logic must be as much above suspicion as Caesar's wife? (R. Omnes, 1990)(1). Abstract. This paper is a series of intertwining observations about the connection.
ALGEBRAIC STRUCTURES AND OBSERVATIONS: QUANTALES FOR A NONCOMMUTATIVE LOGIC THEORETIC APPROACH TO QUANTUM MECHANICS
M. PIAZZA Dip. di Filosofia - Universita di Genova Genova, Italy
What meaning should be given to such results, considering that logic must be as much above suspicion as Caesar's wife?
(R. Omnes, 1990)(1).
Abstract. This paper is a series of intertwining observations about the connection between the logic-theoretic noncommutativity and a logical foundation of quantum mechanics. We will analyze noncommutativity, both from an algebraic and prooftheoretic point of view, w.r.t . the quantum mechanics notion that the order of observation making is central to their description . To this end, we will present the sequential conjunction 0 : A 0 B means "A at time t1 and then B at time t2". The thread running through our discourse is given by qu ant ales , i.e. algebraic structures introduced by Mulvey as models for the logic of quantum mechanics, which offer an appropriate algebraic (and topological) tool for describing noncommutativity.
1. Introduction Quantales ate partially ordered algebraic structures which generalize both locales (also known as frames or complete Heyting algebras) and ideal lattices. They were introduced in the eighties by C. Mulvey in the ambitious aim of providing a possible common setting for noncom mutative C* -algebra, for constructive foundations for quantum mechanics (QM), and for noncommutative logics. The term itself was coined as a combination of "quantum 381
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logic" and "locale". (In terms of quantales, a locale is a commutative, rightsided idempotent quantale)(2) . The idea underlying Mulvey's seminal, albeit programmatic, paper is the assumption of an intimate connection between the Gelfand-Naimark representation of noncom mutative C* -algebra, stated by Giles and Kummer in 1971, and the foundations of QM in terms of propositional theories within a constructive, but noncommutative, logic. This connection, he suggests, appears more fundamental than the formalistic links with C* -algebras within the conventional foundation of QM by way of Hilbert space (Mulvey, 1986)(3). Another intriguing issue concerns the complex numbers: the interdependence between the logic-theoretic noncommutativity and the logical foundation for QM is made closer by the fact that in the set theories based on quantum logics investigated by Takeuti, the complex numbers are noncommutative. For the sake of convenience, this framework can be summarized in Diagram 1.
Diagram 1
We will start from the bottom of the above diagram: indeed, the aim of our contribution is to consider a possible link between the noncommutativity of observations in QM and the noncommutative logics in which the formulas of the language are strictly sequential. In particular, we focus on noncommutative linear logic, i.e. the logic obtained from a sequent formulation of classical (or intuitionistic) logic by rejecting all the structural rules: weakening, contraction, and exchange (Abrusci, 1991). Noncommutative linear logic seems to satisfy Mulvey's request of constructiveness, since it enjoys cut-elimination theorem and consequently it is equipped with an appropriate semantics of proofs. Moreover, quantales give an algebraic
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semantics for linear logic (and in particular, noncommutative quantales do the same for noncommutative linear logic). This paper will be devoted to uncovering the intimate connections between the treatment of noncommutativity in different settings. Some of these connections can be surprisingly subtle. In this sense, even if this paper should seem somewhat rhapsodical, Gershwin-like so to speak, its declared purpose is more to raise questions than answers. 2. The noncommutative paradigm
One of the most natural problems arising out of the investigation of an abstract logic-theoretic framework for QM is that of noncommutativity, that is, roughly speaking, the requirement of a (logic-theoretic) noncommutative operation corresponding more or less to the quantum mechanical notion that the order in which observations are carried out is central to their description. This logical operation "retains a vestige of temporality". Traditional quantum logic appears quite silent on this point. By a quantum logic we mean, as usual, the couple (L, M) where L is an orthomodular a-lattice and M is a strong set of states on L, i.e. the statement {m E Mlm(a) = I} ~ {m E Mlm(b) = I} implies that a ::; b, a,b E L. The well-known Jauch-Piron property in the a-form is also supposed for any state of M. Quantum logic models mathematically the set of all experimentally verifiable propositions about a physical system. The interdipendence between the logic-theoretic noncommutativity and the logical foundation for QM was posed precisely by M.D. Srinivas in the seventies. Previously, in 1957, P. Jordan, still working in a lattice theoretic framework, formulated a theory of skew lattices (i.e. noncommutative lattices) in which the commutativity of the operations is not assumed, in order to provide a quantum propositional calculus (Jordan, 1959)(4). Also Birkhoff in 1959 suggested that the experimental procedure for the proposition A /\ B be given by the sequence of the measurements of A and B in that order (Birkhoff, 1961). And this is an orthodox noncommutative "reading". But, in practice, Birkhoff has exploited the lattice theoretic commutative conjunction in the propositional calculus, since - he argued - "the set of all states that have unit probability for satisfying the experimental procedure {A, B} also have unit probability for satisfying the sequence {B, A}" (Srinivas 1975, p. 1678). Nevertheless - Srinivas observed -: "the set of all states that assign unit probability to a proposition do not completely characterize the corresponding experimental procedure ... Two experimental propositions can have the same set of (all) the states that assign unit probability to them and
