PRODUCT ISOMETRIES AND AUTOMORPHI&MS OF THE CAR ALGEBRA

by

Richard V Kadison*

I. INTRODUCTION.

The methods of multi linear algebra and, in particular, those

of the exterior calculus provide a useful framework for studying the

Fock space. ~~ • of anti symmetrized wave functions and the Fock

representation of the Canonical Anticommutation Relations (CAR) on it.

Wi th the aid of these methods, we study the stmcture of certain

mappings, product isometries, of the n-particle subspace l:In

of l:I5"

With l:I a complex Hilbert space, we denote by f1 ~ ••• ~ fn

_1-the vector (nl) 2L"X(,,)f

o(1) 0 ••• ® f,,(n) in l:I ® .,. ®.I>, the n-fold

tensor product of]:l with itselft where" is a permutation of {t •...• n}

and x(a) is its sign. The space spanned by {fl

/\ ... ~ f n : fj E l:I} is

co denoted by ):jn' and JI,.. (antisymmetricl Fock space, is Ln;O $ Jln , where

Jll

is l:I and l:Io is a i-dimensional space generated by a unit vector, ~O'

the Fock vacuum. A linear isometry V of some subspace Kn (={f1 A ... ~ fn'

f.EXS;l:I} J

is said to be a £roduct isometry when V(ft A

is a product vector (i.e, of the form g1 ~ .,. A gm) in JI,., One of

our principal aims is the following result,

With partial support of NSF,

174

THWREH A. If U is a product unitary tran,formatlon 01 $In onto Sin'

there i,; a unitary transformation U 01 Si ont.o Si Euch ihat

This is proved by a combinatodal-gl"ometric st.udy of the way

in which U transforms the s~bBpace [1 •••• ff ] of Si associated with 1 n

i1 A ••• A fn

• We note that

and that afl

A ... II fn ~ g1 A ... A gn I 0 if and only if thE>

spaces [f1

II '" II fnJ (3[f1

, • fnJ) and [g1' ..• , gn] associated

".'1 th i1 A ••• A fn and 91

A ••• A gn are n-dimensional and ('oincid€.

Applying Theorem A, one can, 'thEns show that:

THEOREM B. If a is an automorphism of the CAR algebra!! whose

transpo,;e a maps thE> s"t <9 of pure, 9au9,,·invariant, quasi-free states

of f! onto itself, then t.hE>rE> is a unitary oI,E:rator U on Ji such that

a(a(f) ) a(Uf), f'or all f in Ji; or there is a <conjugate-linear,

unitary operator W on }I such that a(a(f·)) a (W:f)·. faT all :f in Ii.

where a(:r) is the annihilator for a particle with wave function f.

N'ote that one can r(;ad from this result t.he. fact that 0. trans-

forms the Fock (vacuum) state CPo eithe,r onto itsel:f or onto the anti-

Fock state CPI' though thi 5 :fact is established as a preliminary to

proving Theorem B.

is the adjoint of the annihilator a(f) datBrmin&d by.

175

The mapping f - a(f) is conjugate linear (the inner product on ~

being linear in its second argument,) and satisfies a(f)a(g) + a(g)a(f) O.

a(f)a(g)" + a(gl"a(r) = (rlg)I (the CAR). A conjugate-linear mapping

of» onto operators a(f) on a Hilbert space X satisfying the CAR is

said to be a representation of the CAR (over» on xl. The representing

operators a(f) are partial isometries (with initial and final spaces

orthogonal having sum X), hence. bounded. The particular representation

of the CAR on ~~ which we have described is the ~oCK representation.

The CO-algebra ~. generated by {a(f) ,a(r)*:f E ~}, is the CAR algebra.

Its representations are in one-one correspondence with those of the

CAR.

