Inverse commutation approach to classical and quantum time functions

Abstract

In this paper we propose a formalism in solving the commutation relation [T,H] = iħ without the aid of canonical transformations that gives, after properly defining the domain, a unique solution to it. Formally the time operator can be written as
T = −[H,  ]⁻¹(iħ).
The problem then translates into determining the inverse of the commutator given the Hamiltonian H and then implementing the previous equation. But does the inverse exist? Without qualification the inverse does not exist. To remedy the situation, we note that commutation is linear. The (super)operator [H,  ] then admits decomposition of its domain into the direct sum 𝒟 = 𝒩⊕𝓡, where 𝒩 is the invariant domain of H, and 𝓡 its non-invariant domain. If we restrict the domain of [H,  ] to 𝓡, then we can unambiguously speak of its averse. Restricting the domain to 𝓡 then leads to a unique solution for the time operator T. Hereafter, whenever we speak of the inverse, we mean the inverse in the restricted domain. Now we shall refer to our method as the inverse commutation approach (ICA). In this work we extend the general formalism to determining the classical time function. We will limit ourselves to one dimension, though.