axioms for statistical physical theories and gl-spaces

Vol. 12 (1977) REPORTS ON MATHEMATICAL PHYSICS No. 2

AXIOMS FOR STATISTICAL PHYSICAL THEORIES AND G&SPACES

WAWRZYNIEC Guz

Institute of Physics, Gdarisk University, Gdansk, Poland

(Received April 2, 1976)

The paper contains an axiomatic formulation of statistical physical theories, which may be considered as an alternative to the famous Mackey’s approach. In contrast to the Mackey’s axiomatics, in which the so-called propositional logic appears as a basic object, our approach takes the partially ordered vector space spanned by states of a physical system as a starting point. The propositional logic appears then as a set of positive functionals on this space.

Introduction

In the present paper we formulate a general axiomatic basis for any statistical physical

theory. The basic axioms of the theory are the simplest three Mackey’s axioms [6] (the

Axioms 1-4 in the paper), however, in contrast to the Mackey’s approach, which takes

the so-called propositional logic as a basic object of the theory, in our present approach

the partially ordered vector space spanned by states of a physical system arises as a fun-

damental object. The propositional logic appears then as a set of positive functionals

on this space (see Section 3).

The mathematical apparatus used in the paper is the theory of partially ordered real

vector spaces with a norm additive on the positive cone, developed in our earlier paper [5].

The four facts are of special importance here. These are: the functional representation

of the propositional logic (Section 3) the problem of finding an appropriate norm in the

space spanned by states of the system (Section 4),the general formulation of a time evolu-

tion (Sections 7 and 8), and finally, the problem of finding an ordinary P-algebraic re-

presentation for the set of bounded observables (Section 6).

1. Basic postulates

Let 0 be the set of all observables and S the set of all states of a given physical system.

We do not answer the question what are the observables and states, but, following Mackey

[6], we accept them as primitive notions of the theory.

After Mackey [6] we assume:

152 W. GUZ

AXIOM I. There is a function P: 0 x S x B(R) 4 R, which, for$xcd A E 0 and m E S.

is a probability mca.wse on B(R).

Remark. The notation used here is standard. R denotes the real line, R, := (S E R:

,Y 3 0}, and B(R) is the o-algebra of all Bore1 subsets of R.

The number P(A, m, E), where A E 0, IFI E S and E E B(R), is interpreted as the

probability that a measurement of an observable A for the system in a state 1~ leads to

a value in a set E E B(R); the probability measure P(A, m, .) will be called the probability.

distribution of the obsovablc A irz the state tn.

AXIOM 2. [f P(A, , M, E) = P(A2, 171, E) for all m E S and E E B(R), theta A, = AZ.

The above axiom tells us that different observables must have different probability

distributions in, at least, one state.

AXIOM 3. [f P(A, III, , E) = P(A, tn2, E) f or all A E 0 and E E B(R), then m1 = mr .

This axiom says that our knowledge of the state of a physical system is complete,

when we know the probability distributions of all observables in this state, that is, when

we know the mapping

/>,,,: A + P(A, ~1. .),

which to each observable A E 0 assigns its probability distribution in ~1. Thus /I,,, trans-

forms the set 0 of all observables of the system into the set of all probability measures on B(R).

Note that the identification of 111 with p,,, is obvious from the viewpoint of statistical

mechanics, where we define the state of the system by assigning to each observable the

probability distribution of its values.

2. The GL,-space spanned by states of a physical system

The following notation will be used throughout this and subsequent sections:

V(R) = the partially ordered real Banach space of finite signed measures on B(R)

(see, e.g., [12], see also [5]);

M(R) = the set of finite measures on B(R):

P(R) = the set of probability measures on B(R).

Each state ~1, as we have seen in Section 1, induces the mapping P,,,: 0 --) P(R) defined

by /,,,,(A) = P(z4, 1~1, .),

and, by Axiom 3, the correspondence 111 + pill is one-one.

Therefore the set S of all states of a physical system may be identified with the family

of mappings p,,,: 0 + P(R), i.e. with a subset of the set V(R)‘.’

Let us denote the set V’(R)0 by W; W becomes a partially ordered real vector space,

AXIOMS FOR STATISTICAL PHYSICAL THEORIES 153

f the operations and the partial ordering in W we define in a usual way:

(x*+x&4) := x1(4+x,(A),

(s . x)(A) : = 3. x(A),

x < y :o x(A) < y(A) for all A E 0,

wherex,,x,,x,yE WandsER.

The set M(R)’ is, obviously, the positive cone of W:

M(R)’ = W+.

It can also be easily verified that W = W+ - W+ , i.e. W+ is a generating cone in W. In

fact, let x be an arbitrary element of W; define x+ and x- by

x+(A) := x(A)+,

x-(A) := x(A)-, A ~0,

where x(A)+ and x(A)- are the positive and the negative part of the signed measure x(A),

respectively (see, e.g., [12]; see also [5]). We then have x+, x- E W+ and x = .Y+ -x-,

hence in fact W = W, - W+ .

