT' by M. Shifman there are a number of examples dealing with fermions in 1+1. Another source which might not fit with your interests but was useful to me is 'Noncommutative geometry and particle physics' by van Suijlekom. Chapter 4.1 talks about Clifford algebras in arbitrary dimension and metric. $\endgroup$
    – octonion
    Sep 5, 2015 at 16:05

You can use $\gamma^{5}:= \gamma^{0}$ for the chirality operator in (2+1)D.


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