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Cesaro dephasing for a discrete trace-class quantum model

Physical scope

A trace-class initial density and a self-adjoint quantum Hamiltonian with a complete discrete energy eigenfamily of finite multiplicity under the declared unitary dynamics; the result is a time-averaged state limit, not Gibbs preparation or pointwise relaxation.

Statement

Assume AC and Countable Choice. Let $H$ be a self-adjoint quantum Hamiltonian bounded below with a complete discrete energy eigenfamily of finite multiplicity, and let $ρ_{0}$ be any positive trace-class density operator of trace one. Let $ℏ > 0$ be the assigned action constant in $J s$ , adopt the unitary dynamics $U_{t} = e^{- i t H / ℏ}$ of Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, and assume the trace-class Gibbs and local moment conditions of General quantum Gibbs normalization and canonical identities that supply this discrete eigenfamily. Then, conditional on the current batch-12 draft theorem Trace-norm Cesàro dephasing for a discrete Gibbs spectrum, the Cesaro state ${\overset{ˉ}{ρ}}_{τ} : = \frac{1}{τ} \int_{0}^{τ} U_{t} ρ_{0} U_{t}^{*} d t$ converges in trace norm as $τ \to \infty$ to $ρ_{d i a g} = \sum_{E} P_{E} ρ_{0} P_{E}$ , where $P_{E}$ is the spectral projection onto the full eigenspace of energy $E$ . Coherences inside degenerate energy blocks remain. The limit need not be Gibbs, and no pointwise-in-time convergence or thermalization is asserted.

Facts & Assumptions

Given: A discrete self-adjoint Hamiltonian with finite-multiplicity energy eigenspaces, a trace-class density $ρ_{0}$ , the declared unitary dynamics, and $ℏ$ in $J s$ .

[F1]

Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution and Assigned energy observable, unitary flow and the two pictures of autonomous evolution: the declared closed-system state evolves by $ρ (t) = U_{t} ρ_{0} U_{t}^{*}$ with $U_{t} = e^{- i t H / ℏ}$ and unitary group law.

[F2]

General quantum Gibbs normalization and canonical identities: under the semibounded self-adjoint and trace-class Gibbs hypotheses with the stated local moment envelope, the Gibbs operator has a complete countable energy eigenfamily with finite multiplicity for each energy; this conclusion remains conditional on its current draft mathematical supplier.

[L1]

Trace-norm Cesàro dephasing for a discrete Gibbs spectrum (current batch 12 draft): under AC, a complete finite-or-countable orthonormal eigenbasis of a semibounded self-adjoint operator, finite-dimensional energy eigenspaces, $ℏ > 0$ , and a positive trace-one trace-class density, the trace-norm Cesaro average converges to $\sum_{E} P_{E} ρ_{0} P_{E}$ . The complete current proof was inspected; its Step 1 readiness is missing, so this premise remains conditional.

[A1]

AC and Countable Choice are assumed through [F2] and the quantum preparation postulate; $τ > 0$ is elapsed time in seconds and every phase $(E_{i} - E_{j}) t / ℏ$ is dimensionless.

[A2]

$ρ_{0}$ is positive trace class with trace one, and $ℏ > 0$ is the fixed action constant in $J s$ .

Proof

Given: The trace-class state, the discrete energy basis, the unitary dynamics, and the current draft trace-norm Cesaro theorem.

1.1F1F2L1A1A2

By [F2], the trace-class Gibbs and local moment assumptions supply a complete countable energy eigenbasis with finite-dimensional eigenspaces; [A1] supplies AC and [A2] supplies the positive trace-one initial density and action-unit convention. Together with the semibounded self-adjoint Hamiltonian in the Statement, these are the hypotheses of [L1], and [F1] specifies the same unitary orbit.

2.1F1F2L1step 1.1∎

Apply the current draft result [L1] to the matched inputs in step 1.1. It yields ${\overset{ˉ}{ρ}}_{τ} \to \sum_{E} P_{E} ρ_{0} P_{E}$ in trace norm with the entire degenerate blocks retained. The limit depends on $ρ_{0}$ , so no Gibbs or pointwise-time conclusion follows. This use remains conditional until [L1] has current Step 1 readiness.

Cesaro dephasing for a discrete trace-class quantum model

Statement

Facts & Assumptions

Proof

Depends on

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