Fortschritte der Physik / Progress of Physics: Band 29, Heft 8 [Reprint 2022 ed.] 9783112655986

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 8 • 1981 . B A N D 29

A K A D E M I E - V E R L A G

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EVP 1 0 , - M

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ISSN 0015 - 8 2 0 8

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Zeitschrift „Fortachritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Arthur LSsche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR • 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr.: 114420; Bank: Staatsbank der DDR, Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 1040 Berlin, Hessische StraBe 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Cesamtherstellung: VEB Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15,— I I (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/29/8. © 1981 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AN (EDV) 57618

ISSN 0015 - 8208 Fortschritte der Physik 29, 345—379 (1981)

Conditional Probability and the Axiomatic Structure of Quantum Mechanics W.GTTZ

University of Gdaúsk, Institute of Physics, Gdañsk, Poland1) Abstract The paper, summarizing the results of recent papers of the author, is devoted to a detailed analysis of postulates of nonrelativistic quantum theory appearing within two classical axiomatic frameworks, the "quantum logic" and the "algebraic approach" respectively, which are presently the two main alternatives for quantum axiomatics. The first part of the paper concerns the structure of two important sets, the set of questions (propositional logic) and the set of pure states. I t is shown how the quantum logic axiomatic scheme can be modified and improved in order to overcome the well known troubles connected with the lack of satisfactory physical justification for some postulates of this approach. In particular, an axiom system is developed, related closely to the quantum logic axiomatic scheme, in which the questions of the complete lattice structure, the atomisticity, and the validity of the covering law in the propositional logic do not appear so problematic. The particular attention is directed to the covering law, which is here obtained as a consequence of physically clear properties of the experimental procedures ("filters") associated with the quantum-mechanical propositions. In Section 1.5 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and pure filter. In the second part of the paper the axiomatic scheme presented in Part 1 is developed with the aim to analyse the algebraic structure of some subspaces of the set of simple observables. The quantum logic axiomatic framework is here shown closely connected with the Jordan-Banach algebraic scheme. In the third part of the paper, similarly as it was done in Parts 1 and 2, the correspondence between the propositions and the experimental (measuring) procedures verifying these propositions is put as the basic assumption of the axiomatization of quantum mechanics. As a consequence of this assumption, there is established the structure of a real Jordan-Banach algebra in the set Ob of the bounded observables associated with a physical system, so that we can apply the GNS representation theorem proved recently for such algebras by ALFSEN et al. (1978) to obtain the Hilbert space (or, to be more precise, the C*-algebraic) representation for Ob.

Contents Introduction Part 1 : The Structure of the Sets of Questions and Pure States 1.1. Quantum logic and phase geometry 1.2. Covering law and projection postulate !) 8 0 - 9 5 2 Gdaúsk Ul. Wita Stwosza 57 1

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 8

346 349 349 352

346 1.3. 1.4. 1.5. 1.6. 1.7.

W. Gtrz Superposition principle Conditioning of pure states. The physical interpretation of the covering law Filters on a transition probability space: an alternative to quantum logic Other axioms implying the covering law Bibliographical and historical remarks

354 355 356 358 359

Part 2: The Structure of Some Subspaces of the Set of Simple Observables • 360 2.1. The metric properties of the set of states. The natural norm 360 2.2. Conditional probability on the space of tests 362 2.3. The case of the symmetric transition probability. The Jordan structure in the spaces (Lf), (L0), and U 365 2.4. Bibliographical and historical remarks 366 Part 3: The Structure of the Set of Bounded Observables 3.1. Basic axioms and definitions 3.2. Linear and order structure in the set of bounded observables 3.3. Conditional probability 3.4. Compatibility of filters. The spectral duality between 0b and V 3.5. The Jordan structure of the space 0 b of bounded observables 3.6. Bibliographical and historical remarks