If A is an operat,or on I:! such that 0 ,;: A ,;: I, d"fining

'PA(a(fn )* 00. a(ft)*a(gl) ... a(gnl*l to be det (giIAfj» (= (gl 1\ •••

1\ gnlAfl 1\ ••• 1\ Afn» determines a state 'PA

of ~, the gauge-invariant,

quasi-free state (with one-particle operator A), The state 'PE is pure

if and only if E is a projection on 1:1; and 'PE is equivalent to the

Fock state 'PO if and only if E is a projection with finite-dimensional

range. In case E(I:I) is finite-dimensional and {e1

, ••• , en} is an

!\ e » !\ e (~), for each A in~. The intimate relation between

n Theorems A and B is a consequence of these last comments. for, then.

a unitary operator on I:!~ implementing a transforms product vectors into

product vectors.

n

176

The worK on which thE'S€ not"" an, based is a joint study

conducted with N M Huge>nhol tz. An €xt.(T.ded acc.ount is to be found

in: "Automorphisms and Quasi-free States of the CAR AlgAbra",

Commun. Math. Phys. ~3 (1975), pp 181-197. The author acknowledges

with gratitude the SUPPOI~ of the SHe and the hospitality of the

Department of Mathemat.ics at the Unh-ersity of Newcastle, as well as

t.he support and hospitality of the Cer,tn, Uni\"~rBitaire de Luminy-

Marseille, where parts of this work wt!re completed.

1 I. PRODUCT UNlT AHlES •

If V is a pH)duct isometry of an infinite-dimensional subspace

)( of It (.ell ) into l:I • th€n, with {e.} an orthonormal basis for X. 1 m J

the fact that VeJ

and Vek

are orthogonal pr.:.duc.t vectors in lim and

Ve j • Vek

(~ V(e j+ek ) is also a product vector in lim leads to the con­

clusion that the projections Ej

and Ek with the m-dimensional ranges

[vej

] and rV8kJ commute and [vej

] n [vek

] has dimension m-1. It

follows that 0 E.(~) has dimension m-1, and, hence, n[vx] has dimension J J

m-1.

If W is a product isometry of Kn into M~. isometry considerations

show that W has range in one Mm' If K is infiniie dimensional and

n S; m then [W(f1

" ••• " l' ) ] n [W(g "00. A g )] has dimension at n 1 n

least m-n. To see this, we may a:;snme that ea<-h of {i1,· •• • f

n} and

{91

, •• ,.gn} are orthonormal sets and make use of the mapping

177

f - W(f A f2 A ••• A fnl j which is a product

into Ii III

From the preceding, it follows

isometry of xe [f2 i , ••• f)

that [W(h1 A f2 A .•• A f n}]

and [W(fi A ••• A fn

}] have an intersection of dimension at least m-i.

where hi is a vector in [gl' •••• {;In] orthogonal to [f2 1\ ••• A f n ]·

In the same way. choosing h2 in [g17 •••• gnJ orthogonal to [h1 , fJ

• .. of f ], n

We see that [W(h1

A h2 /\ f, /\ ... /\ fn)] n [W(f1 A ••• A fnl] has dimension

at least m-2. Continuing, we have that [W(hi

A A hn

}] n [W(f1

/\ ••• A f n}]

has dimension at least m,-n, and h1 A ••• 1\ hn : cg1

A ••• 1\ gn'

If {ej } is an orthonormal basis for X and i1' ., •• in' ji' ···.jn

is such that [Wee. A , •. A e1

}] and 11 n

[W(ej

A.,. /\ e. }] have inter-1 I n

section' of dimension precisely m-n then n [w(x A ... 1\ x ) ] has X11tf,jXn 1 n

dimension m-ne This amounts to showing that

for all k1

, •••• kn

, This is effected by arguing inductively on n

the conclusion of the preceding argument allowing us to carry the

hypothesis of an intersection having dimension m-n to one where the

intersection has dimension m-(n-1). For this purpose. we use the mapping

Xl" ••• l\xr _1 "xr + 1 /\ ••• "xn ... W(x1 A , •• "Xr _1 Ae i "xr+1/\ .,."Xn ) r

of (xe [ei )h_1 into Urn' As an intermediate conclusion, we obtain (*)

if at least one of k 1 •••• , k n is in {ii' .,,' in' J1 ' " •• , jn}'

If we know that the intersection. M. of [Wee. A ••• A e. )] and 11 1n

[Wee. 1\ ... /\ e. A e t 1\ J 1 J rt-1

e. /\ ••• /\ e. )] has dimension m-n, when In+1 I n

t f {i1 " ••• i n}; then. if t k ~ {i1,.",! }. we have. from our r n

intermediate conclusion. that [W("k A ••• A "k )] contains M while M 1 n

contains [W(e. A ... A e. )] n [W(e. A '" A e. )]. :11 In Jl, I n

178

that ~1 hci~ dimenfLlon m-n~ note that. r.[~(e. ",.,. 1\ e. ~ J 1 J r - 1

I\et l\e. J r 1

1\ •• , /\ e. )] (=N) has dimension m-1. from our initial I n

obsey'\""at ion. Thus U'dh [we e. J 1

is generated by N and a VH tor gt ori.hogonal to it. Since

{Wee. 1\ ... 1\ e. 1\ et

1\ eJ. 1\ ••• 1\ e. )} is a family o:f ortho-

J 1 J,_1 1-1 I n gonal vectors){gt} is ~u(h a farnilv', and no [Wee 1\ ••• 1\ e 1\ e 1\

