QUANTUM LOGIC AND PROBABILITY

Quantum physics predicts many astonishing physical effects that have been subsequently observed in the laboratory. Perhaps the most significant effect is the violation of Bell's inequality, which implies a failure of classical locality. But the most widely known bit of quantum magic is the experiment of Clinton Davisson and Lester Germer demonstrating interference effects for electrons. Richard Feynman said this is a phenomenon "which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery" (1963–1965, Vol. 3, p. 1-1). As we will see, Feynman somewhat overstates the case, but let us first try to get in his frame of mind.

The interference effect is illustrated by the two-slit experiment. If we send a plain water wave toward a barrier with two narrow slits in it, we find that a circular wave is produced on the far side of each slit. As these two circular waves expand, they eventually overlap and interfere. Where the crest of one meets the crest of the other, we get a crest of twice the height; where the trough of one meets the trough of the other, we get a trough of twice the depth; and where the crest of one meets the trough of the other, the waves cancel out. This creates interference bands : regions of extreme agitation where the waves meet in phase, crest-to-crest and trough-to-trough, juxtaposed with quiescent regions where the waves meet out of phase, crest-to-trough. The pattern of regions of high and low activity is easy to calculate. Notice, in particular, that there are places where one would observe wave motion if either slit alone were open, but where there are no waves when both slits are open, because of destructive interference.

In quantum theory, a wave function represents the physical state of an electron, and for a single electron the wave function is mathematically similar to a water wave. (It is not exactly the same, since it is a complex-valued function. Moreover, this analogy works only for a single electron. The wave function for a pair of electrons is defined on the configuration space of the system, which has more dimensions than physical space.) The dynamics of the wave function is similar enough to the dynamics of water waves to display the same interference effects. That is, in the case of a single electron shot at a screen with a single slit, the wave function that makes it through the slit spreads out on the far side in a sort of circular pattern. And in the case of a single electron shot at a screen with two slits, the wave function that gets through spreads out in two circular patterns, one centered at each slit, and these interfere where they overlap, just like the water waves.

Of course, when we actually look for a single electron, we never find it spread out; we always find it at some localized place. We can use the wave function to make predictions about where the electron will appear by squaring the wave function and interpreting this value as the probability that the particle will be found at a particular location. If we do many identically prepared experiments, we find that the distribution of the electrons matches the square of the amplitude of the wave function, thereby confirming the predictive accuracy of quantum mechanics. But the mystery is this: To get these interference effects, we do not have to send many electrons at the same time. We can send the electrons through the device one at a time, with long gaps between them, and watch the interference bands build up slowly, dot by dot. So it is not that different electrons are somehow interfering with each other ; it is rather that each electron is somehow interfering with itself.

To make the effect even more vivid, consider this fact. We send electrons through the slits one at a time and watch for flashes on a distant screen. There are particular areas on the distant screen where we will sometimes see flashes when only the right slit is open, and sometimes see flashes when only the left slit is open, but never see flashes (because of destructive interference) when both slits are open. So each electron sent through when both slits are open must somehow be physically influenced by the fact that both are open, since that region is only forbidden when both are open. But, to put the question fancifully, how can an electron "know" that both slits are open if (being a tiny particle) it only goes through one slit?

This would appear to be a straightforward physical question that calls for a physical answer. And indeed, two different physical answers are available, corresponding to the two straightforward ways to interpret the quantum formalism. On the one hand, if one thinks that the wave function is complete, that is, that it encodes all the physical characteristics of the electron, then one will simply reject that claim that the electron is a tiny particle that can only go through one slit. If the wave function is complete, then when the wave function spreads out enough to go through both slits, the electron itself spreads out enough to go through both slits, and that is how it can interfere with itself. This leaves a mystery, but the mystery is not why there is interference. Rather, it is why the electron makes a small, localized flash on the far screen. This problem is solved, in this approach, by giving an account of wave-function collapse.

