We extend the Fock and the infinite wedge represen- tations of A to permutative representations of O2

Extensions of representations of the CAR algebra to the Cuntz algebra O₂

—the Fock and the infinite wedge—

Katsunori Kawamura

Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan

Fermions are expressed by polynomials of canonical genera- tors of the Cuntz algebraO2and they generate theU(1)-fixed point subalgebra A ≡ O^U₂⁽¹⁾ of O2 by the canonical gauge action. We extend the Fock and the infinite wedge represen- tations of A to permutative representations of O2. By these extensions, the boson-fermion correspondence is rewritten by canonical generators of O2.

1. Introduction

Let A₀ be the Clifford algebra generated by fermions a_n, a^∗_n, n ∈ N ≡ {1,2,3, . . .}which satisfy the canonical anticommutation relations(=CAR):

(1.1) a_na^∗_m+a^∗_ma_n=δ_n,mI, a^∗_na^∗_m+a^∗_ma^∗_n=a_na_m+a_ma_n= 0 for n, m ∈ N. A₀ always has unique C^∗-norm k · k and the completion A ≡ A₀ with respect tok · kis called theCAR algebrain theory of operator algebras([6]). In [1,2,3,4], we construct several polynomial embeddings of Ainto the Cuntz algebrasO_N. For example, ifs₁, s₂are canonical generators of O₂, that is, they satisfy

(1.2) s^∗_is_j =δ_ijI (i, j= 1,2), s₁s^∗₁+s₂s^∗₂ =I, then

(1.3) a₁ ≡s₁s^∗₂, a_n≡ X

J∈{1,2}ⁿ⁻¹

(−1)^n2(J)s_Js₁s^∗₂s^∗_J (n≥2) satisfy (1.1) wheren₂(J)≡P_k

l=1(j_l−1) ands_J =s_j1· · ·s_jk,s_J^∗ =s^∗_jk· · ·s^∗_j1 forJ = (j₁, . . . , j_k), and C^∗<{a_n ∈ O₂ :n∈N}> coincides with a fixed- point subalgebra O^U(1)₂ of O₂ by the canonical gauge action. Put a linear mapζ on O₂ by

(1.4) ζ(x)≡s₁xs^∗₁−s₂xs^∗₂ (x∈ O₂).

e-mail:[email protected].

Then a_n =ζ(a_n−1) for each n≥2. In this sense,{a_n}_n∈N in (1.3) is called therecursive fermion system(=RFS) inO₂.

In this paper, we extend the Fock and the infinite wedge representations([12, 13]) of Ato permutative representations of O₂ under identification of Aas O^U(1)₂ ⊂ O₂ by (1.3). At first, we give our main theorem for abstract for- mulations of representations of A.

Theorem 1.1. (i) Let (H_F, π_F) be the Fock representation of A, that is, (H_F, π_F) is a cyclic representation with a cyclic vector Ω∈ H_F such that

π_F(a_n)Ω = 0 (^∀n∈N).

Then (H_F, π_F) is extended to an irreducible representation (H_F,π˜_F) of O₂ defined by

˜

π_F(s₁)≡L, π˜_F(s₂)≡π(a^∗₁)·L

where L is the one-sided shift operator on H_F defined by LΩ≡Ω, Lπ_F(a^∗_n1· · ·a^∗_nk)Ω≡π_F(a^∗_n1+1· · ·a^∗_nk+1)Ω for each n₁, . . . , n_k∈Nand k∈N.

(ii) Let (Λ^∞2V, π_∞,+) be the infinite wedge representation of A, that is, (Λ^∞2 V, π_∞,+) is a cyclic representation with a cyclic vector |vac>₊ ∈ Λ^∞2 V such that

ψ_−k|vac>₊=ψ^∗_k|vac>₊= 0 (^∀k∈Z+¹₂, k >0) where

(1.5) ψ_k≡π_∞,+(a_2k+1), ψ_−k≡π_∞,+(a_2k) (k∈Z+ ¹₂, k >0) and Z+¹₂ ≡ {n+ 1/2 :n∈Z}. Then (Λ^∞2 V, π_∞,+) is extended to an irreducible representation (Λ^∞2 V ⊕Λ^∞2 V^∗,Π) of O₂ which satisfies

Π(s₁s₂)|vac>₊ =|vac>₊.

