Example observable types and scoped meanings

Use the framework-local unit and domain dictionary of Quantum field quantities and compatible SI normalizations and the energy, SI/natural-unit, and finite-matrix conventions of QFT example units, metric, gauge, and pairing conventions. A physical quantity is always typed relative to the selected QFT formulation and its declared model data; its name or units alone do not identify it with a quantity in another formulation.

For the exact finite-regulator class used on this page, the state space is the finite-dimensional Hilbert space $\mathcal{H}={\mathbb{C}}^d$ for an integer $d\ge 1$. Model inputs include the selected Hermitian Hamiltonian matrices, their time-dependent Hermitian interaction when present, a dimensionless coupling parameter, an initial state, and an explicitly supplied finite outcome measurement. A pure initial state is a vector $\psi \in {\mathbb{C}}^d$ with $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; a mixed state is a matrix $\rho ={\rho }^{\ast }$ with $⟨\rho v,v⟩\ge 0$ for every $v$ and $\mathrm{tr}\rho =1$. An exact regulated observable on this finite space is a specified Hermitian matrix $O=O^{\ast }$ with a declared physical unit. The matrix and state are model data; the measurement procedure and any physical rule assigning outcomes are additional data. This definition does not say that an operator symbol by itself is a detector outcome or probability.

A finite probability law on the declared outcome set $\{1,\dots ,m\}$, $m\ge 1$, is a vector $(p_1,\dots ,p_m)$ with $p_a\ge 0$ and ${\sum }_{a=1}^m{p}_a=1$. It is a mathematical type. Assigning that law to a particular preparation and measurement is a separate physical postulate. A prediction derived from an actually constructed regulated model is conditional on those inputs and the adopted rule.

A formal coefficient record is a sequence $(c_n{)}_{n\in \mathbb{N}}$ of specified coefficients, sometimes displayed as the symbol ${\sum }_{n\ge 0}{c}_n{\lambda }^n$. Unless convergence and an evaluation map are separately proved and adopted, the symbol records coefficients only: it is not an evaluated function, exact probability, convergent perturbation series, or error bound. A formal coefficient can be assigned a phenomenological meaning only by a separate, model-specific physical adoption with its approximation scope stated.

An empirical estimate is a source-reported number obtained by analyzing a finite record of measurement outcomes under the stated preparation, apparatus, calibration, sampling, and analysis conditions. Its reported uncertainty and conditions travel with the estimate. It is not the exact model value, and it does not prove the framework used to interpret it. The 2022 Fan et al. report is an example of a source-reported electron magnetic-moment determination; this definition supplies no numerical result, uncertainty reconstruction, or new experimental claim.

Primitive and derived status is formulation-relative. In a selected regulated model, the regulator, Hamiltonian data, coupling, initial state, and supplied measurement are declared inputs; an evolution, amplitude, or probability law is derived only when the construction and adopted rule establish it. In a formal calculation, the specified masses, couplings, coefficient rules, and subtraction prescription are inputs, while its listed coefficients are derived within that formal calculation. An empirical estimate is a reported data-analysis result, not a primitive model parameter or a theorem.

The following units are examples of types, not identifications of quantities: a cross section $\sigma$ is area-valued with SI unit ${\mathrm{m}}^2$; a rate $\mathrm{\Gamma }$ is inverse-time-valued with unit ${\mathrm{s}}^{-1}$; an energy width ${\mathrm{\Gamma }}_E$ has unit joule. Only a separately stated convention may identify ${\mathrm{\Gamma }}_E=\mathrm{\hslash }\mathrm{\Gamma }$. The fine-structure parameter $\alpha$, a dimensionless coupling $g$, and a gyromagnetic factor $g_s$ are dimensionless but have different physical meanings. The name “width,” a common symbol, or matching dimensions does not supply an exponential-decay law or an equality between these parameters.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), and use the finite regulated model type of Example observable types and scoped meanings. Let $d\ge 1$, let $\tau >0$ be a time interval endpoint in seconds, let $\mathrm{\hslash }>0$ have unit $\mathrm{J}\mathrm{s}$, and let $\lambda \in \mathbb{R}$ be dimensionless. Let $H_0=H_0^{\ast }\in M_d(\mathbb{C})$ have unit joule and let $V:[0,\tau ]\to M_d(\mathbb{C})$ be norm-continuous, Hermitian-valued, and joule-valued. Set $V_I(t)=e^{it{H}_0\mathrm{/}\mathrm{\hslash }}V(t)e^{-it{H}_0\mathrm{/}\mathrm{\hslash }},B={\sup }_{0\le t\le \tau }\mathrm{\parallel }{V}_I(t)\mathrm{\parallel },z=\frac{\mathrm{\mid }\lambda \mathrm{\mid }B\tau }{\mathrm{\hslash }}\mathrm{.}$ Define $S_0(t)=I$ and $S_{n+1}(t)=-\frac{i}{\mathrm{\hslash }}{\int }_0^t{V}_I(r)S_n(r)dr,S(t)={\sum }_{n=0}^{\mathrm{\infty }}{\lambda }^n{S}_n(t)\mathrm{.}$ For $W\in M_d(\mathbb{C})$, unitary means $W^{\ast }W=W{W}^{\ast }=I$. Write $U(\tau )=e^{-i{H}_0\tau \mathrm{/}\mathrm{\hslash }}S(\tau )$ and, for $N\ge 0$, $U_N(\tau )=e^{-i{H}_0\tau \mathrm{/}\mathrm{\hslash }}{\sum }_{n=0}^N{\lambda }^n{S}_n(\tau ),R_N=e^z\frac{z^{N+1}}{(N+1)!}\mathrm{.}$

