Vol. 16 (1979) REPORTS ON MATHEMATICAL PHYSICS No. 1

SCATTERING THEORY FOR QUASI-FREE TIME OF C*-ALGEBRAS AND VON NEUMANN

HEIDE NARNHOFER

Institut fti Theoretische Physik, Universitit Wien,

(Received March 26, 1977)

AUTOMORPHISMS ALGEBRAS*

Wien, Austria

Relations between scattering theory in L’ space, for quasi-free automorphism groups of C*-algebras and for those of their representations are established and it is shown that wave operators as time limits do not exist in these representations. Finally, the connection of scattering theory and dynamical stability is discussed.

1. Introdnction

One of the main problems in quantum statistical mechanics is the description of time invariant states, i.e. we have to know first, whether an invariant state: can be decomposed uniquely into extreme invariant ones, then, how the extreme invariant states are related to the KMS-condition. We know that the decomposition is unique if the time automorphism is asymptotically Abelian (also weaker conditions work) [l]. The connection between the KMS-condition and extreme invariant states is discussed in [2]. There it is shown that an extreme invariant state with cyclic and separating GNS-vector implies the existence of an automorphism group, commuting with the time evolution and conversely such an automorphism group leads with additional assumptions to the existence of an extreme time invariant state that does not satisfy the KM&conditions with respect to the time evolution.

In classical as well as in quantum mechanics of a finite number of particles scattering theory allows us to find constants of motion [3]. In the same way the existence of the Miiller operator for automorphism groups in statistical mechanics tells us about the stability of ergodic properties under perturbations. In this paper we shall restrict our interest mainly to quasi-free time automorphisms, where scattering theory on the auto- morphisms of the C*-algebra corresponds to scattering theory in L2-space. The central problem will be how perturbation of the automorphism and scattering theory of the C*-algebra is connected with the corresponding problems of the von Neumann algebra in a representation, and it turns out that scattering theory can work for the C*-algebra but does not work for the von Neumann algebra.

* Work supported in part by “Ponds zur Wrderung der wissenschaftlichen Forschung in &.terreich”, Project No. 3016.

2 H. NARNHOFER

2. Scattering theory and an intertwining automorphism

Since for a C*-algebra time automorphisms assume the role of the self-adjoint Hamil- tonian in a Hilbert space, it seems natural to digest whether there exists something like scattering theory for perturbed automorphisms, i.e. does there exist [4], [4

lim 7, z_,A = y* A (2.1) t-&cc

where y is again an automorphism. The same relevance with respect to ergodic problems has the existence of an intertwining automorphism

z,yA = yt,A. (2.2)

Evidently y* defined above would satisfy this relation but y can exist without being a time limit. In scattering theory the existence of such an intertwining unitary operator guarantees the unitary equivalence of {H)’ and (H}‘. Similar we have now an equivalence of com- muting automorphisms. Let y be an intertwining automorphism with try = ~7~ and u an automorphism that commutes with r,. Then

[&, y-lay] = r-‘[J+, ar] = y--l[tt, fJ]r = 0. (2.3)

For the free time evolution we know the class of commuting automorphisms as well as the class of extreme invariant states. If therefore such an intertwining automorphism existed for the time evolution with interaction, then every 0) 0 y with @t-extreme invariant would be textreme invariant. This means that the perturbation does not in this case improve the ergodic properties of the non-interacting system.

Let us assume we are already in a KMSstate with respect to the time evolution t,. If we disturb the time evolution locally, we obtain a new time evolution rr and with ap- propriate assumptions on v’ (e.g. if V E n(d)“) we are able to construct a vector in the Hilbert space of the unperturbed KMS-state, which gives a KMS-state over z(&)” with respect to the new time automorphism. Then rt and $’ are unitarily implemented.

If there exists an intertwining automorphism between the two time evolutions of the algebra, we are able to extend it to z(d)“, since we are still staying in the same representa- tion. The question arises whether the corresponding unitary operator can be obtained as a time limit. In the case of quasi-free time evolution the answer is no, first, since the corresponding Hamiltonians H and H = H+z(V)-~(V)~ have eigenvectors, but even on the continuous subspace X,,,, the limit eiHt eWiat does not exist.