Both representations (H,π˜_F) and (Λ^∞2 V ⊕Λ^∞2 V^∗,Π) of O₂ in Theorem 1.1 are permutative representations([5, 8, 9]) and they are not equivalent each other. Well-known Fock and infinite wedge representations are just realizations of those in Theorem 1.1. The extension for a concrete infinite wedge is given in § 4.

On the other hand, the boson-fermion correspondence on the infinite wedge representation is given by

(1.6) α_n= X

k∈Z+¹₂

ψ_k−nψ_k^∗ (n∈Z\ {0}).

{a_n}_n∈Z satisfies α_−n =α^∗_n,α_nα_m−α_mα_n =n·δ_n,−mI. By identifying s_i and Π(s_i) in Theorem 1.1 (ii) and combining (1.3) and (1.5), we have















ψ_k= X

J∈{1,2}^2k

(−1)^n2(J)s_Js₁s^∗₂s^∗_J

ψ_−k = X

J∈{1,2}^2k−1

(−1)^n2(J)s_Js₁s^∗₂s^∗_J

(k∈Z+¹₂, k >0).

From these and (1.6), we have a direct expression of bosons by canonical generators of O₂ as follows:

(1.7) α_n=X

l∈N

ρ^2l−2(X_n) +B_n (n≥1) where

(1.8) X_n≡ρ(s₁s^∗₂ζ²ⁿ(s₂s^∗₁)) +ζ²ⁿ(s₁s^∗₂)s₂s^∗₁ (n≥1),

(1.9) B₁ ≡ −s₁s₂s₁^∗s^∗₂, B_n≡ρ(B_n−1^∗ )−s₁ζ²ⁿ⁻²(s₂s^∗₁)s^∗₂ (n≥2), ζ is in (1.4) and ρ is the canonical endomorphism of O₂, that is, ρ(x) ≡ s₁xs^∗₁+s₂xs^∗₂ for x ∈ O₂. Furthermore α^∗_n|vac>₊ = B_n^∗|vac>₊ for each n≥1.

In§ 2, we review representations ofAandO₂ and the RFS. In§3, we introduce a branching function system on the space of Maya diagrams and review the infinite wedge space and its dual space. In§4, we show extensions of the Fock and the infinite wedge representations toO₂. A relation between a branching law of a permutative representation of O₂ and the extension of the infinite wedge is concretely illustrated.

2. The recursive fermion system and permutative representations of O₂

Both O₂ and the CAR algebra

CAR≡C^∗<{a_n:n∈N}>

(=Ain§1) are simple, infinite dimensional, noncommutative C^∗-algebras([6, 7]). Remark that a^∗_n ∈ CAR for each n ∈ N by definition of C^∗-algebra.

Unital∗-homomorphisms (specially, unital∗-representations) from these al- gebras to other algebras are always faithful. Algebras which are generated by generators s₁, s₂ in (1.2) and a_n, n ∈ N in (1.1) are unique up to ∗- isomorphisms, respectively. Therefore their representations are determined by only operators on a Hilbert space, which satisfy relations of their gener- ators without ambiguity. In this paper, a representation and an embedding always mean a unital ∗-representation and a unital ∗-embedding, respec- tively.

2.1. Representations ofCARand the RFS.We review representations of CARin theory of operator algebras in [6].

Definition 2.1. Let (H, π) be a representation of CAR.

(i) (H, π) is the (abstract)Fock representation of CAR if there is a cyclic unit vector Ω∈ H such that π(a_n)Ω = 0 for each n∈N. Ω is called the vacuum of (H, π). We denote(H, π) by H_{F ock} simply.

(ii) (H, π) isP[12] if there is a cyclic unit vectorΩ∈ H such that π(a_2n−1)Ω =π(a^∗_2n)Ω = 0 (^∀n∈N).

(iii) (H, π) isP[21] if there is a cyclic unit vectorΩ∈ H such that π(a^∗_2n−1)Ω =π(a_2n)Ω = 0 (^∀n∈N).

For consistency with after statements, any Ω in the above is normalized.

H_{F ock},P[12],P[21] appear in [5] as components of irreducible decomposition of permutative representation of O₂, which are called “atom”. This fact is explained in Proposition 2.6.

Proposition 2.2. All ofH_{F ock}, P[12], P[21]are unique up to unitary equiv- alences and irreducible. Any two of H_{F ock}, P[12], P[21] are not unitarily equivalent.

