Commun. math. Phys. 43, 181—197 (1975) © by Springer-Verlag 1975
Automorphisms and Quasi-Free States of the CAR Algebra N. M. Hugenholtz Instituut voor Theoretische Natuurkunde, Rijksuniversiteit, Groningen, The Netherlands R. V. Kadison
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, USA
Received September 25, 1974
Abstract. We study automorphisms of the CAR algebra which map the family of gauge-invariant, quasi-free states of the CAR algebra onto itself and show (Theorem 3.1) that they are one-particle automorphisms.
1. Introduction The problem discussed in this paper arose from questions regarding equi- librium states of thermodynamical systems. Equilibrium states have been extensively discussed in the framework of C*-algebras of observables (see, for example [6, 15]). Such states are labeled by a very small number of parameters. For example, in the case of a gas of identical particles, the equilibrium states are labeled by the temperature, the chemical potential, and the average velocity of the particles - quantities which relate directly with the conserved quantities - energy, particle number, and total momentum. Since conserved quantities are in one-one correspondence with one-parameter groups of transformations which leave the Hamiltonian invariant, we can describe the situation in a way which remains meaningful for infinite systems. Equilibrium states of thermodynamical systems are labeled by a very small number of parameters which relate directly with one- parameter, automorphism groups of the observable algebra that commute with the time-evolution automorphisms. The fact that there are so few parameters involved, which is related to the fact that there are only a small number of one-parameter groups of automorphisms that commute with the time-evolution, is an aspect of the ergodic nature of most large physical systems. A proof based on the dynamics of the system is still lacking, even though Sinai has obtained very interesting results in this direction for a classical system of N hard spheres. Systems of particles without interaction do not behave ergodically in the above sense. This does not mean, however, that systems of noninteracting particles are uninteresting from the point of view of ergodicity. In [7, 8], the asymptotic time behavior of the free Fermi gas is discussed. It is found in [7] that, for increasing time, primary states of the CAR algebra are asymptotic to gauge-invariant, quasi-free states, provided these states satisfy a certain clustering property. In particular, primary, stationary (i.e. time-invariant) states with that clustering property are quasi-free. Quasi-free states [1-5, 9-14, 16, 17] are particularly simple states in the sense that they lack all except two-point correlations. Some 182 N. M. Hugenholtz and R. V. Kadison basic facts about gauge- invariant , quasi- fre e state s ar e give n i n Sectio n 2 . These states ar e i n one- on e correspondenc e wit h positiv e operator s i n th e uni t ball , the so- calle d one- particl e operators , acting o n th e one- particle Huber t spac e Jf . Stationary, quasi- free state s o f the free fermions ar e not labeled b y a finite number of parameters, as i n the interacting case , but b y one- particl e operators that com- mute with the Hamiltonian H( = - (h2/ 2m) (d2/dx2 + 32/dy2 + d2/dz2) = - (h2βm)Δ). In particular , al l operator s o f th e form , f(hd/idx, hd/idy,hd/iδz) with / positiv e and havin g essentia l supremu m 1 , defin e translationally- invariant , stationar y states o f th e fre e fermio n system . I n thi s case , f(Pi,p2>P3) i s th e momentum distribution o f the particles. The free fermio n syste m i s o f particular interest to us because, as noted before , the se t o f al l primar y state s tha t satisf y a certai n clusterin g propert y i s known . It is a subset o f the gauge- invariant, quasi- fre e states . This resul t i s an important tool in determining the automorphisms that commute with the free- time evolution. Since a clustering propert y tha t i s somewha t stronge r tha n that which hold s fo r all primar y state s ma y wel l b e a conditio n on e mus t impos e upo n a physicall y meaningful state , we shal l restric t ou r considerations t o those automorphisms o f the CA R algebr a whos e transpose s preserv e thi s property . Le t α be suc h a n automorphism that , i n addition , commute s wit h th e free- tim e evolution . It s transpose map s th e se t o f stationary , quasi- fre e states , tha t satisf y th e clusterin g property, onto itself. What ca n on e conclude about α? Before attemptin g t o solv e thi s problem , on e is face d wit h a mor e primitiv e question. What ca n be said about a n automorphism α of the CAR algebra whos e transpose map s th e se t o f quasi- fre e state s (o r a certai n subse t o f it ) onto itself? Our mai n resul t (Theore m 3.1 ) state s that , when th e transpose leave s th e se t o f gauge- invariant, quasi- fre e state s stable , then , either th e Foc k stat e i s mappe d onto itsel f an d ther e i s a unitar y operato r U o n Jf , suc h tha t a(a(f)) = a(Uf), or th e Fock stat e i s mappe d ont o the anti- Fock stat e an d ther e i s a conjugate - unitary operator W on Jf, suc h that α(α(/)) = a(W'/)*, wher e a(f) i s the annihila- tion operator on Fock space. A related resul t i s state d i n Theorem 4.1. A unitar y operato r o n one- particle space defines , i n a n obviou s manner , a unitar y operato r o n π- particl e space . Theorem 4. 1 characterize s suc h unitar y operator s o n n- particl e spac e a s thos e which ma p anti- symmetrize d product s o f one- particl e wav e function s (product vectors) onto product vectors. Section 2 is devote d t o notation and a numbe r o f preliminary results , whic h are use d throughou t th e paper. Th e mai n theore m i s prove n i n Sectio n 3 ; and Section 4 contains som e relate d results .
2. Some Preliminaries An infinit e syste m o f identica l Ferm i particles ca n b e represented , insofar a s their algebraic interrelations are concerned, by a C*- algebra, 91 , the so- called CAR algebra. The abstract algebra 2 1 may be characterized as the norm closure (comple- tion) of an algebra generated by a countably- infinite famil y o f pair wise- commuting, Automorphisms an d Quasi- Free States of the CAR Algebra 18 3
self- adjoint algebra s eac h isomorphi c t o the algebra o f 2 x 2 complex matrice s and eac h containin g th e unit, / , of 91. Each ^- representatio n o f 91 gives ris e to a representation of the canonical anticommutation relations (CAR), and conversely . For ou r purposes , it is more usefu l t o describe 9 1 in itszyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Fock representation.