366 366 370 372 373 374 375

Appendix: Some properties of the partially ordered vector spaces

376

References

378

Introduction We present here a paper (divided into three parts, which will be referred to as Part 1, 2, and 3, respectively) which is aimed to analyse in detail the postulates of nonrelativistic quantum theory appearing in two well known axiom systems, being presently the two main alternatives for quantum axiomatics, which are known today under the name "quantum logic" and "algebraic approach", respectively. However, the purpose of the paper is not only to improve and complete the axiom systems mentioned above. The paper has a synthetic character, and it unifies the results of the previous papers of the author into the one, logically closed, whole, so that in such a way one obtains a review of our previous results in the domain of quantum axiomatics. There are at least three reasons for writing this paper: 1. A formulation of the axiom system (or systems) free of the usual troubles which plague the two "classic" axiom systems mentioned above. 2. A formulation of the model, being less orthodox that the standard Hilbert space model. 3. To make a deeper understanding of the mutual interrelations between the classical and the quantum mechanics. Now we shall describe the points 1—3 above with more details. It is well known that within both the "classic" 'axiomatic frameworks mentioned above (i.e. the quantum logic and the algebraic approach, respectively) there is a possibility to determine partially the ordinary quantum-mechanical formalism based on the theory of a complex Hilbert space, but it is also well known that these axiom systems are plagued by difficulties which still remain to be solved. For instance, it still remains to be proved that the coordinatizing division ring appearing in the representation theorem for the quantum logic is the real, the complex, or the quaternionic number field. Some results in this direction were obtained by ZIERLER [66, 67], CIRELLI and COTTA-RAMUSINO [9, 10], and others, but the assumptions that have been imposed to obtain the desired result

Conditional Prot ability

347

seem to be extremely unphysical. The other, more serious question concerns the physical justification of the complete lattice structure of the propositional logic and the other requirements characteristic to this axiomatic approach, the atomicity of the logic and the validity of the covering law in it. The algebraic axiomatic scheme inaugurated by the Jordan algebra approach of J O B D A N , VON N E U M A N N , and W I G N E R [35, 36, 37, 38, 50] and lately modified and developed by SEGAL [59, 60] has perhaps more serious defects than the quantum logic. The axioms are here formal rather than physical (for instance, there is no physical justification for assuming the distributivity of the Jordan product) and this is, in the author's opinion, the main weakness of the algebraic approach. The other serious difficulty was, up to recent days, the lack of any representation theorem for infinitedimensional Jordan algebras, which clearly is the case of the algebra of quantum mechanical (bounded) observables. Now this difficulty is resolved by the GNS representation theorem proved for real Jordan-Banach algebras by A L F S E N et al. [4]. It is clear that in order to overcome the requirement of the finite dimension of J O B D A N et al. it was necessary to introduce certain topological assumptions for algebras of observables, and this observation was the starting point of the SEGAL'S axiomatic scheme [59, 60]. However, the absence of any representation theorem for an abstract (nonassociative) Segal algebra makes his main assumption identifying that algebra with the real part of a _B*-algebra purely ad hoc (note that there are known many examples of Segal algebras not coming from a _B*-algebras — see, e.g., SHEBMAN [ 6 1 ] , LOWDENSLAGER [ 4 0 ] ) , similarly as ad hoc was the famous MACKEY'S axiom [42] identifying the logic of a quantum-mechanical system with the lattice of closed subspaces of a complex Hilbert space. Therefore, the question how to deduce from physically motivated axioms imposed on an abstract Segal algebra that the latter consists of the self-adjoint elements of some 5*-algebra still remains open. The two "classic" axiom systems described above have been developed and improved by both mathematicians and physicists, but as long as we are within these systems, the difficulties mentioned above still remain without a satisfactory resolution. In particular, much attention has been directed to the justification of the lattice assumption of the quantum logic approach, however, the general conclusion is that there is no empirical basis supporting it (an excellent discussion of this problem can be found, for example, in the papers by MAC L A B E N [43] and SRINIVAS [62]). Nevertheless, it can be shown that the propositional logic is a lattice under some additional assumptions, among them the most remarkable is perhaps the postulate that for any two bounded observables there exists its sum (for details, see MAC L A B E N [43] or G U D D E R [ 1 6 ] ) . However, this additional assumption is in fact beyond the scope of the quantum logic approach (it is characteristic to the algebraic axiomatic scheme, where the basic object under study is the family of the bounded observables); moreover, there still remains to be solved the question of the completeness of the propositional logic and the question of the physical justification of the other assumptions (atomicity, covering law) of the quantum logic approach. But it should be noticed at this moment that there is a possibility to develop another axiom system, closely related to the quantum logic, in which the questions of the complete lattice property and atomicity do not appear so problematic, because they are solved by a suitable extension of the propositional logic (see BTJGAJSKA and B U G A J S K I [9], Guz [20,24]). As regards the covering law, one must say that although many attempts have been made to justify it (see, e.g., J A U C H and P I E ON [34], OCHS [51], BTJGAJSKA and B U G A J S K I [5], Guz [25, 26, 27]), the covering law is still left without a satisfactory empirical justification as long as we remain within the conventional quantum logic axiomatic scheme. In Part 1 we present the results of our recent paper (Guz [25]), in which the covering law has been obtained as a consequence of physically clear pro1*