Proof. See (5.18) in [2], and [5]. In Appendix A, their inequivalences

are shown. ¤

By Proposition 2.2, symbolsH_{F ock}, P[12] and P[21] make sense as equiva- lence classes of representations. Since fermions are often treated as operators on a concrete Hilbert space, any representation which is different with the Fock representation in only permutation of creations and annihilations and their phase factors, are called the Fock representation, too in such situation.

In this paper, we do not call such representation by the Fock representation.

We review a concrete example: Put H ≡ l₂(N) and the completely antisymmetric Fock space F₋(H) ≡ CΩ⊕L_∞

k=1H^∧k, H^∧k ≡ P₋^(k)H^⊗k whereP₋^(k) is the antisymmetrizer onH^⊗k defined byP₋^(k)(v₁⊗ · · · ⊗v_k)≡ (√

k!)^−1/2P

σ∈Sksgn(σ)·v_σ(1)⊗· · ·⊗v_σ(k)fork≥1. We denotev₁∧· · ·∧v_k= P₋^(k)(v₁⊗ · · · ⊗v_k). We seev_σ(1)∧ · · · ∧v_σ(k)= sign(σ)(v₁⊗ · · · ⊗v_k) for each σ ∈S_k. Forf ∈H, define A^∗(f)Ω ≡f, A^∗(f)v ≡f ∧v for f ∈ H, v ∈ H^∧n, n ≥ 1. A(f) is defined by the adjoint operator ofA^∗(f) on F₋(H).

We see that A(f)Ω = 0 for each f ∈H. Then A(f)A^∗(g) +A^∗(g)A(f) =<

f|g > I for each f, g∈ H. For the canonical basis {e_n}_n∈N of H =l₂(N), put π_F(a_n) ≡ A(e_n) for n ∈ N. Then (F₋(H), π_F) is a representation of CAR. (F₋(H), π_F) is the (concrete)Fock representation.

Let s₁, s₂ be canonical generators ofO₂. Define an embedding (2.1) ϕ_S:CAR ,→ O₂; ϕ_S(a_n)≡ζⁿ⁻¹(s₁s^∗₂) (n≥1)

where ζ is in (1.4). For example, ϕ_S(a₁) = s₁s^∗₂, ϕ_S(a₂) = s₁s₁s^∗₂s^∗₁ − s₂s₁s^∗₂s^∗₂. We call ϕ_S by the standard embedding of CAR into O₂. C^∗<

{ϕ_S(a_n)}_n∈N >=O₂^U(1) ={x∈ O₂ : ^∀z∈U(1), γ_z(x) =x} ∼=U HF₂ where γ is the canonical U(1)-action of O₂, γ_z(s_i)≡ zs_i forz ∈U(1)≡ {z∈C:

|z|= 1}andi= 1,2. By identifyinga_nandϕ_S(a_n),a_n’s coincide with those in (1.3) and we have the following intertwining relations:

Lemma 2.3. For n≥1,

s₁a_n=a_n+1s₁, s₁a^∗_n=a^∗_n+1s₁, s₂a_n=−a_n+1s₂, s₂a^∗_n=−a^∗_n+1s₂, s^∗₁a_n+1=a_ns^∗₁, s^∗₁a^∗_n+1 =a^∗_ns^∗₁, s^∗₂a_n+1 =−a_ns^∗₂, s^∗₂a^∗_n+1=−a^∗_ns^∗₂. 2.2. Permutative representations of O₂ and their branching laws.

Permutative representations of the Cuntz algebras are well-studied([5,8,9]).

We introduce two permutative representations of O₂ according to [10].

Definition 2.4. A representation(H, π)ofO₂ isP(1)(resp. P(12)) if there is a cyclic unit vector Ω∈ H such that π(s₁)Ω = Ω(resp. π(s₁s₂)Ω = Ω).

We call Ω by the GP vector of(H, π).

Both P(1) and P(12) exist uniquely up to unitary equivalences, and they are irreducible and not unitarily equivalent each other.

Assume that (H, π) isP(12) of O₂ with the GP vector Ω and α is an automorphism ofO₂ defined byα(s₁)≡s₂,α(s₂)≡s₁. Define an operator U on H by

(2.2) UΩ≡π(s₂)Ω, U π(s_J)Ω≡π(α(s_J)s₂)Ω (J ∈ {1,2}^k, k≥1).

Then U is a unitary which satisfies U² = I and AdU ◦π =π◦α. In con- sequence, (H, π, U) is a covariant representation of a C^∗-dynamical system (O₂, α,Z₂).