With J4? a complex Huber t spac e an d j^n the rc- fold tensor product , so that, for
xί9...,xn9yl9...9yn i n JT, (x^- - ®χn\y1®- (g)yn> = <χί\y1)- '<χn\yn>, le t
S~ b e the projection operato r o n j^n which assigns—rXz(σ)x σ(1)®•• ®x σ(π)
Yl . σ
to xx ® ® xn, wher e σ is a permutation of {1,..., n} and χ(σ) is + 1 if σ is even, — 1 if σ is odd. The range of S~ is the space 3tf$a) of antisymmetric tensors. We writ e 1/ 2 XjΛ Λ ^ fo r (n\) S~(x1®- - <3)xt) (th e "antisymmetrized, n- particl e stat e
with wav e function s x l9 ...,x^). W e have:
Thus, assumin g x 1Λ Λx n and yx A • • • /\yn are not 0, they ar e orthogonal if
and onl y i f there are scalars c 1? ..., cn, not all 0, such tha t
0= Σ < c ixi\yj)=( Σ cixi\yj i= l \ ί = 1
that is , if and onl y i f the space , [x 1? ...,.xj, generate d b y x1? ...,xn, contain s a
non- zero vector (ΣcfXf ) orthogona l to [j;l5..., yj. If, in addition, the intersection,
[xi,..., x w] n !>!,..., yJ, o f the spaces [x1,..., x J an d [^,..., yJ has dimensio n n — 1 (in this case, we say that the spaces ar e "perpendicular"), the projections wit h
ranges [x l5 ...,xj an d [yl5 ...,yj commute . It follows tha t {eiί Λ • • • Λe ίn} is an (a) orthonormal basi s fo r J^n if {ej i s an orthonormal basi s fo r Jf. Moreover ,
x1 A • • • Λxn = 0 i f an d onl y i f x1?...,xπ ar e linearly dependen t (i f and onl y i f
[xl5 ...,x j ha s dimension less than n). Thus ze [xl9 ...,xj, i f zΛ xx Λ • • • Λx tt = 0
and xx Λ Λ xn Φ 0. From this, if xί A Λ xn = y1 A Λ yn + 0 , then [xx,..., x j
= [)Ί » . 9yJ O n the other hand , i f [x 1? ...,xj = [y1? ...,yj, then , expressin g
each ^ as a linear combination of xx, ... , xn, we see that xx Λ Λ xn a.nάy1 A Λ yn are scalar multiple s o f one another. We say that xx Λ Λ xn is a product vector - the exterior (or , wedge) product o f xl5..., xπ. OO The antisymmetric Fock space, Jf£°, i s Σ © ^(α) B Y definition Jf^ α) consists n = 0 of comple x scala r multiple s o f a singl e (unit ) vector Φ o, the Fock vacuum; and Jfi(fl) i s 3f. I f Jf wer e finite dimensional , Jf^ fl) woul d b e the (finite- dimensional ) "exterior" algebr a ove r #?. The mapping , Λ , from th e n- fol d Cartesia n produc t fl) Jf x • • • x tf t o Jfj whic h assign s x1Λ Λx H to (xu ...,xn) i s an alternating, ^- linear mapping. If a is such a mapping of Jf7 x x Jf7 into a space Jf, there is a mapping a oϊ^a) int o JΓ such that ά = ά° A. In particular if Γ is a linear mapping of Jf into JΓ then (xx,..., xn) - • Txx Λ Λ Txn is an alternating rc- linear mapping of J^x x Jf into J^(fl); so that there is a linear mapping f o f Jf|° into JΓj?} suc h thatf (xx A- - - Axn)= Txί Λ Λ TxM. If Tis a unitary transformation of Jf onto Jf, metric considerations appl y an d f i s a unitary transformatio n o f Jfjrfl) ont o JΓjr fl). 184 N . M. Hugenholtz and R. V. Kadison
If ||7J^1 , J T = Jf©^ , / = /θ/,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA P{x,y) = (x,0), an d Q(u, v) = (w,0\ wit h x, 3; in Jf and u, v in X, then there is a unitary transformation U oϊjft ont o JΓ such that QUP(x9y) = (Tx,0) fo r all x,y in J- f. Then t / is a unitary transformatio n of (α {a) {a) (α) #π | ont o £n an d β is a projection o f £n ont o Jf M . Since f i s the restrictio n of QU to J^(fl), w e see that ||T| | rg 1. If T is a positive operato r wit h pur e poin t spectrum, computin g norm s wit h a basi s o f eigenvector s fo r T , we find tha t (fl) II fl J^ || =λί- λn, where λ1,..., λn are the n largest eigenvalue s o f Γ (multiplicity included). An approximation argumen t provide s th e corresponding resul t fo r a general positiv e operator ; an d polar decompositio n provide s a nor m formul a for a general bounde d operator . A simple chec k yield s (T) * = f*.
Since (f1 ,...,/„)- >/ Λ /x Λ Λ /„ i s a n alternating , π- linea r mapping , a) there is a linear mapping , an(f)*9 o f ^ into J^ wit h valu e / Λ fγ A • • • Λ / „ at Λ Λ • • • Λ / „. The famil y {an(f)*} define s a mapping α(/) * on ^\ Wit h {e f} an orthonorma l basis for 3tfΊ a(eγ)* maps {eiχ A - Aein:ίφ {/ 1,...,iw},n = 0 , 1,2,...} onto a n orthonorma l basi s fo r th e orthogonal complement , j^^Qjf; an d
α^)* annihilate s thi s complement . Thus a(e x)* is a partial isometr y wit h initia l space Jf an d final space JίT^QJίT. It follows tha t / = α(e 1)*α(e1) + α(e 1)α(e1)*
(= {fl(e1),α(e1)*} + ). Mor e generall y α(/)α(/) * + α(/)*α(/) = J . Polariza - tion o f thi s yields : {a(f),a(g)*}+=(f\gyi. W e note tha t ou r inner product , , i s linear i n g and conjugate linea r i n /. We have {a(f)*,a(g)*}+ = 0 , as well. A conjugate- linear mappin g f- +a(f) o f Jf ont o operators a{f) on a Hubert space satisfyin g th e relations (canonica l anticommutatio n relations )
