An invertible interface between two finite quantum-mechanical models

Use natural units $\mathrm{\hslash }=1$ on the Euclidean-time interval $[0,T]$, $T>0$. Let the source model ${\mathcal{T}}_{-}$ have Hilbert space ${\mathcal{H}}_{-}={\mathbb{C}}^2$, and the target model ${\mathcal{T}}_{+}$ have Hilbert space ${\mathcal{H}}_{+}={\mathbb{C}}^2$, both with the standard positive inner product and all complex $2\times 2$ matrices as local operator insertions. Their Hamiltonian operators $H_{\pm }\in \mathrm{End}({\mathcal{H}}_{\pm })$ are self-adjoint, with matrices

$H_{-}=\left(\begin{array}{cc}0& 0\\ 0& \mathrm{\Delta }\end{array}\right),H_{+}=\left(\begin{array}{cc}\mathrm{\Delta }& 0\\ 0& 0\end{array}\right),\mathrm{\Delta }>0.$

Here $\mathrm{\Delta }$ and the Hamiltonians $H_{\pm }$ have units of energy; $T$ and each Euclidean-time interval $s$ have units of inverse energy. Propagation is $e^{-s{H}_{\pm }}$. Define the interface map

$V=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right):{\mathcal{H}}_{-}⟶{\mathcal{H}}_{+}\mathrm{.}$

The map $V$ is dimensionless by convention. The target observable corresponding to $O_{-}\in \mathrm{End}({\mathbb{C}}^2)$ is $O_{+}=V{O}_{-}{V}^{†}$. The combined model's boundary data are dimensionless unit vectors ${\psi }_{-}\in {\mathcal{H}}_{-}$ and ${\psi }_{+}\in {\mathcal{H}}_{+}$ for which

$Z=⟨{\psi }_{+},e^{-T{H}_{+}}V{\psi }_{-}⟩\mathrm{\ne }0.$

For each interface position $\tau \in (0,T)$, all declared other insertions are finite ordered lists $(t_i,O_i)$ with distinct times $0<t_1<\cdots <t_n<T$, where $O_i\in \mathrm{End}({\mathcal{H}}_{-})$ if $t_i<\tau$ and $O_i\in \mathrm{End}({\mathcal{H}}_{+})$ if $t_i>\tau$. An insertion at $t_i=\tau$ is not admissible. Each $t_i$ and $\tau$ has inverse-energy units, and the insertion matrices are dimensionless by convention. For every allowed sequence of interfaces, the boundary pair is admissible exactly when its empty-list transfer amplitude is nonzero; all correlators are divided by that amplitude. In particular, for a reverse pair the no-insertion amplitude must have a nonzero identity-interface limit. Define the transfer map from time $s$ to time $t$, $0\le s\le t\le T$, by

$E_{\tau }(t,s)=\{\begin{array}{ll}{e}^{-(t-s)H_{-}},& s\le t\le \tau ,\\ e^{-(t-s)H_{+}},& \tau \le s\le t,\\ e^{-(t-\tau )H_{+}}V{e}^{-(\tau -s)H_{-}},& s<\tau <t\mathrm{.}\end{array}$

For any finite ordered sequence of interfaces with matching source and target spaces and any finite ordered list of other insertions, the unnormalized transfer amplitude is the matrix element between its external boundary vectors of the ordered product of: each interval propagator $e^{-s{H}_{-}}$ or $e^{-s{H}_{+}}$ on the model occupying that interval, each interface map $V$ or $V^{†}$ at its crossing time, and each ordinary insertion matrix at its stated time. Its empty-list amplitude is the same product with all ordinary insertion matrices omitted. The admissibility and normalization rule above therefore defines every declared correlator with a reverse pair and with any finite list of other insertions, not only the single-interface correlator below.

The empty-list transfer amplitude is

$Z_{\tau }=⟨{\psi }_{+},e^{-(T-\tau )H_{+}}V{e}^{-\tau H_{-}}{\psi }_{-}⟩=Z,$

where the equality follows from $H_{+}V=V{H}_{-}$.

More generally, with the ordered list above, it is

$C_{\tau }((t_1,O_1),\dots ,(t_n,O_n))=\frac{⟨{\psi }_{+},E_{\tau }(T,t_n)O_n{E}_{\tau }(t_n,t_{n-1})\cdots O_1{E}_{\tau }(t_1,0){\psi }_{-}⟩}{Z},$

with the empty-list numerator interpreted as $⟨{\psi }_{+},E_{\tau }(T,0){\psi }_{-}⟩=Z$. Thus every finite ordered list of ordinary insertions on both sides, and every admissible boundary pair, has a specified correlator. The source-to-target transport sends each source-side insertion $O_i$ to $V{O}_i{V}^{†}$ and the initial state ${\psi }_{-}$ to $V{\psi }_{-}$; target-side insertions and the terminal vector ${\psi }_{+}$ are already target data. With ${\widehat{O}}_i=V{O}_i{V}^{†}$ for $t_i<\tau$ and ${\widehat{O}}_i=O_i$ for $t_i>\tau$, the corresponding target-only matrix element uses the same transfer normalizer $Z$ and agrees with $C_{\tau }$ by $e^{-s{H}_{+}}V=V{e}^{-s{H}_{-}}$. The usual topology on $\mathbb{C}$ is the value-space topology. This is an invertible topological interface between two specified models. It is an internal symmetry only if the source and target models are identified as one complete model.

Carqueville, §2.1, PDF pp. 5–8