On joint distribution of observables for F-quantum spaces

Fuzzy Sets and Systems 39 (1991) 65-73 North-Holland

65

On joint distribution of observables for F-quantum spaces Anatolij Dvure~enskij Mathematical Institute, Slovak Academy of Sciences, Stefiinikova 49, CS-814 73 Bratislava, Czechoslovakia

Beloslav Rie~an Technical University of Liptovsk~ Mikul6g, Jablohovd 518/1, CS-031 01 Liptovsk~ Mikul6g, Czechoslovakia Received July 1987 Revised February 1989 Abstract: In an F-quantum space model for measurements of quantum mechanical observables based on fuzzy sets ideas, we solve a problem of existence of a joint distribution for a given system of observables in a given F-state of an F-quantum space. Keywords: F-quantum space; P-measure; soft fuzzy a-algebra; observable; F-state; joint distribution; F-ideal; quantum logic.

1. Introduction In the present contribution we shall continue with an application of fuzzy sets ideas to probability theory, especially, to q u a n t u m mechanics, as it has been outlined in the p a p e r [2]. In that paper, the motivation and the definition of an F-quantum space have b e e n given. A usual mathematical description of q u a n t u m mechanics is a q u a n t u m logic in a sense of, for example V a r a d a r a j a n [8]. T w o of the basic notions of q u a n t u m logic theory are state and observable. The state of a q u a n t u m logic L ( - - o o r t h o c o m p l e m e n t e d , o r t h o m o d u l a r lattice [8]) is a m a p p i n g m :L--~ [0, 1] such that (i) if 1 is the maximal element of L, then r e ( l ) = 1; (ii) if {an}~=l is a sequence of pairwise orthogonal elements of L (that is, ~ a~) = ~7=~ m(ai). a i ~-
for any A • B ( R O ,

0165-0114/91/$03.50 ~) 1991--Elsevier Science Publishers B.V. (North-Holland)

(1.1)

66

A. Dvurerenskij, B. Rieran

x (nU__lA n) =

~/ x(An) n=l

ifAnMAm=~forn=/:m, An • B(R1),

n 1> 1,

(1.2)

where A' denotes the complement of A in R~. Let x and y be two observables corresponding to random variables ~ and T/, and let m be a state. Then the expression I~(E × F) = m ( x ( E ) ^ y ( F ) ) = m ( ~ - a ( E ) fq rl-a(F)),

E, F • B(RI),

(1.3)

denotes the probability that simultaneously measured quantities ~ and 7/ lie in Borel subsets E and F, respectively. If we define a random vector T = (~, q) as a mapping from g2 into R1 × R1 = R 2 via T(og) = (~(w), r/(w)), then it is evident that ~ ( E × F) = m ( T - I ( E × F)) and # may be simply extended uniquely to a probability measure on B(Rz), called a joint distribution of x and y in a state m, and a mapping z :B(Rz)---> S defined by z := T -1 is said to be a joint observable of x and y. For quantum logic theory, the notion of joint distribution is very important, but it is well known that it may not exist [1, 6]. The necessary and sufficient conditions for existence of a joint distribution in a given state are given, for example, in [1, 6]. In the present note, within the framework of F-quantum space based on the Piasecki P-measure [4], we solve the problem of the existence of a joint distribution of a given system of observables in an F-state. We show that it always exists although a joint observable may fail.

2. Preliminaries and definitions

According to [2], by an F-quantum space we shall mean a couple (X, M), where X is a non-empty set and M is a subset of [0, 1]x satisfying the following conditions: if l(x) = 1 for any x • X, then 1 • M;

(2.1)

i f f • M, then 1 - f • M;

(2.2)

if ½(x) = ½for any x • X, then ½~ M;

(2.3)

if {fn}~=l c M, then V fn : = s u p f n • M.

(2.4)

n=l

n

In the terminology of Piasecki [4], M is a soft fuzzy o-algebra. If we define the meet A , via An fn := infnfn, then M is a distributive a-lattice satisfying the de Morgan laws with respect to a unary operation _1_ defined b y f ± = 1 - f , f E M. A non-empty subset ~ ~ M is said to be a Boolean algebra (o-algebra) of an F-quantum space (X, M) if

Observables for F-quantum spaces

67

(i) there are the minimal and maximal elements 0~, 1~ • M such that, for any f • ~ , 0~ ~ M such that

x(A') = 1 - x ( A ) ,

A • M,

(2.5)

(here A' denotes the complement of A in M),

X

Ai =

x(Ai)

ifAi^Aj=Ofori4:j,

A1,...,A,•M,

(2.6)

i=1

(x(2 Ao)= o=1x,A.,

for/*,.

We recall that two elements A and B of M are orthogonal if A ^ B = 0. Of great importance for quantum mechanics, is the case when M is a Borel a-algebra of some separable complete metric space Y, in particular, when Y = R1. Since from the descriptive theory of Borel o-algebras [3] it follows that any Borel a-algebra of a complete separable metric space is a-isomorphic to B(RI), we shall deal for simplicity only with B(R 0 observables. It is evident that if x is an M-observable (M-a-observable) of (X, M), then the range of x, that is, the set R ( x ) = { x ( A ) : A • M } , is a Boolean algebra (a-algebra) of (X, M) with the minimal and maximal elements x(0) and x(1), respectively. If x is a B(Rx)-observable (B(R1)-a-observable), we shall say just observable (a-observable). We say that two observables x and y are compatible and write x ~-*y if x(~) = y(~). It is known [2] that a mapping z : B(R2)--> M defined by

z(E x F)--x(E) ^ y ( F ) ,

E, F • B(R1),

(2.7)

may be extended to a B(R2)-observable of (X, M) iff x <-~y. By an F-state on an F-quantum space (X, M) we mean a mapping m :M---~ [0, 1] such that

m(fv(1-f))=l m (/x~/jfi) = ~ m (f/)

foranyfeM, whenever f/~< 1 - fj for i :/: ]

(2.8) (2.9)

ieJ

(where J is countable). In the terminology of Piasecki [4], an F-state is a P-measure. If (2.9) holds only for a finite index set J, m is said to be an F-content.

A. Dvure~enskij, B. Rie~an

68

3. F-ideals In this section, we deal with F-ideals and F-a-ideals of (X, M). These notions will be applied to the problem of existence of a joint distribution of observables in an F-state. Lemma 3.1. Let (X, M) be an F-quantum space. Then M0: = {f e M : f A f J- =0} = {f e M : f v f ± = 1} is a Boolean a-algebra of (X, M) with the minimal and maximal elements 0 and 1, respectively.

Proof. It is evident that 0 , 1 e M o {fn}~=l c Mo. Then O~

^

1

=

and if

n=l

fn^

feMo, then 1 - f e M o .

Now let

1

[] n=l

n=l

A non-void subset I ~ M of an F-quantum space (X, M) is said to be an F-ideal (F-a-ideal) if (i) if f e M and f ~~t is said to be a homomorphism (ahomomorphism) of M into ~ if (i) qg(0) = 0; (ii) qo(f ±) = tp(f)' for a n y f e M; (iii) tp(f v g) = qo(f) v qg(g) for all f, g • M (tp(V~=lfn) = V~=~ qo(f.),

(f.};., c M ) Then Ker ¢p:= {f • M: qg(f) = 0} is an F-ideal (F-a-ideal). A congruence relation (a-relation) on (X, M) is an equivalence relation @ on M such that (i) 3~Ogl, f 2 ~ g z imply f ~ v f E O g ~ v g z ( f n O g n , n >~1, imply V~=~f~@

Vn=l g,,);

(ii) f ~ g i m p l i e s / l O g ±.

Lemma 3.2. Let 0 be a congruence (a-)relation on an F-quantum space (X, M). Then the mapping q~:f ~-* [f ]e : = {g e M :g O f } is a ( a-)homomorphism of M onto a Boolean (a-)algebra [M]e = { [ f ] e : f e M}.

Observables for F-quantum spaces

69

Proof. In the set [M]e, the join v , the meet ^ , and the orthocomplementation

' are well-defined operations defined via

[f]e v [g]e:= [f v g]e, (for the a-case Vn=x [fn]e [f]~:=[f~]e,

: =

L g e M,

[Vn=lfn]e)

f eM.

For any f E M, we have [o1~ = 1o ^ f l e

= [o1~ ^ [ f l ~ ,

[11~ = [1 v f ] ~ = [11o v [ f ] e .