{a(f),a{g)*}+= so that n + α(/)(x1Λ- .ΛxII)= Σ (- iy jjJ j=i With E a projection o n Jf, w e denote by 2lo(iϊ) an d 2ί(£) the *- algebra and α) C*- algebra, respectively , o n jTJτ generated by {a(f) :£/ = / }. We write 9I0 and 91 in plac e of 9I0(/) and 91(7) . The C*- algebr a 9 ί is the CAR algebr a an d its action on fl) J fjr is called its Fock representation. The state φ0 of 9 1 fo r which φo(A) = <Φ 01 AΦ 0> is calle d th e Focfc (vacuum) state o f 21. Note tha t eac h α(/ ) is in its left kerne l Jf ( o(α(/)*α(/)) = 0) ; so that eac h produc t o f annihilators and creators (mono- mial) in which a n annihilator appear s t o the right i s in jf*. No w eac h monomia l is a sum of monomials in which all creators are to the left of all annihilators (we say that such a monomial is Wick- ordered - an d anti- Wick- ordered if all creators are to the righ t o f all annihilators); so that φ0 annihilates al l Wick- ordered monomial s Automorphisms an d Quasi- Free States of the CAR Algebra 18 5 in 2I 0 othe r than / . These monomial s spa n th e null spac e o fzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA φ0 o n 2Ϊ O. If ρ is a state of 21 and ρ ^ 2φ0, the n a(f) i s in the left kerne l o f ρ . Thus ρ and φ0 hav e the same null spac e i n 2ί 0 an d agree a t / . Hence ρ = φ0', and φ0 i s a pure state o f 21. Exactly th e sam e consideration s appl y t o the restriction o f φ0 t o 2I(£) , fo r eac h projection E on Jf. Thu s the restriction o f φ0 t o 2I(£) is pure. The Hubert space Jf, obtaine d fro m Jf b y assignin g a n elemen t / t o each / in Jf , definin g (cf + g) t o b e cf +g an d (f\g) t o b e <#|/ > , produce s #jr α), anti- Fock space, an d φ0 i s the anti- Fock vacuum. Th e mapping /- »α(/)* ( = α(/) ) is a representatio n o f th e CA R (ove r Jf) , th e anti- Fock representation; an d the mapping α(/) - • a(f) extend s t o a ^- isomorphism , A- ^A, o f the CAR algebra 2 1 over J f ont o th e CA R algebr a 2 ί ove r J f . The stat e φι o f 2 1 define d b y A- ^(Φ0\AΦ0} i s th e anti- Fock state of 21 . Each α(/)* is i n the left kerne l o f φj; so that, replacing a(f) b y α(/)* and usin g anti- Wick- ordere d monomial s instead of Wick- ordered monomial s i n the argument above , w e hav e tha t the restriction of φj t o eac h 2I(E ) is pure. Since φ0 i s pure and Φ o is cycli c fo r 2ί , the weak- operator closure , 2 1 ~, of 2ί, fl) } is ^(Jfjr ), th e algebr a o f al l bounde d operator s o n Jfj? . Similarl y 2I(£)~£ 0 = ^([2Ϊ(£)ΦO]), where Eo i s the projection (i n 2l(£)' ) wit h range [2I(£)Φ 0]. I f [7£ is (Γ^2E), the n UEΦ0 = Φ0, a{g)UE=UEa(g), fo r eac h # i n (J- £)(«^) , an d a(f)UE = — UEa(f\ for eac h / i n E(J4?). I f Ao i s a n eve n monomia l in 2l o(J — £) (that is , Ao i s th e product o f a n eve n tota l numbe r o f annihilators an d creators ) and Aί i s a n od d monomia l i n 2I 0(7 — £), then Ao an d A1UE li e i n 2I(£)' . Sinc e 2I0(£) and 2l o(/ - E) generate 2t 0 an d Φ o is cycli c fo r 2l 0 Thus Eo ha s centra l carrie r / i n 2I(£)~ ; an d th e mappin g ί ^ o f 2I(£)"£ 0 ont o 2ί(£)~ whic h assign s ^ 4 to AE0 i s a ^- isomorphism . Now, a(f)Φo=0 and , whe n £/ = 0, α(/ ) (2I0(E)Φ0) = (0) . Thus a(f)E0=0 and JB oα(/)* = 0 , whe n Ef = 0; s o tha t E0AE0 = λE0 whe n A i s i n 2ί o(/- £). α) It follow s tha t B- >E0BE0 i s a (completely- ) positive , linea r mappin g o f ^(J^ ) onto 2I(£)~£0. The composition o f this mapping wit h ιE i s a completely- positive , fl) linear mapping, ψE, o f ^pfjr ) ont o 2I(£)~. B y construction o f ip£, More generally : Proposition 2.1. / / Γ i s α Zm^α r transformation of one Hilbert space, Jf, into another, 3f, and \\T\\ ^ 1 , ίte/t ίte mapping a(xnr..M(xira(y1)..M(ym)- +a(Txn)*...a(Txί)*a(Ty1)...a(Tym) extends (uniquely) to a completely- positive, linear mapping ψτ of the CAR algebra, 21^, over Jf into the CAR algebra, 21^ , o^r Jf . Proo/. lϊ £ = J?®J4?,^jf = jr®jr, P{h,hf) = (h,0) fo r Λ,Λ ' i n JT , β(k,/c r) = (fe , 0) for k, k! in JΓ, and f(h, h') = (Th, 0), then there is a unitary transformation U ofjfi ont o Jf" suc h that Q UP = f. Th e mapping a(f)- >a(U f) extends , uniquely, to a ^- isomorphis m o f 2ίj & onto 21^. The compositio n o f th e restrictio n o f thi s isomorphism t o 2Ij^(P ) and ψQ i s ψτ. 186 N . M. Hugenholtz and R. V. Kadison We note that the characterization ofzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ψτ as the result of distributing T throughout a Wick- ordered monomia l is independent of the ordering only if Tis an isometry; for Ψτ(a(f)a(f)*) = ψτ(I - α(/)*α(/) ) = / - a(Tf)*a(Tf) Φ a(Tf)a(Tff, whe n ||/ || = 1, unless If A € ^pf) and 0 _ A ^ /, we call φ, ° ψΛi th e gauge- invariant, quasi- free state of 9 1 with one- particle operator A. We write φA fo r this stat e and note that there is no conflict betwee n thi s notation and the designation o f the Fock and anti- Fock states o f by φ0 and φ, (i.e. these states are quasi- free wit h one- particl e operators 0 and / , respectively). Not e that φA{a{fn)*- a{fίra{g1)...a{gn)) = detKΛ*/ i I A*gj» = = det« ίjfί IA/ ;» = < . Proposition 2.2. If E is a finite- dimensional projection on Jf with {eγ,..., en} an orthonormal basis for E(JΊP), then φE{T) = (e1 A Λ en | T{e1 Λ Λ en)}. Proof Le t {βj} be an orthonormal basi s fo r Jf, an d T be a Wick- ordere d monomial i n annihilators an d creators correspondin g t o basis elements . The n (eίA~ Aen\T(e1A '/ \eJ> i s 0 unles s T ha s th e for m a{eiσim)*...a(eiσ{i)* . a(eiy..a(eij, wit h {ix...., im} an m- elemen t subset o f {1,..., n}, in which cas e its value and that oϊφE(T) i s χ(σ). If Γdoes not have this form ψE(T) = 0, so φE(T) = 0. Thus our equality holds . 