a non-symmetric transition probability in quantum mechanics

Vol. 17 (1980) REPORTS ON MATHEMATICAL PHYSICS No. 3

A NON-SYMMETRIC TRANSITION PROBABILITY IN QUANTUM MECHANICS

WAWRZYNIEC Guz

lnstitute of Physics, Gdansk University, Gdansk, Poland

(Received June 30, 1978)

(Revised September 8, 1978)

The purpose of this paper is to show the possibility of introducing a non-symmetric transition probability in quantum mechanics, starting from the axiomatic basis of the theory known as the “quantum logic approach”. The quantum-mechanical postulates are later reformulated in the language of the transition probability and the operations (filtering procedures) transforming pure states of a physical system into themselves, and some ad- vantages of this formulation are shown consisting mainly in resolving the old troubles connected with quantum logics (e.g., the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law).

As a consequence of the axioms assumed, the representation theorem for the logic of propositions is deduced.

1. Introduction

The main aim of this paper is to indicate the possibility of a non-symmetric transition probability in quantum mechanics. More precisely, starting from the axiomatic grounds of the quantum theory we show that there is no reason to assume apriori a symmetric form of the transition probability. The quantum-mechanical postulates are here expressed in the language of the well-known “quantum logic” approach, as this very formulation is the most appropriate one here. Incidentally we reformulate all the postulates in terms of the transition probability and operations acting on pure states of the physical system under study. Moreover, we establish a full equivalence of these two ap- proaches to quantum axiomatics, i.e. the “quantum logic” and the “transition probability space” ones (see Section 3). In comparing the “quantum logic” description of the quantum-mechanical postulates with the “transition probability space” one, some advantage of the latter has been shown, as it resolves the old troubles connected with quantum logics: the questions of the complete lattice structure of the logic, its atomicity, and the validity of the covering law in it. A special role is here attached to the so-called I&e- pen&me Postulate (see Section 4), which may be seen as a physical justification of the covering law. Also, as a consequence of the axioms assumed, we deduce here the famous Piron-Mac Laren’s representation theorem for the logic of propositions (Section 5).

386 w. GUZ

2. Postulates and definitions

The following objects are fundamental in the quantum logic approach to quantum axiomatics: the set L of experimentally verifiable propositions (called also questions, events, yes-no experiments, etc.) concerning a physical system, which we call the logic of the system (briefly, a Iogic), and the set S of states of the system. L and S are related to one another by the following (now commonly accepted) postulates:

(Al) L is an orthomodular a-orthoposet, that is, a a-orthocomplete orthocomplemented partially ordered set satisfying the following condition (called the orthomodularity): for any pair a,bEL with a< b we have b = avc for some CEL, c,< a’.

Remark : By ’ we denote the orthocomplementation in L, and the symbol v (A, respectively) is used to denote the least upper bound (the greatest lower bound, respectively) in L. When a < b’, we shall say that a and b are orthogonal and write al_b. The relation of orthogonality is, obviously, symmetric. When aJ_b, we write a+ b instead of a v 6; when a < b, we shall write b-a in place of (b’+u)’ = b A a’.

(A2) S is a a-convex set of probability measures on L.

(A3) There exists a subset P c S, whose members, called “pure states”, are in one- to-one correspondence with atoms of L. More precisely, L is assumed to be atomic, and one postulates the existence of a bijection s: P + A of the set P of pure states onto the set A of all atoms of L such that for every p E P

(9 P(s(P)) = 1,

(ii) p(u) = 1 implies a 2 s(p), u being an element of L.

The atomic proposition s(p) is called the support or carrier ofp, and it is also denoted by suppp or carrp (see [17j, [16], [I]).

Note that the name “pure state” for a member of the set P is fully justified, as one can easily check that any p from P is an extreme point of S,,, the latter being the a- convex set of probability measures on L spanned by P. (Obviously, S, c S.) In fact, as for every p E P one then has (see [7])

(p} = {p>- = {q E P: p(u) = 1 * 4(a) = 1, a E L},

one finds that p= t,ml+t,m,=~sipI (where miES0, Ii >0, tl+tz= 1; piEP,

St > 0, F Si = 1) implies pi E {p}- ‘= (p} f or each i, that is, for all i we have pi = p,

and therefore m, = m, = p, which shows that p is an extreme point of S,,, as claimed. Note that having assumed (Al) and (A2) one can deduce (A3) from the following

assumptions which are perhaps physically more plausible (see [7]):

(a) For every non-zero proposition u E L there exists a pure state p E P such that p(u) = 1.

(b) If for every pure state p E P satisfying p(a) = 1 we have also p(b) = 1, where a,bEL, then a < b.

NON-SYMMETRIC TRANSITION PROBABILITY IN QUANTUM MECHANICS 387

(c) For any pure state p E P there is a proposition a E L such that p(a) = I and q(a) < 1 for all pure states q # p.

