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One-Mode CAR and Its Two Majorana Matrices

Example

For one mode, use the ordered orthonormal basis $e_{\emptyset}, e_{{1}}$ of $F_{1}$ . The annihilator and its adjoint are

$a = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), a^{†} = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}) .$

The Majorana generators are the Pauli matrices

$γ_{1} = a + a^{†} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), γ_{2} = i (a^{†} - a) = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}) .$

Facts & Assumptions

Given: The one-mode CAR algebra, the plus- $Q$ Clifford convention, and the ordered occupation basis $(e_{\emptyset}, e_{{1}})$ .

[F1]

The finite-mode matrix model and Majorana maps are those of The Finite-Mode CAR Algebra Is a Full Matrix Algebra and Finite-Mode CAR and the Complex Clifford Algebra via Majorana Generators.

Verification

1.1givenF1algebra

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

2.1givenF1step 1.1algebra

Substitution into the Majorana formulas gives the two displayed Pauli matrices. Direct multiplication gives $γ_{1}^{2} = γ_{2}^{2} = I$ and $γ_{1} γ_{2} + γ_{2} γ_{1} = 0$ ; both matrices are self-adjoint. Hence they represent the two orthogonal plus- $Q$ Clifford generators.

3.1givenF1algebra∎

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 $M_{2} (C)$ , as asserted by the finite-mode theorem.

Depends on

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