Algebraic Features of Equilibrium States

Daniel Kastler

History (i.e., Maxwell, Gibbs and Boltzmann, as transcribed into

quantum mechanics) provides us with the following prescription to

describe equilibrium statesl ) ~~,~ to the temperature ~-l and the

chemical potential ~:

(1)

Here H is the hamiltonian and N the particle number operator of the

system. This procedure (called "Gibbs' Ansatz") is satisfactory for

(arbitrarily approximate) num~rical results, but inadequate for found­

ational purposes: (1) requires the system to be "enclosed in a box"

(with perfectly reflecting walls, or periodic b0Undary conditions etc.),

whereafter one performs the "thermodynamic limit" (infinite bOx). Before

this limit, the model is highly unphysical (excited states constant in

time instead of "return to eCJ.uilibrium", destruction of invariance under

spatial translations, etc.). Besides, one wishes to develop the nations

of temperature and chemical potential from first principles.

In order to do this, and treat directly the infinite system, we need

a substitute for (1) relevant to the latter situation: this substitute

was found by Haag, Hugenholtz and Winnink (1) to be the Kubo-Martin­

Schwinger (KMS) condition2) formulated in the frame of "C*-systems"

[Cl,R,-r} (Definition 2 below).

1) possibly more than one after the thermodynamic limit, if the latter depends upon boundary conditions (case of phase transitions).

2) proposed by these authors [2] [3] as a boundary condition for the calculation of "Green's functions".

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Definition 1. A C*-system [G,G,,} is the triple of a C*-algebra G,

a locally compact group G and a representation T of G into the

automorphisms of G such that g E G -, (a) g

is continuous for all

a € G. Given a .-invariant state ~ of G, the GNS-construction

is the triple of a *-representation v~ of G,

a continudus unitary representation u~ of G,

vector n~ E ~ cyclic for rr~ such that ~(a)

rr~(Tg(a» = u~(g), rr~(a)u~(g) and u~(g)n~ = n~,

both on :ti~ and a

(n~lv~(a)ln~),

a € G, g € G.

Definition 2. A state ~ of the C*-system {J,R,T} (R the additive

reals) is called [3-KMS fo~ [3 E R, whenever, to all a,b E J,

there is a fUnction Z E C -uab(z) E C holomorphic in the open strip

o ::; Jm Z ::; [3, bounded continuous on its boundary, such that

(2) t € R.

Remark 1. This definition entails that ~ is T-invariant. For ,-

invariant states ~, it is e~uivalent to the following relation between

the Fourier transforms of F ab ' Gab:

a,b € J, E E R.

The KMS condition of mathematicians corresponds to [3 -1 in

Definition 2 [4).

Relation (2) (i.e., essentially w(bTi~(a»; w(ab) for a,b E 3,

a ,-analytic) is easily shown to follow from (1) with

(4) ~ () i(H-~)t i(H-~)t 't a = e a e , t E R, a E :1

Le.

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where t E R .... at and gET' .... 'Y g are the respective groups of time

translations (generated by H) and gauge transformations (generated by

_N)3\

(6) t E R, gET', a E J.

Since (2) with T given by (5) persists through the thermodynamic limit

we can replace the complex Gibbs Ansatz + thermodynamic limit by the

following requirement [1]:

Let (:;,R X Tl,a X 'Y} be the C*-system obtained from the algebra ;; of

local fields (the field algebra) acted upon by the direct product of time

and gauge. The equilibrium states to the temperature ~-l and the

chemical potential ~ are the states of J possessing the ~-KMS

property for the group t .... at'Y~t.

This is now the situation to be explained from first principles.

Remark 2. The gauge groups considered here are gauge groups of the first

kind, the simplest of which is Tl as considered above (one species of

particles). The general case of a compact (non commutative) gauge group

G is of interest in view of groups like SU3

, SU4, etc. Then t .... at'Y~t

has to be replaced by t .... at'Y st' with t .... St a continuous one­

parameter subgroup of G.

Remark 3. The gauge invariant part m of the field algebra4)

(7) m = ;;G = fA E :;; 'Y (A) = A for all g E G} g

3) at'Yg = 'Yga

t , t E R, g E G since H and N commute.

4) We shall denote observables by capitals and fields by low case letters.

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is called the algebra of local observables (non-gauge invariant fields are

in principle unobservable). In the C*-approach to field theory, ~ is

considered as the basic object, the rest of ~ being analytical apparatus

constructible from ~ [4]. Since G acts trivially on ~, (5) reduces

there to (\. Denoting by w the restriction 01' cp to ~ the above

characterization of equilibrium states then splits into

(1) The equilibrium states w of ~ to temperature ~-l are the ~-KMS

states for the time translations t - at 5)

(2) The extensions cp of such states w to ~ are ~-KMS for a one-

parameter mixture t - at'Y 5 of time and gauge. t

These statements (1) and (2) correspond respectively to the notions of

temperature and chemical potential as treated in I and II below.