Example 2.5. (i) Define a representation (l₂(N), π_S) of O₂ by π_S(s₁)e_n≡e_2n−1, π_S(s₂)e_n≡e_2n (n∈N).

Then (l₂(N), π_S) is P(1). We call (l₂(N), π_S) by the standard repre- sentation ofO₂.

(ii) Define a representation (l₂(N), π₁₂) of O₂ by

π₁₂(s₁)e_2n−1≡e_4n−1, π₁₂(s₁)e_2n≡e_4n−3, π₁₂(s₂)e_n≡e_2n (n∈N).

The action of s₁, s₂ on the canonical basis of l₂(N) is illustrated as follows:

e₃ e₄

e₇ e₈

e₆ e₅

π₁₂(s₂) π₁₂(s₁)

π₁₂(s₁) π₁₂(s₂) π₁₂(s₁) π₁₂(s₂)

@@

@ I

@@

@

I

AA AA AKA

e₁ e₂

R

I π₁₂(s₂)

π₁₂(s₁)

This system looks like the Fock representation with two vacuums e₁ and e₂. This diagram appears in § 3 and § 4 again. (l₂(N), π₁₂) is P(12). Remark that π₁₂(s₁s₂)e₁ = e₁ is expressed as a cycle in the above. For this type example, see [11].

Byϕ_Sin (2.1), we identifyCARand a subalgebraϕ_S(CAR) =O₂^U(1)⊂ O₂. For a representation (H, π) of O₂, we have the restriction (H, π|_CAR) of (H, π) on CAR.

Proposition 2.6. ([2]) The following branching laws hold:

P(1)|_CAR =H_{F ock}, P(12)|_CAR =P[12]⊕P[21].

Specially, all of these are irreducible decompositions.

We consider the branching of P(12) onCARmore.

Lemma 2.7. Let (H, π) beP(12)of O₂ with the GP vectorΩ₁ ≡Ωand put Ω₂≡π(s₂)Ω₁. Then we have the followings:

π(a_2k−1)Ω₁= 0, π(a_2k)Ω₁= (−1)^k−1π(s₍₁₂₎k−1s₁s₁)Ω₁, π(a^∗_2k−1)Ω₁= (−1)^k−1π(s₍₁₂₎k−1s₂s₂)Ω₁, π(a^∗_2k)Ω₁ = 0, π(a_2k−1)Ω₂ = (−1)^k−1π(s₍₂₁₎k−1s₁)Ω₁, π(a_2k)Ω₂= 0, π(a^∗_2k−1)Ω₂= 0, π(a^∗_2k)Ω₂= (−1)^kπ(s₂s₍₁₂₎k−1s₂s₂)Ω₁ for each k∈N.

Proof. By π(ζ(x)s₂)Ω₁ = −π(s₂x)Ω₁ for any x ∈ O₂, statements

hold. ¤

Let V₁₂ ≡ π(CAR)Ω₁, V₂₁ ≡π(CAR)Ω₂. ThenH =V₁₂⊕V₂₁ and we see

that V₁₂ V₂₁

vacuum Ω₁ Ω₂

creation a^∗_2k−1, a_2k a_2k−1, a^∗_2k annihilation a_2k−1, a^∗_2k a^∗_2k−1, a_2k wherek∈N. Specially,

(2.3)





π(a₁)Ω₂ =π(s₁)Ω₁, π(a₂)Ω₁ =π(s₁s₁)Ω₁, π(a^∗₁)Ω₁ =π(s₂)Ω₂, π(a^∗₂)Ω₂ =−π(s₂s₂)Ω₂.

Ifαis theZ₂-action onO₂anda_nis the RFS inO₂, thenα(a_n) = (−1)ⁿ⁻¹a^∗_n. Hence U in (2.2) satisfies

UΩ₁ = Ω₂, UΩ₂= Ω₁, U π(a_Ka^∗_L)Ω₁ = (−1)^|K|1+|L|1π(a^∗_Ka_L)Ω₂ where a_K ≡ a_k1· · ·a_kn, |K|₁ ≡ P_n

i=1(k_i −1) for K = {k₁, . . . , k_n} ⊂ N.

Hence U V₁₂=V₂₁.

Example 2.8. (i) In Example 2.5 (i),π_S◦ϕ_S isH_{F ock} with the vacuum e₁. See [1] for more detail.