Let us now consider the subspace X of W, which consists of bounded mappings x: 0

+ V(R), that is

x:= (x. w: \/ ,‘? Il44II G K}, K= K(x)& A EO

where ]I. (I is the standard norm in V(R) (for the definition of 11. II-see [12] or [5]). X

becomes a partially ordered normed vector space, if for the norm in X we put

IMI := ~puIl44)Il.

Also X = X+-X+, where X+ is the positive cone of X.

Furthermore, since IIp,(A)II = 1 for all m E S and A E 0, also ]lprnl] = I, and thus

m --f pm is an injection of S into X, and more precisely, into X+ nS’, where S’ is the unit

sphere of X.

Consider finally the subspace V of X spanned by states of a physical system, that is

the subspace

V : = 13: ,yiPnrr : si~R,mi~S,~~= I,2 ,... )= V+-V+, i=l

where

v+ : = 5‘ ti/&: t, i

i*

;R+,/~~iES,n= 1,2,.. );

V+ is the positive cone of V.

Notice that the norm ]I . II defined above is additive on V+. In fact, if x E V+, i.e.

x = ? tiJJnli, where ti 3 0, Mi E S, then for any A E 0 iY1

lk(A)ll = (-r(A))(R) = ‘;y, ti (P,,,~(A))(R) = 2 ti, i=l isI

154 w. GUZ

hence also

Thus for each x E V+ one can write

II-VII = (-ud)m where A is any fixed observable, and therefore for any pair .r, Y E V+ we have

Il.r+Yjl = [(x+Y)(A,)I(R) = (x(A,))(R)+ (Y(A,))(R) = Ibll+ilyli.

Therefore (V, V+, II. II) E CL,.’

3. The functional form of the logic of a physical system

An ordered pair (A, E) E 0 x B(R) will be interpreted as the following proposition

(see [71, PI): “a measurement of an observable A gives a value lying in a Bore1 set E”,

and the number P(A , m, E)-as the probability that the proposition (A, E) is true for the

system in the state ~7.

In the set 0 x B(R) one can introduce the standard logical operations; the implication

+ and the negation 1 (see [7], [S]):

(A, E) + (B, F) :o P(A, m, E) d P(B, nz, F) for all n7 E S,

T(A, E) := (A, R-E).

DEFINITION. We say that two propositions (A, E) and (B, F) are equivalent, and write

(A, E) E (B, F), when (A, E) -+ (B, F) and (B, F) --) (A, E), i.e. when

P(A, 171, E) = P(B, m, F) for each ~1 E 5’.

In other words, two propositions (A, E) and (B, P) are considered to be equivalent.

if they are equiprobable in any state 1~ ES.

The relation s is, clearly, an equivalence relation in 0 x B(R). The equivalence class

of the element (A, E) will be denoted by /(A, E)I. Furthermore, in the set L : = (0 x

x B(R))/= of all equivalence classes of the relation z one can define a partial ordering

< and an involution ‘: L + L 3 by putting:

I(A, 01 G I(B, F)I :o (A, E) + (B, F),

\(A, E)I’ := Il(A,E)I.

Moreover, in L there exist the greatest element 1 = I(A, It)] and the least eIement

0 = \(A, 0)l, and, obviously. 1’ = 0 and 0’ = 1.

’ Let X be a normed real vector space which is also partially ordered, and let C be the positive cone of X. WC sly that (A’, C, I] . ]I) is a CL,-space and write (A’, C, I/ . 11) E GLo (see [51), if X = C- C ad

if x, .Y E C implies !lx+yll = ]]A/[ + 11~~11. The norm //. 11 is then called the GLo-nornz of X.

3 By an i~vulufion in a partially ordered set L we mean a mapping ‘: L + L such that

(i) a -G b (a, b E L) implicS b’ < cl’,

(ii) n” = a for each a E L


DEFINITION. The partially ordered set (L, <) with the involution ‘: L + L is called

the logic of a physical system (or the logic of the system (0, S, P)-see [7], [8], compare

also [6]).

Using the injection S 5 V one can represent the elements of L as bounded positive

linear functionals on the GL,-space I ‘. More precisely, with each proposition (A, E)

one can associate a linear functional qCa,E,: W + R defined by

qca. E)(X) : = (x(A))(E), x E w.

It can easily be shown that:

(1) q(A,E) is positive.

(2) The set {qCa,a: A E 0, E E B(R)} is total.

Note that for .Y E W,

q(A, E)W < q(A.R)(x) = (x(A))(R) = Ib’+‘f>II- hence, in particular,

q(A. E) G q(A,R,

for all A E 0 and E E B(R).