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W. Grrz

perties of the experimental procedures ("filters") corresponding to the quantum-mechanical propositions. The correspondence between the propositions and the experimental procedures (measurement processes) verifying these propositions has been put by the author as the basic assumption of his axiomatization of quantum mechanics. As a consequence of the application of this assumption to the algebraic-type axiomatics (see Part 3) it has been established (Guz [25]) the structure of the real Jordan-Banach algebra in the set Ob of bounded observables associated with a physical system. The derivation of such a structure is of great importance for quantum axiomatics, because it has recently been proved the GNS representation theorem for Jordan-Banach algebras (ALFSEN et al. [4]), which says that every real Jordan-Banach algebra A has a unique norm-closed Jordan ideal J such that A ¡J has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every irreducible representation of A not annihilating J is onto M3a, the exceptional Jordan algebra of 3 x 3 hermitean matrices over the octonions. It should be emphasized at this moment that up to recent days the possibility of representing Ob in the exceptional algebra M3S has been regarded as a pathology. Today, M38 is considered equally good as the other "well-behaved" Jordan subalgebras of the algebra B(H) of bounded operators on the Hilbert space H, and this is connected with the possibility of applying M3S to the description of colour phenomena in the elementary particle physics (see, for example, GURSEY [31], G U N A Y D I N [SO], and references quoted therein). The Jordan-Banach algebraic scheme is, with no doubt, an important extension of the orthodox Hilbertian quantum mechanics and, at the same time, of the ordinary algebraic axiomatics based upon -B*-algebras. It is remarkable that we may come to the JordanBanach algebras starting from the quantum logic approach, and this is shown in Part 2 of the paper. So, the quantum logic approach, as modified in Part 2, is here shown intimately connected with the Jordan-Banach algebraic scheme, and in this point we are following the pioneering work of GTTNSON [J5], where the connections between quantum logic and Jordan algebras were established for the first time, but, unfortunately, some unprecise statements of Gunson's work (this concerns mainly the section 4 of his paper) made several important conclusions of this paper incorrect. In Part 2 we show how Gunson's results can be improved and generalized; in particular, we establish the structure of the real Jordan-Banach algebra in the closed linear span of the set consisting of the atomic propositions and the greatest element of the propositional logic. As regards the new results and the new concepts introduced into the quantum axiomatics and leading to a deeper understanding of the connections between the classical and the quantum mechanics, the following three must primarily be mentioned, by which both the similarities and the differences between these two theories become more clear: a) the notion of the superposition of pure states and the superposition principle; b) the notion of the transition probability between pure states; c) the notion of the filter associated with a proposition. All the three concepts mentioned above appear trivially in classical mechanics, while in quantum theory they play the fundamental role. In Section 1.5 of Part 1 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and filter, the latter being defined as a transformation of the set of pure states into itself.

349

Conditional Probability

Part 1 The Structure of the Sets of Questions and Pure States

1.1. Quantum logic and phase geometry The primary object of the quantum logic approach to quantum axiomatics is the set L of experimentally verifiable propositions (also called questions, events, yes-no measurements) concerning a physical system, called the logic of the system or the propositional logic ( M A C K E Y [41, 42], M 4 C Z Y N S K I [45, 46]). The states of the physical system and the physical quantities (observables) appear then as secondary objects constructed over L; namely, the set S of states is a cr-convex order determining subset of the set of all probability measures on L, while the set 0 of observables consists of tr-homomorphisms from B(R) to L, where B(R) stands for the 0: —td £td). Note that 1, the greatest element of L, is here identified with the order-unit functional d, that is, d = qt. It can easily be verified (Guz [22]) that the mapping q:L [0, d] defined above is an orthoinjection of the propositional logic (L, ') into ([0, d], '), the latter endowed with the partial ordering inherited from the order dual (Vp, F+* )6) and with the involution / -»• /' = d — f , where / 6 [0, d]; moreover, for an arbitrary a £ L one has ||ga]| sS 1. Finally, the set O of observables whose members are cr-homomorphisms from B(R), the tr-algebra of Borel subsets of the real line R, to L, can be identified with a total family of positive-valued maesures over F (for details, see Guz [22]). So, we come in this way to the conceptual scheme characteristic to the so-called "operational approach" to quantum axiomatics, in which the basic concepts are the partially vector space spanned by states of a physical system and the operation on this space. Remark: The theoretical scheme described above, which is based on the identification: S=