3. Analytic perturbation of quasi-free time automorphisms and their KMS-states

There are essentially two ways of perturbing the automorphism group, either by perturb- ing directly the automorphism group of the C*-algebra [4]

m

7rA = c

(i) s

dt 1 . . . dtn[~JL.. [tf.V, 7t4 -I] (3.1) n=O OC11<...5f”Si~

SCATTERING THEORY FOR QUASI-FREE TIME AUTOMORPHISMS 3

where the expression is well defined for YE I, because the series converges in norm. But we can also start with a KMS-state and the corresponding von Neumann algebra m(d)“. Following [5] we perturb the vector y of the GNS construction obtaining a new vector,

CO

which is also cyclic and separating and therefore defines a new modular automorphism group ry, coinciding with the one above fcr V E z(d) and its KMS-state. In [5] it is shown that yv is well defined and j]yv; 1 ~2 e~llvll for V bounded. The proof can be easily generalized for unbounded V, but with w an analytic vector for all t, V, independent of t, and a similar inequality holds :

The new KMS-state is (3.3)

(3.4)

We can write it as power series in IV

“IV(A) = 2 ftWU”,l,(A, v, .*., V> (3.5) ?I=0

where U,,,, are known as the Ursell functions ([l], 161, 171, 181). (Usually A and Yare the density operators, but the generalization is obvious.) In particular,

U,(A) = <Wl~(A)lW)?

USA, V, . . . . V) = S<ATiylV... ~iy,V)- CIlUkUn_,-

The number of terms that are contained in every Ursell function is of the order (n!)“. This may, however, be misleading. In fact, we know that our perturbed KMS-state is a quotient of two functions that are analytic in the entire A-plane, therefore also OAV is analytic except at the zeros of the denominator, but because of (3.2), (3.3) there are no

zeros for I;21 < A, = -!- ln2. There is of course also an advantage of starting with the Be

Ursell function: if we regard a sequence of V, it may happen that y+ does not stay bounded but the divergence cancels in the Ursell function, so that wAv remahs well defined despite

4 H. NARNHOFER

of passing to a new representation. Unfortunately, this works only if we perturb by a qua- dratic interaction V, so that the automorphism remains quasi-free (the number with connected graphs increases like n!), whereas for real two body interactions (V quartic) the Ursell functions stay finite but increase with n! (we count the contributions of two point functions) so that a more detailed analysis is necessary to obtain Bore1 summability. For the quasi-free perturbation Y = W(X, y)u~aY a direct computation leads to the ex- pected resblt

(3.7)

and the radius of analyticity is determined by the properties of the right-hand side. Higher n-point functions vanish, o 1y being quasi-free. If we stay in the same representation then the region of analiticity is the 1, already mentioned.

4. Scattering theory

Quasi-free time evolution corresponds to the time evolution in an LGpace, since

t, a(j-) = a(f?i”‘j-) . (4.1)

Therefore scattering theory for the automorphisms works if it does in L%pace, namely

r*:a(f) = n lim tt k_,u(f) = nlima(eih7e- “‘f) (4.2) r+*m

iff

st lim ei6Cihff= SJ r-+*m

exists and is a unitary operator. It is not sufficient that Sz, be an isometry, if y is to be an automorphism and therefore should be invertible. In this case, if h has point spectrum and eigenfunctions fi , the perturbed automorphism is no longer asymptotically Abelian, U'Cfi)U(fi) being invariant under time evolution.

Next we want to see how the scattering mechanism can be realized in the KMS-state representation :

Formally we can write the free Hamiltonian (with chemical potential 1~)

HI = (p2+p)&,dp s

or generally (also for lattice systems)

HI = 5 @(p)+,&.