I. Temperature (as obtained from stability).

In that paragraph we consider the C*-system {~,R,aJ defined by the

observable algebra with its dynamics (time translations).

Definition 3. Let h = h* E m. The local perturbation

h of the dynamics t -> at is defined by

(8) t E R, A € m,

with the unitary cocycle determined by

~ p(h) _ 'a (h)p(h) dt t -]J t t'

Remark 4. (9) entails the unitary cocycle property: p(h) = p(h)a (p(h» t+s t s s '

(h)* (h)-l Pt = Pt = at(p~h)), s,t E R; and also the fact that each a-

differentiable B E m is also a(h)-differentiable with

5) restrictions to m of the time translations at on ~ (these

leave m globally invariant since time and gauge commute).

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(10)

Relation (10) shows that Definition 3 amounts to "adding the (local) h

to the hamiltonian'~ whence the name "local perturbation".

Definition 4. An a-invariant state w of [m,R,a} is called stable for

local perturbations of the dynamics whenever there is a map h E ~ _ w(h) E g

from a neighbourhood u of 0 in the self adjoint part of m into the

state space g of ~, such that (i) w(h) 0 a~h) = ,)h), t € R

(11) w(>In)(A) 1=0) W(A), A Em (iii) r)h)(at(A») t=+"') w(A), A E m.

w(h) is interpreted physically as the perturbed equilibrium state,

invariant under a(h), close to w for small h, and returning to

equilibrium.

Theorem 1 (Araki [6]). Each a-invariant state w of lm,R,a} ~-KMS

for a is stable for local perturbations of the dynamics. The perturbed

state w(h), contained in the normal folium of w, is given by the

convergent expansion

(11) ,ih)(A) _ W(AW(h)2 - w(WCh) with

W(h) = I + ~ (_l)n! a. (h) n=l O<s "'<6 <l ~sl

-1 -n

Remark 5. Robinson (7) has shown that expansion [11] can be rewritten

as follows in terms of the truncated expectation values

ITheorem 2 (Haag, Trych-Pohlmeyer, Kastler [8]). Assume the following

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asymptotically abelian property: there is a norm-dense, a-invariant

* subalgebra lUO of lU such that

A,B € lllo'

Let w be a state of m which is (i) a-invariant (u) stab1.e for

local perturbations of the dynamics (iii) hyperclustering of order 4

i.e., to each set Al, ••• ,Ap E lllo, p ~ 4 there are positive

with

(1.4)

Then, if w is not a trace

- either w is t3-KMS for a and some t3 € R

c and k

- or w generates a covariant representation (rrw,uw> with a one-sided

spectrum of Uw (energy spectrum).

Remark 6. The second alternative corresponds to a ground state (limiting

case t3 = 00). Hyperclustering of order 6 was used in [8] - reduced to

order 4 in [9]. Asking w to be coexisting (i.e., local stability for

the system (m,R,a} tensorized by an analogous system in an analogous

state (see [10]), one can reduce condition (iii) above to weak clustering

extremal a-invariance).

Remark 7. One may consider global rather than local perturbations: e.g.,

Definition 3 still makes sense, under relativistic locality, for h

formally replaced by h(f) = J t(X)<:Xx(h)d3X, with ax the space trans­

lations and the function f extending to 00. One can show either on

models [11] or generally via appropriate clustering assumptions (Hugenholtz,

Mebkhout and Kastler, work in progress), the validity of stability for (h )

global perturbations (tor instance far a f with f = 1 at low activity).

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Such perturbations also allow to treat media in motion [10].

II. Chemical potential (as obtained by extending from %l to ;;)

We now work with the C*-system (;;,R X G,a X 'Y}, G compact, m = 1', and investigate extensions to ~ of extremal ~-KMS states W of %l

for a. We know three methods for treating the extension problem: two

presented in [12] the second of which based on [15]), one in [13}.

Theorem 3 (Araki, Haag, Takesaki and Kastler [11]). Let {3',R X G,a X 'Y}

be a C*-system with R X G the direct product of the time axis and a

compact gauge group G and assume {:J,R,a} asymptotically abelian

(15) a,b € :1,

(for the adaptation to Fermi fields see [12]). Then

(i) each extremal a-invariant state w of m has an extremal a.

invariant extension ~ to ~. Furthermore two such extensions

are such that for some g E G.

