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Automorphisms and Quasi-Free States of the CAR Algebra N 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).
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