(4.3)

It is not an operator in the KMS-representation (except /I = co, ~1 = 0, where the rep- resentation is the Fock representation). But we can introduce new creation and annihila- tion operators

UP = s(p)u,--s(p)u;=, & = 3(p)uJ+sW,T, (4.4)

a; = s(p)u;-i(p)u,T, Pi = S(p) a, + s(p) a;=,

SCATTERING THEORY FOR QUASI-FREE TIME AUTOMORPHISMS 5

where a,, a; are the annihilation and creation “operators” (after smearing) of the algebra, aT dr those of the commutant, satisfying the anticommutation relations I’ v

{up, a:> = {up, u,“) = 0

and

scp) = ( 1+&‘),,, s”(P)+s2(P) = 1.

In term of these creation and annihilation operators the Hamiltonian of the representa- tion implementing the time automorphism reads

H = HI--X = ~@~~Md~~~,-~~B,l (4.5)

and the GNS-vector is the corresponding Fock vacuum. If we add now a potential (so far only a formal expression), then

H,+V,-HT-Vf’

1 Wd~~[~~~,-B~Bvl+

+~dp&% q>[s(p>s(q)+s(p)s(qll[a:a,-B~B,l+

+l%, q)rsCp)s(q)-s(q)s(pll[a,B,-_a:ldpdq. (4.6)

This operator is a self-adjoint operator in the GNS-representation (= Fock space over a and /?), if the operator in L2-space

(4.7)

is Hilbert-Schmidt (Berezin [9], III, 7.3). Sufficient conditions on 0 for that are

(a) o itself is Hilbert-Schmidt. Then the corresponding operator V = avu belongs to the C*-algebra so that this result is in agreement with that of [5J

(b) Let v be a multiplication in x-space and h = p2. Then we arrive at

s 4Nfi@-q)2 [( l+l,p,. )‘i’( l+:-P.‘)1’2-( l:&?*z )‘i2( l+:-Py’2]. The integral exists if Ei(p- q) is a L” function or v(x) E L’. Therefore VI does not belong to the C*-algebra, but nevertheless H+ V is self-adjoint on the domain of H.

Further we can make H+ V a well defined operator iff H+ VI is well defined, SO that divergences of VI cannot be cancelled by divergences of VT, and only inner perturbations (eiY1’ E n(e”) of the Hamiltonian are allowed. That we prefer nevertheless H+ V instead of H+ VI, is due to the fact that H+ V also gives the corresponding automorphism for the commutant and if something like scattering theory in Hilbert space should work then there is only a chance for H+V.

6 H. NARNHOFER

Let us assume now that scattering theory works in L2-space

J$nm .&@Jr = 52,) lim &ht,-i%r = 52;‘.

r-r*a, (43)

What do we know about y+ a(./) = a(Q+f) and what can we conclude for ei@‘+~te-r~t? Again Berezin ([9], II, 5.1) tells US that the automorphism is unitarily implementable

in our representation if

where A is some positive and U some unitary operator, and if

is a Hilbert-Schmidt operator, where we considered the automorphism y+ a(j) = a(Q+f) and y+$(f) = a’(Q+f). Considering the automorphism restricted only to the algebra and not to its cornmutant, we obtain

and (&)Li2.(&y’2+ l+;-ah = 28

has to be Hilbert-Schmidt. All these conditions would be satisfied if 0-l itself were Hilbert- Schmidt. But as can be easily shown, this does not happen for a linear perturbation of the Hamiltonian. Instead, it suffices if [H, $21 is Hilbert-Schmidt, and this holds for reason- able ‘u.

Finally we examine eiR’e-iat. We know already that e’“‘ly) = Iy), whereas eiatlyV) = Iyy), so as for these operators, we cannot expect convergence. H has only one eigen- value, the rest of the spectrum is absolutely continuous, so we concentrate on the conver-

gence on 17 a+(Si)Bt(gj)lY>, i.e. vectors that belong to &‘,,,, of H, and are orthogonal to y.

lim eiare-in’L7a+ *(fi)/l+(gj)j y) I-U2

= lim ,i”l~at(e-ih’f)Bt(e_‘h’gj)ly) I-00

so that at most all’we can have is weak convergence. Nevertheless # and H are unitarily equivalent with f(H) U = Uf(H).