V+ n S1,

where F+ is the positive cone of the complete base-norm space (F, ||-||), S1 stands for the unit sphere of the space (F, ||-||), and O is a total family of positive-valued measures over F, can directly be obtained from the well known MACKEY'S axiomatics [42]; moreover, this scheme is in fact equivalent to the MACKEY'S axiomatics (Guz [22, 28]; see also Part 3 of our paper). 2.2. Conditional probability cm the s*pace of tests Here we shall assume, as in Part 1, the validity of axioms (Al), (A 2) and (PP) for the pair (L, P), and we shall use the following notation: (L) = the linear span (in V) of the image q(L) of the propositional logic L under the canonical injection q:L-+ [0, d] £ V. (Lf) = the linear subspace of V' spanned by the set q{Lf), where Lj denotes the set of all finite elements') of the logic L; clearly, (Lf) = (A(L)y (L0) = the linear span (in V) of the set q(L0) with L0 = Lf u Lc, where Lc denotes the set of all co-finite elements7) of L\ obviously, (L0) = (A{L)) + Rd. U = the metric completion of (L0), i.e. the norm-closure of (L0) in V. In the remainder of this paper the propositions a(i L will be identified with their canonical images q„ e [0, d], so we will simply write "a" in place of "qa". In particular, we shall write 1 instead of d. The elements of the vector space (L) will be called, after G U N S O N [J 0) has been verified to be true, then the system is necessarily found to be in the pure state pe = s _1 (e). 8 ) More generally, if the initial state of a physical system is given by the mixture m = E ¿¡pi, where pi £ P, ij > 0, E h = 1> a n d if the atomic proposition e Ç L is verii i fied to be true, then the subsequent (unnormalized) state of the system is clearly (see (2.2) below) E tiPePi = E tiPiie) = i i where t = E hVi(e)> s o that the normalized final state is again s _1 (e). i We thus see that the final state of a physical system does not depend here on its initial state m (we have in fact two possibilities for the final state: s _1 (e) when m(e) > 0, and 0 when m(e) = 0), and this is the reason why we often say that "the pure test L (or, to be more precise, the pure filter Ee corresponding to e) prepares the physical system in the pure state p = s _ 1 (e)." Every convex combination u = E h e i > 0> E h = P u r e tests et- is customarily considered as a mixed test answering the question "Is the physical system in the mixed state m = E hVu where p, = s _ 1 (e,)?". More generally, each linear combination i u = E f i e i with positive coefficients being not subjected to any additional restriction, is «

8

) This is, clearly, a special case of the projection postulate (PP).

2*

364

W. GFZ

meant as an unnormalized mixed test preparing the physical system in the unnormalized mixed state x = £ t^ with the intensity ||x|| = JJ ti, where as before p^ = s~1(ei). i

i

We then alternatively say that the physical system is in the pure state p{ with the probability tJJJ tk. Ik

Now suppose that the physical system is initially in the unnormalized mixed state m — J J tifi, ^ > 0, prepared by the test u = i.e. that the system is initially i

i

found in the pure state p{ = 3 _1 (c i ) with the probability s{ — t¡JJJ tk. Suppose next that after a measurement performed on the system the proposition a £ L has been verified to be true, so that we find as the possible final states of the system the pure states Eapit occuring clearly with the probabilities a/ = SiiPiiEaPrijE sk(Pk-EaPk) = tiPi(a)j£ tkpk(a). In other words, the final (unnormalized) mixed state of the system is m' = £

t

hpiia) EaPi,

(2.2)

and therefore the (unnormalized) mixed test preparing the system in the state (2.2) can be written as «' = E hViifl) s{EaVi) i

= Z hQoei• i

We shall denote this mixed test by @0u. Thus we have « = E hQifii, i

and since for every atomic proposition c £ L we have $ 0 e = Qae, the equality above can be rewritten as

so we have obtained in such a way an affine9) (i.e. additive and positively-homogeneous) extension of Qa onto the cone (Zy)+ = j ^T Mi k ^ 0, e; € A(L), n = 1, 2 , . . .J. Summarizing these heuristic considerations we arrive at the following postulate (Gira[2i]): (Bl) Every Qa(a £ L) can be extended to an affine mapping Clearly, 0 a can easily be extended to a linear mapping Ta:(Lj)

{Lf)+

(Lf)+.

(Lf) by setting

TaU = OaMl — 0a«2

whenever u = uy — u2, where ult u2 6 (Lf)+10). We shall refer to Ta as to the dual filter (or the dual conditional probability

associated with the proposition a £ L (Guz [25]).

mapping)

) By an affine mapping we usually mean a map which preserves convex combinations; here, however, by affine is meant a map (defined on a cone) being additive and positively homogeneous. 10 ) It is an easy matter to check that Ta is well-defined.