Let $\psi \in {\mathbb{C}}^d$ be a unit initial vector. A supplied finite projective measurement is a family $P_1,\dots ,P_m\in M_d(\mathbb{C})$, $m\ge 1$, satisfying $P_a=P_a^{\ast }=P_a^2$, $P_a{P}_b=0$ for $a\mathrm{\ne }b$, and ${\sum }_{a=1}^m{P}_a=I$. Equivalently, the one-Kraus-per-outcome data $M_{a,1}=P_a$ form a normalized finite instrument in the sense of Finite bounded Kraus instrument. Adopt the constructed-model Born interpretation of Physical interpretation of a supplied constructed quantum model only for this actually constructed finite model and measurement. For its physical reading, also adopt the explicit preparation/evolution assumption [A3]: the initial density is ${\rho }_0:=T_{\psi ,\psi }$, and the model density at the measurement time is ${\rho }_{\tau }:=U(\tau ){\rho }_{0}U(\tau {)}^{\ast }=T_{U(\tau )\psi ,U(\tau )\psi }$. Here $T_{u,v}$ is the rank-one map $x\mapsto ⟨x,v⟩u$ of Finite bounded instruments normalize measurement probabilities. The Dyson result proves that this matrix $U(\tau )$ is unitary; identifying it with the selected model's state update is an adopted physical assumption, not a conclusion about nature. Define the exact model probability and the truncated squared norm by $p_a=\mathrm{\parallel }{P}_{a}U(\tau )\psi {\mathrm{\parallel }}^2,q_{a,N}=\mathrm{\parallel }{P}_a{U}_N(\tau )\psi {\mathrm{\parallel }}^2\mathrm{.}$ Then $p_a\in [0,1]$, ${\sum }_a{p}_a=1$, and $\mathrm{\mid }{p}_a-q_{a,N}\mathrm{\mid }\le (2\sqrt{q_{a,N}}+R_N)R_N\mathrm{.}$ The nonnegative number $q_{a,N}$ is a truncation-derived square, not generally a probability.

For the exact finite interaction matrix write $S(\tau ;\lambda )=I+i\mathsf{T}(\lambda )$ and $\mathsf{T}(\lambda )=\lambda {\mathsf{T}}_1+{\lambda }^2{\mathsf{T}}_2+\cdots$. Use the standard basis $e_1,\dots ,e_d$ for matrix entries, so ${\mathsf{T}}_{fi}:=e_f^{\ast }\mathsf{T}{e}_i$ for $1\le f,i\le d$. For a matrix $A$, define its Hermitian imaginary part by ${\mathrm{Im}}_{\mathrm{H}}A:=(A-A^{\ast })\mathrm{/}(2i)$. Exact unitarity gives $2\mathrm{Im}{\mathsf{T}}_{ii}(\lambda )={\sum }_{f=1}^d\mathrm{\mid }{\mathsf{T}}_{fi}(\lambda ){\mathrm{\mid }}^2,{\mathsf{T}}_1={\mathsf{T}}_1^{\ast },2{\mathrm{Im}}_{\mathrm{H}}{\mathsf{T}}_2={\mathsf{T}}_1^{\ast }{\mathsf{T}}_1\mathrm{.}$ These are finite-regulator coefficient identities, not continuum optical or scattering theorems.

In the two-level case $H_0=0$, $\lambda =1$, and $V(t)=g_E{\sigma }_x$ with $g_E>0$ in joules and ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, the exact transition probability from $e_1=(1,0)$ to the projector $P=\mathrm{\mid }{e}_2⟩⟨e_2\mathrm{\mid }$ is ${\sin }^{2}z$, where the previously defined $z$ equals $g_E\tau \mathrm{/}\mathrm{\hslash }$ because $V_I=V$ and $\mathrm{\parallel }{\sigma }_x\mathrm{\parallel }=1$; the first-order squared transition coefficient is $z^2$ and may exceed $1$. For all real $z$, $\mathrm{\mid }{\sin }^{2}z-z^2\mathrm{\mid }\le \frac{\mathrm{\mid }z{\mathrm{\mid }}^4}{3}\mathrm{.}$ The exact survival factor is ${\cos }^{2}z$, which is not $e^{-\mathrm{\Gamma }\tau }$ for any $\mathrm{\Gamma }>0$ on all $\tau \ge 0$.

For the occupation truncation with orthonormal basis $e_0,\dots ,e_N$, define $a_N{e}_n=\sqrt{n}{e}_{n-1}$ (zero for $n=0$), $a_N^{\ast }{e}_n=\sqrt{n+1}{e}_{n+1}$ for $n<N$, and $a_N^{\ast }{e}_N=0$. Then $[a_N,a_N^{\ast }]=I-(N+1)\mathrm{\mid }{e}_N⟩⟨e_N\mathrm{\mid }\mathrm{.}$ More generally, no finite-dimensional matrices $A,B$ satisfy $[A,B]=I$. These exact regulator facts assert neither an infinite-dimensional CCR construction nor a continuum limit.