The Emergence of the Work Operator

In the autonomous description, total energy is conserved. Work therefore has an unambiguous meaning there: it is simply the energy that the clock gives to the system. At the quantum level, this transferred energy is not a number but an operator: the change in the clock Hamiltonian, viewed in the Heisenberg picture.

By translating this transfer onto the system alone, one proves that there exists a unique observable reproducing its statistics — the work operator:

   Ŵ(t,t₀) = Û†(t,t₀) Ĥ_S(t) Û(t,t₀) − Ĥ_S(t₀)
           = ∫_{t₀}^{t} ds  Û†(s,t₀) ∂s Ĥ_S(s) Û(s,t₀)

Read it this way: we carry the final Hamiltonian Ĥ_S(t) "backward" to the initial time (via the evolution Û), then subtract the initial Hamiltonian from it. The result is an observable evaluated on the initial state. The parameter Δt = t − t₀ does not denote the time of a measurement: it sets the duration of the process over which the energy current is accumulated.

This current also has a local expression, the work current:

   ŵ(t) = ∂t Ĥ_S(t)          (instantaneous work current)

Two essential ideas. First, the work operator is not postulated: it emerges from honest energy accounting in the autonomous framework. It was not chosen to satisfy some property; it is imposed by the sole requirement of measuring the transferred energy. Second, since energy is an operator, this work is recorded not only in the energy populations, but also in the coherences — a richness that definitions based on a projective measurement destroy from the outset. Work ceases to be a mere measurement record: it becomes an observable of the system again.