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Countable-Mode CAR Has Inequivalent Irreducible Representations

Statement

Let $π_{F}$ be the countable-mode occupation representation on $F_{\infty}$ and let $α$ be the particle-hole automorphism of Particle-Hole Exchange Is an Automorphism of Countable-Mode CAR. Then $π_{F}$ and $π_{h} : = π_{F} \circ α$ are irreducible representations of ${C A R}_{N}$ , and they are not unitarily equivalent. Here two representations are unitarily equivalent if a unitary operator between their Hilbert spaces intertwines their actions. This proves existence of inequivalent irreducible representations; it does not classify all representations.

Facts & Assumptions

Given: The countable-mode CAR algebra, the occupation representation $π_{F}$ , and the particle-hole automorphism $α$ .

[F1]

The occupation representation is irreducible, and no nonzero vector is killed by every creator $c_{j}^{†}$ (The Countable-Mode CAR Fock Representation Is Irreducible).

[F2]

The particle-hole map $α$ is a surjective $*$ -automorphism with $α (a_{j}) = a_{j}^{†}$ (Particle-Hole Exchange Is an Automorphism of Countable-Mode CAR).

Proof

1.1givenF1F2algebra

Since $α$ is a $*$ -automorphism, $π_{h} = π_{F} \circ α$ is a representation. If a closed subspace were invariant under $π_{h} ({C A R}_{N})$ , surjectivity of $α$ would make it invariant under $π_{F} ({C A R}_{N})$ ; [F1] therefore makes it trivial or all of $F_{\infty}$ . Thus $π_{h}$ is irreducible.

2.1givenF1F2algebra∎

Suppose a unitary $U : F_{\infty} \to F_{\infty}$ intertwines $π_{F}$ with $π_{h}$ . Then for every $j$ , $U c_{j} = π_{h} (a_{j}) U = c_{j}^{†} U$ . Applying this to the vacuum $Ω = e_{\emptyset}$ gives $c_{j}^{†} U Ω = U c_{j} Ω = 0$ for every $j$ . Since $U Ω \neq 0$ , this contradicts the common-creator-kernel statement in [F1]. No such unitary exists.

Remarks

The explicit obstruction is the absence of a nonzero vector in the infinite occupation space annihilated by every creator. In finite mode number such a fully occupied vector exists, so this argument is specifically about countably infinitely many modes.

Depends on

Used by

The Countable-Mode CAR Algebra Has a Non-Fock-Equivalent Irreducible Representation Counterexample

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Sources

Günther Hörmann, C*-Algebras with Aspects of Quantum Physics, Chapter 6, §6.7 (standard reference, not scraped)
Jan Dereziński, Introduction to Representations of the Canonical Commutation and Anticommutation Relations, §§4.1 and 5.7 (standard reference, not scraped)