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A NNALES DE L’I. H. P., SECTION A K RYSTYNA B UGAJSKA S LAWOMIR B UGAJSKI The lattice structure of quantum logics Annales de l’I. H. P., section A, tome 19, no 4 (1973), p. 333-340 <http://www.numdam.org/item?id=AIHPA_1973__19_4_333_0> © Gauthier-Villars, 1973, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Vol. XIX, Section A : Poincaré, n° 4, 1973, Physique théorique. p. 333 The lattice structure of quantum Krystyna BUGAJSKA logics Slawomir BUGAJSKI Katowice, Poland ABSTRACT. - A solution of the question of the lattice structure for quanlogics is proposed. It is achieved by a construction of a natural embeddof ing the logic into a complete lattice, preserving all essential features of the logic. Special cases of classical mechanics and the standard form of quantum mechanics are investigated. tum 1. INTRODUCTION The lattice property for quantum logics is one of puzzling questions of the logical approach. Usually we assume the logic to be an orthomodular, complete ortholattice without any further substantiations. However, a deeper analysis suggest, that this property is neither obvious nor natural. The problem is important, because of the basic role played by the lattice property in the logical approach. Two possible ways out arises: (i) to give a clear phenomenological interpretation to the lattice joins and meets in quantum logics, or (ii) to derive the lattice property from others, more plausible assumptions. Concerning the first possibility, it is probable, that the property in question has no satisfactory interpretation in terms of experimental operations (see Mac Laren’s discussion in [9], p. 11-12); the recent attempt of Jauch and Piron [7] in this direction is not entirely conclusive [11]. The second way is preferred by some authors (for example [2], [9], [6], [12]. Newertheless, usually assumed postulates to imply the desired structure seem to be not more convincing than the derived property itself. In this paper we propose a solution of the problem in question not by forcing the logic to possess a complete lattice structure, but by constructing Annales de l’Institut Henri Poincaré - Section A - Vol. XIX, n° 4 - 1973. 334 K. BUGAJSKA AND S. BUGAJSKI natural extension of the logic into complete lattice. Such extended logic appears to be a good substitute for original one and possesses all its essential features. Our « critical » assumption if any, is the von Neumann’s projection postulate-unquestionable in the common opinion, but criticized by some prominent philosophers and theorists (e. g. [5]). The somewhat similar extension was previously discussed by the authors [3] on a quite different axiomatic basis. In the next section we collect our assumptions. They are formulated in the technical language of the logical approach. Definitions of the used terms one can find in some other papers (e. g. [10], [12]). a 2. ASSUMPTIONS Let J5f denote the logic of given physical system. The fundamental of j~f are discussed in many papers; the reader is referred to the book of Mackey [8] for a justification of the first postulate: properties is an orthomodutar 03C3-ortho-poset. This hypothesis has a good tradition in the quantum logic approach and is not questioned. The next assumption states a connection between the set of atoms d of J~f and the set of pure states of the system. POSTULATE 1. a set g of probability measures on L (the such that: (i) g is separating; (ii) for any b ~ L there of pure states), such that a(b) exists 1; (iii) there exists an one-to-one mapping car : ~ -~ d with property: a(b) 1, implies car (a) ~ b. The following supposition looks very natural: every state is a mixture (countable or not) of pure states. The first assertion of postulate 2 may be treated as a consequence of this not precise, but plausible statement. As regards (iii), this property essentially states, that one can unambiguously identify the pure state in a single yes-no measurement. Observe, that postulate 2 implies the atomicity of J~f and the existence, car (a). of one and only one pure state a, such that a for any a the usual Our next assumption is essentially projection postulate of mechanics : quantum POSTULATE 2. - There exists set = = = 0 then there exists one POSTULATE 3. - Let bE J5f and a E ~. If 1 and a(b) a [car (/?)]