The second physical answer maintains that the electron is indeed a tiny particle that always has a well-defined location, and hence goes through one slit or the other. On this approach, the wave function is not complete, since it does not indicate what that position is. This account is realized in the "pilot wave" theory of Louis de Broglie and David Bohm. As John Bell has written, "While the founding fathers agonized over the question: 'particle' or 'wave', de Broglie in 1925 proposed the obvious answer: 'particle' and 'wave'." That is, in the view of de Broglie and Bohm, there is, in addition to the located particle, a wave function that guides the trajectory of the particle. The state of the wave function is influenced by the fact that both slits are open, in exactly the way the quantum formalism indicates. So each particle "knows" that both slits are open, even though it goes through only one, because the wave function "knows" that both slits are open, and the wave function guides the particle.

These two physical answers to the puzzle of the interference bands are perfectly adequate, and evidently require no adjustments to classical logic or probability theory. The solution of de Broglie and Bohm is even, in certain sense, a classical solution, contrary to Feynman's worry. So there is nothing in the phenomena discovered by modern physics that could require us to abandon or modify classical logic or probability theory.

Nonetheless, there have been many attempts, of various sorts, to argue that a change in logic or probability theory is at least suggested by the mathematical form of quantum theory, or that a change in logic or probability will produce an interpretation that is both physically adequate and somehow preferable to the two physical solutions outlined above.

At this point one would like a clear account of how classical logic or probability theory might be changed, and how the change might help us understand phenomena like the two-slit experiment without recourse to the sorts of physical hypotheses discussed above (hypotheses that, by the way, are already used to solve the measurement problem in quantum theory). Unfortunately, no such clear account is possible, because despite a long history and many attempts, no such account has ever been produced. So in its place, we must search instead for the reasons that anyone ever thought that classical logic or probability theory is responsible for the "mystery" surrounding these phenomena.

There are several different routes that can lead us to call into question classical logic. One, followed by Feynman in his famous Lectures on Physics (1963–1965), proceeds by reasoning about the two-slit experiment. The other, which is the foundation of the technical field of quantum logic, proceeds from an analysis of the mathematical machinery of quantum theory. Let us examine these in turn.

In his analysis of the two-slit experiment, Feynman first introduced proposition A:

Proposition A. Each electron either goes through slit 1 or goes through slit 2.

Feynman then went on to consider what he calls the consequences of this proposition for predictions about the results of the experiment. If proposition A is true, he said, then we ought to be able to calculate the probability that the electron will land at any point of the screen by first determining the probability for electrons that go through slit 1 (by blocking slit 2 and seeing what happens), and then determining the probability for electrons that pass through slit 2 (by blocking slit 1 and seeing what happens). If proposition A it true when both slits are open, Feynman said, then the individual probabilities derived from these experiments should add. With both slits open, there are more ways for any result to come about (since an electron can get to a certain spot either by going through slit 1 or by going through slit 2), and the chance of the result should be just the sum of the chances of each process. This is, of course, not what we see. Because of the interference, there are places on the far screen where electrons appear with either slit open, but where no electrons appear with both slits open. Feynman concluded, "When one does not try to tell which way the electron goes, when there is nothing in the experiment to disturb the electrons, then one may not say that an electron goes through either hole 1 or hole 2. If one does say that, and starts to make deductions from the statement, he will make errors in the analysis." That is, Feynman concludes from considerations of how the probabilities ought to add that proposition A is not true.

We can equally well present Feynman's dilemma using only logic rather than probability theory. There are places on the screen where an electron can appear when only slit 1 is open (and the electron goes through slit 1) and also when only slit 2 is open (and the electron goes through slit 2). So if the electron goes through slit 1, it can appear at a certain point, and if it goes through slit 2, it can appear at that same point. From the premise that the electron either goes through slit 1 or goes through slit 2, it then follows by a disjunctive syllogism that it can appear at that point. But with both slits open, the electron cannot appear at that point. It seems to follow that when both slits are open, the disjunction is not true. It is not the case that the electron went through slit 1 or went through slit 2.

Something must have gone wrong with Feynman's analysis somewhere. For in the theory of de Broglie and Bohm, electrons always have exact locations, and every electron that gets from the source to the far screen goes either through slit 1 or through slit 2. And the de Broglie and Bohm theory makes all the right predictions: exactly the predictions of quantum theory. Where did Feynman go wrong?