7 If follow s tha t φE i s pure whe n E i s a finite- dimensional projectio n o n Jf . More generally, i f E is any orthogonal projection o n jtf* and ρ is a state of 91 such that ρ^2φE the n the restrictions o f ρ to 9I(£) and 21(7 — E) coincide wit h thos e of φI and φ0, respectively. Usin g the fact tha t monomials A and A' in Vίo(E) and $lo(I — E\ respectively , commut e o r anti- commut e an d tha t Wick- ordere d monomials are in the left or right kernels of φ0 while anti- Wick ordere d monomials are in the left or right kernels o f φj, (other than cl, c Φ 0) , we conclude that ρ(AAf) = ρ(A)ρ(A'). Th e same i s true for A in 2Ϊ(£) and A i n 21(7 — E). Thus ρ = φE and φE i s pure. If 0^A0SI wit h Ao( + Al) i n gt(2tf\ using th e Spectral Theorem , there is a one- dimensional projection Eγ o n Jf an d a positive numbe r t such that 0 :g y4x g 7 = and 0 ^ v4 2 ^ 7 , wher e ^4 X = A o + tEί an d ,42 ^ o ~ ~ ^i Computin g wit h a n orthonormal basis {e3) for jf suc h that Ex ex = β x, we have that φAo = \{φAχ + φA2). To see this, note that e a φΛM U* '' (eiι)*a(ejι)...a(ejr)) = (ejί A • • • Λ ejn\Akeh A • • • Λ Akein) , where fc = 0 , 1, 2; and that >l oe7. = >l 1eiy = y4 2eJ , when jφ 1 . Thus φA is pure if and only i f A is a projection. From the foregoing, i f £ is a finite- dimensional projection , φE is a pure, gauge- invariant, quasi- free stat e equivalent to the Fock state. Conversely, i f E is a projec- Automorphisms an d Quasi- Fre e State s o f the CA R Algebr a 18 7 tion o n one- particl e spac ezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 2tf an d φE i s equivalen t t o th e Fock state , then £ i s a finite- dimensional projection . Thi s follow s a s a specia l case o f [12 ; Theorem 2.8] . α) A direc t proo f i s no t difficult . I f φE = ωx\% fo r som e uni t vecto r x o f JfJ , the n 1 = φE(a(e^*a(ejj) = ωx(a(e])*a(e])\ wher e {βj} i s a n orthonorma l basis fo r Thus a(ej)*x = 0, fo r each / I f X C e Λ Λ β Λ e Λ A = Σ iί ..in;jl...jm il '• • in h • • • e'jrn ' where {e)} i s a n orthonorma l basi s fo r ( / — E) (#P), the n 0 = Λ(^.)*χ= ΣCi^^j^^ejA eh Λ Λ^Λ^ Λ Λ e}m so tha t je {/ 1? ...,/„ } unles s C; 1...ΐn;j1...7m = 0. If E(J4?) is infinite- dimensional , w e can choos e j no t i n [iί,..., / „ } an d x = 0, contradicting th e assumption tha t x i s a unit vector. Thus E i s a finite- dimensional projection . 3. Automorphisms Preserving Gauge- Invariant, Quasi- Free States Our mai n resul t i s containe d i n th e followin g theorem . Theorem 3.1. 7/91 / s the CAR algebra in its Fock representation on the complex Hilbert space Jfjrfl) of antisymmetric tensors over J^ and a is an automorphism of 9 1 whose transpose ά carries the set of gauge- invariant, quasi- free states onto itself then, either the Fock state is mapped onto itself by ά and there is a unitary operator U on jf 7 such that a(a(f)) = a(Uf), or the Fock state is mapped onto the anti- Fock state by ά and there is a conjugate- linear, unitary operator W on J f such that <*(a(f)) = a(Wf)*. The proof o f Theorem 3. 1 wil l b e effecte d wit h th e aid o f the following result s (3.2- 3.13). Durin g it s course , w e wil l note , i n th e case wher e ά(φo) = φo, tha t U implements α (see Sectio n 2) . Lemma 3.2. The image of φ0 under ά is either φ0 or φj. Proof Sinc e &(φ0) is pur e and , b y assumption , a quasi- free , gauge- invarian t state o f 91 ; oc(φo) = φEo, fo r som e projectio n Eo o n Jf (se e Sectio n 2) . I f π i s th e representation o f 9 1 o n J^π determine d b y φFo, the n th e mappin g 1 ΛΦo- ^π(a~ (Λ))xEo, wher e xEo i s a uni t vecto r i n J^ π suc h tha t φEo{Λ) = χ π r a fl) ( Eo\ (A)xEoy f° ^ A i n 9ί , extend s t o a unitar y transformatio n U o f ifjr onto Jtπ. lϊE0 i s neither 0 nor / , there is a unit vector/ in E0(Jή?) and a unit vector g orthogonal t o E0(3Ί?). We shal l arriv e a t a contradictio n fro m thi s assumption , so that Eo i s either 0 or 7 and &(φ0) is either the Fock o r anti- Fock state . If E1 i s th e projectio n o n J f wit h {x: Thus φEl i s equivalen t t o φEo; an d ther e i s a vecto r xEi i n Jf π suc h tha t φEl(Λ) = (xEι\π(A)xEί}. Similarly , i f E2 i s th e projectio n o n 3tf wit h rang e generate d by Eγ{$?) an d g, then φE2(Λ) = φEί(a(g)Aa(g)*), fo r al l ^ 4 in 91; and there is a vecto r xE2 i n Jf π suc h tha t φE2{A) = {x 188 N . M. Hugenholtz and R. V. Kadison Since π i s irreducible ,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA xEl = c2π(a(g)*)xEί, and , similarly, xEo = c0π(a(f)*)xEl, where c 0 and c2 ar e scalars o f modulus 1 . Now, 2'HXEO + *E2) = π(α(2- *(c0/ + c2g))*)xEι( = xF) I so that φF(Λ) = (xF\π(A)xFy, fo r all A in 21, where F is the projection o n Jf wit h range generate d b y E^Jί?) an d cof + c2g. I f t/ X x = x F and Ux2 = xEl then , since 1 i UΦo = xEo, Φ 0 + x 2 = 2± x1. Notin g tha t UAU' =π(oc~ (A)\ fo r each X in 9ί, x Λ 1 1 we have ωX2191 are pure, gauge- invariant, quasi- fre e state s whic h transfor m unde r ά into φF an d φEl. Sinc e φF and φ£2 are equivalent t o φEo, ωXί |9I and ωX2|9I are equiva- fl) lent t o φ0. From Propositio n 2.2 , x1 an d x2 ar e product vector s i n JfJ . But = Φo + * 2 2*x l5 and each product vecto r lie s in an n- particle space . Thu s xί and x2 ar e multiples o f Φo an d φEi = φ0, contrary to the choice of E2 differen t fro m 0. f In case &(φ0) = φj9 6t (φ0) = φ0, where α ' = a ° σ and σ(a(f)) = a(Wof)* wit h Wo a conjugate- unitar y operato r on Jf7 (obtained , for example, b y transforming eac h linear combinatio n o f elements i n an orthonormal basi s fo r Jf 7 int o th e linear combination resultin g fro m replacin g eac h coefficien t b y its complex- conjugate) . Since σ determines an automorphism o f 91 which interchange s the Fock and anti- Fock state s an d which map s th e set of gauge- invariant , quasi- fre e state s ont o itself; ά' maps th e set of gauge- invariant, quasi- fre e state s ont o itself . If we prove that ther e i s a unitar y operato r Uo o n #f suc h tha t *(a(Wof)*) = a{Uof), an d α(α(/)) = α(l7 0W?