CAR Algebras, Majorana Generators, and Clifford Comparison: Examples and Counterexamples

The displayed matrices act by $a{e}_{\mathrm{\varnothing }}=0$, $a{e}_{\{1\}}=e_{\mathrm{\varnothing }}$, $a^{†}{e}_{\mathrm{\varnothing }}=e_{\{1\}}$, and $a^{†}{e}_{\{1\}}=0$. Thus $a^2=(a^{†}{)}^2=0$, while $a{a}^{†}=\mathrm{diag}(1,0)$ and $a^{†}a=\mathrm{diag}(0,1)$, so $\{a,a^{†}\}=I$. This verifies the one-mode CAR relations.

Substitution into the Majorana formulas gives the two displayed Pauli matrices. Direct multiplication gives ${\gamma }_1^2={\gamma }_2^2=I$ and ${\gamma }_1{\gamma }_2+{\gamma }_2{\gamma }_1=0$; both matrices are self-adjoint. Hence they represent the two orthogonal plus-$Q$ Clifford generators.

The four matrices $a{a}^{†}$, $a$, $a^{†}$, and $a^{†}a$ are respectively the matrix units $E_{00},E_{01},E_{10},E_{11}$, so the one-mode algebra is all of $M_2(\mathbb{C})$, as asserted by the finite-mode theorem.

Reading columns of the displayed matrices gives $a_1{e}_{\{1,2\}}=e_{\{2\}}$ and $a_2{e}_{\{1,2\}}=-e_{\{1\}}$. Applying the operators in the reverse order gives $a_2{a}_1{e}_{\{1,2\}}=e_{\mathrm{\varnothing }}$ and $a_1{a}_2{e}_{\{1,2\}}=-e_{\mathrm{\varnothing }}$, while both products vanish on the other three basis vectors. Hence $\{a_1,a_2\}=0$. The adjoint matrices give the corresponding creation relations, and their same-mode products satisfy $a_j{e}_S=0$ when $j\mathrm{\notin }S$, $a_j^{†}{e}_S=0$ when $j\in S$, and $\{a_j,a_j^{†}\}=I$ on each of the four basis vectors.

For the mixed distinct-mode relation, on $e_{\{1\}}$ one has $a_1{a}_2^{†}{e}_{\{1\}}=-e_{\{2\}}$ and $a_2^{†}{a}_1{e}_{\{1\}}=e_{\{2\}}$; on $e_{\mathrm{\varnothing }},e_{\{2\}},e_{\{1,2\}}$ both terms sum to zero by the displayed matrices. Thus $\{a_1,a_2^{†}\}=0$. This confirms the sign convention for modes in the order $1<2$.

By [F1], $P_{\mathrm{\varnothing }}=\mathrm{diag}(1,0,0,0)$. The operator $a_1^{†}{P}_{\mathrm{\varnothing }}{a}_2$ sends $e_{\{2\}}$ to $e_{\{1\}}$ and kills the other three basis vectors, so it is the matrix unit $E_{\{1\},\{2\}}$. Together with the general formula in [F1], this is one explicit off-diagonal matrix unit inside the algebra $M_4(\mathbb{C})$ of [F2].

The automorphism $\alpha (a_j)=a_j^{†}$ is defined on the same countable-mode CAR algebra, so composing the Fock representation with $\alpha$ gives another representation of that algebra.

By [F1], this representation is irreducible but not unitarily equivalent to ${\pi }_F$. It is therefore a counterexample to the universal claim that every irreducible representation is Fock-equivalent.