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Finite-Mode CAR and the Complex Clifford Algebra via Majorana Generators

Statement

Let $n \geq 0$ , let $V = C^{2 n}$ with ordered basis $e_{1}, \dots, e_{2 n}$ , and put $Q (\sum_{r} z_{r} e_{r}) = \sum_{r} z_{r}^{2}$ . Use the plus- $Q$ convention of Universal Clifford algebra of a quadratic space. Give ${C l}^{+} (V, Q)$ the conjugate-linear anti-involution that fixes each $e_{r}$ ; it is defined by conjugating coefficients, reversing words, and fixing the basis generators. Then there is a $*$ -isomorphism

${C l}^{+} (V, Q) ≅ {C A R}_{n}^{a l g}$

specified by

$e_{2 j - 1} ⟼ γ_{2 j - 1} = a_{j} + a_{j}^{†}, e_{2 j} ⟼ γ_{2 j} = i (a_{j}^{†} - a_{j}) .$

The inverse sends $a_{j}$ to $(e_{2 j - 1} + i e_{2 j}) / 2$ and $a_{j}^{†}$ to $(e_{2 j - 1} - i e_{2 j}) / 2$ . These are finite-mode algebra maps; they assert no relation between CAR and CCR.

Facts & Assumptions

Given: An integer $n \geq 0$ , the finite-mode CAR $*$ -algebra, and the complex quadratic space $(V, Q)$ with its ordered basis and plus- $Q$ convention.

[F1]

In ${C A R}_{n}^{a l g}$ , ${a_{i}, a_{j}} = 0$ , ${a_{i}^{†}, a_{j}^{†}} = 0$ , and ${a_{i}, a_{j}^{†}} = δ_{i j} 1$ ; its quotient presentation gives the unique $*$ -homomorphism for every tuple satisfying these relations (Finite-Mode Complex CAR Algebra).

[F2]

The plus- $Q$ Clifford algebra is the quotient with canonical generators satisfying ${\overset{ˉ}{v}}^{2} = Q (v) 1$ (Universal Clifford algebra of a quadratic space).

[F3]

Its polarized relation is $\overset{ˉ}{v} \overset{ˉ}{w} + \overset{ˉ}{w} \overset{ˉ}{v} = 2 B (v, w) 1$ (Polarized relations in a Clifford algebra).

[F4]

Every linear map into a unital associative $k$ -algebra satisfying $f (v)^{2} = Q (v) 1$ extends uniquely to a unital algebra map from $C l (V, Q)$ (Universal property of the Clifford algebra).

Proof

1.1givenF2constructalgebra

On $T_{C} (V)$ define $e_{r}^{*} = e_{r}$ , conjugate scalars, and reverse each tensor word. For $v = \sum_{r} z_{r} e_{r}$ , let $\overline{v} = \sum_{r} \overline{z_{r}} e_{r}$ . Then the adjoint of the defining relation $j (v)^{2} - Q (v) 1$ is $j (\overline{v})^{2} - \overline{Q (v)} 1 = j (\overline{v})^{2} - Q (\overline{v}) 1$ , another defining relation. The generated two-sided ideal is therefore $*$ -stable, so this operation descends to the stated anti-involution on the Clifford quotient.

1.2F1algebra

By [F1], $γ_{2 j - 1}^{*} = γ_{2 j - 1}$ and $γ_{2 j}^{*} = γ_{2 j}$ . The same-mode relations give $γ_{2 j - 1}^{2} = a_{j} a_{j}^{†} + a_{j}^{†} a_{j} = 1$ and $γ_{2 j}^{2} = - (a_{j}^{†} - a_{j})^{2} = a_{j}^{†} a_{j} + a_{j} a_{j}^{†} = 1$ . Also ${γ_{2 j - 1}, γ_{2 j}} = i {a_{j} + a_{j}^{†}, a_{j}^{†} - a_{j}} = 0$ . For distinct modes every term in the expanded anticommutator vanishes by [F1]. Thus ${γ_{r}, γ_{s}} = 2 δ_{r s} 1$ .

2.1givenF4step 1.2construct

For $v = \sum_{r} z_{r} e_{r}$ , linearity and step 1.2 give $(\sum_{r} z_{r} γ_{r})^{2} = \sum_{r} z_{r}^{2} 1 + \sum_{r < s} z_{r} z_{s} {γ_{r}, γ_{s}} = \sum_{r} z_{r}^{2} 1 = Q (v) 1$ . Consequently [F4] gives a unique unital algebra map $Φ : {C l}^{+} (V, Q) \to {C A R}_{n}^{a l g}$ with $Φ (e_{r}) = γ_{r}$ . Since each $γ_{r}$ is self-adjoint, $Φ$ preserves $*$ .

2.2givenF1F3step 1.1construct

Put $b_{j} = (e_{2 j - 1} + i e_{2 j}) / 2$ and $b_{j}^{†} = (e_{2 j - 1} - i e_{2 j}) / 2$ in the Clifford algebra. By [F3], ${e_{r}, e_{s}} = 2 δ_{r s} 1$ . Expanding the three CAR anticommutators for the $b_{j}, b_{j}^{†}$ therefore gives ${b_{i}, b_{j}} = 0$ , ${b_{i}^{†}, b_{j}^{†}} = 0$ , and ${b_{i}, b_{j}^{†}} = δ_{i j} 1$ . The anti-involution from step 1.1 gives $b_{j}^{*} = b_{j}^{†}$ . The quotient definition in [F1] now yields a unique $*$ -homomorphism $Ψ : {C A R}_{n}^{a l g} \to {C l}^{+} (V, Q)$ with these generator images.

3.1step 2.1step 2.2algebra∎

On each generator, $Φ Ψ (a_{j}) = (γ_{2 j - 1} + i γ_{2 j}) / 2 = a_{j}$ and $Φ Ψ (a_{j}^{†}) = (γ_{2 j - 1} - i γ_{2 j}) / 2 = a_{j}^{†}$ ; likewise $Ψ Φ (e_{2 j - 1}) = e_{2 j - 1}$ and $Ψ Φ (e_{2 j}) = e_{2 j}$ . The two composites are identities because each algebra is generated by the displayed generators and both maps preserve its operations. Thus $Φ$ and $Ψ$ are inverse $*$ -isomorphisms.

Remarks

The scalar-product pairing convention is irrelevant to these generator relations: the modes are indexed by an orthonormal basis and the CAR use only $δ_{i j}$ . The sign is fixed by $e_{r}^{2} = + 1$ .

Depends on

Used by

One-Mode CAR and Its Two Majorana Matrices Example
The Finite-Mode CAR Algebra Is a Full Matrix Algebra Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Günther Hörmann, C*-Algebras with Aspects of Quantum Physics, Chapter 6, §§6.1–6.4 (standard reference, not scraped)
Jan Dereziński, Introduction to Representations of the Canonical Commutation and Anticommutation Relations, §§4.1–4.2 (standard reference, not scraped)