Operator-Observable Correspondence

Abstract

Many of the problems inherent in quantum mechanics arise out of considerations concerning correlated systems. For example, if A ^M is a macroscopic observable of a system M with eigenstates φ ₁, φ ₂,... and ℬ^S an observable of a system S with eigenstates ψ ₁, ψ ₂,..., one can ask questions about the value of A ^M for the joint system M + S when it is in a correlated state Σc _i (φ _i ⊗ψ _i ). Problems such as Schrödinger’s Cat, Wigner’s Friend, or the celebrated ‘paradox’ of Einstein, Rosen, and Podolsky depend upon questions about correlated states. Another group of problems arise in the context of quantum field theory. For, in dealing with the phenomena of radiation it has become commonplace to treat divergent sums as though they possessed well-defined, albeit infinite, values and to perform mathematically fictitious operations upon equations containing such terms. However, one can find difficulties in more elementary quantum mechanical contexts and it is one aspect of these which I would like to discuss here.