Theorem: Exact Hamiltonian evolution

The interaction (eq:HI) has the unitary propagator

It satisfies the composition law and the Schrödinger equation on the common field-number form domain in the doubled Fock realization.

After the source has switched off,

Its adjoint action depends only on the free test class of $F$; the time-ordering phase is common to both source branches.

Proof. All integrals in (eq:phase) are finite by Lemma lem:instant and $|\sigma (k,l)|\le 2\Vert k{\Vert }_{\mathrm{U}}\Vert l{\Vert }_{\mathrm{U}}$. Weyl operators are unitary, so the displayed formula is unitary.

A chronological product of short interactions has factors $W(-sg{k}_{t_j}\Delta t_j)$, with later times on the left. The exact Weyl product gives the phase

Smooth weighted continuity makes its limit (eq:phase); the summed coefficients converge to $K(t,t_0)$. Strong continuity of the Weyl representation gives (eq:propagator). Splitting the same integrals proves the propagator composition law.

The number form domain is invariant under Weyl operators. Indeed the displacement identity for annihilators and the creation/annihilation estimates bound its graph norm after a finite displacement. On this domain, a differentiable one-particle path $r(t)$ obeys

This follows from the Weyl relation and the Stone derivative of $W(\delta r)$; approximation by coherent finite-particle vectors extends it in the form-domain graph topology.

For $r=-sgK$, the scalar derivative term is $\mathrm{i}{g}^2\sigma (k_t,K)/2$. It cancels $\mathrm{i}\dot{\gamma }$, leaving ${\partial }_{t}U=-\mathrm{i}{H}_I(t)U$. This also establishes uniqueness on that domain by comparing two solutions and differentiating their unitary relative propagator.

Finally $\varphi (Pu)=0$, so replacing $F$ by $F+Pu$ does not change $k_F$. A different path phase is an overall phase and leaves the adjoint action unchanged.