J j1 jl'-1 t

e, A ... 1\ e. ) ] i& <:or~t.:.dEcd in a union of the other~" If M has J 1,.1, I n

dimension greate.r than m-n it has a vector orthogonal to [W(e. ,,~~. '1,

1\ e. ) J n [Wee, 1\ 1 J 1 r.

and lw(c. J 1

1\ ,. . 1\ " n+l 'Vec!orE: (takl ng

1\

J y-l b 10

" ) J I n

1\ e, 1

S

be 1,

at:-. ao the int€.l'i-.-2c-tions 01 [Wee. "II ,,. 11

1\ e 1\ ." /\ e. )]. Each of these J r ',l I n

~"" ~n) are not in the m 1 dimerndonal

spa(,"t7 N, for j otb~!'wl;s~, thE-yare in [W(e. /\ .•• 1\ e. )], hence in

Lw( J 1 I n

1\ ••• /\ e. ) J, cont.rary to choi ce. Thus I n

<'arh of th,,~e vecTors generates wit.h N its corresponding m-dimem;ional

spac.e [W'(e. /\ .. , /\ e. /\ IL A e. J 1 J r,-l 115 J tc;.l

/\ , .. 1\ e, ) J (or [W( e. A, .. I n J 1

A e. /\ e k "e. ". ,. /\ e. ) J). A J r "l J r ."l J r:

linear ]'elation among these

vectors would imply, therefore, that one of these spaces is contained

i,n the union of the others - cont.rary to what we have noted. But these

n+1 v€("iors and the m~n-dimen5ional space [W(6. '1

I\ ••• /\e.)]n 1

n II

II

A E. )] orihogo~al to th~m are all containEd I n

1\ €, ) js an m d:im-snsional &pace~ Thus thE.-re 1

n

in

must be a

linear r£"lation among th€t.E:' n-t'1 vt::;ctor5~ from this contradic..tion» we

conclude that M has dimen~ion mot,.

179

Summarizing, to this point~ we have proved:

Proposition C. If W is a product isometry of Xn into ~m' where

n ~ m and K is an infinite-dimensional subspace of M. and the inter-

section of [w(ei

A .,. A e. )] and [W(e. A ••• A e. )] has dimension l.n J 1 I n 1

m-n ~ fOI' some e ~ , 11

,,,, e. ,e, ~ ... , e. ,then n [W(x1

A •• b A x)] ln J 1 I

n xl'" , ,xn n

has dimension m-n)

It follows. without difficulty, from this proposition. that a

product unitary defined on ~~ maps each ~n onto ~n' If U is a product

unitary on ~n then [x • "'j x J n t n

[U(x1 A .,. f\ xn) J n ". n [O(Z1 A

."j z ] and n

A z )] have the same dimension n

(for infinite, as well as, finite intersections). In par~icular.

A xn

_1 A el] has dimension 1 for each unit vector

orthonormal basis for ~ and f~ is a unit vector J

A x A e.)] then {f:} is an orthonormal basis for~. n-l J' J

To see this, we note f; E Mk when k I j, where Mk ; [u(e1

A ••• A e k _1

A ek

+1

A ,., A e 1)]' and f: is orthogonal to M.> From Proposition C n+ J J

and the consequences noted following it, M1 V M2 is an n+l-dimensional

space containing each (n-dimensional) Mj

• j 1, ., •• n+i. Thus

f~' ••• , f~+i is an orthonormal basis for M1 V M2 •

It follows. now, that U (e. A, ., A e. ) ; c. 11 1n 11

, " f. 1n 11

A ••• A < ; Wri ting c: for

J

for cof' J J

j we have

n

(:1 ... j-1j+1 ... n+1' c j for

180

for j ~ 1, •• ,' n+1. Using the fact that. U is a proouct unitary on

)In' it follows that D(e t " •.• A e j _1 " e j ",1 " ... A e n+1 A e n+2 )

cf1 A ••• A f j _1 " fj+t " ••• A fn+1 A f~ .. 2' where the phase factor

c is the same for j ~ 1 •••• , n+1. Taking fn+2 to be cf~+2' we

conetI~ct, inductivel)" an orthonormal' basis {r.} for If such that J

U(e, A ••• A e. ) ~ 1:. " ... A 1:. for all i1

, ... , in' Theorem A 11 "n 11 1n

n,,;;ult.s flom letting U be the 'Unitary operator on If dEotermined by'

Ue, f" J J

Ill. THE AUTOMORPHISM.