For an arbitrary x E W one has

I4 (A,E)(d = lq(A,E)(X+)-CI(A,E)(X-) < q(A,E)(.X+)+(l(A,E)(X-)

G Ilx’GOII + lb- WII (see (3.1))

= II-~(4+lI+llxCWIl = Ilx(A>II~ hence, for x E X,

I4 (A,E)b)I Q lbll~

which means that

c3) q(A,E) E x; : = x+* n x’ and j)q(A, B) II < 1forallAeOandEEB(R).4

(3.1)

If q(A, E) are regarded as functionals on the subspace V spanned by states of the system,

then for any .Y E V, x = 2 Sipm,(SiE R, mi E S) we get i=l

II n n

q(a,~,(-Y) = 2 siq(A, e)(pmi) = 2 Si * (pm,(A))(R) = y si, i=l i=l i=i’

which, as we see, does not depend on A E 0. The functional qCA,Ej: V + R, which is the

same for any A E 0, will be denoted by a. By (2) we see that d is total.4

Finally, notice that

(4) (A. E) = (By F) if and only if q(A,E) = Y(B,F) on V, and therefore the mapping

9: I(A > ml --f 4(.4E, E IO,4 E v5_ is one-one.

4 By C* we denote the set of all positive linear functionals on the partially orderedvector space (X, C). A positive functional fe C* is said to be total (see [S]), if the set [O,f] : = {g E C*: g < f} is total.

Every total functional f has the following property: .x E C and f(x) = 0 implies x = 0.

156 W. GUZ

Thus, the elements of the logic L may be identified with the functionals qCa,E,: V + R.

which will be called propositional functionals.

Let us note that the identification map q preserves also the algebraic structure of the

logic L. This is the content of the following statement:

(5) TJze set [0, d] c V; may be endowed with the partial ordering Q inherited from the

order dual (VP, VT) of the partially ordered vector space (V, V+),5 and witlt tlte involution

defined by

I-’ := d-f, .fE P,dl.

Then, tlze mapping 4: L -+ [0, d] is an iyjection of the logic (L, 6, ‘) into ([0, d], < , ‘),

that is

I(A, E)I G l(R, F)I *4(&E) G q(B,F),

4: I(AT a’ 4 &,E),

and

q(0) = 0. q( 1) = d.

4. The natural norm in the space V

Using the total functional d one can define another norm in V being additive on V+ ,

which is also closed.6 It is defined as follows:

ljxlld := inf{d(x,)+d(x,): x = .Y, -.Y,; .yl, x2 E V+).

Note that llxlld = d(x), whenever s E V+ .

(6) II * Ijd is the greatest element in the set of all norms II . (I’ in V for wlliclz tlze states

pm 7 are the elements of the unit ball K’ : = (x E V: llxll’ < I }. ln particular, for all s E V

lbll G IMId.

wlzere II . II is the norm defined on page 153.

Proof‘: Let )I + 11’ be an arbitrary norm in V such that IJp,,,lI’ < 1 for all m E S. Then

for each x E I’+ (X = 2 tipm,, ti > 0,m~S) we have i=l

IIXII’ 6 d(x) = II.dld,

hence (see [5], Thm. (17)) Ilxll’ < Ilxll” for all s E V.

(7) The norms I I . I I and I I . IId coincide on V+

Proof: It suffices to note that llp,,,l I = Ijp,,,Jl” = 1 for all m E S.

’ By definition VP: = V$- V$.

6 A G&norm /J* jJ in (X, C) is said to be rhred (see [5]), if J/xji = inffl~x,ll+lix~ll: x = XI--.Yz;

.Y, , x2 E C} for all x E X. ’ We use the name “state” for pm owing to the identification rn++p,,,.


DEFINITION. The norm ]I . IId in V, owing to its property described in (6) will be

called the natural norm of V.

Following Mackey [6] we now assume:

AXIOM 4. For each pair m, , m, of states and each number t E (0, 1) there is a state

m E S such that

P(A,m,E) = t*P(A,m,, .@+(I--t).P(A,m,,E)

for all A E 0 and E E B(R).

Using Axiom 3 we find the state m to be uniquely determined by m, , m2 and t; we shall

denote it by tm, + (1 - t)m2. Furthermore, it can easily be shown by induction that having

two finite sequences (mi} c S and {tj} E R, , where i, j = 1,2, . , n and 5 tj = 1, I=1

one can form exactly one (by Axiom 3) state m E S such that

P(A,m,E)= gti.P(A,mi,E) i=L

for all A E 0 and E E B(R). We shall denote it by 2 timi. Physically the state m = 9 timi

i=I i=l

we interpret as the mixture of states mi in the proportion t, : tz : . . : tn.

DEFINITION. If a state m E S cannot be written as a mixture of two other states differ-

ent from m, then we say that m is pure. Otherwise, we say that IIZ is mixed.

Let us note that

( miES,

i=l

which means that the set s^ : = {p,: m E S} is convex.