Moreover, one can readily verify that (a), (b) and (c) together are equivalent to the following assumption which may be seen as a stronger version of (A3):

(A39 L is atomistic, i.e. L is atomic and every non-zero a from L is a least upper bound of atoms contained in a, and there is a bijection s: P --) A qf some subset P of S (whose members are called pure stutes) onto the set A of atoms of L, which sati!fies conditions (i) and (ii) from (A3).

The implication from (a), (b), (c) to (A3’) was shown in [7], thus only the converse one needs to be shown. Assume therefore (A3’), and let a E L, a # 0. Then, by atomicity, a 2 c for some atom e, hence p(a) = 1 for the pure state p defined by s(p) = e. This proves the validity of (a). To show (b) let us assume p(u) = 1 to imply p(b) = I (all p) for some a, b E L, and let e be an arbitrary atom contained in a. Then, for p = s-‘(e) we get p(a) = 1 by (i); hence also p(b) = 1 by our assumption, which leads to e < b by (ii). We thus have shown that any atom contained in a is also contained in b; hence we find a < b by atomisticity of L, which proves (b). The statement (c) is satisfied trivi- ally. as the desired proposition is then a = s(p).

DEFINITION. We say that two states m, and m, are orthogonal [S], and write m, Im,, if for some proposition a EL one has ml(a) = 1 and m2(a) = 0.

This orthogonality relation J_ is, of course, symmetric. The pair (P, I), where J_ denotes the above-defined orthogonality restricted to the set P, called the phase space of the system (compare [6], [7]), plays a very essential role in quantum axiomatics.

Let ;ld c P; define Ml to be the set of all pure states p E P such that plM (that is, p _Lq for all q E M), and write M- instead of M 11. Obviously, M c M-. If h4 = M-, we call the set IV closed, and the family C(P, I) of all closed subsets of P we call the phase geometry associated with a physical system [6]. One can easily check [6] that under set inclusion C(P, I) becomes a complete atomistic lattice with joins and meets given by

‘)/ ~~ = (U M,)- and r\ M, = (7 Mj i i j

((Mj i, being an arbitrary family of closed subsets of P) and with the orthocomplementation defined by M + M-i (ME C(P, J_)).l

The significance of the phase geometry C(P, I) has been clarified in [7] (see also [6]), where the following embedding theorem was stated:

For ever>* a E L the set P, := (p E P: p(a) = 1) belongs to C(P, I), and the correspondence a --f P, d@nes an orthoinjection oj’the logic L into the phase geometry C(P, I).

The theorem above sheds some light on some controversial postulates assumed for the propositional logic L, like the complete lattice property of L or atomisticity. However,

’ For the empty set 0 we put, by definition, 01 = P, which leads immediately to 0, P E C(P, I).

388 W. GUZ

there is a very essential property of L, which is still left without a clear physical justification, although there have been many attempts to justify it (see, e.g., [Ill, [14], and references quoted therein). This is the so-called covering law, and it is an aim of the present paper to give a clear physical significance to it.

Let m, and m2 be two arbitrary states of a physical system; one then defines the number

(ml:m2) := inf{m,(a): a E L, m2(a) = I}

and calls it the degree of dependence of ml on m2 (see [8]). Note that when m, and m2 are the usual quantum-mechanical pure states (that is,

the rays in a Hilbert space), the number (m, :m2) gives us the transition probability between ml and m2. Moreover, if ml and m, are mixed states (density operators), the number (m,:m,) coincides then with the so-called semi-inner product between m, and m2. More precisely, consider ml and m, as the elements of the Banach space T,(H) of the trace-class operators on H, H being the Hilbert space corresponding to the quantum- mechanical system under study, then (ml :m,) = [ml, m,] (see [S]), where [ * , . ] is the semi-inner product in T,(H) defined by Kossakowski [12]:

[m, , m21 := llm211tr(~~, signm,),

where 11. 11 stands for the trace norm in T,(H) defined by 1 lrnll : = tr(m m*)‘12, and

signm (m E T’(H)) is defined by

+53 signs := s sign t E(d),

-m

E being the spectral measure of M. In the more general framework of the axiomatic quantum mechanics described by

the postulates (Al), (A2), (A3) one easily finds for any two pure states p, q E P that (p:q) = p (s(q)).’ Indeed, (q:p) = inf {q(a): a E L, p(a) = I} = q(s(p)), as p (s(p)) = 1 and p(a) = 1 implies s(p) < a.

The transition probability (p:q) is here evidently non-symmetric with respect to the variables p and q.

One can easily show the following properies of the transition probability:

(i) 0 G (p:q) < 1 for all p, q E P, (ii) (p:q) = 0 iff p_Lq,

(iii) (p:q) = 1 iff p = q.

3. Covering law and the operational form of quantum-mechanical postulates

Let L be an atomic a-orthoposet, that is, we assume that for every non-zero a E L there exists an atom e EL such that e < a.

* In this very form the transition probability was defined by Bugajska and Bugajski in [ll.


We shall say that the covering law holds in L, or that L possesses the covering property

[91, if (A) for any a E L and any atom e E L there exists a v e in L,

(B) a v e covers a, when e Q a, that is, a ve 2 b 2 a implies either b = a or b = a ve.