The solution is not hard to seek. Feynman considers first doing an experiment with slit 1 open and slit 2 closed, and then an experiment with slit 2 open and slit 1 closed. So the experimentally confirmed propositions are that if the electron goes through slit 1 with slit 2 closed, it can appear at a certain spot, and that if it goes through slit 2 with slit 1 closed, it can appear at that spot. The relevant disjunction for using disjunctive syllogism is the following: The particle either goes through slit 1 with slit 2 closed or through slit 2 with slit 1 closed. From this disjunction it does indeed follow that the electron can appear at the spot. But this disjunction tells us nothing at all about what can happen with both slits open.

Feynman's thought, evidently, is that if the electron goes through one slit, then it cannot make any difference whether the other slit is open. This is a reasonable conjecture, supported by classical intuitions. But this conjecture is false, and quantum theory shows why it is false: The state of the wave function is influenced by the state of both slits. Indeed, one consequence of quantum mechanics is that the state of the wave function is influenced by the presence or absence of detectors at either slit. Even when both slits are open, a detector at one slit will cause the interference to go away even when the detector does not fire. This is a straightforward mathematical consequence of the dynamics of the wave function. The ultimate physical moral is that one must take account of the entire experimental arrangement when considering what quantum mechanics predicts. As John Bell put it, "When one forgets the role of the apparatus … , one despairs of ordinary logic… . Hence 'quantum logic.' When one remembers the role of the apparatus, ordinary logic is just fine." And the apparatus in question is the whole experimental situation, including elements (such as the presence or absence of detectors that do not fire ) that would be deemed irrelevant in classical physics.

Feynman's argument is a physical argument: It proceeds solely from the observation of experimental results to the (incorrect) conclusion that proposition A cannot be true, since one could deduce a false consequence from it. The field of quantum logic takes the opposite tack. Quantum logicians want to maintain that something like proposition A is true when both slits are open. But since false claims can apparently be deduced from proposition A using classical logic, this requires a change in logic itself.

Quantum logicians tend not to start from experiment, as Feynman does, but from observations about the form of the mathematical apparatus used in quantum theory. In particular, they begin with the observation that the space of all wave functions is a complex vector space. This means that given any pair of wave functions |ψ₁〉 and |ψ₂〉, and any two complex numbers α and β, there exists another wave function of the form α|ψ₁〉 + β|ψ₂〉. Such a wave function is called a superposition of |ψ₁〉 and |ψ₂〉.

Suppose that the wave function of an electron that goes through slit 1 with slit 2 closed is |ψ₁〉, and the wave function of an electron that goes through slit 2 with slit 1 closed is |ψ₂〉. Then when both slits are open, the wave function will be of the form α|ψ₁〉 + β|ψ₂〉, a superposition of |ψ₁〉 and |ψ₂〉. (In particular, in the usual experimental configuration, it will be (1/√2)|ψ₁〉 + (1/√2)|ψ₂〉.) This wave function is evidently neither |ψ₁〉 nor |ψ₂〉. It would not be correct to say, with classical logical connectives, that the electron is either in state |ψ₁〉 or in state |ψ₂〉. So if we allow |ψ₁〉 now to stand for the proposition that the electron is in state |ψ₁〉, and |ψ₂〉 to stand for the proposition that the electron is in state |ψ₂〉, then the classical proposition "|ψ₁〉 or |ψ₂〉" is false when both slits are open.

What the quantum logician does, though, is to introduce a new connective, usually written ∨, that is defined so that |ψ₁〉 ∨ |ψ₂〉 is true whenever the electron is in a superposition of |ψ₁〉 and |ψ₂〉. If one tries to think of ∨ as a sort of disjunction, one can then have a disjunction that is true even though neither disjunct is true—a circumstance that violates classical truth conditions.