9

Conditional Probability

365

2.3. The case of symmetric transition probability. The Jordan structure in the spaces (Lf), (L0) and U It is a common belief of physicists that in the microworld there is no reason, as yet, for assuming the asymmetry of the transition probability between pure states, so that, according to this conviction we accept as a postulate the following (GTJZ [ 2 5 ] ) : (B 2) For each pair p, q of pure states we have (p:q) — (q'p). Axiom (B 2) has several interesting consequences, among them the most important is perhaps that the space TJ, defined as the norm-closure of (L0) = (Lf) + -Rl in the orderunit space (F', 1), becomes then a distributive Segal algebra. More precisely, the following statement holds (GTJZ [28]): If (L, P) satisfies axioms (Al), (A 2), (PP), (B1) and (B2), then the pseudoproduct o can be extended to a commutative product on TJ such that (TJ, o, 1) becomes a distributive Segal algebra with 1 acting in it as the unit. Now we shall briefly describe the way of introducing the structure of distributive Segal algebra in TJ. By using the product o we define the squaring operation u u? in (L0) by u2 = u o u; obviously, no v = l/4((w + vf — (u — v)2) = l/2((w + v2) - u2 - v2). It is not difficult to verify (GTJZ [25]) that for all u,v£ (L0) we have (i) iitt'H = IMP (ii) I]«2 - v2|| max (Mi, |H|) (iii) ||«ot>|| iSIMHMI. Moreover, it can easily be shown that the usual rules for operating with polynomials in a single variable are here valid, i.e. that if f , g, h are polynomials with real coefficients such that f[g{t)) = h(t) for all real t, then f(g(ufj — h(u) for all u € (L0), where by definition /(«) = s 0 l + £ skuk if f(t) = JJ sktk, and uk is defined inductively as uk — u Finally, by (iii) the product o is easily seen to be norm-continuous, so it can be extended by continuity to U = (L0), and U is then readily shown to be a distributive Segal algebra with respect to this product and with 1 acting as the unit element in it. At the same time, being a subspace of the order-unit space (F', ||-||, 1), (TJ, ||-||, 1) becomes an order-unit space too. If we additionally assume one of the Pool's axioms on conditional probability, namely (POOL [54], see also Part 1, Section 1 . 6 ) : (B 3) If a ^ b and p(b) > 0, where a, b £ L, p E P, then pb(a) = p(a)/p(b); then we are in a position to prove a stronger result (Gtrz [28]): U = (L0) endowed with the product o and with the order-unit norm inherited from V, where V' is the base-norm space spanned by states of a physical system, becomes a real Jordan-Banach algebra. Remark: In course of deriving the Jordan structure in U = (L0) we have shown, by the way, that (Guz [25]): (a) (Lf) endowed with the product o and with the order-unit norm inherited from V is a normed real Jordan algebra; moreover, if 1 £ (Lf), then 1 € Lf. (b) (L0) is a normed real Jordan algebra (with respect to the product o and the same norm). The derivation of the structure of a real Jordan-Banach algebra in the space TJ on the basis of a set of more primitive assumptions is of great importance for quantum axioma-

366

W. Grz

tics, because it has recently been proved the GNS representation theorem for such algebras (Alfsen et al. [4]), which says that every real Jordan-Banach algebra A possesses a unique norm-closed Jordan ideal J such that A\ J has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every irreducible representation of A not annihilating J is onto MSB, the exceptional Jordan algebra of 3 X 3 hermitean matrices over the octonions. We shall close this section with a remark about the physical interpretation of the axiom (B 3), which is in fact very simple. The number pb(a) = (Ebp) (a) gives us, as we know, the conditional probability that the " e v e n t " . a £ l will occur, provided the "event" b £ L was found to occur for the system being initially in the pure state p, and therefore the number p(b) pb(a) — (Pbp) {a) gives us the probability that the event b and then a will occur, provided the initial (pure) state of the system was p. It should be emphasized at this moment that the order in which b and a are expected to occur is very essential, because we in general have (Pbp) (a) #= (P„p) (&). The axiom (B 3) tells us that if a ^ b, then this probability does not depend on the order of occurrence of a and b: p(b) pb(a) = p(a) = p(a) pa(b) (here we have used the fact that pa(b) = 1, whenever a ^ b), and this can equivalently be rewritten as follows (B 3') a 52 b^Vp,p\\PaPbp\\ = \\PbPap\\ = \\Pap\\ or in the form (Guz [28]) (B 3") a ~ 6

\/nP\\PaPbp\\ = \\PbPap\\.

