Is Quantum Logic Really Logic

Is Quantum Logic Really Logic? Michael R. Gardner

Philosophy of Science, Volume 38, Issue 4 (Dec., 1971), 508-529. Stable URL: http://links.jstor.org/sici?sici=0031-8248%281971 12%2938%3A4%3C508%3AIQLRL%3E2.O.CO%3B2-U

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IS QUANTUM LOGIC REALLY LOGIC?* MICHAEL R. GARDNERt Haruard University Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truthfunctionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering (“implication”) to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially ordered set of propositions as a logic-i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way.

1. Logic and Necessity. There is a sense of “necessary” according to which a sentence is a necessary truth just in case any evidence whatever would confirm it, i.e., just in case it could never be rational to abandon belief in its truth. The laws of logic are surely the prime candidates for necessity in this sense. If it could be shown that we already possess good grounds for abandoning some of the standard logical laws, they would be debarred from candidature for necessity. It would then become doubtful that any sentences at all are qualified aspirants for that lofty status. Some physicists and philosophers have in fact suggested that certain quantum-mechanical paradoxes have refuted the distributive laws, and that quantum theory ought therefore to embody a nondistributive logic. Their claims seem to support Quine’s famous suggestion [24] that the laws of logic are not necessary (or “analytic”).

10

Light source

2-

I 1

CR

1

Let P(A,, R) be the probability that a photon which passes through slit i (i = 1,2) strikes the small region R. Let P(A,) be the probability that a photon emitted by the

* Received January,

1971.

f Present address: Dept. of Philosophy, Mount Holyoke College, South Hadley, Ma. This

paper is based upon part of a Ph.D. dissertation, Harvard University, 1971. I wish to acknowledge helpful conversations with Jeffrey Bub, Imre Lakatos, Sir Karl Popper, Hilary Putnam, Abner Shimony, and Lawrence Sklar.

508

IS QUANTUM LOGIC REALLY LOGIC ?

509

source passes through A t , and suppose the emission pattern is symmetric so that P(A1) = P(A2) = 1/2P(A1 V AS). Then

We can measure P(A, v A2, R ) by counting the number of photons striking R in a long time-interval and dividing by the number striking the entire screen-i.e. the number passing through either 1 or 2. Since the value of P(A,, R) should presumably be unaffected by closing 2, we can measure it by counting the arrivals during another interval during which only 1 is open. P(A2, R ) is measured with only 2 open. The paradox is that the values thus obtained do not satisfy the equation derived above (following Putnam [21]).Graphed as a function of x (the position of R) the two distributions look approximately like this:

Fig. 2

Fig. 3

The second paradox relates to the quantum-mechanical phenomenon of tunneling, or barrier-penetration. Suppose that the potential energy of a particle is a

510

MICHAEL R. GARDNER

function of position similar to that shown below :

i I Fig. 4

and that a particle with total energy E less than Vo (the value classically required to escape) is initially inside the well. According to quantum mechanics, the probability is positive that the particle will later be found outside the well, even though to arrive there it must pass through a region in which its potential energy exceeds its total energy, which is impossible. The escape of alpha particles from uranium nuclei provides experimental confirmation of this result. The energy E of the escaped particles may be measured in the region V N 0 (where E = the kinetic energy) and the paradoxical relation E < Vo thereby verified (cf. Reichenbach [27], p. 165). The last of our initial set of paradoxes is due to Heisenberg ([lo], p. 33). The potential energy of an atom with a single orbital electron looks roughly as follows : V

Fig. 5

If the system is in an eigenstate of the Hamiitonian and the eigenvalue is E, the electron presumably cannot move out beyond the point xo at which V = E. But according to quantum theory, the probability is positive that the electron will be found at any arbitrarily large distance from the nucleus. In any sufficiently large sample of atoms with energy E,there will therefore be some in which the potential exceeds the total energy, just as in the second paradox. When he first introduced the paradox, Heisenberg sought to resolve it by arguing that any measurement which could localize the electron in a region beyond xo would introduce additional energy into the system and thereby prevent a violation of conservation of energy. But as Reichenbach points out in connection with ana-

51 1

IS QUANTUM LOGIC REALLY LOGIC?

