Calibrated unitary groups
Abstract
An operator-norm differentiable unitary group determines a Hamiltonian after independent calibration.
Theorem 1.1 (The generator determines the exponential evolution).
Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/CalibratedUnitaryGroup.calibrated_group (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Vittorio Giovannetti; Seth Lloyd; Lorenzo Maccone (2015). Quantum Time. DOI: 10.1103/PhysRevD.92.045033. URL: https://arxiv.org/abs/1504.04215.
Commentary.
Let H be a complete complex inner product space. Let U(t) be a continuous complex linear operator for each real t, with U(0)=I, U(s+t)=U(s)U(t), and U(t)*U(t)=I for every t. Assume the operator-valued map has real derivative G at zero in the operator norm. These assumptions include the original finite-dimensional differentiable unitary group, since finite-dimensional complex inner product spaces are complete. They do not assume a Hamiltonian or its equation of motion.
The group law writes U(s)=U(s-t)U(t). Differentiating at s=t proves U’(t)=G U(t) for every real t. Differentiating U(t)U(t)=I at zero gives G+G=0. The derivative of F(t)=exp(-tG)U(t) is therefore zero, and the mean-value constancy theorem gives F(t)=F(0)=I. Multiplying by exp(tG), whose product with exp(-tG) is I by the commuting exponential identity, proves U(t)=exp(tG). These steps construct the generator relation and the exponential representation from the original assumptions.
For an independently supplied nonzero real hbar, define Hamiltonian=i hbar G. Skew adjointness of G makes Hamiltonian self-adjoint. Scalar normalization gives U(t)=exp((-i t/hbar) Hamiltonian) and U’(t)=(-i/hbar) Hamiltonian U(t). Applying the continuous evaluation map at any initial vector psi gives the Schrodinger equation for U(t)psi; the evaluation is an exact compiled application, not another retained theorem. The original finite history, clock conditioning and all-input restoration applications remain in FinitePropagationHistory. Neither a continuous time parameter nor hbar follows from discrete gate count, and this theorem makes no claim about an unbounded generator of a merely strongly continuous group.
Giovannetti, Lloyd and Maccone discuss conditional-clock dynamics. The bounded, norm-differentiable generator argument here is a formal reuse and synthesis of standard differentiation, adjoint and exponential results, not a literature novelty claim.