Having assumed property (A) for L, L being an orthomodular atomic a-orthoposet, it can easily be established [9]” that the covering property (B) is equivalent to the following Jauch-Piron condition :

(C) For all a E L, a # 0, and e E A one has either e v al-a’ E A, when e 4 a’, or

e V a’ - a’ = 0, wlre~i e < a’.

Owing to the above-mentioned equivalence, one can assign to each non-zero a E L

(L being an orthomodular atomic o-orthoposet with the covering law holding in it) the mapping f, : Au (0 > + Au (0) defined by [9]:

Jo(e) := eva’-a’, e E Au(O),

which induces, in an obvious way (by using the bijection s: A c* P), the mapping E,,

from Pu {O> to Pu {O}.4 It is defined by [9]:

E *= s-lfas, 0 .

or, more explicitly,

E,,p = s-l (s(p) va’-a’),

where the bijection s is here meant to be extended onto Pu {0} by putting s(O) = 0, and thus it now transforms Pu {0} onto Au (O}.

For the properties of the transformations E, we refer the reader to paper [9]. Note that when we assume for the pair (L, P) the stronger axiom (A3’) instead of

(A3), we are then in a position to prove that the correspondence a + E, is one-one. In fact, suppose that E, = Eb for some a, b EL, i.e. that

for all p E P.

for all p E P;

s-l (s(p) va’ -a’) = s-‘(s(p)vb’-b’)

This leads, on applying s to both the sides of (3.1), to

s(p)va’-a’ = s(p)vb’-b’

hence, by applying p to (3.2), we find that

1 -p(a’) = 1 -p(b’), all p E P,

(3.1)

(3.2)

or that p(a) = p(b) for all p E P, which implies a = b by (b). Having therefore assumed (A3’) for (L, P), one can identify any proposition a E L

with E,. The mapping E, is interpreted as the filtering procedure corresponding to the proposition a.

3 For the case where L is a lattice such a statement was proved by Bugajska and Bugajski [2]. 4 Here 0 denotes the improper “pure” state (called the zero state) adjoined to P and defined as the

zero function on L. i.e., O(u) = 0 for all a E L.

390 W. GUZ

CONVENTION. For the sake of brevity we shall write in the sequel pII instead of E,p and PO instead of Pu (0).

It is convenient to extend the transition probability function onto the whole set P, by putting (0: p) = (p: 0) = 0 for all p E PO. Note then the following properties of the filters E. :

(1) p,(u) = 1, provided pa # 0 (p E PO, a E L) ; (2) any E,, is idempotent, i.e. E,” = E,,; (3) (p:p,) =p(a)jor allpEP,, and aEL; (4) (p: pa) = 0 implies pII = 0 and (p: qJ = 0 for all q E PO ;

(5) U (P: pa) = (p : aJ Z 0, t&n p. = a. Indeed, from the definition of E, we get s(p,) = s(p) va’-a’ < a; hence p.(a)

2 po (“CJ%>) = 1, as p. # 0, hence p,(u) = 1, as claimed in (1).

Next, (P : P,) = P (4~~)) = P (S(P) vu’--a’) = 1 -~(a’) = p(u), which proves (3). By using (3) and (1) one finds for p,, # 0

(p&?&7) = p,(u) = 1;

hence we get p,, = (p,,), for all p E PO, which means that E,’ = E,,. Suppose now that (p:p,,) = 0; hence p(u) = 0 by (3), which leads to s(p) < a’ and

therefore p. = s-l (s(p) vu’--a’) = s-‘(O) = 0, which proves the first part of (4). To prove the second part, let us note that s(p) < a’ implies s(p) < s(q)’ for all q E P, with q(a) = 1, which means that pIq, or that (p: q) = 0. This proves the second part of (4), as q(u) = 1 is equivalent to q = q..

Finally, let (p:pJ = (p: q.) = p(s(qJ) =p(s(q) vu’-u’) = p(s(q) vu’)-p(u); hence one gets by using (3) that p(s(q) v a’) = p(u)+p(a’) = 1, hence s(p) < s(q) v a’, which leads immediately to s(p,) = s(q,,), as s(p,) # 0. (Indeed, p(s(p,)) = p (s(p) v a’ - a’) = 1 -~(a’) = p(u) = (p: p,) # 0; hence s(pJ # 0.) Hence p. = q,,, as claimed.

Note now that the covering postulate can be expressed in terms of the properties of filtering transformations E,,, as stated in

PROPOSITION 3.1. Suppose (L, P) to be a pair sutisjying the axioms (Al) and (A3). Then the logic L possesses the covering property if and only if to each non-zero proposition a E L there corresponds an idempotent transformation E, : PO + PO (E,: p + p.) satisfying the folloGng conditions :

(i) (p:po)=p(u)jor allpEP undaEL;

(ii) (p:p.) = (p: q.) # 0 implies p. = qe.