Hence, [0]e ~< [ f ] e ~< [1]e, and the mapping (p : f - + [ f ] e is that in question.

[]

L e m m a 3.3. Let I be an F-(a-)ideal of (X, M). If we put

fOtg

iff f A g ± e l , f ± A g ~ l

(3.1)

for f, g ~ M, then 01 is a congruence (a-)relation on (X, M). Proof. It is evident that f O t g

implies f " O j g ~. Now let f / O z g i , i = 1, 2. Then

(fl V f2) ^ (gl V g2) ± = (f, V rE) ^ gi L m g~ = f l ^ g i L ^g~- ^f2 ^ g i u ^g~- e l . Similarly, (gl v g2) ^ (fl vf2) ± E I. Let f e M be given. Then

ff ^ / ~ ) ^ 0 1 = f

^f~I,

0^ (/^p)=0~/,

which implies f ^ f ± 0 1 0 . In order to prove that a t is a congruence o-relation, we may use the same arguments as that for a congruence relation. [] T h e o r e m 3.3. There is a one-to-one correspondence between the F-( a-)ideals and

the congruence (a-)relations on an F-quantum space (X, M). Proof. For simplicity we prove the assertion only for F-ideals and congruence relations. Let # ( M ) and O ( M ) be the sets of all F-ideals and congruence relations, respectively, on (X, M). A mapping ~ : # ( M ) - - + O ( M ) defined via ~ ( I ) = C)j, where a t is from (3.1), is that in question. Indeed, let 114:I~ and let f ~ l l , fckI2. Then f = f ^ O ± v f ± ^ O e l l which entails fO~, 0 and f is not in a relation with 0 by O~2. Now let O be a congruence on (X, M). Define l e = {f • M : f 0 0 } . Then I e is an F-ideal. This is true, since by L e m m a 3.2, a mapping qg:f~--~[f]e = {g e M: g O f } is a homomorphism of M onto [M]e. Hence, Ker q9 = {g: (p(g) = 0} = {g: g O 0}. For any f e M we have

o = ~o(f) ^ ~o(f)' = ~o(f) ^ ~o(f l ) = q~(f ^ f i )

which entails f ^ f i e Ie-

A. Dvure~enskij, B. Rie6an

70

We assert ~ ( l e ) = ~ . In fact, f O g implies f ^ g ± e g ^ g ± G 0 , g ^ f ± O f ^ f ± O 0, consequently, f @ l e g. Conversely, let f @ / e g" Then f ± ^ g O 0, f ^ g i G 0. Hence, f@fv(f

± Ag)=(fvfz)A(f

gOgv(g±Af)=(gvg-L)^(f

A g ) O 1A f A g = f A g , ^g)C)l^f Ag=f ^g.

[]

Let I be an F-(a-)ideal of (X, M). Then by M / I we mean the Boolean (a-)algebra [M]e,, where GI corresponds to I via (3.2). Theorem 3.4. Define two relations ~o and e ~ on M as follows:

f Oog iff there are al, • • • , an • M such that { f ^ gX > ½} U { f x A g > ½} c 0 {ai = 1} i=l

(i)

and

fGoog iff there are al, a2 . . . . • M such that {f ^ g ± >½} U {f± ^ g > ½ } c 0 {a, = ½}. i=1

(ii)

Then 0o is a congruence and @o~is a a-congruence, respectively, on M. If we put Io = {f • M : f O o 0},

(3.2)

Io = {f • M : f O ~ 0 } ,

(3.3)

then Io (Io) is a proper F-(a-)ideal of (X, M), and Io = Io. Moreover, if I is any F-(a-)ideal of (X, M), then Io c L Proof. It is straightforward to verify that 0 o and 0 ~ are a congruence and a a-congruence. In order to prove that I0 and I, are proper F-ideals, it is sufficient to take into account that, for all {f/} c M, Ui {f//= ½} 4: X. It is clear that f e o g implies f@o~ g. Conversely, suppose f ~ ® g; then we may find a sequence {ai} of fuzzy sets from M such that (ii) holds. Put c = A i (aiv alL), then U {ai = ½} = U {ai v a~- = ½} ~ {c = ½} i i

so that f 0 o g. In other words, e o = Oo~ a n d / o = Io. Let now I be an arbitrary F-ideal of (X, M). If a • M, a ~<½, then a • Io and a • I . Define a N b iff a ^ b-L, a ± ^ b ~<½. If a •Io, then we can find a fuzzy set c • M, c t> ½, such that {a :/: ½} ~_ {c -- ½}. It is simple to verify that a - a ^ c, a ^ c - 0 which entails a • I. [] Lemma 3.5. Let M1 c M.

