@ I* Ch06-N52870.h Page 215 Monday, March 26,2007 10:29 AM HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES 215 Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved QUANTUM LOGIC AND PARTIALLY ORDERED ABELIAN GROUPS David J. Foulis and Richard J. Greechie 1 INTRODUCTION Our purpose in this expository article is to give an account of the connection between quantum logics1 and partially ordered abelian groups. In what follows, we use the word "logic" in the sense of a partially ordered algebraic structure L that could be interpreted as a semantic model for a formal symbolic logic L. This usage is customary in the literature of algebraic logic [Halmos, 19621, and it is consistent with the nomenclature of Birkhoff and von Neumann in their seminal article on the logic of quantum mechanics [~irkhoffand von Neumann, 19361. Our emphasis is on the mathematical structure of L rather than on the interpretation of its elements as propositions pertinent to a physical system. An account of L and C from the point of view of logic as an instrument of reasoning, as well as more comprehensive treatment of the general notion of a quantum logic, can be found in [~allaChiara et al., 20041. We begin, in Sections 2-5 below, with a brief sketch touching on some of the ba- sic notions of orthodox quantum mechanics (QM). By orthodox QM, we mean non- relativistic Hilbert-spaced-based QM without superselection rules and for which the observables are represented by projection-valued (PV) measures. Readers al- ready knowledgeable about quantum physics can skim these sections rapidly to familiarize themselves with our notation. The relevant quantum logic (QL) for orthodox QM is the complete atomic orthomodular lattice P(4j) of projection op erators on a Hilbert space Jj. The QL P(4) is a subset of the partially ordered abelian group G(4) of Hermitian (i.e., bounded self-adjoint) operators on 4. A comprehensive and authoritative account of the standard quantum logic P(4j) in the context of orthodox QM can be found in [~eltramettiand Cassinelli, 19811. In Section 6, we turn our attention to the much more general and flexible positiveoperator-valued (POV) measures for which the corresponding QL is the secalled effect algebra IE(4) of all positive semidefinite Hermitian operators on 4j that are dominated by the identity operator. Evidently P(4) C IE(4) C_ G(5). As "quantum-logical propositions", effect operators in IE(4) can manifest "fuzzi- ness" or "un~harpness'~,while the projection operators in P(4) can be regarded as l~efollow common usage in which the term "quantum logic" refers not only to a logic associated with a quantum-mechanical system, but indeed to any physical system whatsoever. @ I* Ch06-N52870.h Page 216 Monday, March 26,2007 10:29 AM 216 David J. Foulis and Richard J. Greechie "sharp," even though their "truth values" might be subject to statistical fluctua- tions. For an exposition of POV measures and the contemporary quantum theory of measurements, see [~uschet al., 19911. The mathematical structures of P(Ej), IE(fj), and G(4) are so rich and the quantum-mechanical interpretations of these structures are so intriguing that one cannot resist the temptation to formulate and study more general triples P C E c G as abstractions or analogues of P(4) C IE(4j) c G(4). Thus, in Sections 7 through 16 we conduct an abstraction process, motivated and guided by suit- able examples, that will ultimately result in the notion of a so-called "CB-triple" P c E c G consisting of a regular orthomodular poset P, an effect algebra E, and a CB-group G, i.e., a partially ordered abelian group enriched by a so-called "compression base." The notion of a CB-group is both very general and mathe- matically attractive, and may be considered as an appropriate basis for the general study of quantum logics. In Sections 17 and 18, we introduce and study a class of CB-groups, called archimedean RC-groups, for which a spectral theory has been developed. Because of the expository nature of this article, we give prooh only when we regard them as being illuminating or when they could be difficult to locate in the existing literature. Background material on Hilbert spaces can be found in [Halmos, 19981, a source for basic facts in regard to a-fields, measurable functions, and so on, is [Halmos, 19501, and [Halmos, 19631 is an authoritative exposition of the theory of Boolean algebras-all three of these classic references were authored by P.R. Halmos. The monographs of R.V. Kadison and J.R. Ringrose ri ad is on and Ringrose, 19831 provide comprehensive expositions of both the basic and ad- vanced theory of operator algebras, and the treatise of G. Emch [~mch,19721 is an excellent reference for applications of operator algebras to physics. In our treatment of partially ordered abelian groups, we follow the monograph [~oodearl, 19851 of K.R. Goodearl. For general lattice theory, the classic [~irkhoff,19791 of G. Birkhoff is the standard reference. 