Proof: Only the “if” part of the statement remains to be shown. Suppose that for every a E L there exists an idempotent mapping E,,: P,, + PO satisfying (i) and (ii). Properties (i) and (ii) may readily be rewritten as the following single condition:

(*> Ij’p(a) # 0, then there exists a unique q E P such thot q(u) = 1 and (p: q) = p(u), namely q = pO.


This is exactly the well-known projection postulate [3] (being an abstract version of the famous von Neumann’s projection postulate), from which condition (A) required in the definition of the covering property follows, as was shown in [3].” It remains to be shown that also condition (B) follows from the projection postulate.6 We will prove that (*) implies the Jauch-Piron condition, as we found the latter to be equivalent to (B). To do it, we shall adopt the method used in [2] for the case where L is a lattice.

Let a E L and e E A, and let e = s(p). One can assume without any loss of generality that 0 <p(a) -C 1 (as p(a) = 1 implies P = s(p) G a, and then, by orthomodularity, e vu’-a’ = e E A; and similarly p(a) = 0 leads to e Q a’; hence e vu’-a’ = 0), i.e. that p(a) # 0 and ~(a’) # 0. Applying now twice the projection postulate (*), one finds that

P(Q) z 0 = &oPq(a) = 1 & p(a) = (M) = P(+I))V

~(a’) # 0 => LP r(u’) = 1 & ~(a’) = (p:r) = p(s(r)),

hence s(q)_Ls(r), as q(u) = 1 and r(u’) = 1 imply s(q) < a and s(r) < a’, respectively, and therefore

p(s(q)vs(r)) = p(s(q))+p(s(r)) = p(a)+~(a’) = 1;

hence e = s(p) < .s(q) v s(r), which leads to

evu’--a’ < s(q)vs(r)va’-a’ = (s(q)+a’)-a’ = s(q) EA.

(To derive the last equality we have used the orthomodularity of L.) Thus, we have shown that that either e vu’- a’ = 0 or e vu’- a’ E A, which is precisely the Jauch-Piron condition.

This completes the proof of the proposition. We shall now state some facts about the orthogonality and the partial ordering in

L, L being an orthomodular atomic o-orthoposet with the covering law holding in it. (For (L, P) we assume the validity of (A3’).

PROPOSITION 3.2. For a, b E L the fOlIowing statements are rnutuully equivalent:

(i) u Ib; (ii) E, Eb = 0;

(iii) R, _L Rb, where R, denotes the range of E,.

Proof: Let us assume (i) and then prove (ii), that is one must show that (p& = 0 for all p E PO. Obviously, it is sufficient to prove that s((p,),) = 0, which follows directly

s ((PA) = s(P*) va’--a’ = (s(p)vb’-b’)va’-a’ = 0

since s(p) vb’- b’ < b < a’. We shall now show the implication (ii) =S (iii). Let us assume (ii), and let p E Rg.

Then p = pb, and therefore p. = (p& = 0 by (ii); hence, by (4) (p: qJ = 0 for all q t P,, that is, p _LR, . This proves that Rb IR,, as claimed.

392 w. em

It finally remains to show that (iii) implies (i). Assume therefore that R, IR,. To conclude that a.Lb, it is sufficient to write a and b (see [7]) as

a = //’ {s(P): PEP, P(o) = l} = v (s(p): PER,),

b = v {s(p): PEP, p(b) = 1) = ‘,/ {S(P): PE&,};

alb follows then from the fact that s(p)Is(q) for p l-q.

This completes the proof of the proposition.

PROPOSITION 3.3. For a, b E L the follon+ng conditions are mutually equivalent:

(9 a < b,

(ii) E,E, = E,,,

(iii) R, c Rb.

Proof’: Let us assume (i) and show that (P& = p. for all p E PO. One can assume without any loss of generality that p. # 0; then p.(a) = 1 (see (l)), which implies p,,(b)

= 1 by (i). But p,(b) = (p,: (pJb), and we thus get p. = (p.)* by property (iii) of the transition probability (see page 388). The implication (i) =+ (ii) is therefore shown.

Assume now (ii), and let p E R,. Then p = pa = (p,Jb, the last equality holding by (ii); hence p E Rb, which shows the implication (ii) * (iii).

To prove the last implication, (iii) * (i), it is sufficient to note that Rx = {p E P:

p(x) = 1)~ {0} for every x E L, and then apply (b) (see page 386). The proof of the proposition is thus complete.

PROPOSITION 3.4. For every a E L one has R,,, = Ri.

Proof’: Let p E R,. ; then p = p.,; hence (p: p.) = 0. Indeed, if p = po, # 0, then we have (p: p.J = 1, which implies (p: p,) = 0 by (3); if p = pp, = 0, one has also (p: pJ = 0. By using (4) we now find that (p: q,,) = 0 for all q E PO, which means that p IR,. We thus have shown that R,. E Rt;.