logous attempts to dismiss the tunneling paradox, the problem is to explain how the electron could get far enough from the nucleus to be detected by an apparatus well beyond xo-e.g. a mile away. The energy it has after it interacts has no bearing on the question of how it got to the apparatus at all. 3. Propositions and Subspaces. The proposal to adopt a nondistributive logic for quantum theory was originated by Birkhoff and von Neumann [4], and has more recently been supported and developed by such writers as Mackey, Jauch, Piron, Varadarajan, Finkelstein, and Putnam. Essentially, the proposal is that we postulate an imbedding (order-preserving map) of the system of “experimental propositions” of quantum theory into an appropriate Hilbert space. Certain relations and operations are defined for the system of subspaces and then correlated with logical relations and operations. The laws of the new logic may then be, as it were, read off from the Hilbert space. As it turns out, the analogs of the classical distributive laws,

a A (b v c) t+ (a a v (b A c) f-) (a

A V

b) v (a A c) b) A (a V c)

are false in the system of subspaces. Accordingly, quantum logic differs from standard logic most conspicuously in its lack of distributivity. The Hilbert spaces $I used in quantum mechanics are separable, complete linear vector spaces over the complex numbers, possessing an inner product (x, y ) and norm (x, x). Separability means that there is a sequencef, E H such that each vector in H is a sum of the form C:=, a, f,. Completeness means that any Cauchy sequence of vectors of H converges (in the norm) to a unique vector of H. A subset M of H i s a linear manifold if it is closed under addition and scalar multiplication, and a subspace if in addition it contains any vector to which any sequence of members of M converges. If M is a subspace, its orthocomplement M is the subspace containing all vectors orthogonal to all members of M . The subspaces are partially ordered by the subset relation. This ordering permits the following definitions: if ( M i ) is a family of subspaces, its span UzM iis the least subspace containing all the M ilub {Mi).Its set-theoretic intersection M iis the greatest subspace contained in each of the M , - glb (Mi}. It can be shown that any subspace and its orthocomplement span the entire space. The set L of subspaces of H forms a complete lattice: i.e. it has a maximal and a minimal element, and Mi and M iexist for all subsets {M,} of L. The quantum-logical operations are not defined truth-functionally, as in standard and even Reichenbachean (three-valued) logic, but in terms of a relation 0, to be defined later, which is a partial ordering of certain formulas. The atomic formulas are expressions of the type “A = a” or “A E E,” where A is an observable, a is a real number, and E is a half-open interval on the real line. The formulas are thought of as predicates of quantum-mechanical systems. “p, = 6” is true of the system S iff the y component of S’s momentum is 6 . Two formulas are equivalent iff they bear @ to each other. The equivalence classes of formulas are called ‘(experimental) propositions’ by quantum logicians, and the formulas they contain

ni

ui

nt

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MICHAEL R. GARDNER

are their names. They might better be called ‘properties’, since they are classes of predicates rather than of sentences. Using schematic letters in place of formulas, we can define an ordering < ,called implication, of the propositions:

p < =&f ‘p’ @ ‘q’, Since propositions are equivalence classes of formulas, mutual implication of propositions is actually identity:

p =q

EE

( p < q and q < p ) .

The disjunction v f p tof a class of propositions is its least upper bound, and the conjunction AtpLis its greatest lower bound. We assume that these bounds always exist. It follows that there is a proposition, called ‘l’, which is implied by every other, and one, called ‘O’, which implies every other. (‘0’is also used to denote the subspace consisting solely of the zero vector.) We assume there is an operation ‘-’ satisfying the usual properties of negation : - -a=a - a ~ a = O a < b = -b c -a. We can now describe the imbedding 12 of the quantum propositional calculus into the lattice of subspaces. We begin with the propositions which assert that an observable has a particular value : A = a. If a is not an eigenvalue of A , even if it is in the (continuous) spectrum of A , h(A = a ) = 0. If a is an eigenvalue of A corresponding to a single eigenvector $, h(A = a) is the one-dimensional subspace {CI) I c complex}. If a corresponds to several eigenvectors, h(A = a ) is the span of them all. h(A E (c, b]) = the range of the operator (Eb - E J , where {Ex} is the spectral family corresponding to A (Jordan [12], p. 42). We proceed recursively to define the imbedding for molecular propositions :

u*

h(ViPJ = NPi) NA, P J = N - P ) = h(P)l. For reasons to be explained later, we postulate that h is order-preserving :

ntmi)

a