(ii) In Example 2.5 (ii), we considerπ₁₂◦ϕ_S. Then (l₂(2N−1), π₁₂◦ϕ_S) is P[12] and (l₂(2N), π₁₂◦ϕ_S) is P[21]. If we identify a_n and (π₁₂◦ ϕ_S)(a_n), then

a_2n−1e₁=a^∗_2ne₁=a_2ne₂=a^∗_2n−1e₂ = 0,

a_2ne₁ = (−1)ⁿ⁻¹e₄n−1·6+1, a^∗_2n−1e₁= (−1)ⁿ⁻¹e₄n−1·3+1, a_2n−1e₂= (−1)ⁿ⁻¹e₄ⁿ⁻¹_·3+2, a^∗_2ne₂ = (−1)ⁿe₄ⁿ⁻¹_·6+2

for eachn∈N. These statements are shown by using (π₁₂(s₁s₂))^me_n= e₄^m_(n−1)+1 for each m, n∈N. Specially, when n₂> n₁,

a_2n1a_2n2e₁= (−1)ⁿ²⁻¹e_3·(22n2−1+2²ⁿ¹⁻¹)+1.

3. A branching function system on the infinite wedge We review a representation of the fermion algebra which is called theinfinite wedge space([12,13]) according to notation in [13]. In order to extend this representation toO₂, we introduce the dual infinite wedge space at once and a branching function system on them.

3.1. Maya diagram.DenoteZ+¹₂ ≡ {n+¹₂ :n∈Z}. Put

Z_+/2 ≡ {n+ 1/2 :n∈Z, n≥0}, Z_−/2 ≡ {n−1/2 :n∈Z, n≤0}.

For a subsetS ⊂Z+¹₂, define ∆_±(S)⊂Z+¹₂ by (3.1) ∆_±(S)≡(S\Z_∓/2)∪(Z_∓/2\S).

Remark the sign of both sides.

Definition 3.1. An element in M_± ≡ {S ⊂ Z+ ¹₂ : #∆_±(S) < ∞} is called a Maya diagram. Specially, Z_∓/2 ∈ M_± is called the vacuum in M_±. We see thatM₊∩ M₋=∅and M_±={−S :S∈ M_∓}where−S ≡ {−k: k ∈ S}. There are maxS for any S ∈ M₊ and minS for any S ∈ M₋, and #S = ∞ for any S ∈ M_±. Therefore we can always parameterize as follows: S ={t_i :i∈N} such that t_i > t_i+1 for i≥1 when S ∈ M₊, and S ={t_i :i∈N} such thatt_i< t_i+1 fori≥1 whenS ∈ M₋.

We illustrate S ∈ M_± by a two-sided infinite sequence consisting of symbols ◦ and • along the lattice Z+¹₂ as follows: For S ∈ M_±, put ◦ at k∈Z+¹₂ whenk∈S, and put •atk∈Z+¹₂ whenk6∈S. For example, if {−5/2,−1/2,3/2,7/2} ⊂S and {−7/2,−3/2,1/2,5/2} ∩S=∅, then

· · · −7/2 −5/2 −3/2 −1/2 1/2 3/2 5/2 7/2 · · ·

· · · • ◦ • ◦ • ◦ • ◦ · · ·

By this illustration, M_± are the following sets:

M₊={· · · ◦ ◦ ∗ ∗ ∗ ∗ ∗ ∗ • • · · · }, M₋ ={· · · • • ∗ ∗ ∗ ∗ ∗ ∗ ◦ ◦ · · · } where∗ ∗ ∗ ∗ ∗∗is taken any finite sequence consisting of ◦and•. Specially,

vacuum Z_−/2 ∈ M₊ · · · ◦ ◦ ◦ ◦ • • • • · · · dual vacuum Z_+/2 ∈ M₋ · · · • • • • ◦ ◦ ◦ ◦ · · ·

3.2. A branching function system on the space of Maya diagrams.

Put the space of all Maya diagrams

M ≡ M₊∪ M₋.

We give a branching function system on M. Put S_± ≡ S∩Z_±/2, S₊₁ ≡ {k+ 1 :k∈S} and S_±,+1≡(S_±)₊₁. ForS ∈ M, define

(3.2) g₁(S)≡ −(S_+,+1∪S₋∪ {1/2}), g₂(S)≡ −(S_+,+1∪S₋).