To prove the inverse inclusion, assume that p E Rf; then, of course, (p:p.) = 0;

hence (p: p,.) = 1, which leads immediately to p = po, E R,. . This shows that Rj; E R,,

and the proof is complete. The statements proved show that the mapping a + R, (from L to C(P,, 1)) is, in

fact, an orthoinjection of L into C(P,, I), and therefore the logic L may be identified with some sublogic of C(P,, I). Moreover, taking into account the properties of the mapping a + E,,, we see that it is an orthoisomorphism of L with some “logic” of filters F, the latter consisting of the mappings E,. Indeed, by using Proposition 3.4 we know that F may be endowed with the orthocomplementation defined by Ei := E,,., as one finds E,, to be the greatest element of F orthogonal to E,, and the rest follows directly.

Thus, we have passed from the quantum logic language to the operational description which involves the concepts of the transition probability and the transformations from P,, to PO, the latter being interpreted as the filtering procedures corresponding to the propositions from L. This is the “operational” form of the quantum logic. \

NON-SYMMETRIC TRANSITION PROBABlLlTY IN QUANTUM MECHANICS 393

Obviously, we are now in a position to translate all the axioms of quantum mechanics from the quantum logic language to the

: )) consisting of a set PO together with a real function ( : ): PO x PO --t R is said to be a transition probability space (briefly, t.p.s.) if the following requirements are fulfilled:

(TPl) 06 (p:q)< I ,jbr allp,qGP,.

(TP2) There exists p0 E PO such that (p,, : p) = (p: pO) = 0 for all p E PO.

(TP3) For every p # p. we have (p: p) = 1.

(TP4) (p: q) = 1 implies p = q.

The members of the set P,, will be called pure states, and the function ( : ) will be called the transition probability in P,,.

Note that the state p. in (TP2) must necessarily be unique, as by (TP3), p. is the unique state with (po:po) = 0. Denote it by 0 and call it the zero state. Thus the set PO is of the form PO = Pu {Oo>, where the set P consists of pure states # 0; they will be called proper pure states.

By using the transition probability ( : ) one can define the orthogonulity in the set PO; namely, we put by definition

Plq if and only if (p: q) = (q:p) = 0.

Note that axioms (TP2), (TP3) and (TP4) imply the following property of the transition probability:

(TP5) Zf (p: q) = (p:r) for all p E PO, then q = r, and similarly, if (q:p) = (r:p) for eachpEPo, then q=r.

Proof: Assume (p: q) = (p : r) for every p E PO. Two cases will be now considered. If q = 0, then obviously r = 0 = q, as r # 0 would imply, after substituting p = r, (r : q) = (r :r) = 1 by (TP3); hence q # 0 by (TP2). If q # 0, then after substituting

P = q, we find 1 = (q:q) = (q:r), hence q = r by (TP4). The second part of (TP5) can be shown in exactly the same way.

DEFINITION. A set F of mappings from PO to PO will be called a logic offirters if the following conditions are all satisfied:

(Fl) Every a E F is idempotent, that is, a2 = a.

(F2) For each a E F its range R, determines a uniquely, that is, R, = Rb implies a = b.

394 W. GUZ

(F3) If (JI:PJ = 0, then pII = 0 and (p: q,,) = (qO :p,,) = 0 for all q E PO. ’

(F5) +‘&-F~/EF~~GP~. .u+o(P:P.) + (P :Pb) = 1.

(F6) For any sequence (ai}?=, G F of pairwise orthogonal $lters’ there exists a ,filter

a E F such that

(P:p.) = y (p:p,,) i

for allpEP,.

(F7) The mappings I and ep (all p E PO), where Z stands for the identity transformation

of the set PO into itself; and e, are defined by

i

P if e,(q) :=

4 non IP,

0 if 4lP9

all belong to F.

Note that the filters b in (F5) and a in (F6) are unique owing to (TP5). We denote

them by a’ and c ai, respectively. Note also that previously the role of e,, was played

by the filters I&,~~.

in We shall now establish some facts about the orthogonality and the partial ordering F. Before doing this, however, one has to prove a lemma.

LEMMA 3.5. For every a E F we have:

(a) R, = {PEP,: P = P.>,

(b) R: = (p E PO: p. = 0} = R,. .

Proof: Statement (a) is,obvious by the idempotency of a. To prove (b), let us assume that p E Rt. This leads, in particular, to (p:pJ = 0; hence p. = 0 by (F3). Conversely, -p,, = 0, which is equivalent to (p:p,) = 0, implies by (F3)

(P?L) = (4&P) = 0

for all q E PO, which means that p E Rt.

= Tne second part of statement (b) is almost obvious: p E R,. iff p = p., by (a) iff (p:p.,)

1 iff (p:po)=Oiffp,=O.

The lemma is therefore shown. PROPOSITION 3.6. For a, b E F the following conditions are pairwise equivalent:

(i) aIb,

(ii) (p:pu)f(p:pb) < 1 for all p E PO,

(iii) R,J_R,.

’ We write, as usually, p. instead of up. * Two filters 4, b E F are said to be orthogonal if ub = ba = 0, where by 0 we denote the mapping

from PO to PO identically equal zero. Note that it belongs to F, as 0 = eo.