/ )*, wit h W the conjugate- unitar y operator Uo W$ on Jf. We assume , henceforth , tha t a(φQ) = φ0 and use the notation o f Theorem 3.1 throughout the remainder o f this section . Wit h thi s assumption , U, constructed in Lemm a 3.2 , is a unitary operato r o n Jίf^ whic h carrie s produc t vector s ont o product vectors. Although th e components of the argument provin g tha t are to be found i n the proof o f Lemma 3.2 , we make th e statement an d proof explici t in Lemma 3.3 . Note that the automorphism σ , above, i s a special case o f the large r class of Bogoliubov transformations . In Section 2 , we introduced the notation f to denot e a certai n transformatio n o n 34?^ arising fro m T defined o n Jf. In Lemma 3. 3 and the results following , we construct a unitary operato r U on ^J~°. We wil l eventually locat e a unitary operato r ί/on Jf fo r which U is the transfor- mation on Jfjrα) arisin g fro m i t - justifyin g thi s notation . Lemma 3.3. There is a unitary operator U on J^jrfl) which implements a and maps product vectors onto product vectors. Proof. Since ά(φo) = φo, th e mapping AΦ0- ^a(A)Φ0 extend s t o a unitar y operator V on jfj?> suc h tha t UAU* = aι(A). We show tha t U(x1 Λ • • • Axn) i s a product vector, for all xl5 ... , xn in Jf. Sinc e xί A Λ xn is a scalar multipl e of the wedge- product o f an orthonormal se t of vectors ( a basis fo r [xl5 ...,xj, whe n xx Λ • • • Λ xΠ=f=O), we may assume tha t {xί9..., xn} i s an orthonormal set. If Eo is the projection wit h rang e [x l5..., xj, φEo = ωXl Λ ...ΛXn, fro m Propositio n 2.2 . As α) α is implemented b y a unitary operato r o n Jf^ , oc(φEo) is a vector stat e of 9ί. B y assumption, όc(φEo) is a gauge- invarian t quasi- fre e stat e o f 91 (equivalent t o φ0, from th e precedin g remark) . Thu s oί(φEo) = ωyί A...Ay |9 l wher e yι A - • Aym = ϋ*(xiA >- Axn). Automorphisms an d Quasi- Fre e State s o f the CAR Algebr a 18 9 A linea r transformatio n (suc h a szyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA U9 above ) whic h i s define d o n a subspac e o f J)fjrfl) an d maps produc t vector s ont o product vector s wil l be sai d t o be a product linear transformatio n (o r product unitary , etc.) . Lemma 3.4. The intersection of an infinite, commuting family of m- dimensional projections each pair of which has m- ί dimensional intersection is m- ί dimensional Proof Sinc e eac h projectio n i s m- dimensional , eac h i s th e su m o f m one - dimensional projections. Sinc e the family i s commutative, we can find an orthogo- nal set of one- dimensional projections suc h that each projection o f our commuting family i s the sum o f m of them. In this way , ou r problem reduce s t o showing tha t i f S = \J Sa wher e {Sa} i s a n infinit e famil y o f set s eac h o f whic h ha s m element s a and suc h tha t eac h pai r ha s intersectio n wit h m- 1 elements , the n f] Sa ha s m- ί a elements (in other words, each Sc contains SanSb, whe n a φ b). Suppose Sc does not contain SanSb. The n SanSc an d SbnSc ar e distinct m- 1 element sets ; s o that their union has at least m elements. As thi s union is contained i n Sc i t coincides wit h Sc and ScQSauSb. B y the same token, with c + d, Sd does not contain both Sa n Sc and 'SbnSc; s o that Sd i s containe d i n eithe r SaκjSc o r SbvSc. Bu t thi s contradicts the assumption tha t {Sa}, and , hence , u5fl ar e infinite . Proposition 3.5. / / V is an isometric, linear mapping of an infinite- dimensional subspace ^ of ^ onto a set of product vectors in J^a\ then f] [Fx ] has dimen- sion m- 1. ([Fx] is the subspace of H determined by Fx. ) Proof'. Wit h {βj} a n orthonorma l basi s fo r Jf , Veί=xί A - Axm9 e = V i y\ Λ Λ ym9 an d V(e1 + e2) = z1 A Λ zm9 w e hav e Some Zj, sa y z 1? i s no t i n [x 1?...,xm]. Thu s z1 Axί A - AxmAyj = 0; an d ; sinc e b>i> • • >) m]£l>i>Xi> . oX m]. Veγ an d Ve2 ar e no t 0 , [x l5...,xj an d ar e Lyi9 J yml ^- dimensiona l subspace s oϊ[zί,xί9..., x w], a spac e o f dimension n ; ; m+ 1 . Thus [x l5 ...,^ m] C) i5 • • ?J m] has dimension a t least m — 1; and Ve1=xAv1 A • • • Avm_l9 Ve2 = yAv1 A- Aυm_ί9 with {x , v1 ,..., vm _ x} an d {y9 υ1 ,..., vm _ x} orthonorma l sets . I n addition , 0= infinite, commuting famil y o f projections o n Jf suc h that EjEk(J^) ha s dimensio n m— 1 when j+ fc. From Lemm a 3.4 , P ) Ejffl) ha s dimensio n m — 1 . Lemma 3.6. / / V is a product isometry of J^ (fl) into ^fj? }, with J Γ α subspace of &?, ί/ ieπ F /zα s rαn^f^ m som ^ J^α). If n^m and Jf is infinite dimensional then [_V(xι A Λ xj] π [F(y1 Λ Λ yj] has dimension at least m — n. Proof. I f {ej} i s a n orthonorma l basi s fo r Jf an d {il9...9it}9 {/ U JΛ } are disjoint set s o f indices, 190 N . M . Hugenholt z an d R . V . Kadiso n ReplacingzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA V(eh A • • • Λ eit A ekt + 1 Λ • • • Λ ejby V{ehAei2 A - ΆeitAekt+1A . AeJ in the preceding computation, it follows that both (product) vectors li e in the same a) space ^ a s V{{eiχ + ej Λe J2Λ Λe ίtΛ^t+ iΛ Λ eJ does . Applyin g thi s result t o successiv e replacements o f i2, i3, ...,it by j2j3, - ..,jt9 one concludes tha t {a) fl) V(eh A • • • Λ ejt A ekt + 1 A • • • Λ ekj lie s i n ^\ Thu s F maps Jfn into J^ , sinc e a) {etί A • • • Λ etj i s a n orthonorma l basi s fo r ^ . Since [x 2, ...,xj ha s dimensio n n— 1 , it s orthogona l complemen t an d ; are no t DΊJ • • • '} n] disjoint . Le t z A b e a uni t vecto r i n thei r intersection . The n [_V(z1 Λx2Λ AxJ]n[V(xί A • • • ΛxJ] ha s dimensio n a t leas t m— 1 , fro m Proposition 3.5 ; fo r x- >F(xΛx 2Λ Λx J i s a n isometri c linea r mappin g o f ^θ[*2> • • • >*« ] onto a se t o f product vector s i n Jί^^Let z 2 b e a uni t vecto r i n D>i> "- >yΔ orthogona l t o [z 1}x3, ...,xj. The n [F(z1Λx2Λ Λx /ί)]n[F(z1Λz2Λx3Λ Λx, 1)] has dimensio n a t leas t m — 1 s o tha t its intersection wit h [F(z1 Ax2 A - ΛxMnlVfa A • • • ΛxJ] has dimensio n a t leas t m — 2 (bot h ar e subspace s o f th e m- dimensiona l spac e [V(z1 A x2 A Λ xπ)]). Thu s [F^ Λz2Λx3Λ Λ xj] n [F(xx Λ Λ xn)] ha s dimension a t leas t m — 2. I f w e hav e foun d mutuall y orthogona l uni t vector s zi,...,z*- i i n [)Ί,...,}>„ ] suc h tha t [V{zγ A • • • Azk_ί A xkA • • • ΛxJ] an d [F(x: Λ • • • Λ xj] hav e a n intersectio n o f dimensio n a t leas t m — k+ί, choos e a unit vecto r zk i n [JΊ , ...,y j orthogona l t o [z 1? ...,zk_l5xfc+ 1, ...,xj. The n \_V(zγ A Λ zk_! Λ xk Λ Λ xj] an d [F(z x Λ Λz tΛxk+ 1Λ Λ xπ)] hav e intersection o f dimensio n a t leas t m — 1. Thus [F(z x Λ Λ zk A xk+ x Λ Λ x,,)] and [F(x ! Λ • • • Λx π)] hav e intersectio n o f dimensio n a t leas t m—k. Finally , z1 A- - Azn = cy1 A - • ' Ayn an d lV{xί A Λ xj] n [V(yi A Λ yj] ha s dimen - sion a t least m — n. {a) {a Lemma 3.7. Fo r eαc/ z n , 1 7 maps J^n onto Jt?n \ Proof. Fro m Lemma 3.6 , 1 7 maps J^ (α) into som e JfJ[ α). Sinc e 17 * satisfies the same hypothese s a s U, ί7* maps ^α) int o JfJa) (again , fro m Lemm a 3.6) . Thus U (fl) a maps J^ M ont o ^ \ Fo r some orthonorma l set s {x 1? ...,x n} and {yί9 ...,yn} i n JίT, Uix^A' - Axn) = eiA--- Aem an d t)^ Λ • • • Λj; n) = ^w+1 Λ • • • Λ e2m, wher e {e1, ...,e2m} i s an orthonormal se t in Jίf. lϊn^m, fro m Lemm a 3.6 , has dimensio n a t leas t m — n. Thus m^n. Applyin g thi s conclusio n t o U*, n^m; so that m = n. We denot e the dimension o f a (finite- dimensional ) subspac e E o f Jf b y Proposition 3.8. / / V is a product isometry ofjf}a) into J^fl), where n^m and is an infinite- dimensional subspace of 34?, and d(\V{eh A • Λ βtj] n IV(eh A Λ eJ]) = m - n for some eiί9 ...,et , ejl9..., ^ 7 9 w/ ier e {^ } i s α? i orthonormal basis for Jf , 1 Withou t los s o f generalit y w e assum e tha t {xί, ...,x j an d {yu ...,yn} ar e orthonorma l sets . Automorphisms and Quasi- Free States of the CA R Algebra 19 1 Proof. W e write Ek fo r [V(ekι A • • • ΛeJ], Ekt fo r lV(ekι A ». Λ e^ Λ et A e,r+l Λ • • • Λ ej] , andFrforEtnEjr Ifzliesi n P ) Efc, then, withx 1,..., x,,i n Jf ,z A F(xx Λ • • • Λxπ)=0, k since z Λ V(ekι Λ Λ Sincex1 Λ • • • Λxr_! Λx r+1 Λ • • • Axn- ^V(xι Λ • • • Ax,.^ Λ eirAxr+1 Λ • • • ΛxJ fl nto α) is a n isometric , linea r mappin g o f (^θ[^i r])i -i i ^ ? d(Fir)^m~n+1, from Lemm a 3.6 . I f d(Fir)>m — n+ 1 , then , sinc e d(EjΓ\Ejίr) = m— 1 (fro m Proposition 3. 5 - a s argued i n Lemma 3.6) ; d(EiC\E^ >m — n, contrary to assump- tion. Thus d(Fir) = m — n+ 1 . Our inductive hypothesi s applies , and dί f] Eki\ = m- n+ί. Sinc e f] EkirQEJir; U,...,/c π / But / Π \ £ Rr nEjQEtnEj. Thu s In particular , w e hav e establishe d (unde r th e inductio n hypothesis ) tha t i f d(EinEj) = m — n the n EinEjQEk provide d on e o f th e kί9...9kn i s i n In ou r {iί9..., in9jl9 ...,7«} « ( argumen t kr = ir). Having prove d tha t d(Fir) = rn — n+ 1 and EiΓ\EjQFir, i t follow s tha t Fir i s generated b y EinEj an d a uni t vecto r fr orthogona l t o it . Moreover {/ l5..., / „} are linearl y independent . To se e this , note that f] Ejt i s m — 1 dimensional, fro m t Proposition 3.5 , so that Ejt i s generated b y f] Ejt an d a unit vector gt orthogonal to f P) EJt. I n addition, Thus n o £j t i s containe d i n th e unio n o f th e other s (fo r gt i s orthogona l t o tha t union). Now fr i s not in f] Ejt; for , otherwis e fr i s in Ej9 hence , in EiΓ\Ep contrary t to th e choice of/ ,.. Thus fr an d f] Ejt generat e Ejir; s o that a linear relation amon g t {/n ••• ' fn) woul d entai l tha t som e E t i s contained i n the union o f the others. B y the sam e token, if Ft ha s dimension m — n + 1 or greater and t is not in {i1,..., i n}, Ft contains a uni t vecto r f0 orthogona l t o EtnEj an d linearl y independen t o f {/i> • • • ?/ «} • Recalling tha t Eir^EjQFt (a s establishe d before) , an d FtQEι (b y 192 N . M. Hugenholtz and R . V. Kadiso n definition o fzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ft); w e se e tha t d(£ f) woul d excee d m. Thu s d(Ft) = m — n whe n tφ {il9 ...,/„} ; so tha t EinEjQFkrQEk. (W e no w hav e tha t eiί A '"SeiJ\n\V(eh A • • • Λ^ Λ^ rΛejV+l Λ • • • when fcr ^ {il5..., in}; and, from th e first par t o f thi s proof , since kr £ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA {z1?..., ι fV j1, ...,j r-1? k r> Jr Lemma 3.9. 