Suppo"e, now, that U is a product unitary on If:J' that induc .. s

an automorphi5m of' 21" The·n U maps 14n

onto ):fn for ea-r'n n, and the<i-e

is a unitary oper'ator Un on J:l such that V(x1

" ••• A xn) ~ Unx1. " .,' /\ Unxn'

It follows that Ua(f)a(f) -U* and a(U f)a(U f) * have the same restriction n n

to )jn' for bach f in If. A calculation (see Appendix I) shows that

U f = Co. U is for all f: n and nt, in this case, An (easy) algebraic n Inm rn

lemma (see Appendix III) allows us to conclude that Un = cnmUm• Hence,

U(f A U"'g:!./\ ••• IIU"'g ) n n n

so that Ua(f)"fl* and c a(U r)" hav€ tne sam", restriction too If. Another n 1 n

calculation ("ee Appendix 1I) ",how.;; that Dalf)"U* ~ "ra(U1f)·, Applying

our alg<>braic lemma. again. t;a(f)*U~ ~ <"a (U:;.f) .. (on 11:J" the phase factor

c is no longer dE-pend~nt on f). Finally. as

Uf

181

Writing U for Ui

, it follows that

Ua(ft

)*

for all f1!

a(fn)*U*~o U(f1 A ••• A fn)

••. , l' in ~ and all n. n

If ex is an automorphism of 21 such that a(~) = ~ and t,;«(flo) = (flO'

then C( is implemented by a unitary operator fl on~. Since the states

in & equivalent to (flO are precisely those vector states of 21 corres-

ponding to product vectors,

alW f A .•. A l' ) 1 n

Since 21 acts irreducibly on ~, we conclude that U(fi

A ••• A fnl is

a product vector (a scalar multiple of 91

A ••. A gnl. From the

preceding section. Oa(f)U* = alUf) , where U (a unitary operator on ~)

is the restriction of U to ~.

If ~(~o) ~ ~I' V is a conjugate linear unitany operator on ~,

and cr is the automorphism of 21 determined bya(a(f» a(Vf) *, then

~a(~o) = ~O and ~cr(G) =~, Hence there is a unitary operator U

on It such that

where W is the conjugate-linear unitary operator UV-1 on ~.

Theonem B follows once we show that ~ maps (flO onto (flo or (flI'

In any event, a(~o) E ~, so that a«(flO) ~E1 for some projection El

on It. We derive a contradiction from the assumption that El is

neither 0 nor I. With this assumption, there are unit vectors 1'1

1 and f2 in and orthogonal to E1(~)' If f) = /2 (f1+f2 ) and EO' E2 , E3

are the projections with ranges E10lle (ft], Eo(l:!) @ [f

2]. EO()!) @ [f

3].

then ~E (Al = ~E (a(f.)*Aa(f.», j o j J J

valent, the states l\:--l«(flE.) are in i!)

J

1, 2~). Since the ~E are equi­j

and equivalent to ~O(=~-1(~E ». 1

182

w I?l, wh€re 9. is a unit product vector in ll,.. Now, gj J

where le.1 J

1. As a(fj

) is a partial i80metry and

l\o:(a(f.l)g.1! = lIe.goll ; 1, od&(:f.» is J J J J

a partial isometry with g. in e J

in its final space. Thus 93

_l[a(a(f )*)g .(2 1 0

vectors distinct from. hence orthogonal to. ~O.

a contradiction. Thus Cx(q:>o) is either 'PO or 'P r '

APPENDIX I

Lemma. If!l is the CAR algebra in it.s Pock representation. on Pock

space l!~ and A is an operator in fl such that All! = a(f la(f )*Il! J n n n n

for each n, then A = a(f1

1a(f1)*.