NON-SYMMETRIC TRANSITION PROBABILITY 1N QUANTUM MECHANICS 395

Proofi The validity of the implication (i) * (ii) easily follows from (F6), as by applying (F6) one gets

a -Lb => VDEPO (P:Pa)+(P:Pb) = (P:Pa+b) G 1.

To prove the next implication, assume (ii) and let p E R,. We shall show that pl.Rb. One can assume without loss of generality that p # 0 (as 0 J_Rb); then, since p E R, means that p = p., one gets (p:p,,) = 1; hence (p:p& = 0 by (ii), hence by using (F3) we find that (p: qb) = (qb :p) = 0 for all q E P, , or that p _LRb. This proves that R, _LRb, as claimed.

The last implication, (iii) * (i), is almost obvious, as having assumed (iii) we find pa E R, c Ri for all p E P,; hence (p& = 0 (all p) by Lemma 3.5, which means that ba = 0. By symmetry one finds that R,iR, implies also ab = 0. This completes the proof of the statement.

DEFINITION. Let a, b E F; we shall say that a is stronger than b (or that a implies b) and write a 6 b if ba = a.

PROPOSITION 3.7. For a, b E F the following statements are mutually equivalent:

6) a < b,

(ii) For every p E P, for which (p:p,) = 1 +ve have also (p:ph) = 1,

(iii) R, E Rb.

Proof: Let us assume (i), and let (p:pJ = 1, i.e., p = pO. As a < b means, by definition, that ba = a, we have pu = pb. = (P~)~ = pb; hence (p:&+,) = (p:p.) = 1, which proves the implication (i) =S (ii).

The next implication is obvious, since R, = {pePO: (p:px)= l}u(O}foranyx~F. Assume, finally, (iii) and then prove (i), i.e. p. = pb,, = (p,)b for all p E PO. But this

iS triVid Since p. E R, E Rb implies immediately pa = (p,& by Lemma 3.5 (a). As an immediate consequence of the Proposition 3.7 we get:

PROPOSITION 3.8. The following holds for F: (a) ,< is a partial ordering in F, and a < b i# (p:po) < (p :pb) for all p E PO; (b) alb iflu < b’.

Proof: The first part of statement (a) is obvious since a < b and b G a imply R, = Rb by Proposition 3.7; hence a = b by axiom (F2).

Also (b) is obvious, as alb if and only if R,l_RI, (by Proposition 3.6) iff R, c R$ = Rb. (see Lemma 3.5) iff a < b’ by Proposition 3.7.

To prove the second part of (a), let us note that a G b iff a_Lb’ (by (b)) iff (p:p.)+ + (p:pb?) d 1 for all p E PO (by Proposition 3.6) iff (p:& < 1 - (p:pb’) = (p :pb) for all p E PO. The proposition is thus proved.

Keeping in mind the results of Propositions 3.6-3.8 one can easily deduce from axioms (Fl)-(F7) assumed for F that F, the logic of filters, becomes an orthomodular a-orthoposet, that is, an orthomodular o-orthocomplete orthocomplemented partially ordered

396 W. GUZ

set, with the partial ordering defined by

a<b iff ba=a,

and with the orthocomplementation given by the mapping a + a’. The a-orthocompleteness and the orthomodularity of F are here shown in a standard

way. Namely, for a sequence {al}$il of pairwise orthogonal filters we easily find its least m cc

upper bound ‘/ ai to be the sum c ai defined in axiom (F6), and to prove the ortho- i-1 i=l

modularity, we show that a = b+ (a’+ b)‘, when a 2 b.’ Indeed, for an arbitrary p E P,, we find

hence

(P:P~.~+b)‘) 7 1-(P:P,*+*) = 1 -(P:Pd~-(P:Pb)

= (P:PJ-(P:PbL

(P:P.) = (P:Pd+(P:P@,+b,J = (P:Pb+c.*+ay)

for all p E PO and the desired result follows from Proposition 3.8 (a). Furthermore, every non-zero pure state p E P can be identified with the function

MP. . F + [0, I] defined by

m,(a) := (P:P,),

which is a probability measure on F, as m,(O) = 0, m,(Z) = 1 by axiom (F6).

In fact, suppose that mp = m4 for some p, q # 0. Then,

mP@)q = QeJ = (4:d = (4:d =

but

and m, is o-orthoadditive

in particular,

1,

(p:q) mp(e,) = (P’Pe4) = 0

i

if p nonlq, if

Plqv

and we thus find (p: q) = 1; hence p = q. Finally, one easily finds the pair (F, {mp},p) to satisfy all the axioms postulated

for the pair (L, P) in Section 2, that is, axioms (Al) and (A3’). Indeed, note for instance that for any a # 0 there exists a pure state p E P such that

mp(4 = 1, as a # 0 implies qa # 0 for some q E P, and therefore m4.(a) = (4.: qal) = (q,,:qJ = 1. This shows that axiom (a) (see page 386) holds for (F, (mp}pEP). Axiom (b) (page 386) is also fulfilled owing to Proposition’ 3.7. The last axiom, axiom (c), can also easily be checked to hold for (F, {mp}p,p). In fact, we have for p f 0

mp(ep) = (p:p,J = (P:P) = 1, and

m,(ed < 1

p Note that then bla’, as b ss n, and thus n’+ b exists by (F6); also bl(u’+ b)’ since b f a’+b, which implies the existence of b+ (u’+b)‘.

