Thawing the Frozen Formalism: The Difference Between Observables and What We Observe

Summary

Abstract

In a parametrized and constrained Hamiltonian system, an observable is an operator which commutes with all (first-class) constraints, including the super-Hamiltonian. The problem of the frozen formalism is to explain how dynamics is possible when all observables are constants of the motion. An explicit model of a measurement-interaction in a parametrized Hamiltonian system is used to elucidate the relationship between three definitions of observables—as something one observes, as self-adjoint operators, and as operators which commute with all of the constraints. There is no inconsistency in the frozen formalism when the measurement process is properly understood. The projection operator description of measurement is criticized as an over-idealization which treats measurement as instantaneous and non-destructive. A more careful description of measurement necessarily involves interactions of non-vanishing duration. This is a first step towards a more even-handed treatment of space and time in quantum mechanics.

There is a special talent in being able to ask simple questions whose answers reach deeply into our understanding of physics. Dieter is one of the people with this talent, and many was the time when I thought the answer to one of his questions was nearly at hand, only to lose it on meeting an unexpected conceptual pitfall.