2 ORTHODOX QUANTUM MECHANICS Orthodox quantum mechanics (QM) is founded on the assumption that there is a correspondence S -+ fi assigning a complex separable Hilbert space fi to a QM- system S. Let (. , a) be the inner product2 on fj and denote the additive abelian group of all bounded Hermitian operators on 4j by G(fi). The tenets of orthodox QM speclfy that bounded (real) observables for S are represented by operators A E G(fj), (pure) states for S are represented by unit vectors $J E 4, and the expectation value of an observable A when the system S is in the state $J is given by (A$J,$J). DEFINITION 1. If $J is a state vector in 4, i.e., ll$Jll = 1, and W denotes the system of real numbers, we define the expectation mapping w$: G(4j) -+ IR by 2~hysi~istsusually use the Dirac bra-ket notation (. I .) for the inner product. @ I* Ch06-N52870.h Page 217 Monday, March 26,2007 10:29 AM Quantum Logic and Partially Ordered Abelian Groups 217 w+(A) := (A$, $) for all A E G(J~).~ The expectation value w+(A) is usually interpreted as the limit as n + oo of the arithmetic mean of n independent measurements of the bounded observable A on the QM-system S, or replicas thereof, in the fixed state $. The expectation mappings w+ are used to organize the Hermitian group G(4) into a partially ordered abelian group [Goodearl, 19851 as follows: if A, B E 6(4), then by definition A 5 B iff4 w$(A) 5 w+(B) for every state vector $J E 4. In fact, under the partial order 5, G(4) is an order-unit-nomed real Banach space with the identity operator 1 E G(4) as the order unit [Alfsen, 1971, page 691. F'urthermore, the order-unit norm coincides with the uniform operator norm 11 . 11 on G(fi), and therefore, for A E G(a), (1) IlAll = inf{m/n I 0 5 m E Z, 0 < n E Z, and - ml 5 nA 5 ml), where, as usual, Z is the ring of integers. If 0 is the zero operator in G(*), then (by a slight abuse of language), observables A E G(4) with 0 5 A are called positive. We note that the norm of a positive bounded observable is the maximum expectation value of that observable for all state vectors, i.e., The positive operators A in G(4) are exactly the operators of the form A = B2 for B E G(4). Indeed, if B E G(4), then for every state vector $, whence 0 5 B2. Conversely, if 0 5 A E G(4), then there is a unique operator aE G(4) such that 0 5 and A = [~ieszand Nagy, 1955, page 2651. If A E G(4), then 0 5 A', and one defines IAl := a,A+ := l(IAl2 +A), and A- := i(IAI - A). Then Thus, every bounded observable is the difference of two bounded positive observ- ables. Suppose that A1 5 A2 5 . is an ascending sequence of bounded observables. Then the sequence is bounded above in norm (i.e., there is a real number P with llAnll 5 P for all n E M := {1,2,3, ...)) iff it is bounded above in G(4) (i.e., there is a bounded observable B E G(4) such that A, 5 B for all n E M). Furthermore, if the sequence is bounded above in either sense, then by Vigier's theorem [Riesz and Nagy, 1955, page 2631, there is a uniquely determined bounded observable A E G(4) such that, for every $ E 4, IIA,$ - A611 + 0 as n + oo, i.e. the sequence A1, A2,... converges to A in the strong operator topology. But, if A, -t A in the strong operator topology, then A, + A in the weak operator 3~henotation := means "equal by definition." 4We use "iff' as an abbreviation for "if and only if!' @ I* Ch06-N52870.h Page 218 Monday, March 26,2007 10:29 AM 218 David J. Foulis and Richard J. Greechie topology, and it follows that w+(A,) converges monotonically up to w+(A) for each state vector $ E 4j. The latter condition implies that A is the supremum (least upper bound) of the sequence All A2, ... in G(4j); hence the Hermitian group G(4j) is (Dedelcind) monotone a-complete, i.e., every bounded ascending sequence in G(4) has a supremum.