77z e equality, 1 Λ ... AxJln- nCϋίz! A • • • A zπ)]) is valid for finite and infinite intersections. Proof. We establis h th e assertio n o f th e lemma , first , fo r th e intersectio n [xl9 ...,xjn[z 1? ...,z j o f two n- dimensiona l subspace s o f J f. I f {vί, ...,vk} i s an orthonormal basi s fo r thi s intersection , changin g notation , w e ca n writ e x1A"Άxn_kAvίA'"Avk an d zί A • • • Λz n_fe A υί A • • • A vk i n plac e o f (α) xίΛ - Λxn an d zx A • • • Λz π. I n Lemm a 3. 7 w e note d tha t U map s J^, int o (α) Jfn ; s o tha t yx A • • • Ayn_k^U(yί A • • • Λ 3^ Π_Λ A v1 A • • • A v^ i s a produc t { (α) isometry o f (J^θ[vu ...9vj) "lk int o Jf M . Fro m Lemm a 3.6 , [ί7(x1A. AxJ]n[ί7(z 1A...Az/J)] has dimensio n a t leas t k. Applying thi s t o U~1, w e se e tha t l A .- . A xn)- ]nlϋ(Zl A - Λzj]) = k = d(lxl9 ...,xjn[z l5 ...,zj) . Let w l5..., w r b e a n orthonorma l basi s fo r [x 1}...,xjn π[z l5 ... ?^J; an d let M 19 ..., wπ_r, wi,..., w^_ r b e a n orthonorma l se t o f vector s i n J^θ[wl9..., wj . Then, from th e preceding , Moreover};! A • • • /\yn-r^U(y1 A • • • Ayn_r A wί A • • • A wr) is a product isometr y fl (α) of (JfθEWi,..., w r])i lr int o Jf n . Fro m Propositio n 3.8 , 1 Thus d([U(x1 A " Λx n)]n n[C/(z 1 A • • • ΛzJ])^r. Applyin g thi s t o ί/ " , we hav e r = d([Xi A A xJ n n [zx A A zj) from whic h ou r lemm a follows . Corollary 3.10. With e0 a unit vector in Jf , d( Π [ί>(x 1Λ Λx 1I_1Λeom Automorphisms an d Quasi- Fre e State s of the CAR Algebra 19 3 Lemma 3.11.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA For each unit vector e0 in Jήf, Ue0 lies in ft [U(x, Λ ΛviΛe 0)]. Xl,...,Xn- i Proof. From Corollar y 3.10 , there i s a uni t vecto r f1 whic h generate s Now d / Π [t/ *^ ! Λ • • • Λ j;π_! Λ fj]\ = 1, from Corollar y 3.1 0 an d ί=d( Π [C/ *ί/ (x 1Λ Λx II_1Λeo)] so that each [U*(y1 Λ • • • Λ yn_ί A fj\ contain s e0. Thus U*(yx Λ »• Λ^.j Λ / !) = /! Λ - Λy ^Λ^ . Suppose /x is not a scala r multipl e o f Ueo(=eί). The n φ0, wher e e2 i s a unit vector i n [el5 /J orthogonal to e^ Let {^} J=1?2,... be an orthonormal basis for Jf an d le t A be #α(eo)*α(έ?o)ί7*. Then A{f, A ej2 Λ Λ ej = Ua{e0)* a(e0) (e0 A e'h A Λ Finite sums , Σc ilJpijliiιiίfl(gi^.fl(gjJ*fl(g.,,fl(gJί)(=B), for m a norm - an dense subset of 21. Let ε be K/Ί |e2>|/5, ^ choose 5 such that \\A — B\\ <ε. Then (examining the coefficient o f e2), where c0 is the coefficient of / in the sum representation of B and all ex, e,- appearing in thi s su m ar e amon g el9..., em_1 (s o tha t ^ 4 and al l terms of B other than col map e m to 0 or a multiple of a basis element other than em)9 and 2 2 2 s >\\(A- B)e2\\ ^\c0 + c2;2\ (examining the coefficient o f e2). Thus |c o|<ε, \c2;ι\<ε an d \c2;2\<2ε. The onl y term s in the sum for B that yield multiples o f e2 A em+2 A Λ em+n when B acts on fi^em+2A- - - Aem+n 194 N . M. Hugenholtz and R. V. Kadison arezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA col, c2.Λa*(e2)a(e1\ an d C2.2a*{e2)a(e2). Thu s SO that ^ K/ i k2>l - ko l K/i k2>l - |c 2;il K/ i IOI - k2;2| K/ i a contradiction . Thus /x = c'e1 fo r some scala r c'. Lemma 3.12. There is a sequence {cn} of complex scalars of modulus 1 such that U{xι A • • • Axn) = cn(Uxι A • • • Λ Uxn), where U is the restriction of U to one- particle space. Proof Le t {βj} be an orthonormal basi s for jf. The n for all y and n. Thus t/(^ 7l Λ Λ e^) and ί/^Λ Λ [7^ π diffe r b y a phase factor . Say Λ (7^^ Λ Λ ^.J = c l/ e^ Λ Λ ί/β jn and ^ I7(^fl Λ^Λ Λ ^ J = c' l/ e^ Λ 17^.2 Λ Λ Uejn. Writing / fo r {eh+eiχ)l}/ 2 an d /' fo r {ceh+c'eh)l]/ 2, U(f A eJ2 A • • • Λ ejn) = Uf'AUeJ2A Ά Uejn. Sinc e U(f A eh A Λ ^n) = c" Uf A Ueh A- - AUejn and Uf — c"Uf i s orthogona l t o UeJ2, ...,Uejn; w e hav e Uf =Uc"f. Thu s / // / ce^ + cβίl = c"eh + c"eiχ\ an d c = c = c . Changing basi s element s successively , we conclud e tha t U(ejί A • • • Λ ejn) = cUejι A • • • Λ £/e,n, fo r all j1? ...,./„ . Writin g cn fo r c, we have (J^ Λ • • • Axn) = cn(Uxί A • • • Λ UX). Of course c x = 1 ; an d Lemma 3.13. For all xί9...,xn in Jf and n = 09ί,2, ...,U(x1 A - - Axn) = UxιA"ΆUxn. Proof. From Lemma 3.12 , there is a sequence {cn} of scalars o f modulus 1 such that U(xί Λ Λx w) = c n(Ux1 A Λ l/ x^). I f {^} i s an orthonormal basi s for Jf, so is {Uβj}. Write/ , - for C/ ^ . Since Ua(ej)U*(fjA fJ2A • • • Λ/ ,- ) = cnϋ(eJ2Λ'~ Aβj) = cn- icn(fj^'" A /J and Ua(ej)U*(fh A ... Λ /J = 0 if7^ Ui, -. J«}ί we have that C/α(^)£7 * | ^ = cn_ xc^ί/j) | JfnH Writing c ^ fo r cn_xcn (s o tha t c i = 1) , w e sho w tha t c ^ = c^,_ t = = 1. Let Σc il,..ip;il...i2α(/il)*...a{fip)*a{fh)...a(fjq) (=β ) b e chose n suc h tha t ||^4~5|| <ε, where A=Ua(e1)U*. Suppos e il9 ...,ip'Jl9 ...,jq ar e less tha n m. Then, examinin g th e coefficient o f fm+2 Λ • • • Λ /m+n ? This inequalit y hold s fo r n=l,2,..., so tha t |c ^ —c^J_1|<2ε fo r al l positive ε . Thus l = cΊ = c' 2 = ,andc π = cB_1 = = c 0= 1 . Automorphisms an d Quasi- Fre e State s o f the CAR Algebra 19 5 Proof of Theorem 3.1. W e kno w tha t U implement s α (Lemma 3.3 ) and i s th e α) product unitar y operato r o n JfJ correspondin g t o U o n jf , whe n &(φ0) = φ0 (Lemma 3.13) . Th e cas e i n whic h &(φo) = φI i s reduce d t o thi s case followin g th e proof o f Lemm a 3.2 . T o conclud e th e proof , w e not e tha t ot(a(f)) = a(Uf), or , equivalently, tha t α(α(/)* ) = α(C//)* . Fo r this , observe tha t = C>fl(/ )* for al l fl9..., / „ in Jf. Thu s α(α(/)*) = α(C//)* . 