From the convexity of S it follows readily that S = V+ n S’, where S’ := {x E V:

]]x]ld = I}. In fact, if x E V+, x = 2 tiPmi tti 2 O)Y and J]x]ld = I, then 2 ti = I, and i=l

therefore x = p,,,, wherem = 2 timi. I=1

Thus the set S of states may be identified (by the mapping m + pm) with the (convex)

set V+ nS’, where S’ is the unit sphere of the space (V, )I . II”). Let us note, finally, that

V, = R, . i The norm I] * IId, in addition to the advantages of purely mathematical nature (see (6))

has also a clear physical meaning, since the metric induced by I] * IId in the set S of states

is equivalent to the following metrics

a(m,,m,) := inf{tE(O, 1): (I-t)m,+tm; = (1-t)m,+tm;;nl;,m;ES),

the physical significance of which is obvious.

a The metric n has been introduced by S. Gudder [3].

Note that a(m, , mz) < $ for all ?n,, mz E S.

158 w. GUZ

In order to prove the above-mentioned equivalence, we shall first show that the metric

induced in S by the norm 11 . IId coincides (up to a constant multiplier) with the metric

p(177L,n72) := infit. (I-r)-‘:tE(O, 1),(1-t)m,+W7; = (l-~)n7,+tm~;m,,m,~S}.

One can assume, without any loss of generality, that 117, # ~7~~ and let

I) Ml - /‘!?I 2 = .\‘, -.Y2, where .Y ,..\.zEV+;

then t/(x, ) - fl(.\-2) = d(p ,,,,) -d(p,,,J = 0. Let us put

d(x,) = L/(Sz) = .Y.

(4.1)

Note that s > 0. In fact, .s = 0 implies .v, = _yz = 0, hence p ,,,, = P,,,~, which leads im-

mediately to 117, = rn2.

Instead of (4.1) one can thus write

Pm, - PIUL = s . (/I,$ - /‘,n;). s > 0.

where p,,,; : = .s-‘.Y, , p,“; := s-~.Y~, and therefore

IIp,“, ---/I ,,,* IId = inf(2s: s > O,P,,,~ -/I,,,, == .s. (IT,,,; -pm;): m,, 177~ E Si-

= 2. infjt. (I--t)-‘: t E (0, I), (I-t)p,,,+tp,,,;

= (1 - f)p,,,, + fpPmL ; rn; , in; E S}

= 2.inf{r. (I-t)-I: t E (0, I), (1 -t)w7, ft177:

= (I -tt)/z-ttn7;; m;, 117; E s> = 2. !,(i,7, ) /H2).

But for all 777, , 177, E S

and thus, finally,

,o(I77,. 7772) = U(I77, , i772) (I --o(m,, /772))m1.

II/l,,,, -p”l,lld = Zo(r77,, /77?) (1 -cT(/I?, , 777,))-‘,

hence. since (7 < J. we find the metrics n and cl, where cl(/n,, 777,) := II~),,,,-/J,,,~\~~

(In L , 177? E S). to be equivalent.

5. A representation theorem for the triple (0, S, P)

Let (X, C, (/ . 11) be an arbitrary CL,-space.

DEFINITION. A mapping /I: B(R) -+ C’:’ is said to be a spectral t77~asure with values

in X, if for all s E C one has

(i) [/7(0)](s) = 0 and [/~(R)](S) = ~~.Y~~.

(ii) [“(;Q &)I (.Y) = ;$7 [/7(E7)](.\-), whenever l?j n Ek = 0 for,j # k.

Note that any spectral measure /7 possesses the following property:

E c F (E, FE B(R)) 3 /7(E) 6 h(F), i.e. /z(F) -h(E) E C’::.


DEFINITION. A family {/Ij]jeJ- of spectral measures is said to be total. if the set

{lZj(E) : j E J, E E B(R)} c X* is total.

Let US note that for any fixed A E 0 the mapping qfa,,) : E --t qca, El is a spectral measure

with values in (V, V+, II . II), the GL-space spanned by states of a physical system. More-

over, the map A + qca,.), which to each observable A E 0 assigns its spectral measure

q(A..,, is one-one, hence the set 0 of all observables may be identified with the family

(464. .) : A E 0} of spectral measures with values in V. From (2) it follows easily that the

family {q(A,.) : A e o} is total.

Summing up the results that we obtained one can write:

THEOREM. Let (0, S, P) be a triple satisfying Axioms l-4. Then, by using the mapping

m + pm the set S may be ident$ed with V+ AS’, the intersection of the positive cone V+

and the unit sphere S1 of the CL-space (V, V+ , 11 . 11) p s anned by S, the set 0 may be identi-

jied with a totalfamily {q(A,.,: A E 0} of spectral measures with vahles in V, and jnally

P(A) n12, E> = [qci~,.,Wl(p,)

for all A E 0, m E S and E E B(R).

Conversely, let (V, V+ , ) I * 1 I) be an arbitrary GL-space, S = V+ n S’, let 0 be some

total family of spectral measures with values in V, and let

P(A, m, E) := [A( for all A f 0, m E S, E E B(R). Then the triple (0, S, P) satisjes all Axioms 1-4.

6. The problem of the representation of the set of bounded ohservables

Let A be an observable, and let E be a Bore1 subset of the real line R.

DEFINITION.’ We say that E has the measure zero with respect to A, if P(A , tn, E)

= 0 for all m E S.