SCATTJZRING THEORY FOR QUASI-FREE TIME AUTOMORPHISMS 7

5. Remarks on dynamical stability

In IlO] it is, shown that for asymptotic Abelian time evolution dynamical stability implies the ELMScondition. We consider what this result tells us in scattering theory.

An extreme invariant state o is called dynamical stable if (i) it satisfies some cluster properties and if (ii) V E d and z!’ is the corresponding perturbed time automorphism. Then it is possible to perturb the state o so that

(1) %(rY4 = 44,

(2) o,(A) = o(A) + hJ’(A) + 0(12) where o’ is normal with respect to CI).

We start with the free time automorphism r, and consider a commuting automorphism group os. Let o be a KMS-state with respect to d and z:’ a perturbation to which algebraic scattering theory applies, i.e. ya exists. Then the property of o’ being normal is equivalent to o 0 y;? o Ye+ = o, i.e. a dynamical stable state is invariant under the scattering auto- morphism. For reasonable conditions on V (V being Hilbert-Schmidt, i.e. V E &)Qnn is analytic in il (in the weak sense) for lill < 1, so that w,(A) : = o(y). A) is again an analytic function of 1. We want to examine whether o’ is normal for w = l~(k))(~(k)l which is typical for quasi-free perturbations. Then [ll]

k?(P) e(4) <Pll--J&q) = ___ (P2--q2+iW+(q2) (5.1)

and it is not Hilbert-Schmidt, so that in general oL neednot be normal. Taking the derivative with respect to 1, one finds that the divergence that must cancel for o’ to be normal is

1

(P” - q2Y [S(p2)2S(q2)2+S(p2)2~(q2)2- 2s(p2)s(q2)~(p2)S(q2)] (5.2)

which in fact does for s(p”) chosen appropriately, for instance s(p’) = (1 +@f(p2))-1 and f ‘(p’) bounded. In general

.n; = i dteihtve-iht

0

and if @(p, q) (s(p2) - s(q2))/(p2 - q2) is square integrable then our o1 satisfies all the conditions on dynamical stability with respect to this V. This shows especially that the restriction that o’ be normal is not too serious and not a peculiarity of KMS-states. There is no contradiction to the equivalence of dynamical stability and the KMS property because the state o must be dynamically stable with respect to all perturbations, also non quasi-free ones. Here there are no results on scattering theory and in fact we do not expect that in general such a ya will exist.

But we can also consider a state invariant under free time evolution not satisfying (5.2) and such that perturbation of state and time automorphism is possible but not in the sense of dynamical stability breaking the condition of normality.

8 H. NARNHOFER

Let w(ut(f)a(g)) = (jVlg> with [I, h] = 0 but 1 not a function of k. Then

lim 0 0 ti” (ut(f)a(g)) = 0:” (WMd) = <flfJt, m,lg) r+*w

differ for &-co if 1 does not commute with the scattering operator. Writing the perturbation series in ;Z up to first order we obtain

~+(t+-A) = limo(t:Yr_t-A) = limo(z_t~:Yt_t-A) I-+rn f-r*

0

= w(A)-il 1 o([zt, V, t_,-Al)&’ -CO

_- ? t-em

= o(A)-iA i o([+V, ADdt’ _- w *

--a0 I -b--o0

The limits i + f 00 coincide iff w+ is normal with respect to ti and also iff fcO

s o([r,Y, A])dt’ = 0. -03

We know they can differ for some 1. Let us see why then o* are really not normal. Choose typically (~111 q) = l(p2) * 8(p- Uq), U being a rotation. Berezin tells us that

must not be Hilbert-Schmidt, i.e. zl(p, Up) # v(Up, p), which is equivalent to noncommu- tativity of the corresponding scattering matrix with I.

Acknowledgments

I am grateful to W. Thirring and A. Wehrl for stimulating discussions and critical remarks.

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