(ii) let w be a state of m extremal ~-KMS for a and assume W

to be faithful (<p(A*A) = 0, A E %l A = 0). If <p is an extremal

a·invariant extension of w to 3, ~ is ~-KMS for a one-parameter

group of the type t ... at 'Y ~ , t ... ~t a continuous one-parameter t

subgroup of the center of the stabilizer G~ of ~

{g E G; ~ 0 'Y = <p}) g

(iii) in addition to the assumptions of (ii) take G = Tl (so that

gt = ~t for some ~ E R) and assume %l to possess a relativistic

local structure. The chemical potential ~ is characterizable as

follows: with p a localized automorphism of m and vt E %l the

6) The automorphism p of m is localized (in the region R) (see [5]) if it reduces to the identity on the algebra corresponding to the points space like to all points of R.

152

unitary cocycle relating p to its time conjugates:

(16) t € R,

one has that (a) wop is quasi-equivalent to W (b) the cocycle

Radon-Nikodym derivative of wop w.r.t. W (see [14]) is related with

V t as follows:

( ( ) inF\(f./.+c)t ( ) D wop :Dw t e Jr w v -F\t '

where c is a real constant independent of w and Jrw is the represen­

tatJ.on generated by w.

Theorem 4 (Araki, Kishimoto [13]).7) Let l;j,R X G,a: X y} be a C*-

system with G compact abelian. Set f y(g)dg, 'Y E G, and assume ....

the existence of a generating set 6 of G such that:

(A) For each y E 6 there is a norm-bounded sequence [bn } C Ey(:J)

with the properties: the a:t(bn ) are equicantinuous in t;

b~bn + bnb; ? 1 for all n; the commutators [bn,A], (b~,a:t(bn),Al,

[a:t(bn)b~,A] all tend to 0 in norm n ~oo.

Moreover, for a state w of m ~ we consider the following assumption

(B) For each y € 6, sup s (rr,,(b*b)} sup s (rr,,(b b*)] = 1, where n c ~ n n n c ~ n n

lbn } is the sequence considered in (A) and Sc denotes the central

support in If w( m)" •

Then, with E the set of one-parameter subgroups of R X G of the form

t ~a:t'Y£t' t ~ St a continuous one-parameter subgroup of G, denoting by

- ~(m) the set of states of m satisfying (B) and (3-KMS for t - a:t.'

- Ext a(3(m) extremal (3-KMS for t - a:t'

- ~ (;;:) the set of F\-KMS states of :J for a one-parameter group in L:,

7) We modified the notations of [13) for uniformity. Our :J, resp G ... m, 0:, 'Y, ~, w is their m, resp m, p, 0:, ~ and ~.

153

G - St3(J:) the subset of 'Y-invariant states in

- Ext ~(3') the set of extremal elements of

we have that

(ii) cp E St3(3') implies cp I ~ E Sf' ( ~) • Further if cp is primary,

cpl~ E Ext St3(~); if cp E ~(3'), cp = cpl~ 0 81

; and if cp E Ext sg( 3'),

cpl~ E Ext gt3(~).

(iii) one has a bijection w E Ext gt3(~) <-> cp € Ext sg(3') by taking

w = cpl~ and cp = w = w 0 8 1 •

(1v) for any W E Ext ~(~) there is an extremal t3-KMS state for

t -< Pt for some P E Z with qJl~ = W. If qJ 0 'Ygl of CVYg2

' gl,g2 E G,

are disjoint. The central decomposition of

w = W 0 81

is given by

w = J qJ 0 'Ygdg.

Remark 7. The faithfulness assumption in Theorem 3(ii) (automatic if ~

is simple) and assumption (B) in Theorem ~, are for ruling out the

occurrence of "vacuum like properties" in gauge, see [12] and [13].

Remark 8. In [2] we treat the more general case ·of an asymptotically

abelian group ~ commuting with a (e.g., space translations) and an

extremal T-invariant state of ~. We give here the version of [12] which

seems adapted to relativistic fields, sL~ce [13] seems preferable for non

relativistic models (e.g., spin systems).

Remark 9. For an investigation of the representation ITw of 3' generated

by the "average" w we refer to [13], Theorem 2, and Section III of [J2).

154

References

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Quantum Statistical Mechanics, Corom. Math. Phys, 215 (1967).

[2] R. Kubo, Statistical-mechanical Theory of Irreversible Processes.

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[3] P. C. Martin and J. Schwinger, Theory of Many Particle Systems,

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[4] M. Take saki , Tomita's Theory of Modular Hilbert Algebras and its

Applications. Springer Lecture Notes in Math., No. 128.

[5] S. Doplicher, R. Haag, J. Roberts, Fields, Observab1es and gauge

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