Let us denote by n(A) the set union of all open subsets of R that have the measure

zero with respect to A. It is easy to see that n(A) is also an open subset of R of

the measure zero with respect to A.

DEFINITION.~ The set R-n(A) is called the spectrum of the observable A, and is

denoted by SPA.

It can easily be shown that the spectrum spA of A is the smallest element in the

family of all closed subsets F of R such that P(A, m, F) = 1 for each m E S.

DEFINI-HON.~ An observable A E 0 is said to be bounded, if its spectrum spA is

a bounded set. The number sup (Is]: s E SPA} we then call the spectral norm of A, and denote

it by 1141,,. The set of all bounded observables will be denoted by Ob.

’ See Mackey [6].

160 w. GUZ

Making some physically reasonable assumptions concerning the set O,, of the bounded

observables of the system, we may come directly (see, for example, [l], Pt. 1, Ch. 2) to

the identification of Ob with a subspace of the Banach dual (V’, 11. IId) of (V, 11 . [Id), where

V is the GL-space spanned by the set S of states of the physical system.

The problem of the representation for Oh is thus closely related to that problem for V’.

Let us note that (V’, Vi) is a GM-space (see, e.g.. [5]), and hence the representation problem

may be formulated as follows:

Under what assumptions about (A’, C), a GM-space (X, C) may be identified with

a real part of some C*-algebra (or, equivalently, of some B*-algebra)?

The question above is not easy, and up to now there is no satisfactory answer to it.

There are only partial results, which, owing to their significance for quantum axiomatics,

will be now presented.

Let ~2 be a *-algebra (i.e. a complex algebra with an involution *: ~2 -+ ~2) with the

unit e, and let

H(d) = the real vector space consisting of the self-adjoing elements of SZ’,

C,(d) = the cone in H(.zZ) consisting of all finite sums of the form

7 xi” Xi) where xi E lal,

C(.ol) = {.Y E If(&): f(x) 3 0 for all f E C,(&)*}.

DEFINITION. We say that ~2 is a D-algebra, if there is a *-monomorphism j of z?

into some B*-algebra g such that

(i)j(e) is the unit of a’,

(ii) every linear functional defined on j(d) and nonnegative on j(C(&‘)) extends

to a positive functional on 27.

The following theorems hold:

(a) THEOREM (Naimark [IO]). .01 is a D-algebra if and only if the following conditions

are sa tisjied:

(i) (x E &:f(x*x) = 0 for allf E C(d)*) = (0);

(ii) for each x E _vI

sup(f(x*x): f E C(d)*, f(e) = 1 > < + co.

(b) THEOREM (Miles [9]). ~2 is a D-algebra ifand only if(H(&), C(d)) is a GM-space

with e acting as an order unit. Then the mapping j restricted to H(d) is an isometry with

respect to the GM-norm induced bJj e.

We notice also the following fact (see [9], p. 229):

(c) If ~4 is a D-algebra, then LA? has the GNS-representation as a dense subalgebra of

some closed (in the operator norm) *-algebra B c .%?(H) of bounded linear operators in

a Hilbert space H, i.e. as a dense subalgebra of some F-algebra.


DEFINITION. We say that d is a DO-algebra, if the unit e of & acts as an order unit

for (H(4, C(4).

Our final conclusion is:

It is sqficient to assume that

V’, K) = (K4, C(d))

for some D,-algebra d in order to obtain the representation theorem (c) being, in fact,

the GNS-representation.

The above postulate, although similar assumptions were accepted by physicists (com-

pare, e.g., [l], Pt. 1, Ch. 2), cannot be considered as physically necessary. It is, mathe-

matically, the regularity condition for V’, and therefore it should be regarded also as the

regularity condition for the GL-space V spanned by states of a physical system.

7. A general description of a time evolution

Let for any state m E S and any pair of real numbers t, s, where t > s, W,, S(m) denote

the state of a physical system at the moment t, whenever at the moment s the system has

been in the state m. Then, for any fixed pair t, s E R with t > s, W,, s is a mapping of the

set S of states of the system into itself. Moreover, according to the definition above, one

has for all u > t > s and m E S

W,.,(KPs(m)) = W,,,(m), hence

WU., w,,, = WU,.% for all 24 > t > s.

Thus, the family of all W,,, (t > s) forms a two-parameter semigroup of transformations

of the set S into itself. One may complete the set { W,, s : t > s} by extending the domain

of the transformation (t, s) + W,,, and adjoining to it the pairs of the form (t, t); for

each t E R we put by the definition

w*,, = 1,

where I stands for the identity transformation from S to S.

Thus, finally, the time evolution of any physical system is described by some two-

parameter semigroup { W,, s : t 2 s} (with the unit I = W,, J of transformations of the

set S of states of the system into itself. We call it the dynamical semigroup of the system.

If each mapping W,,, is one-one and onto, we write W<f = W,,, and say that the

physical system is reversible. (Otherwise, the system is said to be irreversible.) The family

of mappings Wf,J: S + S, where now (t, s) runs over the set Rx R of all pairs of real

numbers, is then a group; we call it a dynamical group of the (reversible) system.