Proof. We shall show that f = cf with lei = 1; so that n m

a(f la(r )* n n

established that fa' f1

, ••• , fm

_1

differ from each other by scalar

multiples 01 modulus 1. If B = LC •••.. . are. )* ••• ale. )*a(e. ) i1 lp'Jt···Jq 11 lp 31

•.• are. ) and r is an integEr laIger than all the indices occurring Jq

•

in this sum, 'W"here {e,} is an or.thonormal basis for l:I such that J

f t = Il ft li e

l and fm E [e1 ,e:J, th .. n

183

and

2 2 1 12 !lA-BII > II(A-B)(e t 1\ e 1\ ... 1\ e lll:> Co 0 + c 1 1 ' - r+2 r+n ; ;

when n < m; so that I (e If> J ,;; 21IA-BII, for each such B; and 2 m

<e2 I f

m> ~ O. Since fm E [el

,e2 ]. fro ~ ael

allf111-1f1' Moreover

and

IIA-nI12 ~ II( A-B)(e2 1\ er+2 1\ .,. 1\ e r +n )1I 2 ?: lllf1112 -(cOjO+c2 ;2) 12

when n < m; so that Jllf1112-1 <e1Ifm>12J = Illf1112 - lal2

1 ,:; 211A-BII

APPENDIX II

Lemma. If ~ is the CAR algebra in its Fock representation on

Fock space ~~ and A is an operator in ~ such that AI~ = a(r )*I~ J' n n n

for each nl then all f are equal (to r) and A = a(r)". n

Proof. Suppose we have proved that fO f1 = f2 ~ ••• ~ f m_1

(=fl. Lot {OJ} be an orthonormal basis for ~ such that Ilfl(1f = e 1

and fm E [e1 ,e2 ) (so that fm ~ (elifm>e1, + (e2 Jfm>e2 )· If

B ~ l::c. " j aCe. ) ..... aCe. )*a(e. ) ... aCe. ) and r is an 1. ••• l.p1J 1 '·· q 11 lp Jt' J q

integer larger than any of the subscripts appearing in this finite

sum then:

IIA-B!l2

::t II(A-B) (e 1 1\ r+

184

when n < m, and where a sub5~ri.pt ~O' before the 5£-micolon refers t.o

the absence of creator's and after the semJ,colon ref<>rs to the absence

of annihilators (c 0 is the coefficient of I in the sum for B). We 0, have. too.

IIA_BI1 2 ~ II(A-B) (e 1\ ••• 1\ e )11 2 r+l r+m

Thus

and

Since B may be chosen so that IIA-BII is arbitrarily small. (e2Ifm) O.

As f E [e ,e ]. f ae. In addition. IIrli = (ellr ) = a. Thus m 12m 1 m

fm = IIrlle1 = f; and A a(r)*.

APPENDIX III

Proposition. If V andJ1Yare vector spaces, A and B are linear

transformations of'V into )fsuch that for Each v in V there is a

scalar c for which Bv V

cvAv; then B = cA for some scalar c.

Proof. Let n be the null space of A. From the hypothesized

relation between A and B, n is contained in the null space of B.

'rhus A and B induce linear transformations A and B of the quotient

space y of V by n into)f such that A = AoT! and B Bo,!), where T! is

185

the quotient mapping of V onto V. With Vo in n, Bv = c AvO ='0; o vo so that we may assume that c

Vo = 0 when Vo E n. With this assumption,

if v ~ V and Vo E n. then B(v+vO) = c A(v+vo) c Av v+vO v+vO

Bv = c Av. v

If v ~ n then Av I 0 so that c v

c v+vO

c v+va

o. Thus, defining cv to

v = v + n. we have Bv = Bv = c· Av = v

If v E n then v + Vo E n

be c • for v in V. where v

Note that the null space

of A in r is (0). If we show that B = cA, for some scalar c then

Bv = Bii' cAv cAv. for all v in V. so that B = cA. We may assume.

from this discussion. that n (0). With v and v' in V. we have

B(v+v') =c ,A(v+v')=c ,Av(+ c 'Av'=Bv + BV' = c Av + c ,Av'. v+v v+v v+v v v

are linearly independent. Let (v ) be a linear basis for V. Then a

Bv cAy for all at where c = c for all a. Thus B = cA. a a v a

Department of Mathematics E1 University of Pennsylvania Philadelphia Pennsylvania 19104 USA