Then the minimal F-ideal, I(MO, and F-a-ideal, Iv(M1), of (X, M) generated by M1 are given respectively by I(M1) = [ f • M: there exists a c • M, c >i ½, f ^c<" 9 (ging~-)v 9 h,, w h e r e g i • M , h , • M 1 } i=1

j=l

(3.4)

Observables for F-quantum spaces

71

and Io(M 0 = {f • M: there exists a c • M, c ~ ½, f Ac<~ ~/ (giAg~-)v ~/ hi, wheregiEM, hjEM1 }. i=1

Proof. It is evident.

(3.5)

j=l

[]

Theorem 3.6. Any Boolean o-algebra is a o-homomorphic image of M for some

F-quantum space (X, M). Any F-quantum space may be a-homomorphically embedded onto some Boolean o-algebra. Proof. Let M be a Boolean o-algebra. Due to the Loomis-Sikorski theorem [7], there is a measurable space (X, S) and a a-homomorphism h from S onto M. Define M = {IA: A • S}, where IA is the indicator of the set A. Then (X, M) is an F-quantum space, and a mapping Cp(IA)=h(A) is the a-homomorphism of M onto M. Conversely, let (X, M) be an F-quantum space, and let Iv be the minimal F-o-ideal of (X, M) of the form (3.3). Defining the congruence o-relation O via

f @ g iff f A g i v f ± ^ g • l o , we have, due to Lemma 3.2, a Boolean o-algebra [M]e in question.

[]

Theorem 3.7. Let (X, M) be an F-quantum space. Then on (X, M) there is an

F-content. Proof. Let I0 be the minimal F-ideal of (X, M) given by (3.2), and let Oo be the congruence relation corresponding to I0 by Theorem 3.3. They determine a Boolean algebra M/Io as well as, in accordance with Lemma 3.2, a homomorphism q9 from M onto M/lo. The Stone theorem [7] implies that M/Io is isomorphic to some Boolean algebra of subsets of a some set. Therefore, M/lo possesses a probability content (= finitely additive measure)/~. If we define m ( f ) =/~(qg(f)), f • M, then m is an F-content of (X, M). [] Remark. We note that the problem of describing all F-states, or to prove the existence of at least one F-state on any (X, M), seems to be open.

4. Joint distribution

In the present section, we apply the results of the previous ones to the problem of existence of a joint distribution of o-observables in a given F-state. Definition 4.1. Let m be an F-state on an F-quantum space (X, M). We say that: (i) a finite system {xl . . . . . xn} of o-observables of (X, M) has a joint distribution in an F-state m if there is a probability measure/~ =/Zx~..... on B(Rn) such that

#(El x . . . x En) = m

xi(E,) ,

El . . . . . En • B(R~);

(4.1)

A. Dvure~enskij, B. Rie~an

72

(ii) an infinite system {x,: t • T} of a-observables of (X, M) has a joint distribution in m if {x,: t • a~} has a joint distribution in m for any finite non-empty subset a~ c T. We recall that a mapping m on a Boolean (a-)algebra ~t is a strictly positive probability measure if (i) m ( a ) = l iff a = l ; (ii) m ( V , a ~ ) = ~ , m ( a n ) if ai A aj = 0 for i #:j. The following lemma has importance for building up the probability calculus on F-quantum spaces. 4.2. Let m be an F-content (F-state) of (X, M). Then Im := {f • M: m ( f ) = 0}/s an F-(a-)ideal of (X, M) and M/Im is a Boolean (a-)algebra with a strictly positive probability measure rh defined by Lemma

rh(f) = m ( f ) ,

f • M,

(4.2)