DEFINITION. A dynamical semigroup { Wt,s: t > s} is said to be homogeneous, if

for all pairs t, s, where t > s, and for all r E R one has

W t+r,s+r = W*,,.

162 w. GUZ

Then Wt., = W~t_-s~+s.O+s = K-s,o~ and thus W,,, is a function of the difference t-s

only. Let us put by definition

v, := wt.,. t 3 0:

then V,V, = W,., W,,, = Wf+S,F (Ys,, = Wf+s.o = V,+s, and V,, = W,,, = 1, that is

{V, : t > 0} is a one-parameter semigroup with the unit I = V, . Conversely, if ( V, : t > O]

is a one-parameter semigroup of transformations from S to S such that V, = I, then

by putting

w,,,% : = K-s for t > .s.

we define a homogeneous two-parameter semigroup W,,,,: S --f S with W,., = I for all

t ER. We thus see that any homogeneous dynamical semigroup W,,.s: S --f S may be re-

placed by some one-parameter semigroup Vt: S + S such that V0 = I. Such a semigroup

will be also called the dynamical semigroup of a (temporally homogeneous) physical

system. In the sequel we restrict ourselves to the case of homogeneous dynamical semi-

groups only, that is to the case of one-parameter semigroups Vr: S + S (with I’, = 1).

The homogeneity of the dynamical semigroup { Wtes: t > .r} of a physical system is,

in fact, a reflection of the time-translational invariance of the system. Indeed, the homo-

geneity of W,,, guarantees us that the final state m(t), appearing t units of time after the

initial state m, does not depend on the choice of the initial moment .F for the preparation

of the system and the execution of an experiment:”

Suppose now that at the initial moment t = 0 a physical system is in the mixed state

172 = t, rn! + t, m, (t 1 , t, > 0, t, +tz = 1), i.e. that at the moment t = 0 the statistical

ensemble described by 1~ is a mixture of the ensembles corresponding to m, and m, in

the proportion t, : t, .

Owing to the homogeneity of the dynamical semigroup it now becomes clear that

the ensemble V,(m), being simply a time-translated ensemble m, must contain the en-

sembles V,(m,) and P’,(m,) in the same proportion t, : t,. In other words, if at the moment t = 0 a physical system is in the state

m = t,m,+t,m, (tl, t, > 0, t, + t2 = I),

then at any moment t > 0 the state of this system is

Vt(m) = t, V,(m,)+ t, V,(m,).

Our considerations about the time evolution may be now summarized by formulating

the following postulate concerning temporally-homogeneous physical systems:”

lo Such a conclusion may be not true, of course, for a system in an external field.


AXIOM 5.” Time evolution of any temporally-homogeneous physical J’ystem is described

by a one-parameter semigroup {V,: t > O> (with V, = I) of transformations of the set S

of states of the system into itself such that

K(tl mJ. + t,mJ = tl V&Q> + t2 V,(mJ

for any pair m, , ma of states and any pair t , , t, of nonnegative real numbers with a sum 1.

It seems to be reasonable to assume (see [6], Ch. 2, Sec. 3, Axiom 9) that the pro-

bability P( A, Vt(m), E) should be only little changed during a small time interval, i.e.,

strictly speaking, that

(C) For each triple (A, m, E) consisting of an observable, a state, and a Borel subset

of R, respectively, the function

t -+ P(A, Vz(m), E)

is continuous on R, .

However, although for most physical situations the above assumption seems to be

completely justified, we find the following assumption to be more plausible and, at the

same time, less restrictive:12

(M) For every triple (A, m , E) E 0 x S x B(R) the function

t -, P(A, Vt(m>, E),

where t 2 0, is Lebesgue-measurable.

DEFINITION. Any dynamical semigroup { Vt : t 2 0} satisfying the condition (M)

will be called P-measurable; analogously, every dynamical semigroup satisfying the condi-

tion (C) will be called P-continuous.

DEFINITION. A dynamical semigroup {V, : t 2 O> is said to be measurable in the mean

(continuous in the mean, respectively), if for any pair (A, m) E O,, x S the function

t + (A, V,(m)) : = i sP (A, Vr(m), ds) -cc

is Lebesgue-measurable (continuous, respectively).

The following theorem connects the two above-introduced concepts of continuity:

THEOREM. If a dynamical semigroup Vr: S + S, t > 0, is P-continuous, it is also

continuous in the mean.

The proof of the theorem above is simply a repetition of the proof of Theorem 3.1

from [4].

I1 Compare [6], Axiom 9. I2 Replacing the continuity assumption (C) by the less restrictive assumption (M) may be supported

by the fact that for classical dynamical semigroups the assumption (C), in general, does not hold.