The latter means that for any pair of compatible propositions a, b 6 L the intensity of each pure beam passing on through the filters Pa and next Pb does not depend on the order of the filters. Therefore, we have obtained here an interpretation of the Pool's axiom (B 3) in terms of intensities of pure beams. 2.4. Bibliographical and historical remarks The connections between quantum logics and Jordan algebras were shown, for the first time, in the work of Gttnson [15] ; however, unfortunately, some unprecise statements of Grunson's work (mainly, in Section 4) made several important conclusions of this paper incorrect. Gunson's results were corrected and generalized in recent paper by Gtjz [28], where it has been established the structure of the real Jordan-Banach algebra in the closed linear span of the set consisting of the atomic propositions and the greatest element of the propositional logic. Part 3

«

The Structure of the Set of Bounded Observables

3.1. Basic axioms and definitions The mathematical structure of the set of bounded observables can obviously be investigated within the framework of the quantum logic axiomatic scheme; here, however, we should like to formulate a more general approach, being in fact a continuation of the well-known Mackey's axiomatics [ál, 42], but in final count belonging rather to the class of axiomatics of an algebraic type.

367

Conditional Probability

Let O be the set of all observables, and S — the set of all states of a given (but arbitrary) physical system. We do not answer the question what are the observables and states but instead accept them, following M a c k e y [41, 42], as the primitive concepts of the theory we will build up. After M a c k e y [42] we assume (B denotes here, as usually, the real line, R+ its nonnegative part, and B(B) stands for the cr-algebra of all Borel subsets of B): Axiom 1

There is a function p\ 0 x S x B(B) -> B+ which, for fixed 4 e 0 andTOe S, is a probability measure on B(B). Axiom 2

If p(Au m, E) = p(A2, m, E) for all m € 8 and E 6 B(B), then A1 = A2. Axiom 3

If p(A, mx, E) = p(A, m2, E) for all A 6 0 and E £ B(B), then m1 = m2. Axiom 4

For each sequence {TO¿}~J of states and each sequence {ijl^j of positive real numbers oo satisfying £ t¡ = 1 there exists a state m € S such that 1= 1

p(A, m,E)=Z

oo

i=1

t¡p(A, m¿, E)

for all A £ 0 and E 6 B(B). The physical interpretation of axioms 1—4 introduced above is standard. The number p{A, m, E) is interpreted as the probability that a measurement of an observable A for the physical system being in a state m yields to a value in a Borel set E. The probability measure p(A, m, •) is called the probability distribution of the observable A in the stateTO.Axiom 2 says that probability distributions associated with different observables must differ in, at least, one state. Axiom 3 tells us that our knowledge of the state of a physical system is complete if we know the probability distributions of all observables in this state, so that each state m can be identified with the mapping pm:A ->p(A,m,

•),

which with every observable A £ 0 associates its probability distribution in the state m. Finally, the stateTOdefined in Axiom 4, being uniquely determined by the sequences {m¿}»~i and {ijJ^j, is customarily interpreted as the mixture of the states m¡(i — 1, 2,...) 00 in the proportion í : í : . . . , and denoted by JJ í ¿ t o ¿ . The definition of the "purity" of j=i a state is now standard: TO is said to be pure if it cannot be written as a nontrivial mixture of two other states; otherwise we call m mixed. An ordered pair (A,E) £ OxB(B) is customarily identified with the experimentally verifiable proposition ( M 4 . C Z Y Ñ S K I [45, 46]) which says that " a measurement of an observable A yields to a value in a Borel set E", and then the number p{A, m, E) is interpreted as the probability that the proposition (A, E) is true for the system in the state to. In the set O X B(B) one can define two logical operations, the implication and the negation respectively ( M ^ c z y n s k i [45, 46]): 1

2

(A, E)

(B, F) o VmesP(A, TO, E) = p(B, TO, F), (A, E) — (A,B

\

E).

