Compatibility and probability

The notion of “incompatibility” of system variables arose with quantum mechanics; it is inconceivable in classical physics. Two expressions of incompatibility from the earliest days of quantum theory are the uncertainty principle of Heisenberg (measurement of one variable causes uncontrollable disturbance in other variables) and Bohr’s “complementarity” (not all variables have simultaneous “reality”). Interference — the non-additivity of probabilities of disjoint alternatives — arises only when a preparation of indistinguishable alternative values is followed by observation of a variable incompatible with those values. In quantum mechanics, incompatibility is expressed by the non-commutativity of the operators corresponding to the variables; very little emphasis has been placed on the expressibility — or meaning — of incompatibility in classical terms. Of course, variables in classical (deterministic dynamics) physics must be compatible — but variables of systems which obey a classically probabilistic dynamics may very well be incompatible, in the sense that they satisfy a probability expression for incompatibility — a probabilistic formula which, in the quantum setting, is equivalent to the non-commutativity of the corresponding operators. Not only have such expressions been known since Lüders’ 1951 paper (at the latest), we now (Kirkpatrick, 2003a,b) have explicit examples of classical systems with incompatible variables — incompatibility is not an exclusively quantal phenomenon. After a very brief review of necessary (and well-known) matters of probability and quantum mechanics (Sec. 2), we present in Sec. 3 improved treatments of two classical expressions of compatibility due to Lüders and one more due to Davies (1976). In Sec. 4, considering all these, we present a favored classical definition of compatibility, and, in Sec. 5, we discuss a classical example of incompatibility which exhibits interference.