Physical interpretation of a supplied constructed quantum model

Adopt the following physical reading only for an explicitly selected model whose mathematical data have actually been constructed. Let $\mathcal{H}$ be a nonzero complex Hilbert space, let ${\rho }_t$ be the model's density operator at a time $t$ at which a measurement is made, and let $\{M_{a,r}:r=1,\dots ,q_a{\}}_{a=1}^m$ be a supplied normalized finite bounded Kraus instrument on $\mathcal{H}$ in the exact sense of Finite bounded Kraus instrument. Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ when using the normalization result of Finite bounded instruments normalize measurement probabilities. Its hypotheses include that ${\rho }_t$ is a density operator and that the instrument really satisfies ${\sum }_{a,r}{M}_{a,r}^{\ast }{M}_{a,r}=I_{\mathcal{H}}$; the supplier's result is a draft with a complete conditional proof, not an independent approval or an apparatus-realization theorem.

Put

$E_a:={\sum }_r{M}_{a,r}^{\ast }{M}_{a,r},{\mathcal{J}}_a(X):={\sum }_r{M}_{a,r}X{M}_{a,r}^{\ast }\mathrm{.}$

As the Born interpretation postulate for this selected model, read

$p_a(t):=\mathrm{tr}({\mathcal{J}}_a({\rho }_t))=\mathrm{tr}({\rho }_t{E}_a)$

as the model probability for outcome $a$. The mathematical supplier proves $p_a(t)\in [0,1]$ and ${\sum }_{a=1}^m{p}_a(t)=1$ under the hypotheses just listed. When $p_a(t)>0$, adopt ${\mathcal{J}}_a({\rho }_t)\mathrm{/}{p}_a(t)$ as the model's conditional state after outcome $a$; at zero probability no normalized update is assigned. These are model-level probability assignments, not observations.

If the selected construction supplies a self-adjoint observable in the sense of Symmetric, self-adjoint and essentially self-adjoint operators and its spectral projection-valued measure as in Projection valued measure, a declared finite Borel partition may be used only after its projections have been constructed and verified as a finite projective instrument. In that case the projections are the single Kraus operators for their outcomes. A supplied bounded effect is read as an outcome effect only as part of a supplied normalized instrument; this item does not construct square roots, a spectral measure, or an instrument from an arbitrary formal observable.

If a selected construction supplies a self-adjoint Hamiltonian $\widehat{H}:D(\widehat{H})\subseteq \mathcal{H}\to \mathcal{H}$ in the sense of Symmetric, self-adjoint and essentially self-adjoint operators, and an evolution family $U(t):\mathcal{H}\to \mathcal{H}$ whose maps are surjective complex-linear and preserve the Hilbert inner product, as specified in Quantum field quantities and compatible SI normalizations, for $t\in I\subseteq \mathbb{R}$ (with $t$ in seconds), adopt the Schrödinger convention $U(t)=\exp (-it\widehat{H}\mathrm{/}\mathrm{\hslash })$ and ${\rho }_t=U(t){\rho }_{0}U(t{)}^{\ast }$ when that construction establishes that ${\rho }_t$ is a density operator. The unitary convention and SI dimensions are those of Quantum field quantities and compatible SI normalizations, where $U(t{)}^{\ast }:=U(t{)}^{-1}$ and $\widehat{H}$ has unit joules. For any supplied complex-linear operator $O:D(O)\to \mathcal{H}$ with specified linear domain $D(O)\subseteq \mathcal{H}$, define its Heisenberg conjugate by $O_H(t):=U(t{)}^{\ast }OU(t),D(O_H(t)):=\{\xi \in \mathcal{H}:U(t)\xi \in D(O)\}\mathrm{.}$ The model supplies these domains and the evolution. This postulate neither constructs the Hamiltonian, evolution family, state, or field representation, nor any model-specific operator domain.

The model declaration identifies the selected species, field normalization, regulator, preparation and measurement data using Quantum field quantities and compatible SI normalizations. It does not assert that every classical action has a quantization, that every formal state is positive, or that any cutoff can be removed. In particular it supplies no interacting continuum four-dimensional QED Hilbert space, scattering operator, asymptotic completeness, LSZ limit, or nonperturbative convergence. A formal coefficient is not an exact probability without an additional controlled approximation map and a constructed model. Agreement with reported data would test the selected approximation and empirical inputs; it would not prove this postulate or the model. No reported data are asserted here.