This talk is about quantum logic, and before we engage in the subject proper let me say how much I appreciate the invitation to deliver it that was extended to me by the Japanese Society for the Philosophy of Science.In particular because I feel that such a meeting is really the place that is most adequate for the discussion of these matters.Let us introduce the subject by mentioning a few general points.First of all, logics deals with propositions.Hence we shall be concerned with propositions.But in fact our interest will be exclusively focused on what is-or what should be the simplest type of propositions: the propositions by means of which a physicist expresses what he knows about a physical system.A priori, our subject should then be very simple.Indeed the elementary books about logic tell us that the rules concerning these propositions can be formalized.That, for example, the propositions can be combined by means of two rules, corresponding to and and to or, and that the resulting structure is just that of a Boolean Algebra.We would expect this to hold for any system.However, as you know, a major difficulty already creeps in at this point when we want to deal with atomic systems.This is because a very simple word such as "and" may well have no obvious meaning in that case.And when we try to force a meaning to it was sometimes encounter quite unpleasant surprises, exemplified by the Heisenberg relations.Among the founders of quantum physics the reactions to such a fact were quite varied as is well known.Some of them sort of hoped that the difficulty just came from our using the wrong concepts.For example, it is quite obvious that if we try to describe as a particle something which is really a wave, we are heading for trouble and we are bound to formulate some meaningless propositions.However, this remark did not lead to a satisfactory solution in the end.Other physicists notably the Copenhagen group took a very different standpoint.Substantially, what they said boiled to this: since we cannot "see" the elementary particles, let us refrain from formulating directly propositions bearing on them.Ultimately, there fore, all their propositions bear on the instruments; or should be interpreted in terms of such propositions.This is -it turned out a very good recipe.Of course, propositions bearing on instruments are propositions of classical, macroscopic physcis.As such, they fall -57-
This talk is about quantum logic, and before we engage in the subject proper let me say how much I appreciate the invitation to deliver it that was extended to me by the Japanese Society for the Philosophy of Science.In particular because I feel that such a meeting is really the place that is most adequate for the discussion of these matters.Let us introduce the subject by mentioning a few general points.First of all, logics deals with propositions.Hence we shall be concerned with propositions.But in fact our interest will be exclusively focused on what is-or what should be the simplest type of propositions: the propositions by means of which a physicist expresses what he knows about a physical system.A priori, our subject should then be very simple.Indeed the elementary books about logic tell us that the rules concerning these propositions can be formalized.That, for example, the propositions can be combined by means of two rules, corresponding to and and to or, and that the resulting structure is just that of a Boolean Algebra.We would expect this to hold for any system.However, as you know, a major difficulty already creeps in at this point when we want to deal with atomic systems.This is because a very simple word such as "and" may well have no obvious meaning in that case.And when we try to force a meaning to it was sometimes encounter quite unpleasant surprises, exemplified by the Heisenberg relations.Among the founders of quantum physics the reactions to such a fact were quite varied as is well known.Some of them sort of hoped that the difficulty just came from our using the wrong concepts.For example, it is quite obvious that if we try to describe as a particle something which is really a wave, we are heading for trouble and we are bound to formulate some meaningless propositions.However, this remark did not lead to a satisfactory solution in the end.Other physicists notably the Copenhagen group took a very different standpoint.Substantially, what they said boiled to this: since we cannot "see" the elementary particles, let us refrain from formulating directly propositions bearing on them.Ultimately, there fore, all their propositions bear on the instruments; or should be interpreted in terms of such propositions.This is -it turned out a very good recipe.Of course, propositions bearing on instruments are propositions of classical, macroscopic physcis.As such, they fall -57-
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
For many years the physical interpretation of quantum theory has been dominated by the “wave-particle duality” attitude of the “Copenhagen school.” This point of view is eloquently described in Bohr's collection of essays on the subject. Despite persistent concerns with apparent paradoxes and limitations to this interpretation (as exemplified by Schrodinger's cat, the paradox of Einstein, Podolsky and Rosen, among others), the Copenhagen view persists de facto in the daily life of the modern physicist. Traditional quantum theory has so successfully explained such a vast amount of data in atomic and molecular physics, solid state physics (and to a lesser extent, elementary particle physics) that little doubt can exist concerning its essential validity. As a consequence of the overwhelming practical success of the “Copenhagen interpretation, ” the latter has acquired the status of dogma. For many years, therefore, most physicists have found it expedient to relegate the puzzling aspects of the theory to philosophers, and mathematicians. Nevertheless, a persistent interest in this subject has produced a significant and fascinating literature, reviewed by Jammer. In recent years, concerns over the proper meaning to be ascribed to quantum theory have produced an increasingly deep and incisive series of investigations. These investigations fall generally into two categories: either (1) discussion of the philosophical content of the theory, or (2) analyses of the mathematical variants of the theory and their connection with differing interpretational schemes. Physicists tend to be detached from a commitment to philosophical issues, because of their realization of the transient character of the meanings attached to theories of the day. As a simple illustration of this we can mention the profound differences between nonrelativistic and relativistic quantum mechanics.
Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.
Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to highlight some rather bold points about the `logic' aspect in quantum physics, necessarily restricting when and which logic may be admissible. And one may understand why that path is hardly followed in the literature. The mathematics/formalism of quantum, compared with classical, physics, may be fairly basically characterized by non-commutative algebras replacing commutative. These classically appearing, in fact, in dealing with systems of possibilities (say, all possible planetary motions under gravity of which one is the actual one). In particular, contrary to too common usage, the quantum non-commutativity should make it impossible to simply `transcend' the `system of possibilities' aspect into a `yes-no' logic essential for an `actual world'. One may have the latter only in a `haven' of approximately commutative algebras of `quasi-classical macroscopic observables', and moreover that `yes-no actual world' would plainly be an `extra ingredient' to the base quantum theory itself.
Standard Boolean logic is based on ideas taken from classical, first-order set theory. The first-order logic of quantum mechanics introduced by von Neumann is not a classical one. It is therefore natural to ask whether it is possible to formulate the corresponding theory of quantum sets, their (quantum) topology and higher-order structures. Once we have quantum sets and quantum topologies we can try to quantize essentially anything: phase space, groups, spacetime, etc. In his book Finkelstein reports results of some 40 years of studies in this direction. The book is not addressed to specialists, but to `thoughtful students of physics, mathematics and philosophy', which may justify the fact that this review is written by a person who is not actively working in the field of quantum logic. From this layman's position I must say that the book does not read very well. First of all, the standard language of physics is not naturally suited for speaking about such problems, so Finkelstein invents a new one. Obviously, it is not easy to read a difficult text which, in addition, is written in a language we do not know. It seems that the author should have tried to depart as little as possible from the standard terminology, but he could not resist coining new terms even for things we already know under different names. When I first looked into the table of contents I felt both curiosity and some intimidation since I did not understand the words. To give an example, we find there spinors, cospinors, anti-spinors, anti-cospinors, ambi-spinors, ambi-cospinors, etc. It turned out that they are simply upper- and lower-index, primed or unprimed ordinary spinors, or their second-rank tensor products. Fortunately higher-rank spinors were not needed. The notation also does not help to understand the text. For example, on p 148 the symbol denotes a spacetime index, but three lines later, in equation (11), it is a gauge group parameter, and no comment is made on this. On the other hand, the same object is often denoted in different ways (see, for instance, equation (1) on p 44 and (7) on p 47) and if we add a number of typographical errors the first reading of the book is really tiring. A general philosophy advocated in the book is to use a kind of two-time description which replaces continuous dynamics with discrete algebraic `activity'. The construction begins with a bi-modal logic of initial and final actions and the preclusion (or `occlusion' ) relation ` ' (essentially iff a transition between and is forbidden). A Hilbert space and a field of coefficients are derived from . Discretization of time goes hand-in-hand with the idea that spacetime should be quantized and, finally, an outline of its construction is given. An intriguing consequence of the Finkelstein formalism is the appearance of internal local gauge groups of the standard model just from the spacetime structure. It seems that some results presented in the book are subject to limitations which are not mentioned by the author. For example, the construction based on will not work correctly if the dynamics between the initial and final acts is nonlinear. This is so because mutually precluding acts do not have to correspond to orthogonal vectors if we use nonlinear eigenvalue problems. So details of evolution between initial and final acts may be more important than one would expect naively. The same concerns the proof that transition probabilities must be of the form . Such subtleties can be ignored in standard quantum mechanics but here we aim at generalization. I find some parts of the text misleading. A canonical example illustrating mathematical and physical meanings of the Finkelstein formalism is a Malusian experiment (i.e. the one involving photons and polarizers), but the Bell theorem given on p 251 cannot distinguish between its single-photon and two-photon versions. Physically the two systems are essentially different and a purely classical Aerts model that gives the Malus law for a single particle is known, even though the theorems and discussion on pp 251 - 3 suggest that this is impossible. Perhaps the original work the theorems were taken from clarifies this point but a reader will not find it for two reasons. First, the name of its author is misprinted. Second, there is no reference to the paper. This is quite typical of the book; the results which are regarded as inessential from the viewpoint of the main argument are oversimplified and quoted without references to original works. The clarity of the presentation would be improved if the author did not try to briefly summarize what he knows about physics (and certainly he knows a lot) but, instead, were more detailed about the material that is really new. In spite of all these minor critical remarks I think the book may be inspiring and opens an entirely new perspective on physics. Many of the problems discussed by Finkelstein remain unsolved but the strategy he proposes looks at least promising and, hopefully, some readers will join the author in his brave attack on the most difficult questions of modern science.