368

W. Guz

We say that propositions (A, E) and (B, F) are equivalent, and write (A, E) ~ (B, F), if (A, E) -> (B, F) and (B, F) (A, E), i.e. if p(A, m, E) = p(B, m, F) for all m € 8. In other words, two propositions (A, E), (B, F) are considered equivalent if they are equiprobable in all states. The relation ~ defined above is, clearly, an equivalence relation in the set 0 X B(R), and the set L — [0 X of the equivalence classes of this relation, called the logic of a physical system or the logic of propositions (MACKEY [42], M^CZYNSKI [45]), is shown to be a partially ordered set with an involution, provided we define (here |(A, E)| stands for the equivalence class of the proposition (A, E)) : 1(4,^)1^1(5,^)1 .iff

(A, E) -> (B, F),

\(A,E)\' = \-^{A,E)\. Moreover, L has the greatest element 1 = |(A, 22)1 and the least element 0 = |(A, 0)\ (A being an arbitrary observable), and obviously 1' = 0 and 0' = 1. The elements of the logic L are called propositions or, after MACKEY [42], questions. We shall say that two propositions a = \(A, E)\ and b — \{B, i*)[ are mutually exlusive or orthogonal, and write a _L b, if a ^ b'. Note that the relation J_ is obviously symmetric, and under some additional assumption it can be shown that when restricted to L \ {0), is also irreflexive, i.e. a ¿ a whenever a d L\ (0). We shall use the following notation: = the partially ordered real Banach space of bounded signed measures on B(R) (see, e.g., YOSIDA [65]), M+ = the positive cone of M consisting of the bounded (nonnegative) measures on B(R), Mp = the convex set of probability measures on B{R). M

The set 8 of all states of a physical system, after we identify it with the family of mappings pm: 0 Mp, becomes a subset of the partially ordered real vector space M°, where M° stands, as usually, for the set of all mappings from 0 to M; the vector space operations and partial ordering are defined in M° in an obvious way. The set M+° is, clearly, the positive cone of M°, i.e. M + ° = (M°) + ; moreover, M + ° generates M°, that is, M° = M+° - Jf + °. | Now let us consider the subspace W S M° consisting of all the bounded mappings x:0 -t-M, that is

W = {x £ Ma : 3KxíR

\/mo 11^)11^^),

where ||-|| denotes the standard norm in M (see, e.g., YOSIDA [65]). If for x 6 W we will set by definition INI = sup |[^)||, then W becomes a partially ordered normed vector space being positively generated, that is, W = W+ — W+, where W+ is the positive cone oiW(W+= W n M+°). Furthermore, since ||pm(^4)|| = 1 for all A £ 0, we have \\pm\\ = 1, and therefore m pm is in fact an injection of S into W+ n S1, where S1 is the unit sphere in W. Consider finally the subspace V S W spanned by states of a physical system, that is

Obviously, F = F+ — F+, where F + = B+ • È, è = {pm : m £ £).

369

Conditional Probability

Note that the partial ordering in F induced by the proper cone F+ via the formula x^y

iff y — x € F+

(x, y € F)

is obviously stronger than the one inherited from M°. It can also easily be verified (Guz [22]) that the above-defined norm |]-|| is additive on F+, so (F, F+, II-1|) is a space of the type OL0.n) However, F+ is not, in general, norm-closed. If, on the other hand, we take into consideration the usual positive cone of the space F, F + = F n M+° 3 F+, it possesses all the desired properties, namely (Guz [2«]): (i) F + is closed with respect to the norm |[-1[; . (ii) F + is generating, i.e. V = F + — F + ; (iii) The norm ||-1| is additive on F + . Moreover, (iv) (F, F+, ||.|[) is a space of the type GL12), and Int F + = F+ \ {0). However, for the purposes of physical axiomatics the space (F, F+, ||-1|) is better than (F, F + , ||-H); the reason is that the equality F+ = R+ • $ means that the canonical image $ of the set of states is the base of the cone F+, so it can be used to define the socalled base norm in the space F, which is the most natural norm from the physical point of view (see Part 2, Section 2.1). This norm can equivalently be defined by using the previous norm |> || as follows: IMIi = inf {IM + IM :x1-xi-=x,

x2 € F + }.

Obviously, 6 = V+ n S1, where S1 = {x £ F : \\x\h = 1). Moreover, it can be proved (Guz [25]) that (F, || - Hi) is complete. Thus we have the following theorem (Guz [25]): (Y> II"Hi)) the real vector space spanned by states of a physical system and endowed with the norm | | - i s a complete base-norm space with a generating proper cone F+ = R+ • S, and with 0 such that — te sS x sS te. Obviously, e £ G. With every order unit e £ G one can associate the seminorm ||-|[„ on V defined by llxll, = inf {i > 0 : — te ^ x ^ te}. I t was shown that ||-||e is a norm if and only if (F, G) is almost Archimedean ordered, in which case (F, G, e) is said to be an order-unit space. Note that if (F, G) admits an order unit e £ G, then G generates F. Note also that if d is another order unit for (F, C), then ||-||e and ||-|[tf are equivalent seminorms.