. and only one pure state f3 such that f3(b) The projection postulate (in many, seemingly non-related formulations) is one of the most popular axioms in the quantum logics approach, see [4] for further remarks and examples. Postulate 3 completes our list of needed assumptions. = = Annales de l’Institut Henri Poincaré - Section A LATTICE STRUCTURE OF 335 QUANTUM LOGICS 3. THE EMBEDDING Let us introduce some notations. If M is a subset of orthocomplement of a E 2. In the sequel simple properties of operations~, ~’ collected in with a’-the The application of the of closure operation the we then shall use some afte the operationv gives a kind of H. C. Moore [1]. It is the content of operation D. one We define the extended logic Ef as the set of all « closed » (in the sense of above closure operation) subsets of 2. It is easy to see (the reader is referred to [1] for the general theorem), that Ei is a complete lattice under the partial ordering by inclusion, with lattice joins and meets defined (M, N e F) as (M u N)B7~ and M n N resp. One can introduce a natural orthocomplementation in J~. Indeed, the possesses needed properties: operation M - M’~ for not exist any element a of 2 such that Thus e-the greatest element of except Proof: (i) Obviously, there does a E and a E MV, Is a consequence of lemma 1. (iii) (M’~)’6 We summarize the obtained results in (ii) = (M")V~ = M~ = M. D COROLLARY 1. - The extended logic ~ is a complete, atomic ortholattice. The relation between 2 and !P is clear. Namely, the set a° for a E 2 is a close subset of hence is a member of 2. The mapping of 2 into J~: Vol. XIX, n° 4 - 1973. 336 K. BUGAJSKA AND S. BUGAJSKI a ~ a° preserves all essential features of extension of 2. LEMMA 4. - Let a, b, a~ E 2, i E 2, thus that is the mentioned I, I-some set of indices. Then: (provided left hand side exists) . Proof - The first assertion as well . regards (iii), b6 = bV6 for any bE 2. the second The equality as one is obvious. As is easy to prove. Thus So the above constructed embedding 2 -+ ,;l preserves the partial ordering, the orthocomplementation and all existing joins (it also preserves all meets, of course, as a result of this). The extended logic would be therefore a good substitute for the original one if only it were orthomodular. The orthomodularity is commonly accepted as a basic feature of quantum logics and an eventual lack of it makes our construction unuseful. The next section is devoted to a demonstration of the orthomodularity of J~. 4. THE Before QUESTION OF ORTHOMODULARITY we come near LEMMA 5. Proof - - The to the problem, we examine some If a ~ A and b~L then there exists a projection postulate implies the existence properties of 2. V b E 2. of the atom al , sucn mai Moreover, Let c a a, b. We would like to demonstrate V ai. Indeed, by the orthomodularity, there is in 2 an element, say b1 , such that bi1 1 b and c bl V b. If one. applies the projection postulate to a and b, then one can find the atom a 1 with the properties: = Annales de l’Institut Henri Poincaré - Section A 337 LATTICE STRUCTURE OF QUANTUM LOGICS Thus 311 al and c ~ ai, b. It means, that b in the set {a, b ~~ , i. e. b V a 1 b V a. D = is the least element V al = If c e (b~ n then two following Obviously, b e (b~ n c° b~ c~ or n ~ ~ b° n d. Let us possible: consider the first one. By the projection postulate, there exist, for given a [(x(&) 5~ 0, 0], atoms ci and bl such that a(ci) a(c), But a(b) ~ a(c 1 ) a(c) and a(c) ~ a(b). Thus a(b) a(c) for all a E f!jJ, and the separating property of g assures b = c. If b0394 n A, Proof cases - are = = = = then c > b by lemma 5. Thus n = ~)° implies c,> b, i. e. b a. 0 terms of [9]). = aeb n.r~ The proved lemma states, that ~ is atomistic (atomic in Moreover, COROLLARY 2. Proof atomistic too. - - It suffices to note, that if M then b E M is equivalent to Let BM denote a maximal set of pairwise orthogonal atoms, contained in M E orthobasis of M). We demonstrate M to be determined by BM, namely. If M ~ and BM is an orthobasis of M, then B~0394M = M. Proof - BM M, then B~6 MV6 = M. Let a be an atom belonging to M B B~, and let a car (a). One can prove, that if b E BM then x(b) ~ 0 LEMMA 7. - c c = only for some countable subset of there is an atom c, such that BM, say {bl, b2 Atom a is orthogonal to all b E b2 whole BM. Hence, it is no atom in M ’ to COROLLARY 3. Proof Bi - extend Let - M1 to some Vol. XIX, n° 4 - 1973. 5i is c ... }, ... thus }. By c is lemma 1 orthogonal and, by the atomicity orthomodular. M2 and orthobasis Bi be an orthobasis of MI. One B2 of M2. Obviously, B2. can Let 338 B3 i. K. BUGAJSKA AND S. BUGAJSKI All elements B2" Bl and M3 B3 c B1° , By applying lemmas 1 and = e. thus = M3 is orthogonal to of B3 are orthogonal to 2 we obtain B1, Mi. Moreover, This proves the orthomodularity 0 The above corollary solves the question of orthomodularity for J~. 