164 w. GUZ

8. The linear extension of a dynamical semigroup

Let (V, I’,, 11 . 11) be the GL-space spanned by the set S of states of a physical system,

that is (see Sections 2 and 4) V = V+--V+,

where V, = R,.S*, i= {p,: m E S}, II . II-the greatest element in the family of all

norms 11. 11’ in V such that IIp,,,ll’ < 1 for all m ES.

The set S of states of the system, which is identified with the set S by the correspondence

m + pm, becomes then (see Section 4) the intersection of the positive cone V+ and the

unit sphere S’ of the space (V, V+ , II . II) :

s+-+i= V+nS’.

Each member Vt of the dynamical semigroup of the system may be considered, in an h

obvious way, as a mapping from S to S. Thus, in the sequel, we shall always write m

instead of P,,,; also the set S will be denoted simply by S and its elements will be called

states.

We shall now show that every one-parameter dynamical semigroup Vt : S -+ S (t 3 0)

may be uniquely extended to a one-parameter semigroup {T,: t 2 0} of positive endo-

morphisms of the space (V, V, , I I .II) preserving the norm on the positive cone I’+. We

shall first show that any mapping W: S + S which preserves convex combinations of states

may be uniquely extended to a positive linear transformation T: V + V preserving the

norm on V+ . The extension is defined as follows:

(I) T(Y) := IIYII WIIYII-‘Y> for Y E V+ - 10); (II) T(0) := 0;

(III) T(x) : = T(x,)- T(xJ for an arbitrary x E V’, whenever x = x, -x2, x1 , x2 E V+ .

Let us note that T is well-defined, i.e. that T(x) in (III) does not depend on the choice

of xi and x2 in the decomposition x = xI -x2. However, before proving this we must

show that T, defined on V+ by the formulae (I)-(H), is additive. Indeed, let x and y be

two arbitrary elements of V+ . One can assume, without any loss of generality, that X, y

# 0. Then, according to (I),

T(x+y) = Ilx+yll W(llx+ylI-‘(x+.19)

= II-y+.vII w(llx+YII-’ . ll.dl . (ll~~lI-lx)+II~+YII-l . IIYII . (IlYlr’Y))

= II~~+YlI(II.~+J~II-’ . ll4l wlxlI-‘-~)+lIx+Yll-’ . IIYII JwlYll-‘Y))~ since Ilx+yll-’ . llxll+ Ilx+yl)-’ . I(yII = 1, i.e.

T(x +Y) = T(x) + T(Y).

Let us now suppose that .Y = x, -x2 = _Y; -.Y> , where xi, _Y*, xi, XL E V+ ; then x1 +.Y;

= x;+x,, which leads to T(x,)+ 7Y.Q = T(x’,)+ T(x,), hence T(x,)- T(x,) = T(4)-

- T(.x;), which shows that T is well-defined indeed.

Let now x, y be arbitrary elements of V, and let

x = x, -X2 and Y = Yl--Yz,


where x1, x2, y, , y, E V+ . Then, according to (III),

T(x+J4 = T(x, +y1) - T(x2 +Yz) = (T(x,) - T(x2)) + (T(Y,) - W))

= T(x) + T(Y) 2

which shows that T is additive on the whole space V.

In order to show the homogeneity of T, let us first note that T(sy) = ST(Y) for all

s~Oandy~V+.Letnowx~Vandletx=x,-x,,wherex,,x,~V+;then,forall

s 2 0 we have

T(sx) = T(sx, -sxJ = T(sx,) - T(sx,) = s * (T(x,) - T(x,)) = ST(X),

and for all s < 0 also

T(sx) = T((- ) s x2-(--$x1) = (--I). (T(x,)-T(x,)) = ST(X).

The positivity of T follows directly from (I)-(II) and from the fact that W transforms

states into states. Hence follows also the preservation of the norm on the cone V+ by T. In fact, let y E V+ and y # 0 (the case when y = 0 is trivial); then, by (I),

IIT(Y)II = llll~ll Wllvll-‘~111 = Ilull, since W(lIyll-‘y) is a state.

We shall now prove the boundedness of T. Let x E V and let x = x1 -x2, where x, , x2

E V+ ; then

IITWll = IIT(TWII G IITWll+IIT(x,)lI = IIx~II+IIxAl, hence, of course,

IlW4ll < inf{llxIIl+ll~zll: x = xl-x2;xl,x2 E V+> = llxll,

which means that T is contracting: II TIJ < 1.

Moreover, if V+ # {0}, then IJT(I = 1. Indeed, let y E V+ and y # 0; then

IITII = ,,s;$, IITWll 2 Il Wlull-‘r>ll = IIIIYII-‘YII = 1, x

which together with the opposite inequality leads to II TII = 1. Note, finally, that the uniqueness of the extension of W to the linear transformation

T: V + V is obvious.

Let now {T,: t 2 0} be the collection of the unique linear extensions of elements of

the dynamical semigroup Vt: S -+ S, t 3 0. It can easily be shown that (T,: t 2 0} is

also a semigroup. It is also easy to see that T,, = I = the identity operator in V.

