Fock and non-Fock states on CAR-algebras CLAUDE-ALAIN PILLET CPT-CNRS, UMR 6207 Université du Sud Toulon-Var B.P. 20132 83957 La Garde Cedex, France [email protected] In the formalism of second quantization a system of fermions is described by creation and annihilation operators ∗ a (f), a(f) on the anti-symmetric Fock space Γa(h) over the one-particle Hilbert space h. For systems confined in a finite volume, this Hilbert space description is sufficient and states of finite positive density can be represented by density matrices in Γa(h). The situation changes when taking the thermodynamic (infinite volume) limit. There is no density matrix in Fock space describing a positive density state of an infinitely extended system of fermions. For such systems a more sophisticated description involving the C∗-algebra CAR(h) is needed (see [The C∗-algebra approach]). 1 Gauge invariant states on CAR(h) Global U(1)-gauge symmetry is a fundamental property of quantum mechanics. Its implementation on CAR(h) is given by the gauge group R ∋ ϕ 7→ ϑϕ, the group of Bogoliubov automorphisms defined by ϑϕ(a∗(f)) = a∗(eiϕf) = eiϕa∗(f), ϑϕ(a(f)) = a(eiϕf) = e−iϕa(f). As a Banach space, CAR(h) has a direct sum decomposition into charge sectors CAR(h) = CARn(h), n Z M∈ where CARn(h) is the closed linear span of monomials of the form ∗ ∗ a (f1) · · · a (fj)a(gk) · · · a(g1), with j − k = n. In terms of the gauge-group one has ϕ inϕ CARn(h) = {A ∈ CAR(h) | ϑ (A) = e A}. ∗ If A ∈ CARn(h) and B ∈ CARm(h), then AB ∈ CARn+m(h), A ∈ CAR−n(h). In particular, the zero charge ∗ ∗ sector CAR0(h) is a C -subalgebra generated by I and elements of the form a (f)a(g). Physical observables of a system of fermions are gauge invariant and hence elements of CAR0(h). ϕ A state ω on CAR(h) is gauge-invariant if ω ◦ ϑ = ω for all ϕ ∈ R. A state ω0 on CAR0(h) has a unique extension to a gauge-invariant state ω on CAR(h), given by ω(⊕nAn) = ω0(A0). Thus, a gauge-invariant state on CAR(h) is completely determined by its restriction to the gauge-invariant sub- algebra CAR0(h). When dealing with fermionic systems it is often convenient to work on the full algebra CAR(h) and to restrict the states to be gauge-invariant. 1 2 Characteristic functions Denote by U the group of unitaries u on h such that u − I is finite rank. For each u ∈ U there exist finite rank self-adjoint operators k such that n ik k = κjfj(fj | · ), u = e . (1) j=1 X Moreover, the unitary ∗ i P κ a (f )a(f ) U(u) = e j j j j ∈ CAR(h), only depends on u, not on the particular choice of the representation (1). The Araki-Wyss characteristic function of a gauge-invariant state ω on CAR(h) is defined as E : U → C u 7→ ω(U(u)). It satisfies 1. For any u1,...,uN ∈U and z1,...,zN ∈ C, N ∗ E(uj uk)¯zj zk ≥ 0. j,k X=1 2. For any u,v ∈U, f ∈ h and λ ∈ R E(ueiλ(f|· )f v) − E(uv) 2 , eiλkfk − 1 is independent of λ. Reciprocally, any function E : U → C satisfying the above two conditions is the characteristic function of unique gauge-invariant state ω on CAR(h) (see [AW]). 3 Vacuum state and Fock representation The vacuum state vac(·) on CAR(h) describes the system in absence of any fermion. If {ei | i ∈ I} denotes an ∗ arbitrary orthonormal basis of h then ni = a (ei)a(ei) is the number of fermions in state ei and we must have vac ( i∈J ni) = 0 for any finite J ⊂ I (note that [ni, nj ] = 0). It immediately follows that the characteristic function of the vacuum state is Evac(u) = 1. Q The GNS representation associated to the vacuum state is the Fock representation (HF , πF , ΩF ) where HF = Γa(h) is the fermionic Fock space over h, πF (a(f)) = aF (f) is the annihilation operator on Γa(h) and ΩF is the Fock vacuum vector. For fi, gj ∈ h one has vac ∗ ∗ ∗ ∗ ∗ ∗ (a(g1) · · · a(gm)a (fn) · · · a (f1)) = (aF (gm) · · · aF (g1)ΩF |aF (fn) · · · aF (f1)ΩF ) = δnm det {(gi|fj )} . Special features of the Fock representation are: # (i) πF (CAR(h)) is irreducible, i.e., any bounded operator on Γa(h) commuting with all aF (f) is a multiple ′′ ∗ of the identity. Equivalently, the enveloping von Neumann algebra πF (CAR(h)) is the C -algebra of all bounded operators on Γa(h). 