377

Conditional Probability

A nonempty subset K £ G, where C is a cone in F, is said to be a base of C if for every nonzero x £ C there is a unique positive real number t > 0 such that x £ tK. It is not difficult to show that K is a base of G if and only if there is a strictly positive linear functional e on V (i.e. satisfying e(G\ {0}) > 0) such that K = {x £ G\ e(x) ¡= 1). Moreover, for each base K there exists exactly one such a functional e. Now, if for x £ F we set by definition II^IIK = inf {e(x! + x2): xlt x2 £ G, xr — x2 = x), then, provided we assume that G generates F, we obtain a semmorm || • || x on F which can equivalently be defined by ||x||£ = inf {t > 0 : x £ t conv [K u

(-K))}

where conv (K u {—K)} denotes the convex hull of the set K u (-K). A triple (F, G, K), where C is a generating proper cone in F with the distinguished base K, is said to be a base-norm space, provided || • is a norm on F. I t was shown that the Banach dual of a base-norm space is an order -unit space and conversely, the Banach dual of an order-unit space is a base-norm space. More precisely, if ( V , C , K ) is a base-norm space, then the partially ordered vector space (V',C), where V' is the Banach dual of F and C stands for the cone of all positive j|*||x — continuous linear functionals on F, has an order unit e £ C' such that K = {x £ G: e(x) = 1); namely, e is defined by e{y) = Wy^ — ||«/2||x, where y £ F, ylt y2 £ G, yx — y2 = y, and it is easily seen that this definition does not depend on any particular choice of positive elements ylt y2 in the decomposition of y € V. Moreover, the order-unit norm || • ||e of V coincides with the standard norm of V dual to ||-|| x . Conversely, if (F, C, e) is an order-unit space, then (F', C', K) with K = {/ £ C : /(e) = 1} becomes a base-norm space, with K being a base of C', and the base norm [[• |(K is identical with the standard Banach norm of V dual to || • ||e. Now let (X, X+) be an arbitrary partially ordered real vector space with X+ being its positive cone. We say that two positive projections P , Q on X are quasicomplementary if (im P ) n X+ = (ker Q) n and (im Q) n X + = (ber P) n X+. Let (X, X+) and (Y, Y+) be two partially ordered vector spaces which are in separating order duality, i.e. we assume that there is defined a nondegenerate bilinear form (•, •): XxY ^ R such that for x € X, y 6 Y

{

x ^ 0 iff {x, y) ^ 0 for all y ^ 0, y^ 0 iff (x, y) ^ O f o r a l l a ^ O .

(A 2)

We say that two weakly continuous positive projections P , Q on X are complementary if P , Q are quasicomplementary and if so are the dual projections P*, Q* acting on the space Y. (Other equivalent definitions of complementarity can be found in [3]). I t can be shown that in a pair P , Q consisting of two complementary projections the second member is uniquely determined by P and vice versa, so we write Q = P ' or Now let us consider as a particular case an order-unit space (A, A+, e) and a base-norm space (F, F+, K) and assume that they are in separating order and norm duality, i.e. we assume (A 2) together with the following requirement (in which a £ A, x £ V):

{

f [[ce|| ^ 1 iff { ||«|| ^ 1 iff

3

|(a, x)\ g 1 for every Z with ||z|| S 1, |(a, a;)| g 1 for every a with- ||a|| ^ 1.

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 8

(A3)

378

W.Guz

Let P be a weakly continuous positive projection on either A or V with norm a t most 1. We say t h a t P is a P-projection if P admits a complement with norm a t most 1. Clearly, P' is then a P-projection too, since P" = P. I t has been proved that a weakly continuous positive projection P acting on one of the spaces A or V is a P-projection if and only if its weak dual P * is a P-projection on the other space; then we clearly have P * ' = P ' * . For a given P-projection P on V the set FP = (im P ) n K is a face of K (i.e. it is a convex subset of K satisfying the following: if x, y £ K, 0 < t < 1, and tx (1 — t) y 6 FP, then x, y £ FP), and the faces of K of this form are called the projective faces oiK. A P-projection Q on A is said to be compatible with an element a £ A if Qa Q'a = We say that a projective face FP is compatible with a £ A if P * is compatible with a. The spaces A and V are said to be in weak spectral duality if for every a £ A and every t £ R there exists a projective face F of K compatible with a such t h a t (a, x) ^ t for x £ F and (a, x) > t for x £ F', where F' = FP