More technically, the quantum logician associates propositions with subspaces of Hilbert space, the vector space of the wave function. The "conjunction" of two propositions (written A ∧ B ) is just the intersection of the associated subspaces, and the "disjunction" of two propositions (A ∨ B ) is the span of the subspaces, that is, the subspace consisting of all vectors that can be formed by adding vectors from the two given subspaces. A proposition is true just in case the wave function of the system lies in the associated subspace. So if the wave function of the system is (1/√2)|ψ₁〉 + (1/√2)|ψ₂〉, then the proposition |ψ₁〉 is not true, and the proposition |ψ₂〉 is not true, but the proposition |ψ₁〉 ∨ |ψ₂〉 is true.

With this terminology in place, one can easily show that the set of "quantum propositions" form a non-Boolean (nondistributive) lattice under the operations ∨ and ∧. This is a straightforward mathematical fact about the structure of subspaces of Hilbert space under these operations. There is no nonclassical logic or probability theory here, just standard mathematics.

Of course, if one starts to pronounce ∨ "or" and ∧ "and," then matters can get somewhat confusing. Because the lattice of quantum propositions is nondistributive, (A ∨ B ) ∧ C can be true while (A ∧ C ) ∨ (B ∧ C ) is false. If one presents this fact by saying that "(A or B ) and C " is true while "(A and C ) or (B and C )" is false, then it appears that de Morgan's laws have failed. Hence the supposed need for quantum logic.

If quantum logic is just the study of the structure of subspaces of Hilbert space, then it is a perfectly legitimate, but badly named, enterprise. It is not an alternative to, or replacement for, classical logic, since it studies connectives that are not the classical connectives. Nothing has been shown to be wrong or misleading about classical logic. Rather, the problem lies with our intuitions about experimental conditions, which lead us incorrectly to expect that whether the second slit is open is irrelevant to the behavior of the electron at the other slit. Quantum mechanics shows not that there is anything wrong with classical logic, but rather that the physics of the quantum world is very unlike the physics of Isaac Newton and James Clerk Maxwell. The surprising relevance of experimental conditions is shown by experiments like the two-slit experiment, and the appropriate way to reason about these experiments is, of course, classically.

What about Feynman's proposition A? With both slits open, is it correct or incorrect to say that the electron either went through slit 1 or slit 2? The answer to this question once again depends on physics rather than logic. If the de Broglie and Bohm theory is correct, then the electron always goes through one slit or the other. Retrospectively, one can even tell which slit it went through. Proposition A is therefore true. If one adopts an interpretation according to which the wave function is complete, then the wave function is all there is to the electron, and the wave function "goes through" both slits. Part of it goes through each slit, so it goes neither entirely through slit 1 nor entirely through slit 2. On a truth-functional reading of "or," proposition A is false. As long one is clear about the exact content of any proposition and about the interpretation of quantum theory at issue, classical logic and probability theory work just fine.

See also Bell, John, and Bell's Theorem; Bohm, David; Hilbert, David; Logic, Non-Classical; Maxwell, James Clerk; Newton, Isaac; Non-locality; Quantum Mechanics.

Bibliography

Birkhoff, Garrett, and John von Neumann. "The Logic of Quantum Mechanics." Annals of Mathematics 37 (1936): 823–843.

Feynman, Richard, Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics. Reading, MA: Addison-Wesley, 1963–1965.

Kochen, Simon, and Ernst P. Specker. "The Problem of Hidden Variables in Quantum Mechanics." Journal of Mathematics and Mechanics 17 (1967): 59–87.

Maudlin, Tim. "The Tale of Quantum Logic." In The Philosophy of Hilary Putnam, edited by Yemima Ben Menahem. New York: Cambridge University Press, 2005.

Putnam, Hilary. "Is Logic Empirical?" In Boston Studies in the Philosophy of Science. Vol. 5, edited by Robert S. Cohen and Marx Wartofsky. Dordrecht, Netherlands: D. Reidel, 1968. Reprinted as "The Logic of Quantum Mechanics," in his Mathematics, Matter, and Method. New York: Cambridge University Press, 1976.

Van Fraassen, Bas. Quantum Mechanics: An Empiricist View. New York: Oxford University Press, 1992.

Von Neumann, John. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer-Verlag, 1932. Translated by Robert T. Beyer as Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press, 1955.

Tim Maudlin (2005)