2 (ii) The second quantization Γ(U) of a unitary operator U on h provides a unitary implementation of the associ- ated Bogoliubov automorphism γ(a(f)) = a(Uf), ∗ πF (γ(a(f))) = Γ(U)πF (a(f))Γ(U) . In particular, the gauge group ϑt is implemented by a strongly continuous unitary group whose generator N = dΓ(I) is the number operator. A Fock state on CAR(h) is a state ω which is normal with respect to the vacuum state vac. Such a state is therefore defined by ω(A) = tr(ρπF (A)) where ρ is a density matrix on Γa(h). The GNS representation of a Fock state ω is a direct sum of Fock representations, i.e., there exists a Hilbert space K such that Hω = HF ⊗ K and πω(A) = πF (A) ⊗ I. Typical examples of Fock states are finite volume, grand-canonical Gibbs ensembles e−β(HΛ−µNΛ) ρ = , tr(e−β(HΛ−µNΛ)) for Fermi gases with stable interactions (see [BR2]). Thermodynamic limits of such states yield non-Fock states with finite density. It is usually impossible to describe explicitly the GNS representations of these infinite volume KMS states. Notable exceptions are the ideal Fermi gases which lead to the Araki-Wyss representations. Since there exists a self-adjoint (and hence densely defined) number operator N = dΓ(I) on the Fock space HF , Fock states describe systems with a finite number of fermions. A number operator can be tentatively defined in the GNS representation of any state ω as follows. For any finite J ⊂ I denote by nJ the quadratic form associated to the operator i∈J πω(ni). For Ψ ∈ Hω set nω(Ψ) = supJ nJ (Ψ). It can be shown that nω is a closed, non-negative quadratic form on the domain Dω = {Ψ ∈Hω | nπ(Ψ) < ∞}. If this domain is dense then nω is the P quadratic form of a self-adjoint number operator Nω and the state ω is a Fock state (see [BR2] for details). 4 Anti-Fock representation A state full(·) describing a completely filled Fermi sea must satisfy, for any orthonormal basis {ei | i ∈ I} and any full finite J ⊂ I, ( i∈J (1 − ni)) = 0. It can be obtained using the particle-hole duality. Denote by · an arbitrary ∗ ∗ complex conjugation on h and define the ∗-automorphism α by α(a(f)) = a (f¯). Since 1 − ni = a(ei)a (ei) = ∗ Q α(a (¯ei)a(¯ei)) we can set full = vac ◦ α. It follows that ∗ ∗ vac(a(f¯1) · · · a(f¯n)a (¯gm) · · · a (¯g1)) = δnm det {(gi|fj)} . For u ∈U one has α(U(u)) = det(u)U(¯u), hence the characteristic function of the filled Fermi sea is Efull(u) = det(u). The corresponding GNS representa- tion is the anti-Fock representation (HF , πAF , ΩF ) where πAF = πF ◦ α. If h is finite dimensional then the states vac and full are mutually normal and the Fock and anti-Fock repre- sentations are equivalent. By fixing an orthonormal basis {e1,...,en} and setting aJ = i∈J a(ei) the unitary ∗ ∗ operator defined by UAJ ΩF = AI\J ΩF intertwines πF and πAF . If h is infinite dimensional these two represen- tations are inequivalent and full is not a Fock state. Q 5 Jordan-Wigner representation The equivalence of the Fock and anti-Fock representations of CAR algebras over finite dimensional spaces is a consequence of a more general fact about such algebras which we discuss briefly in this last section. We refer the reader to [D] for a more detailed discussion. 3 If h is finite dimensional then CAR(h) is ∗-isomorphic to the full matrix algebra Mat(2dim h). An explicit representation is provided by the Jordan-Wigner transformation described below. Since it maps fermions into quantum spins this transformation is also quite useful in many applications to statistical mechanics. (1) (2) (3) Let {e1,...,en} be an orthonormal basis of h and denote by σ , σ , σ the Pauli matrices. On the n-fold n tensor product H = C2 ⊗ · · · ⊗ C2 ≃ C2 define (α) (α) σk = I ⊗ · · · ⊗ σ · · · ⊗ I, where σ(α) acts on the k-th copy of C2. Clearly, this operators generate the full matrix algebra B(H) ≃ Mat(2n). One easily checks that the operators (3) (3) (1) (2) ak = σ1 · · · σk−1(σk − iσk )/2, ∗ satisfy [ak, al]+ = 0 and [ak, al ]+ = δk,l. The Jordan-Wigner representation of CAR(h) is defined by aJW zkek = z¯kak. k ! k X X The inversion formulas (3) ∗ (1) (3) (3) ∗ (2) (3) (3) ∗ σk = 2akak − I, σk = σ1 · · · σk−1(ak + ak), σk = iσ1 · · · σk−1(ak − ak), show that CAR(h) is isomorphic to Mat(2n). The Jordan–Wigner representation plays for fermions the same role as the Schrödinger representation of the CCR: If dim h < ∞ then any irreducible representation of CAR(h) is equivalent to the Jordan–Wigner representation.