5. CONCLUSION Thus we have proved the following. THEOREM 1. satisfy postulates 1-3, then there exists an orthoand a natural embedding of L modular, complete, atomic ortholattice into fl, preserving the partial ordering, the orthocomplementation and all joins of ~. One can treat !i as a new, extended logic of the system, satisfying all The described extension regularity conditions usually assumed for procedure takes, however, into considerations some new elements, with no counterparts in « experimental questions » concerning the system. It is the usual price paid for an application of a regular mathematical structure to a description of some features of reality. Theoretical physics provide numerous examples of that. An application of the obtained results to the problem of linear representation of quantum logic will be a subject of subsequent paper of the authors. 6. T W O SPECIAL CASES The problem of lattice structure for 2 becomes remarkably simpler in two special cases of interest: (i) any two elements of 2 split (the case of classical mechanics); (ii) 2 is separable, i. e. any set of pairwise orthogonal elements of 2 is countable (the case of the standard quantum mechanics). In the first case, 2 appears to be a Boole’an algebra, hence it is a lattice. If 2 is assumed to be not complete, then the described extension is not trivial. Our embedding theorem holds in this case with weaker assumptions : THEOREM 1 (i). any two elements of If 2 split, then - an atomic, orthomodular orthoposet and the thesis of theorem I holds. Annales de l’Institut Henri Poincaré - Section A 339 LATTICE STRUCTURE OF QUANTUM LOGICS of ~ and the embedding are quite analogous above ones. We must prove the orthomodularity of !l only. Any of J~f, b, c E 2 split, i. e. there exist pairwise orthogonal elements b 1 , then bi = 0 and b c. such that b = bl V d, c ci V d. Let c E (b~ n Thus 2 is atomistic. Now the orthomodularity of 4 readily follows. D In the second case we also do not need the set Observe, that the without formulated making any reference to projection postulate may be transition probabilities: Proof The construction to the = then there exist unique atoms and a POSTULATE 3’. a2 such that and a b’, al V a2 . This is the Varadarajan’s version [13]; for further discussion see [4]. The embedding theorem takes now the form : THEOREM 1 (ii). - If 2 is separable, orthomodular, atomic and the projection postulate 3’ holds in 2, then 2 is isomorphic is in itself a complete lattice. 03C3-orthoposet to 2, i. e. ~ Proof One can prove, in an analogous manner as for lemma 5, that if a exists in Moreover, let { cl, c2 ... ~ atom, then a V b with any b E If be an orthobasis of By the orthomodularity, c=Ci1 V c2 V is an .... and !e is atomistic. Now a2 ... ~ be an orthobasis of M E 2. The existence of such ao E M that a2 V ...)ð. is contrary to M Hence demonstrated the just (at V a2 V ...)~. D properties = ACKNOWLEDGEMENTS The authors express their kind interest in this work. [1] [2] [3] [4] [5] [6] gratitude for Professor A. Trautman for his G. BIRKHOFF, Lattice Theory, New York, 1948. K. BUGAJSKA and S. BUGAJSKI, On the axioms of quantum mechanics (Bull. Acad. Pol. Sc., Série des sc. math. astr. et phys., vol. 20, 1972, p. 231-234). K. BUGAJSKA and S. BUGAJSKI, Description of physical systems (Reports on Math. Phys., vol. 4, 1973, p. 1-20). K. BUGAJSKA and S. BUGAJSKI, The projection postulate in quantum logics (Bull. Acad. Pol. Sc., Série des sc. math., astr. et phys., vol. 21, 1973, p. 873-877). M. BUNGE, Foundations of Physics, Springer Verlag, Berlin-Heibelberg-New York, 1967. S. GUDDER, A superposition principle in physics (J. Math. Phys., vol. 11, 1970, p. 1037- 1040). Vol. XIX, n° 4 - 1973. 340 [7] [8] [9] [10] [11] [12] [13] K. BUGAJSKA AND S. BUGAJSKI J. M. JAUCH and C. PIRON, On the structure of quantal proposition systems (Helv. Phys. Acta, vol. 42, 1969, p. 848-849). G. W. MACKEY, The mathematical foundations of quantum mechanics, W. A. Benjamin Inc. New York, Amsterdam, 1963. M. D. MACLAREN, Notes on axiom for quantum mechanics, Argonne National Laboratory Report ANL-7065, 1965. M. J. MACZY0143SKI, Hilber space formalism of quantum mechanics without the Hilbert space axiom (Reports on Math. Phys., vol. 3, 1972, p. 209-219). W. OCHS, On the foundations of quantal proposition systems (Z. Naturforsch., vol. 27a, 1972, p. 893-900). J. C. T. POOL, Baer-semigroups and the logic of quantum mechanics (Commun. math. Phys., vol. 9, 1968, p. 118-141). V. S. VARADARAJAN, Geometry of quantum theory, vol. I, Van Nostrand Co, Princeton, N. J., 1968. (Manuscrit reçu le 3 mars 1973) . Annales de l’Institut Henri Poincaré - Section A