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QUANTUM. LOGIC Gjj QUANTUM. LOGIC gJJ DAVID J. FpCLlS , Ulliversily o{ Massa ι'hmells ， Alldlt!rSI. ,Hassacllllsl'llS, L:.5..-\. (5ecs. 1-3) Rr CHARD J. GREECHIE, Louisial1 a Tech Ulli\'ersil γ ， RU5(Oll, Louisialla, U. 5.A. (5 1' ι'5. 1-3) MARIA LOUISA DALLA CHIA民主 AND ROBERTO GIUNTINI, Ullivt!rsità di Firω， 1: 1'， Flore l1 ce, llalv (5ecs. 4一 10) Introduction ...... ................. 229 3.4 Cartesian and T ensor Products 1. A Brief History of Quantum of Orthoalgebras .................. 238 Logic .................................. 230 3.5 The Logic of a Physical - 'EA - The Origin of Quantum Svstem ................................ 239 Logic .................................. 230 3.6 The Canonical Mapping ........ 240 1.2 The Work of Birkhoff and von 3.7 Critique of Quantum Logic ... 240 Neumann ...……-………..….. 230 4. The Logician's Approach .... 241 1.3 The Orthomodular Law ........ 231 S. Algebraic and Possible- 1.4 The Interpretation of Meet World Semantics ................ 242 and Join ..................... ......... 232 6. Orthodox Quantum Logic ... 243 2. Standard Quantum Logic .... 232 6.1 Semamic Characterizations of 2.1 The Orthomodular Lattice of QL ...................................... 243 Projections on a Hilbert An Axiomatization of QL ...... 244 Space .................................. 232 6.2 2.2 Observables ......................... 233 7. Orthologic and Unsharp 2.3 States ...........…..………........ 234 Quantum Logics ................. 245 2.4 Superposition of States ......... 235 8. Hilbert-Space Models of the 2.5 Dynamics ............................ 235 Brouwer-Zadeh Logics ....... 247 2.6 Combinations of Standard 9. Partial Quantum Logics ...... 248 Quantum Logics ...... ............ 236 9.1 Algebraic Semantics for 3. Orthoalgebras 臼 Models for WPaQL ............................... 249 a General Quantum Logic '" 236 9.2 An Axiomatization of Partial 3.1 Orthoalgebras ...................... 236 Quantum Logics .................. 249 3.2 Compatibility, Conjunction, 10. Critique of Abstract and Disjunction in an Quantum Logics ................. 250 Orthoalgebra ............... ........ 237 3.3 Probability Measures on and Gloss缸-y ............................. 250 Supports in an Orthoalgebra . 238 Works Cited ....................... 254 INTRODUCTION ing logical formalisms, each with its own domain of applicability. Among these formal­ In forτnulating and studying principles of isms are Boolean-based propositional and valid reasoning, logicians have been guided predicate calcu!i, modal and multivalued log­ not only by introspection and philosophical ics, intuitionistic logic, and quantum logic. reflection, but also by an analysis of various Our purpose in this article is to outline rational procedures commonly employed by the history and present some of the main mathematicians and scientists. Because these ideas of quantum logic. In what follows , it principles have a multitude of disparate wi11 be helpful to keep in mind that there are sources, efforts to consolidate them in a sin­ four levels involved in any exposition of logic gle coherent system have been unsuccessful. and its relation to the experimental sciences. Instead, philosophers, logicians, and mathe­ maticians have created a panoply of compet- 1. Philosophical: Addresses the epistemology Encyclopedia of Applied Physics , Vol. 15 。 1996 VCH Publishers, Inc. 3.527-28137-1/96/$5.00 + .50 229 飞 230 Quantum Logic of th.: experiment.:t l sciences. Guides .:t nd tions affiliated with qu.:t ntum-mech.:t nicaJ moti\. .:t t 出 the activities at the remaining entltles. Icvels 认 hile assimibting and coordinating According to von Neumann, a quantum_ the insights gained from these activities. mechanical system c;f is represented mathe­ 2. S:..l/1 tacrÎC: Emphasizes the fOI寸n .:t l struc­ m.:t tically by a separable O.e. , countable di­ ture of a general calculus of experimental mensional) complex Hilbert space 贺， proposlt1ons. obsen'ables for c;f coπespond to self-adjoint 3. SemallfÎc: Focuses on the construction of operators on 坷， and the spectrum of a self­ classes of mathematical models for a log­ adjoint operator is the set of all numerical ical calculus. \'alues that could be obtained by measuring 4. PraglllatÎc: Concentrates on a specific the corresponding observable. Hence, a self­ mathematical model pertinent to a pa 口 ic­ adjoint operator with spectrum consisting at ular branch of experimental science. most of the numbers 0 and 1 can be re­ garded as a qlLantwn-meclzanical propositio J1 For instance, studies regarding the logics by identifying 0 with "false" and 1 with associated with classical physics could be "true." Since a self-adjoint operator has spec­ categorized as follows: trum contained in {O,I} if and only if it is an 1. Philosophical writings extending back at (orthogonal) projection onto a closed linear 况， least to Aristotle. subspace of von Neumann (1 955, p. 253) 2. Propositional and predicate calculi. observed that 3. The class of Boolean algebras. the relation between the properties σalgebra 4. The Boolean of all Borel sub­ of a physical system on the one hand, sets of the phase space of a mechanical and the projections on the other, system. makes possible a sort of logical calcu­ lus with these. Likewise, for quantum logic, we have 1. Philosophical writings beginning with 1.2 The Work of Birkhoff and Schrödinger, von Neumann, Bohr, Ein­ von Neumann stein. et al. 2. Quantum-logical calculi. In 1936, von Neumann, now in collabora­ 3. The class of orthoalgebras. tion with Garτett Birkho 匠， reconsidered the 4. The lattice of projection operators on a matter of a logical calculus for physical sys­ Hilbert space. tems and proposed an axíomatic foundation for such a calculus. They argued that the ex­ expositorγreasons ， su 凹ey For our proceeds perimental propositions regarding a physical roughly in the order 1, 4, 3, 2. Thus, we give system c;f should band together to fonn a lat­ historγof a brief quantum logic in Sec. 1, tice L (Birkho匠， 1967) in which the meet outline the standard quantum logic of pro­ and join operatíons are fonnal analogs of the jections on a Hilbert space in Sec. 2, intro­ and and or connectives of classical logic (al­ duce orthoalgebras as models for quantum though they admitted that there could be a logic in Sec. 3, and discuss a general quan­ question of the experimental meaning of tum-logical calculus of propositions in these operations). They also argued that L Sec.4. should be equípped with a mapping carτymg each proposition a ε L into its negation a' ε L. In present-day tenninology, they proposed 1. BRlEF HISTORY OF QUANTUM that L fonns an orthocomplemented lattÎce LOGIC with ^. V. and a • a' as meet, join, and or­ thocomplementation, respectively (Kalm­ 1.1 The Origin of Quantum Logic bach, 1983; Pt址 and Pulmannovã, 199 1). Birkho 征 and von Neumann observed that The publication of John von Neumann's the experimental propositions conceming a Mathematische Gnmdlagen der Quantenme­ classical mechanical system c;f can be identi­ chanik (1 932) was the genesis of a novel sys fied with members of a field of subsets of the tem of logical principles based on proposi- phase space for C;f, (01', more accurately, with QuantU l11 Logic 231 dements of a quotient of such a field bv an fail to salísf)' the mudular la 川 E 飞 iJently ， ideal). In any ca妃， for a c1 assical mechanical von Neumann consid..:red this to be a po 吕 SI­ system ':1, they concluded that L forms a ble serious fla 飞 v of the Hilbert- 革 pace formll­ Booleal1 algebra. lation of qtl~lntum mcchanics as proposed in An orthocomplemented lattice L is a Bool­ his own Cnllldlage'l. Much of von Nell­ ean algebra if and only if it satisfies th 巳 dis­ mann's 叭 ork on continuolls geometries tribwive law (1 960) and rings of operators (Mun-ay and von Neumann, 1936) 飞vas nlotl 飞 ated bv his concret 巳 complemented x^(yVz) = (x^y)V(x^z). (1) desire to constluct modlllar lattices can丁ing an "a priori An example in which a ε L denotes the ob­ thermo-dynamic 飞 \'eight of states," that is , a servation of a wave packet on one side of a continuous dimension or trace function. plane, a' ε L its observation on the other ε L side, and b its observation in a state 1.3 The Orthomodular Law symmetric about the plane shows that Although the projection lattice of an infi­ b = b ^ (αVa') ;;é (b ^ a) V (b ^α') 0, nite-dimensional Hilbert space fails to satisfv the modular law in Eq. (2) , it was discovered so that the distributive law of classical logic by Hllsimi (1 937) that it does satisfy the fol­ breaks down even for the simplest of quan­ lowing weaker condition, now called the or­ tum-mechanical svstems. As Birkho旺 and thomodlllar law: von Neumann observed, If Z ::5 x , then x = (x ^ z') V 乙 (3) . whereas logicians have usually as­ sumed that properties of negation were If Eq. (2) holds, then so does Eq. (3) in 札ew the ones least able to withstand a criti­ of the fact that x x ^ (z' V z). The same cal analysis, the study of mechanics condition was rediscovered independently by points to the distributive idel1 tities as Loomis (l955) and Maeda (1 955) in connec­ the weakest link in the algebra of logic. tion with their work on extension of the Invok.ing the desirability of an "a priori Muπav-von Neumann dimension theorv of thermo-dynamic weight of states," Birkho旺 rings of operators to orthocomplemented lat­ and von Neumann argued that L should sat­ tices. An orthocomplemented lattice satisfy­ is 鸟r a weakened version of Eq. (1), called the ing Eq. (3) is called an orthomodular lattice. modular law, and having the following form: ln 1957, Mackey published an expository article on quantum mechanics in Hilbert space based on notes for lectures that he was If z ::5 x , thenx^(yVz) = (x^y)V 乙 (2) then giving at Harvard.
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