4. Product Unitary Operators on w- Particle Space In th e precedin g section , afte r showin g (Lemm a 3.3 ) tha t U i s a produc t unitary operato r o n Fock space , w e mak e no further us e o f the hypothesis tha t U induces a n automorphis m o f 9 1 unti l Lemm a 3.11 . Wit h thi s hypothesis , i t i s proved (Lemm a 3.13 ) tha t U i s induce d b y a unitar y operato r o n one- particl e space. The fact tha t this same result i s valid fo r a product unitary operato r define d only o n rc- particle space i s proved i n the theorem that follows . Theorem 4.1. If U is a product unitary operator on n- particle space ^a\ there is a unitary operator U on one- particle space J f such that U(x1Λ- Λxn) = [/ x 1Λ ΛL7χ π. We prov e thi s theore m wit h th e ai d o f th e followin g lemma s (notatio n as i n Theorem 4.1) . Again, th e argumen t wil l justify th e us e o f th e notation U. Lemma 4.2. If {ej} is an orthonormal basis for Jf , there is a unit vector fj in Π \fi{xι A • • • Axn_ί Λtfj) ] such that {fj} is an orthonormal basis for 2tf. Xl,...,Xn- l Proof T o sho w tha t < ( fj \ /fe ) = 0, when j Φ k, we ma y assum e tha t j an d k are in {1,... , n + 1} . Let Ej b e the π- dimensional space , [U{eί Λ • • • Λehl Aej+1 Λ • • • Λ en+1)]9j= 1 , ...,n+ 1 , and E b e the spac e E1 V£ 2 Fro m Lemm a 3.9 , EίnE2, E^nEp an d E2c\Ej ar e n — 1 dimensional; an d E1nE2nEj i s n — 2 dimensional, whe n jΦ 1,2 . Thu s E i s n+ 1 dimensional; an d Eί9 E2 contai n uni t vector s vί9 v2, respectively , tha t li e i n Ej bu t no t i n E1nE2nEj. I t follow s tha t EιnE2nEj an d υγ generat e a n n— 1 dimensional subspace o f E1nEj (whic h is, therefore, EίnEj). A s v2 i s in E2 bu t not in E1ΓΛE2nEj, υ2 i s no t i n EίnEj; s o tha t EίnE2nEj,v1 an d v2 generat e a n rc- dimensional subspace o f Ej9 whic h i s therefore , Ej. Thu s Ej i s a subspac e o f E. Let fj b e a unit vector i n E (unique up to a phase factor ) orthogona l t o Ej. From Section 2 and Lemm a 3.9 , Ej an d Ek ar e perpendicular , whe n jφk; an d bot h ar e subspaces o f E. Hence fj i s i n Ek an d fk i s i n Ej9 whe n j φ k. Since / } i s orthogona l 196 N . M. Hugenholtz and R . V. Kadison tozyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ej'9 = 0, whenj + k. Thus {/ 1?..., fn+ί} i s an orthonormal basis for £, and Ej=ifί9..., fj_u fj+1,..., fn+i~]. Henc e / , - generates f] Ek. From Corol - - fcφ / lary 3.10 , f ] [ί7(x i Λ • • • Axn_1 A e,.)] i s a one- dimensiona l spac e [contained in f] Eλ so that / , - generates it. V fcφj / Lemma 4.3. If U is a product unitary operator on Jf^ and {βj} is an orthonormal basis for Jf, there is an orthonormal basis {fj} for Jf such that U{eii Λ • • • Aeir) = fiιA - AfίJoralliu...Jn. Proof Fro m Lemm a 4.2 , we can find an orthonormal basi s {fj} suc h that/; e Π [^(*i Λ Λ *n- iΛ^ )] for a1 1 J . Thu s t/ ^ Λ Λ βj inin C ii...J=1 Le t Cj be c l...j- lj+ l...«+ l» and let Cj be n + 1 \ 1 In ( Π4 .With/ , as cjfj, k=ί ) U{eίA"Άej_1Aej+1A'- - Aen+1) = Cj{c1...Cj_1cj+1...cn+1)fιA.'- Afj_ιAfj+1A - Afn+ι = / iΛ Λ/ J._1Λ/j+ 1Λ...Λ/l l + 1. Suppose, now , tha t w e have chose n fί9..., fm,m>n, s o that /7 is a multiple of /; and U(eiί Λ Λ g/n) = /fl Λ Λ fn whe n 1 ^ zx < /2 < < in S w Suppos e n ϋ(eίA- . Aen_1Aem+1j=c'f1A ' Af n_ίAf^ +1 an d U(e2A- AeHAem+1) = c" f2 A Λ fn A fή+!. From Lemma 4.2 , there is a vector / suc h tha t #(*! Λ ΛVJ Λ (β m+1 + e Λ+ 1))= Λ Λ • • • Λ /„_! Λ / and Since / ί/(^1Λ Λ^_ 1Λ(em+ 1+^+ 1)) = /1 Λ Λ/ n_1Λ(c /;+ 1+/M+ 1) and / C/(β2Λ Λe IIΛ(ew+ 1+eII+1)) = /2 Λ Λ/ IIΛ(c 7w+i+ / Λ+ i); f=έfm+i + fn+i an d cf = c"&+1+fn+1. Thu s (cc'- c")/^!+(c- l)/ Λ+ 1=0; and c= 1, d = c". Applying thi s conclusio n t o step- by- ste p replacements , w e have tha t U(ehA- ~ AeinAem+ι) = cfiϊA ' AfinAfή+ι fo r one phase facto r c and al l i1,...,in les s tha n m+ 1 . Definin g fm+1 t o be c/^+ 1, ou r induction yield s the basis {/,}. Proof of Theorem 4Λ. Wit h {^ } and {/,} as in Lemma 4.3, let U be the unitary operator o n Jf fo r whic h V βj = fj9 j = 1,2,.... The n t/(e tl Λ Λ ein) = Ueil A • • • Λ C7ein; so that [/ ( q Λ • • • Λx rt)= t/ x j Λ • • • Λ C/χn fo r all xl9 ...,x n in J f . Acknowledgment. Th e authors ar e grateful t o the National Scienc e Foundatio n (USA ) fo r it s support an d als o t o the Mathematic s Departmen t of the Universit y o f Britsh Columbi a fo r its kind hospitality an d the National Research Council of Canada for support durin g the final stage of this work. Automorphisms an d Quasi- Fre e State s o f the CAR Algebra 19 7 References 1. Araki , H.: O n th e diagonalizatio n o f a bilinea r Hamiltonia n b y a Bogoliubo v transformation . Publ. Res . Inst . Math. Sci. Kyot o University , Ser . A4, 387—41 2 (1968 ) 2. Araki , H.: On quasi- fre e state s o f CA R an d Bogoliubo v automorphisms . Publ . Res. Inst. Math. Sci. Kyot o University , Se r A6 , 385- ^4 2 (1970 ) 3. Balslev,E. , Verbeure,A. : State s o n Cliffor d algebras . Commun . math . Phys . 7 , 55—7 6 (1968 ) 4. 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