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still correspond to experimental procedures if the associated measurement transformations are different" (Ibidem). In his work Srinivas dismissed the lattice-theoretic logics as models for the calculus of quantum experimental propositions: the "logic of experimentally verifiable proposition" leads now to a "logic of operations (or measurement transformations)" on physical systems. This logic of operations arises out of the analysis of the relations among experimental procedures, and it differs radically from the usual lattice-theoretic logics which represent each experimentally verifiable proposition in quantum theory as the set of all states assigning probability one to the proposition. This representation fails to characterize completely the experimental procedures corresponding to a proposition. In order to correlate the conjunction more closely with empirical procedure, Srinivas defines the proposition E1 1\ EE as follows: "the experimental procedure corresponding to the proposition Ell\EE is the procedure in which the system is subjected to the sequence of procedures E1 , EE in that order" (Srinivas 1975, p. 1679). Schematically:
Let V be the set of self-adjoint trace class operators on a Hilbert space, and let 0 denote the composition of operations. The operation El 1\ E2 is thus defined such that:
Obviously, 1\ is associative and, in general, it is not idempotent. Two propositions are compatible, (written E1 H E2), if E1 1\ E2 = E2 1\ E1. In the next section we will see how this noncommutative operation 1\ and the noncommutative linear operation 0 resemble each other. 3. Quantum proof theory?
It is well-known that measurements in a quantum system, carried out in order to find its current state, influence the measured value noticeably, and that if the interaction between the instrument (measuring device) and the system is weak, then this influence is less. We have to increase the period of observation adequately if we want the uncertainty in the estimate of the state of instrument to decrease. The measurements within a system in a pure state (i.e., a state that can be described by a wave function) are reduced to the identification of the initial conditions, whereas if the interaction between the instrument and the
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system is strong, the pure state of the system becomes mixed (i.e., only the probability that the state occurs is known) in the process of measurement. It is interesting to note that if the observable (the measured physical quantity) is defined by an operator commuting with the Hamiltonian of the system, "the originally determined state cannot be perturbed by any subsequent measurements" (Butkovskiyand Samoilenko, 1990 p. 5). Can we describe logically the participation of observers in the process of measurements? The aim of this section is to present a possible prooftheoretic approach to the quantum theory of measurements: this kind of approach to QM can be completely justified by the question raised by Mulvey: with what rules of deduction are physical observations naturally manipulated? From the standpoint of the syntactical manipulability of physical observation, Mulvey's question can be strengthened by another question: what is, if any, "the physical meaning" of cut-elimination in a sequent calculus for quantum logic? The latter question boils down to a question of the meaning of the subformula property in such a sequent calculus(5) . In QM, both the aspects that we would like to observe (the state of the particle) and the act of observation itself are modelled. Classical logic is not a "logic of observations" since connectives are static operations, manipulating frozen objects oblivious to observation. This is why classical logic cannot formalize precisely any description or reasoning associated with quantum world: in other terms, there is no definite direction of time. For example, in classical logic A A B means that the observer sees both A and B at the same time. From a proof-theoretic viewpoint, in a Gentzen-type sequent calculus, the structural rule (Le. a rule of inference which does not involve any connective), which is harmless in both classical and intuitionistic logic but responsible for the simultaneity of observations, is the exchange rule(6):
where A, B are formulas and rand Ll finite sequences of formulas of the language C. This rule says that the order of formulas on the left side (antecedent) and right side (succedent) of the sequent symbol =} is immaterial. In terms of physical observation, in a weak reading, the rule (E, L) represents the commutativity of observations: the order of such observations is immaterial to the outcome. In a strong reading this rule says that the observations are simultaneous. Another structural rule of classical logic that is misleading from the
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point of view of observation is the contraction rule:
r :::} ~,A, A (C R) r :::} ~,A '
A,A,r:::} ~ (C L) A, r :::} ~ ,
.
