The first-order kinetic term makes $\eta^a$ and $\bar\eta_b$ canonically conjugate <Grassmann variables>. Canonical quantization gives the <canonical anticommutation relations>
$$
\boxed{\{\widehat\eta^a,\widehat{\bar\eta}_b\}=\delta^a{}_b,
\qquad\{\widehat\eta^a,\widehat\eta^b\}=0,
\qquad\{\widehat{\bar\eta}_a,\widehat{\bar\eta}_b\}=0.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 2 / a / Solution

{title2=CAR}

Fermionic creation and annihilation operators satisfy $\{a_i,a_j^\dagger\}=\delta_{ij}$ and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.


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