Then g ={g₁, g₂} is a branching function system on M, that is,g₁ and g₂ are injective maps on M, g₁(M)∩g₂(M) =∅ and g₁(M)∪g₂(M) = M.

Furthermore g₁(M) ={S ∈ M : −1/2 ∈ S}, g₂(M) ={S ∈ M :−1/2 6∈

S},

g⁻¹₁ (S) =−{(S₋\ {−1/2})₊₁∪S₊}, g⁻¹₂ (S) =−(S_−,+1∪S₊).

If θ(S) ≡ Z+¹₂ \S, then θ² = id, θ(M_±) = M_∓, θ(Z_±/2) = Z_∓/2 and g₂=θ◦g₁◦θ.

Lemma 3.2. Denote S_±n ≡ {k±n: k∈ S}, S_±,+n ≡ (S_±)_+n. Then the followings hold:

(i) (g₁◦g₂)ⁿ(S) =S_−,−n∪S_+,+n∪ {−1/2, . . . ,−(n−1)−1/2} forn∈N.

(ii) (g₁◦g₂)⁻ⁿ(S) = (S_−,+n)₋∪S_+,−n for n∈N.

(iii) Put h_n(S)≡(g₁₂ⁿ⁻¹◦g₁◦g⁻¹₂ ◦(g₁₂ⁿ⁻¹)⁻¹)(S) and k_n(S)≡(g₁₂ⁿ⁻¹◦g₁◦ g₁◦(g⁻¹₁₂)ⁿ)(S) for n∈N. Then

h_n(S) =S∪ {−(n−1)−1/2}, k_n(S) =S∪ {n−1 + 1/2} (n∈N).

We illustrateg by Maya diagrams:

S g₁(S) g₂(S)

Z_−/2 Z_+/2∪ {−1/2} Z_+/2

Z_+/2 Z_−/2 Z_−/2\ {−1/2}

Z_+/2∪ {−1/2} Z_−/2∪ {1/2} (Z_−/2\ {−1/2})∪ {1/2}

Z_−/2\ {−1/2} (Z_+/2\ {1/2})∪ {−1/2} Z_+/2\ {1/2}

... ... ...

Z_+/2∪ {−1/2} Z_−/2\ {−1/2}

Z_−/2∪ {1/2} Z_+/2\ {1/2}

(Z_−/2\ {−1/2})∪ {1/2} (Z_+/2\ {1/2})∪ {−1/2}

g₂ g₁

g₁ g₂ g₁ g₂

t t d d d d d d t t t t

t t t t d d d d d d t t

d d t d t t t t d t d d

@@

@ I

@@

@

I

AA AA AKA

d d d t t t t t t d d d

R

I g₂

g₁

Z_−/2 Z_+/2





◦ (k∈S)

• (k6∈S)

3.3. The infinite wedge representation of CAR and its dual.We introduce the infinite wedge space by a Hilbert space of Maya diagrams.

For a set Λ, l₂(Λ) is a complex Hilbert space with a complete orthonormal basis{e_λ}_λ∈Λ and diml₂(Λ) = #Λ.

Definition 3.3. ForM_± in Definition 3.3,

Λ^∞2 V^#≡l₂(M), Λ^∞2 V ≡l₂(M₊), Λ^∞2 V^∗ ≡l₂(M₋)

are called the bi-infinite wedge space, the infinite wedge space and the dual infinite wedge space, respectively.

We see that Λ^∞2 V^# = Λ^∞2V ⊕Λ^∞2 V^∗. By definition, Λ^∞2 V^#, Λ^∞2 V and Λ^∞2 V^∗ have canonical basis {e_S :S ∈ M}, {e_S :S ∈ M₊} and {e_S :S ∈ M₋}, respectively. Usually, the symbol Λ^∞2 V means a subspace ofl₂(M₊)

consisting of finite linear combinations of {e_S : S ∈ M₊}([12, 13]). We denote

|vac>_±≡e_Z∓/2.

Since there are maxS for any S ∈ M₊ and minS for any S ∈ M₋, and

#S=∞ for anyS ∈ M_±, we can denote

t₁∧t₂∧ · · ·=e_S when S ={t_i : ^∀i∈N, t_i> t_i+1} ∈ M₊, t₁∧t₂∧ · · ·=e_S when S ={t_i : ^∀i∈N, t_i< t_i+1} ∈ M₋. Then we see that

|vac>₊ = (−¹₂)∧(−³₂)∧(−⁵₂)∧ · · ·, |vac>₋= ¹₂ ∧³₂ ∧⁵₂ ∧ · · ·. For a permutation σ∈Sk k≥2, define

t_σ(1)∧ · · · ∧t_σ(k)∧t_k+1∧ · · · ≡sgn(σ)·t₁∧ · · · ∧t_k∧t_k+1∧ · · ·. By these definitions, “∧” seems the exterior product of infinite vectors.