This rule says that the number of times we carry out an observation has no effect on the outcome. The first step for a logical foundation of QM is then given by removing the exchange rule and contraction rule from our sequent calculus: this means that the 2-place logical operation @ (/\ and & in resp. Srinivas's and Mulvey's notation)(7), which corresponds to the comma in (E, L), and in (C, L), is no longer commutative and idempotent. The validity of A@B, as in Srinivas' idea, is to be regarded as "we have verified A, and then we have verified B". Thus, we have the noncommutative (and not idempotent) conjunction @. The sequential conjunction A @ B means "A at time tl and then B at time t2". The sequent calculus rules for @ are the following: r 1 :::} ~I,
A, ~2
r 2 :::} ~3,
B, ~4
r 1 , r 2 :::} ~3, ~l' (A@ B), ~4' ~2
R)
(
@,
if ~2 = ~3 = 0, or ~2 = r 1 = 0, or ~3 = r 2 = 0. Dually, we have the noncommutative (and not idempotent) disjunction * (par) which corresponds to the comma in (E, R) and in (C, R) : A*B means sequentially "A at time tl or then B at time t2". The rules for are:
*
A, r
2 :::} ~l
r 3 , rI,
(A*B)'
r l,
f3,
B, r
4 :::} ~2 (
*,
L)
~l' ~2 if r 2 = r3 = 0, or r 2 = ~1 = 0, or r3 = ~2 = 0. We might also think of r, A, B as events (or experimentally r 4, r 2 :::}
verifiable propositions of a statistical physical theory): the happening of the events r, A, B in that order causes ~ to happen. Beyond any particular kind of informal interpretation of the formulas, it should be stressed that, in absence of the exchange rule, a sequent may represent time-ordered strings of observations on a sub-microscopic scale, as well as events or single-time propositions describing what may occur in a quantum system: we shall read A @ B as "A precedes B" (or "B follows A"). If it a pair of formulas A, B exists such that A @ B ~ B @ A then we have a sort of closed timelike loop(8). The full removal of the exchange rule from classical logic (and contraction and weakening also) leads to different negations in noncommutative linear logic: the postnegation and the retronegation .L(_) (see Abrusci, 1991)(9):
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(_)1. -rules:
r => ~,A
r, A1. =>
~
((_)1. L)
,
A, r => ~ ((_)1. R) r => A1., ~ "
.
1. (_) -rules:
De Morgan's laws
= c1.*b1.
e
(b*c)1. = c1. 0 b1.
e 1.(b 0 c) =1. c*1.b
e
1.(b*c)=1. c0 1.b
e (b 0 c) 1.
Thus, we obtain also two different implications: the postimplication (or right implication): A - e r B = A1.*B and the retroimplication (or left implication): A ---I B = B*1. A. Intuitively, since noncommutativity has an intrinsic temporal meaning the postimplication ~r may denote forward causality, whereas the retroimplication ~I "retrocausality", i.e. the causal influence back in time, from future to past. The postimplication assumes a situation A at time tl and predicts a situation B at a later time t2; the retroimplication assumes the situation A at time t2 and reconstructs the past by concluding with the situation at an earlier time t l . As Omnes stresses in a different context "this kind of results only holds for a system having a regular dynamics" and "dynamically regular systems behave in an (almost) deterministic way and one has recovered many cases where determinism holds, although one never left QM" (Omnes, 1990). The phenomenological interpretation of postimplication may be summarized in this way: A post-implies B when the probability measure of B is equal to 1, once A is granted to happen. Since probability does not have an inherent direction in space or time A retro-implies B is interpreted analogously. It is worth stressing that retrocausality which describes a time-reversed process, is crucial in QM since retrocausality is a way of reacting to a consequence of Bell's theorem: any interpretation of QM in terms of classical logic must be nonlocal. If we remove the exchange rule from our sequent calculus, it would nevertheless be absurd not to recover it in some way since otherwise we would have a logic less expressive than classical logic. To get the strength of classical logic back the exchange modalities: I> (read everywhere) and - rules:
