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Take $\boxed{c=1}$. Indeed the conjugate transpose isso the anticommutator is a Hermitian matrix. The original PDF, p. 6, supplies the factor $BA$ that is corrupted in the local OCR.

The anticommutator condition and invertibility give the matrix similaritiesSince matrix trace is invariant under matrix similarity,Hence $\boxed{\mathrm{Tr}A=\mathrm{Tr}B=0}$. Taking determinants in the first matrix similarity also givesSince $A$ is an invertible matrix, $\det A\ne 0$, and $\boxed{n\text{ is even}}$. In terms of eigenvalues, the same matrix similarity pairs each nonzero eigenvalue $\lambda$ with $-\lambda$, with the same algebraic multiplicity, so an odd number of them is impossible.

Expand the anticommutators using the relations in part (a). The terms with two creation or two annihilation operators vanish, leavingEach Majorana fermion operator is Hermitian and squares to the identity. In particular, $({\gamma }_i^A{)}^{†}{\gamma }_i^A=1$, so this expression cannot distinguish an empty mode from an occupied one. A single Majorana operator is its own adjoint and does not supply distinct creation and annihilation operators for such an occupation.

An occupation can instead be assigned after pairing Majorana operators into a fermionic mode. For two distinct Majorana operators ${\gamma }_a,{\gamma }_b$,This is a nontrivial projection operator with eigenvalues zero and one. Thus occupation belongs to a chosen pair of Majorana operators, not to an individual Majorana operator.

For any two distinct normalized Majorana fermion operators $\eta ,\chi$, let $f=(\eta +i\chi )/2$. ThenandThe same applies to $f^{†}$. Both specified pairs have this form. Moreover, the Majorana operators entering $f_1$ are all distinct from those entering $f_2$, so every cross anticommutator is zero. ThereforeThus $f_1,f_2$ are ordinary fermionic annihilation operators, constructed by a different pairing of the four Majorana operators.

For self-adjoint operators and an admissible pure state, the quantum covariance is the real, symmetrized second moment of two centered observables. The displayed expression uses their anticommutator; equivalently $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$, which requires only the individual finite second moments. It reduces to ordinary covariance for commuting observables. For noncommuting observables, its definition does not assert that both have simultaneous sharp values drawn from a joint classical probability distribution.

At rest the two complex supercharges have anticommutators $2m{\delta }_{ab}$. A normalized scalar Clifford vacuum generates four states at occupation levels $0,1,2$: two spin-zero bosonic states and one spin-half pair. Products of three creation charges vanish. The one-creation normalization is $(2m{)}^{-1/2}$ and the two-creation normalization is $(2m{)}^{-1}$. The resulting on-shell content is a complex scalar and a massive fermion with equal masses.

For left Grassmann derivatives, choose $D_{\alpha }={\partial }_{\alpha }+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{D}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}+i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Differentiating the other operator's odd coefficient gives two equal contributions to the mixed anticommutator; coefficient-coefficient terms cancel by the Grassmann algebra. Thus the mixed anticommutator is $2i{\sigma }^{\mu }{\partial }_{\mu }$, while equal-chirality ones vanish. Also $\bar{D}(x+i\theta \sigma \bar{\theta })=0$, giving the coordinates used in the chiral-superfield component expansion. Negating the barred derivative negates the mixed algebra.

With left Grassmann derivatives, define ${\mathcal{P}}_{\mu }=-i{\partial }_{\mu }$, ${\mathcal{Q}}_{\alpha }=-i{\partial }_{{\theta }^{\alpha }}-{\sigma }_{\alpha \dot{\gamma }}^{\mu }{\bar{\theta }}^{\dot{\gamma }}{\partial }_{\mu }$ and ${\bar{\mathcal{Q}}}_{\dot{\beta }}=i{\partial }_{{\bar{\theta }}^{\dot{\beta }}}+{\theta }^{\gamma }{\sigma }_{\gamma \dot{\beta }}^{\mu }{\partial }_{\mu }$. The anticommutators of the derivative parts and of the coordinate parts vanish. Each mixed derivative/coordinate pair contributes $-i{\sigma }^{\mu }{\partial }_{\mu }$ by the graded product rule, giving the displayed mixed anticommutator. Equal-chirality anticommutators vanish, and all these operators commute with translations. Under Lorentz transformations their spinor indices transform in conjugate Weyl spinor representations, completing the minimal Super-Poincaré algebra. A convention for left derivatives is essential to the signs.

If $A$ is diagonalizable, the anticommutator map $B\mapsto AB+BA$ has eigenvalues $2{\lambda }_1$, ${\lambda }_1+{\lambda }_2$ twice, and $2{\lambda }_2$. Their product gives the displayed characteristic polynomial. The density of diagonalizable complex matrices and continuity of polynomial coefficients prove the formula also for defective matrices.