Define a family {ψ_k}_k∈Z+¹

2 of operators on Λ^∞2V^#by ψ_ke_S≡





(−1)^dS(k)·e_S∪{k} (k6∈S),

0 (otherwise)

(S ∈ M)

where d_S(k) ≡ min{#{x ∈ S : x > k},#{x ∈ S : x < k}}. We simply denote

ψ_ke_S = (−1)^dS(k)·χ_S^c(k)·e_S∪{k}

whereχ_S^c is the characteristic function onS^c≡(Z+¹₂)\S. We can easily check that the definition ofψ_k coincides with the following ordinary defini- tion:

ψ_kv=k∧v (v∈Λ^∞2 V^#, k∈Z+¹₂).

Lemma 3.4. (i) The adjoint ψ_k^∗ of ψ_k is given by

ψ_k^∗e_S= (−1)^dS\{k}(k)·χ_S(k)·e_S\{k} (k∈Z+ ¹₂, S∈ M).

(ii) ψ_kψ_k^∗e_S =χ_S(k)·e_S for k∈Z+¹₂ and S ∈ M.

(iii) ψ_kψ_l^∗+ψ^∗_lψ_k=δ_klI fork, l∈Z+¹₂ and other anticommutators vanish.

We see that

ψ_−k|vac>₊=ψ_k^∗|vac>₊= 0,

ψ_k|vac>₊=k∧ |vac>₊, ψ_−k^∗ |vac>₊= (−1)^k−12 ·e_{Z−/2\{−k}}

fork∈Z+¹₂,k >0. In the same way, we see that Λ^∞2 V Λ^∞2 V^∗ vacuum |vac>₊ |vac>₋ creation ψ^∗_−k, ψ_k ψ_−k, ψ^∗_k annihilation ψ_−k, ψ_k^∗ ψ_−k^∗ , ψ_k

wherek∈Z+ ¹₂,k >0.

Definition 3.5. A representation(Λ^∞2V^#, π_∞) of CAR defined by (3.3) π_∞(a_2n−1)≡ψ_−n+1/2, π_∞(a_2n)≡ψ_n−1/2 (n∈N) is called the bi-infinite wedge representation ofCAR.

On the other hand,

(3.4) ψ_k=π_∞(a_2k+1), ψ_−k=π_∞(a_2k) (k∈Z+¹₂, k >0).

Proposition 3.6. (i) The following irreducible decomposition of repre- sentations of CAR holds:

Λ^∞2V^#= Λ^∞2V ⊕Λ^∞2V^∗. (ii) If we denote

π_∞,+≡π_∞|_Λ^∞_{2 V}, π_∞,−≡π_∞|_Λ^∞_{2 V∗}, then (Λ^∞2 V, π_∞,+) is P[12] and (Λ^∞2 V^∗, π_∞,−) isP[21].

(Λ^∞2 V, π_∞,+) and (Λ^∞2 V^∗, π_∞,−) are called theinfinite wedge representation and thedual-infinite wedge representationof CAR, respectively.

4. Standard extensions of representations of CAR

In order to show extension theorems, we prepare a notion, “standard exten- sion” of a representation of CARtoO₂ as follows:

Definition 4.1. Let ϕ_S be the standard embedding of CAR intoO₂ in (2.1).

For a representation (H, π) of CAR, ( ˜H,π)˜ is the standard extension of (H, π) toO₂ if H is a closed subspace ofH˜ such that

(4.1) (˜π◦ϕ_S)|_H=π.

4.1. Standard extension of the Fock representation.

Theorem 4.2. Let (H, π) be the Fock representation ofCAR with the vac- uum Ωin Definition 2.1. Put two operatorsπ(s˜ ₁),π(s˜ ₂) onH by

˜

π(s₁)Ω≡Ω, π(s˜ ₁)π(a^∗_n1· · ·a_n^∗_k)Ω≡π(a^∗_n1+1· · ·a^∗_nk+1)Ω,

˜

π(s₂)Ω≡π(a^∗₁)Ω, ˜π(s₂)π(a^∗_n1· · ·a_n^∗_k)Ω≡π(a^∗₁a_n^∗₁₊₁· · ·a^∗_nk+1)Ω for n₁ < n₂ <· · · < n_k, n_j ∈ N, j = 1, . . . , k, k≥1. Then the followings hold:

(i) (H,π)˜ is a representation of O₂. (ii) ˜π◦ϕ_S =π.

