Time-local interaction-picture convention

PDF p. 10: Eqs. (S2.4)–(S2.6)

Assume the Axiom of Choice and Countable Choice. Work in units with $\mathrm{\hslash }=1$. Let ${\mathcal{H}}_S$ and ${\mathcal{H}}_B$ be complex Hilbert spaces, called the system and bath spaces, and let the real time $t$ belong to an interval containing $0$. A unitary on a Hilbert space is a bounded linear operator from that space to itself whose Hilbert adjoint, denoted by $†$, is its inverse. Suppose a prescribed system Hamiltonian $H_S(t)$ has a specified unitary propagator $U_S(t)$ on ${\mathcal{H}}_S$ with $U_S(0)=I$, conventionally written $U_S(t)=\mathcal{T}\exp [-i{\int }_0^t{H}_S(s)ds]$; $\mathcal{T}$ means that later times appear to the left in the ordered expansion. This notation assumes the propagator exists; in particular, it is defined by the norm-convergent ordered series when $H_S$ is a norm-continuous family of bounded self-adjoint operators on a compact interval containing $0$ and $t$.

For any time-local linear system operator $O(t):D(O(t))\to {\mathcal{H}}_S$, with linear domain $D(O(t))\subseteq {\mathcal{H}}_S$, time-local meaning specified at that same time, define its interaction-picture representative by $\tilde{O}(t)=U_S(t{)}^{†}O(t)U_S(t)$; the original operator is recovered by $O(t)=U_S(t)\tilde{O}(t)U_S(t{)}^{†}$. This convention includes time-dependent jump operators, which are operators used to represent dissipative terms, and the Lamb-shift operator, a Hamiltonian correction. For an unbounded $O(t)$, the conjugated operator has domain $U_S(t{)}^{†}D(O(t))$; the inverse identity is an equality of operators with their domains.

Let the bath Hamiltonian $H_B$ be self-adjoint on ${\mathcal{H}}_B$, possibly unbounded, and let $B_{\alpha }:D(B_{\alpha })\to {\mathcal{H}}_B$ be a linear bath operator with linear domain $D(B_{\alpha })\subseteq {\mathcal{H}}_B$. The bath convention is ${\tilde{B}}_{\alpha }(t)=e^{i{H}_{B}t}{B}_{\alpha }{e}^{-i{H}_{B}t}$, where the exponentials are the unitary group supplied by the self-adjoint functional calculus. If $B_{\alpha }$ is unbounded, its transformed domain is $e^{i{H}_{B}t}D(B_{\alpha })$. These formulas specify a change of representation; they do not assert a physical evolution law or establish existence of a propagator for every time-dependent unbounded Hamiltonian.

Norm Dyson series and uniqueness

Stone's theorem: unitary groups and self-adjoint generators

Hilbert space

A bounded linear operator between normed spaces

The Hilbert-space adjoint of a bounded operator

Symmetric, self-adjoint and essentially self-adjoint operators

Unbounded linear operators: domain, graph and extension

Given: The Hilbert spaces, times, specified system propagator, operators and domains in the definition, together with the choice assumptions.

Goal: Verify that the stated conjugations and inverse are well defined, including the stated sufficient case for the system propagator.

1.1 In the norm-continuous bounded self-adjoint case, [F1] applies on the compact interval with $\mathrm{\hslash }=1$ and initial time $0$: its unique unitary solution is the latest-time-left ordered series denoted by the time-ordered exponential. In the more general case, existence and unitarity of $U_S(t)$ are explicitly assumed. By [F2], the bath exponentials exist as bounded unitaries even when $H_B$ is unbounded. [given, A1, F1, F2]

2.1 For a unitary $U$ on a Hilbert space $\mathcal{H}$ and a linear operator $O:D(O)\to \mathcal{H}$ with linear domain $D(O)\subseteq \mathcal{H}$, $U^{†}OU$ is defined exactly on $U^{†}D(O)$. Conjugating it back gives domain $U(U^{†}D(O))=D(O)$ and action $U{U}^{†}OU{U}^{†}x=Ox$ there. Applying this to $U_S(t)$ proves the system formulas. Applying it to $U=e^{-i{H}_{B}t}$ gives the bath formula and its stated domain. Thus all displayed conjugations are well defined on their stated domains. [given, step 1.1, algebra] ∎