where f = {g aM: m ( f A g ± v g A f l) =0}. Proof. Due to Piasecki [4], any F-state has the following properties: (i) m(f)<~m(g) i f f <-g, f , g • M ; (ii) m ( 1 - f ) = 1 - m(f), f • M; (iii) m ( f v g) + m ( f A g) = re(f) + re(g), f, g • M; (iv) fn/~f implies m(Vn=lfn) limn m(fn), fn • M, n >>-1. Therefore, Im is an F-(a-)ideal of (X, M). The canonical mapping tp :f ~ ] " is a (a-)homomorphism of M onto the Boolean (a-)algebra M/Im. The mapping rh(f) := m ( f ) , f • M, is defined well: if g • f , then, due to the property (iii) and fAg+fAg±<-l, g ^ f + g A f-L---<1, we have =

re(f) = m ( f ^ (g v gi)) = m ( ( f A g) V ( f A g±)) = m ( f ^ g ) + m ( f ^ g ± ) = m ( f Ag) + m ( f I Ag) =m(g). It is clear that rh(1) = 1 and ~(O) = 0. Let now {fn}~=l be a sequence of orthogonal elements in M/Im. Define the fuzzy sets g~ in M by /n--I

gl=fl,

g~=f~AIVgi)

\ &

,

n>t2.

It is simple to see that gi +gj<~ 1 for any i ~ej, gn e f t , for any n 1---1, and Vn g~ • V~ fn. This entails

~, rh(f~)= ~, m(f~)= ~ m(g~)= m ( V g~)/ n

n

n

Moreover, rh(f) = 1 iff re(f) = 1 i f f f = 1.

\

n

[]

Before presenting the main result of the paper, we note that, in [2], it has been proved that if {x,: t • T} is a system of mutually compatible a-observables of

Observables for F-quantum spaces

73

(X, M), then they have a joint distribution in any F-state. The proof of that result depended on the notion of a joint o-observable (for details see [2]). But there are simple examples of (X, M) with two non-compatible observables and with a non-empty set of F-states. For example, let X 4: 0, M = {c: 0 ~< c ~< ~ or 3 ~
(X, M), {x, :t • T}, has a joint distribution in any F-state m of (X, M). Moreover, there is a unique probability measure l~ on l-lt~rB(R 0 such that /z (,f~-~ :~-'(E,,) = m(,Axt(E,,)

(4.3,

for any Et • B(RO, t • o¢ and any finite subset el ~ T, where :~, :RT--->R1 is the t-th projection function. Proof. According to (X, M) and th is a .~t:B(RO--~M/I,~ via homomorphism from defined by

Lemma 4.2, Im = { f • M : m ( f ) = 0 } is an F-o-ideal of strictly positive measure on M/I,. Define a mapping E~-~xt(E), E•B(R1), for any t • T . Then ~t is a oB(Rx) into the Boolean o-algebra M/I,,. The f u n c t i o n s / ~

P'~ (,9~ n [ X(Et))= rh ( t ~ $,(Et) ) = mQAxt(Et)) for any Et E B(R1), t • cr, and any finite subset o~ ~ T, are finitely additive on whole l-lt,~ B(R1), and a-additive in any fixed coordinate. Therefore, due to [5], they may be uniquely extended to a probability measure /~ on liter B(R1) with (4.3) which proves the theorem completely. []

References [1] A. Dvure~enskij and S.P. Pulmannov~i, On joint distribution for observables, Math. Slovaca 32 (1982) 155-166. [2] A. Dvure~enskij and B. Rie~an, On joint observables for F-quantum space, Busefal 35 (1988) 10-14. [3] K. Kuratowski, Topology 1 (Academic Press, New York, 1966). [4] K. Piasecki, Probability of fuzzy events defined as denumerable additivity measure, Fuzzy Sets and Systems 17 (1985) 271-289. [5] J. Pfanzagl and W. Pierlo, Compact Systems of Sets, Lecture Notes in Mathematics No. 16 (Springer-Verlag, Berlin-New York, 1966). [6] S. Pulmannov~i and A. Dvure~enskij, Uncertainty principle and joint distribution of observables, Ann. Inst. Henri Poincard Phys. Th~oret. 42 (1985) 253-265. [7] R. Sikorski, Boolean Algebras (Springer-Verlag, Berlin-New York, 1964). [8] V.S. Varadarajan, Geometry of Quantum Theory (Van Nostrand, New York, 1968).