(iii) (H,π)˜ isP(1) with the GP vector Ω.

This proof is given by direct computation and Lemma 2.3. For more detail, see§3.3 in [1]. Clearly, ( ˜H ≡ H,π) in Theorem 4.2 is the standard extension˜ of the Fock representation. Theorem 1.1 (i) about an operatorLfollows from Theorem 4.2 as another expression of this extension.

4.2. Standard extension of the infinite wedge.For g = {g₁, g₂} in (3.2), define a representation (Λ^∞2 V^#,Π) of O₂ by

Π(s₁)e_S≡(−1)^d+(S)e_g1(S), Π(s₂)e_S≡(−1)^d+(S)e_g2(S) (S∈ M₊), Π(s₁)e_S≡(−1)^d0−(S)e_g1(S), Π(s₂)e_S ≡(−1)^d−(S)e_g2(S) (S ∈ M₋) where d₊(S) ≡ #(S ∩Z_+/2) + #(Z_−/2 \S) and d⁰₋(S) ≡ #(Z_+/2 \S), d₋(S)≡#(Z_+/2\S) + #(S∩Z_−/2).

Lemma 4.3. When K ={k₁, . . . , k_n} and L={l₁, . . . , l_m} ⊂Z_+/2 satisfy k₁>· · ·> k_n and l₁<· · ·< l_m,

Π(s₁)|vac>₊ =ψ_−1/2|vac>₋=e_{Z+/2∪{−1/2}} Π(s₂)|vac>₊ =|vac>₋, Π(s₁)|vac>₋=|vac>₊, Π(s₂)|vac>₋ =ψ_−1/2^∗ |vac>₊=e_{Z−/2\{−1/2}}.

Π(s₁)e_{Z−/2∪K\(−L)}= (−1)^n+me_Z+/2∪(−K^∗

+1)\L, Π(s₂)e_{Z−/2∪K\(−L)}= (−1)^n+me_{Z+/2∪(−K+1)\L}, Π(s₁)e_{Z+/2∪(−K)\L}= (−1)^me_{Z−/2∪K\(−L+1)}, Π(s₂)e_{Z+/2∪(−K)\L}= (−1)^m+ne_{Z−/2∪K\(−L}^∗₊₁₎ where K₊₁ ≡ {k+ 1 :k∈K} and K₊₁^∗ ≡K₊₁∪ {1/2}.

Proposition 4.4. (i) (Λ^∞2 V^#,Π) isP(12).

(ii) If π_∞, π_∞,± are in Proposition 3.6, then Π◦ϕ_S =π_∞. Specially, (Λ^∞2 V, π_∞,+) = (Λ^∞2 V,(Π◦ϕ_S)|_Λ^∞_2V)∼P[12],

(Λ^∞2 V^∗, π_∞,−) = (Λ^∞2 V^∗,(Π◦ϕ_S)|_Λ^∞_{2 V∗})∼P[21].

Proof. (i) By Lemma 4.3, Π(s₁s₂)|vac>₊=|vac>₊. By definition of g₁, g₂, (Λ^∞2 V^#,Π) isP(12).

(ii) Identify ϕ_S(a_n) anda_n for eachn∈N. By Lemma 3.2 and Lemma 4.3, we can check the followings:

Π(a_2n−1)|vac>₊= Π(a_2n)|vac>₋= Π(a_2n^∗ )|vac>₊= Π(a_2n−1)^∗|vac>₋= 0, Π(a_2n)|vac>₊=ψ_n−1/2|vac>₊, Π(a_2n−1)|vac>₋=ψ_−n+1/2|vac>₋,

Π(a^∗_2n−1)|vac>₊=ψ_−n+1/2^∗ |vac>₊, Π(a^∗_2n)|vac>₋=ψ_n−1/2^∗ |vac>₋ for each n∈N. By Lemma 4.3, Π(a_n) =π_∞(a_n) for each n∈N. ¤ The branching law Π|_CAR = π_∞,+⊕π_∞,− is illustrated by Maya diagrams as follows: