Quantum mechanics presented as harmonic analysis

Abstract

This paper grew out of a desire on the part of its author to be able to explain, for philosophy, the significance of Quantum Mechanics. The traditional formulation, based on Hilbert spaces and operators on them, leaves much to be desired if, for example, one is interested in the theory of physical measurement. (It may fairly be asked what the operation on a state function of partial differentiation really has to do with the actual business of measuring the momentum associated with that state function.)

Of great interest here, of course, are the uncertainty relations. They have done much to still the deterministic belief that nature is, at least theoretically, ultimately controllable. They have conjured up again the venerable questions about form versus substance in the theory of matter. They are, then, as fundamental to philosophy today as any other contemporary issue. And yet their explication is deeply problematical because of the relatively complicated form of the usual presentation of these relations. There will be more on this topic in the second section, together with what I believe to be a ‘simplest-possible’ reformulation.

The reformulation of Quantum Mechanics which is presented below takes its inspiration from the point of view that measurement (or physical knowledge) requires conservation laws, and that these in turn invariably involve symmetry, and hence groups. Thus, as with the uncertainty relations, a concomitant part of knowledge is an associatedinability to know which arises out of a symmetry group.

Less than that is accomplished here. However, Quantum Mechanicsis brought significantly closer to group theory, and ease of interpretation. Further, the techniques are sufficiently simple, mathematically, and in terms of justification of the methods used, that they might well be substituted for the traditional approach for the teaching of this subject to undergraduates.