NON-SY MMETRK TRANSITION PROBABILITY IN QUANTUM MECHANICS 397

for all q # p, q # 0, as we have shown above that mq(ep) = 1 implies q = p. Therefore, we have shown the validity of all axioms (a), (b), (c) (pages 386-387), or, equivalently, axiom (A3’) for (F, {wz,,}~~~), as claimed.

It can also be easily established that ep (p E P) are the atoms of the filter logic F, and, moreover, that every atomic filter is of this form.

Moreover, we also find that F possesses the covering property since by axioms (F3) (and to be more precise, by its first part) and (F4) we guarantee the validity of the pro-

jection postulate for (F, {m,)p,p). Alternatively, one can easily establish the validity of statements (i) and (ii) of Proposition 3.1 for the pair (F, {m,},,EP); hence the desired result follows.

Summarizing the results obtained in this section, one can write:

THEOREM 3.9. Let (L, P) be a proposition-phase space structure; then there exists a transition probability space (PO, ( : )) such that L is orthoisomorphic to some logic of filters acting on PO.

Converse/J,, jbr a given logic F ofJilters acting on a transition probability space (PO, ( : ))

there exists a o-orthoposet L, coinciding actually with F itself, and a set P of probability measures on L, whose elements are in one-to-one correspondence with pure states from P,\ {0), such that (L, P) satisfies all the axioms required for a proposition-phase space structure, and therefore any $lter from F may be identified with the corresponding Sasaki projection on the logic F.

The theorem above may be seen as giving a characterization of the concept of the Sasaki projection in terms of operations (filtering procedures) acting on some transition probability space.

4. The independence postulate as the justification of the covering law

Impose the following restriction on the transition probability between p and q,,: (F8) For all p, q E PO one has

6) (p:qJ = (p:~.) (p,:q&

(ii) (q.:p) = (q.:p.) (p,:p).

The first part of this assertion was assumed as a postulate by Deliyannis [4] for the case of a symmetric transition probability, and it expresses the spontaneity of transitions between pure states, as it says (see Deliyannis [4]) that the probability (p:q.) of passing from an arbitrary pure state p to a pure state p’ = q,,, in which a occurs with certainty, is the product (independence !) of the probability (p:p@) of the occurrence of a in the state p and the probability (p.:q.) of the subsequent transition from p,, to q.. For the general case, where the transition probability may be nonsymmetric, it is reasonable to assume also condition (ii), being the “left-handed” counterpart of (i).

For reasons which we gave above, we shall call (F8) the “independence postulate”. Note that (F8) is obviously satisfied in the conventional models of classical (phase space) and quantum mechanics (Hilbert space model)-for details see [4].

398 w. GUZ

Let us note that conditions (i) and (ii) together are equivalent to the following one:

(iii) (P:PJ(p.:4) = (P:4J(qa:4), allp, 4 E P0.

Indeed, by using (ii) and next (i) one gets

(P:P.)(PAI) = (P:PlJ(_A:4.)(4o:4) = (P?A(4.:4).

Conversely, assuming (iii) and putting in (iii) qO in place of q and next pU instead of p, we find

and (P:PcJ(Pa:4a) = (P%7)Gz.%7) = (PXJ

(Po:d = (P.:P.)(P.:4) = (Pn%)Gf.:d~

which, after interchanging p $ q in the second equality, become identical with (i) and (ii), respectively.

Note that postulate (F8) includes, as a special case, a part of axiom (F3) and axiom (F4), from which the covering law follows, thus (F8) may be seen as a physical justification of the validity of the covering law in F.

We shall show that also (F2) follows from (F8). More precisely, assume for the tilter logic F axioms (Fl), (F5), (F6), (F8) and the first part of (F3) together with its -‘left- handed” counterpart, i.e. assume that

(F3’) If p. # 0, then (p:pJ # 0 and (pu:p) # 0.

Then the following statement holds, from which axiom (F2) follows as a corollary:

PROPOSITION 4.1. For a, b E F the following two conditions are equivalent:

(i) R, C &,

(ii) a = ba = ab.

Proof: SUppOSe R, E Rb. For every p E PO we then have p. E Rb; hence pII = (p& = pb., which means that a = ba.

To prove that R, C Rb implies also a = ab, we adopt here the arguments used in [4;. Let us take two arbitrary pure states p, q E PO, and write using (iii):

(P:!d(b:d = tP:Pa>h:d = (P:P~)(P=:(Pa)b)((P~)b:q)

since a = ba and (p.: (p,&) = (p.:p.) = 1 or 0. Hence, by using (iii) again, one gets

(P:4.)~4~:4) = (P:PJ(Pdl:%)(qb:d

= (&b)a) ((qb).:qb)kb?d = (P:q.b)(q.b:qb)(qb:q). (4.1)

Now two cases need to be regarded. In case qO = 0 we have also q,& = (&)a = 0 = qU,

as desired. Indeed, (qb). # 0 would imply &, # 0, and therefore for p = (qb)a one has

r.h.s. of (4.1) = ((qb)o:hk) (hh?b)(qb:d

= ((~bh?b)(~b??h

which is non-zero by (F3’), but the 1.h.s. of (4.1) = 0, a contradiction.


(4.2)

In case qa # 0 we have (q.:q) # 0 by (F3’) and (qbjo # 0, for otherwise the r.h.s. of

(4.1) is zero for allp, while the 1.h.s. is not, as, e.g., for p = qa the 1.h.s. of (4.1) = (q,,:q) # 0. Thus. in this case we obtain from (4.1) by putting p = qa and next p = qab that

and (40 : 4) = (Ya: qod Gh: al) (4/J : 4) G ka* : %) (4b: 4)

hence (%lfY:%)(%:q) = (4&4J(4a:4) G (qlI:q),

(.q.:q) = (CL7b%J)(4LJI) z 0,

and. therefore. by dividing both sides of (4.1) by (4.2) we find that

(P:4J = (P%J for all p ; hence qa = qub .

The proof that a = nb is thus complete. Now. for the converse, assume that a = bu = ab, and let p E R,. Then p = q0 = qba

= (q,J,, for some q, that is, p E Rb. This shows that R, E Rb, as claimed, and the proof

of the proposition is complete.

COKOLL4RY 4.2. R, = Rb implic.~ a = b.

5. Conclusion: the embedding theorem

The filter logic F may easily be embedded in an orthocomplemented complete lattice by forming the so-called completion by cuts ofF (see, e.g., [3] for details). Alternatively, this embedding can be realized as follows [lo]. For any M c F define W to be the set of all N E F which are orthogonal to M (that is, such that a_Lb for all b E M), and let F : = [A4 c F: M = M-Ll}; obviously, A4 E A4 IL for every M c F. Under the set- theoretic inclusion, F” becomes a complete lattice with joins and meets given by

\i” M, = (u Mj)Il and A Mj = f! M,

( (Mj t being any faiily of subse/ts of F satisfying Mt = M+l ‘for each .i), and with the orthocomplementation given by M -+ Ml.

It can easily be shown [IO] that i coincides with the usual completion by cuts of F, and that the mapping a + {u} 11 has all the properties of an orthoinjection, thus it realizes the desired embedding of F into l?

Moreover, f is shown (see [lo]) to be atomistic, orthomodular, and satisfying the covering law, and therefore our axioms (Fl)-(F7), or (Fl), (F3’), (F5)-(F7), implying the above-mentioned properties of F, are sufficient to deduce the well-known representation theorem for F ([15], [13]), and thus for F also, if, of course, we assume that f is irreducible and of the projective dimension 2 4.‘O

” Note rllat the irreducibility of P is not a severe restriction, as in case it does not hold, one can take into consideration any irreducible part of F in place. of the whole I? Note, on the other hand, that rhe irreducibility of F”can also be justified on physical grounds (see [lo]), as expressing the superposition principle for pure states.

w. GUZ

REFERENCES

[l] Bugajska K., and S. Bugajski: Rep. Math. Phys. 4 (1973). 1. [2] -: Bull. Acad. Polon. Sci., Ser. math., 21 (1973), 873. [3] -: Ann. Inst. H. Poincare’ 19 (1973), 333. [4] Deliyannis, P. C.: J. Math. Phys. 17 (1976), 653. (51 Gudder, S. P.: /. Math. Phys. 11 (1970), 1037. [6] Guz, W. : Rep. Math. Phys. 7 (1975), 313. [7J -: Ann. Inst. H. Poincare 28 (1978), 1. [E] -: Rep. Math. Phys. 8 (1975), 49. [9] -: ibid., 16 (1979), 135.

[lo] -: Ann. Inst. H. PoincarP 29 (1978), 357. [II] Jauch, J. M., and C. Piron: Helv. Phys. Acta 42 (1%9), 842. [12] Kossakowski, A.: Bull. Acad. Polon. Sci., Sk. math., 20 (1972), 1021. [13] Mac Laren, M.D.: Pacific J. Math. 14 (1964), 597. [14] Ochs, W.: Commun. math. Phys. 25 (1972), 245. [15] Piron, C.: Helv. Phys. Acta 37 (1964), 439. [16] Pool, J. C. T.: Commun. math. Phys. 9 (1968), 212. [17] Zierler, N.: Pacific J. Math. 11 (1961), 1151.

Added in proof: After the submission of this paper to Reports Math. Phys. it has been recognized by the author

that the definition of transition probability given in this paper coincides in fact wth that given some years ago by Mielnik (see Commun. math. Phys. 15 (1969), 1; ibid. 37 (1974). 221). Also the possibility of introducing the nonsymmetric tratxition probability has been indicated by Mielnik in the two papers mentioned above.

a non-symmetric transition probability in quantum mechanics

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