Formalization of the distinguishability heuristics

“The concept of interfering alternatives is fundamental to all of quantum mechanics”[1]; the issue of the (in)distinguishability of these alternatives is equally fundamental. That which distinguishes these alternatives is traditionally called welcher weg (“which way”) information. Unfortunately, this seems to have no formal basis: neither distinguishability nor indistinguishability have commonly recognized definitions, and their effects are dealt with by several rather informal heuristics (RIA and RDA, below). The incompatibility of interference with welcher weg information is sometimes said to be the result of irreducible physical disturbance of observation. Physical disturbance is certainly a possible cause of a loss of coherence—but the failure to observe fringes with an interferometer on a wobbly table is not interesting. It is the irreducibility of this disturbance, not the disturbance itself, which is significant, and that irreducibility appears directly in the formalism of quantum mechanics: “the loss of coherence in measurements on quantum states can always be traced to correlations between . . . the measuring apparatus and the system.”[2] (See also Ref. [3]). A pure state has no correlations with the exterior, and is coherent; this is suggestive of a system which is uncorrelated with any exterior system is in a pure state. In the following section, we show that this statement is, in fact, equivalent to the distinguishability heuristics, and thus formalizes them. In the next section, we present clear statements of the accepted heuristics. In Sec. 3, we define hermeticity, distinguishability, and indistinguishability in terms of probability (independent of quantum mechanics). In Sec. 4, we present, in precise terms, a postulate and show that this postulate exactly replaces the distinguishability heuristics. In Sec. 5, we examine the general case of less-than-full distinguishability. And, in Sec. 6, we obtain an equivalent formalization of the distinguishability heuristics, in terms of the state of the universe.