Summarizing now the results that we have obtained one can write:

THEOREM. Each dynamical semigroup V, : S + S (t > 0) of a temporally homogeneous

physical system may be uniquely extended to a one-parameter semigroup {T,: t 2 0} of contracting positive endomorphisms of the GL-space (V, V+ , I I * 11) spanned by states qf

166 w. GUZ

the system such that

(i) ]]T,y]] = ]]JJ]] fur all _v E V+ and t 3 0,

(ii) To = I = the identity operator in V.

Remark I. Owing to the theorem above, every one-parameter semigroup { T, : r > 0 j

of linear operators in the space V satisfying the conditions (i) and (ii) of the theorem may

be also called the dynamical semigroup.

Remark 2. The most important results of the theory of one-parameter operator

semigroups have been obtained with the assumption of the completeness of the space,

in which the semigroup acts. Notice, however, that the incompleteness of our space 1’

cannot be considered as an essential trouble, since we can always extend the dynamical

semigroup T,: V + V onto the metric completion p of V in such a way that the extended

semigroup ?, : p + c preserves the essential properties of the dynamical semigroup T, ,’ 3

that is

(a) for each t > 0 the operator T, is contracting in v,

(b) ?, = I”= the identity operator in r?.

Remark 3. For both cases of classical and quantum mechanics the states of a physical

system may be identified with probabilistic measures on the logic L of the system (see,

for example, [6] or [I I]). The set S of states of the system becomes then a subset of the

partially ordered real vector space X = X(L, < , ‘) of bounded signed measures on L

(for the definition of X, see, e.g., [5]), which is a Banach space with the norm

l/s]] = sup{s(a): a E L)+sup{-s(a): a E Lj, .Y EX.

It is also convenient to restrict ourselves to the subspace V 5 X defined by

(8.1)

v:= X+-X+,

where X+ is the positive cone of X.

In the case of classical mechanics, where L = B(S) = the u-algebra of all Bore1 sub-

sets of the phase space S of the system, one has V = V(S, B(S)), where V(S, B(S)) is

the space of all finite signed measures on B(S). For the quantum case, where L = L(H)

= the (complete) lattice of all closed subspaces of a separable complex Hilbert space H

corresponding to the physical system under study, the states of the system may be identified,

by the famous Gleason’s theorem [2] (see also [I I]), with density operators in H.14 The

set S of states of the system becomes then a subset of the partially ordered real vector

space V(H) of self-adjoint trace-class operators in H. Moreover, since the Gleason map

identifying states with density operators preserves also convex combinations of states,

I3 This follows from the fact that after the complttion V + ? the space V becomes a dense subspace of v.

l4 By a density operator (in a Hilbert space) we mean such an operator Q of trace-class, for which ~=@,e>Oandtre=l.


it may be extended, in an obvious way, to a linear isomorphism of V onto V(H). However,

if we consider V and V(H) as endowed with the norms (8.1) and ]] * 11 = tr I * 1 (the trace-

norm), respectively, then the Gleason’s isomorphism is, in general, not an isometry. This

trouble may be, however, easily overcomed. Note that in contrast to the case of the classical

space V = V(S, B(S)), 1 w lere the norm (8.1) is appropriate, since it is closed (see [5],

Example (d)), for the quantum space I’ = V(H) the norm (8.1) may be not closed. If,

however, instead of (8.1) we take its closure, then the Gleason isomorphism becomes

an isometry. Thus, up to Gleason’s isomorphism, the trace-norm in V(H) may be con-

sidered as the closure of the norm (8.1). This clarifies the significance of the trace-norm

of the space P’(H).

According to what we have said above, any dynamical semigroup Tt : V -+ V (t 3 0)

can be regarded:

(a) in the classical case-as the dynamical semigroup in the CL-space (Y(S), V+(S),

]I . II), where I I * II is the standard “sup-inf” norm,

(b) in the quantum case-as the dynamical semigroup in the CL-space (V(H), V+(H),

I( . II), where I I . 1) is the trace-norm.

REFERENCES

[I] Emch, G. G.: Algebraic methods in statistical mechanics and quantum field theory, Wiley, New York 1972.

[2] Gleason, A. M.: J. Math. Mech. 6 (1957), 885. [3] Gudder, S. P.: Int. J. Theor. Phys. 7 (1973), 205. [4] Guz, W.: Rep. Math. F’j7y.v. 6 (1974), 455. [5] -: Rep. Math. Phys., in print. [6] Mackey, G. W.: The mathematical foundations of quantum mechanics, Benjamin, New York 1963. [7] Mqczyriski, M. J.: Bull. Acad. Polon. Sci. Math. 15 (1967), 583. [8] -: Rep. Math. Phys. 2 (1971), 135. [9] Miles, P. E.: Trans. Amer. Math. Sot. 107 (1963), 217.

[IO] Naimark, M. A.: Normed rings, Moscow 1956 (in Russian). [II] Varadarajan, V. S.: Geometry of quantum theory, Vol. 1, Van Nostrand, Princeton 1970.

axioms for statistical physical theories and gl-spaces

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