Quantum field quantities and compatible SI normalizations

Fix the oriented, time-oriented affine Minkowski spacetime and an inertial chart of Oriented time-oriented affine Minkowski space (the spacetime primitives) and Inertial frames and inertial charts of the affine Minkowski model. Use signature $\eta =\mathrm{diag}(-1,1,1,1)$, length coordinates $x^0=ct$ and $x^i$ in metres, and coordinate time $t$ in seconds as in SI units and coordinate conventions of the special-relativity framework. Greek spacetime indices run from $0$ to $3$; spatial indices run from $1$ to $3$ and are contracted with the Euclidean spatial metric. The coordinates $x^a$ have unit metre and ${\mathrm{\partial }}_a:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$ has unit ${\mathrm{m}}^{-1}$.

The positive constants $c$, $h$ and $e_{\mathrm{S}\mathrm{I}}$ have the exact SI defining values; write $\mathrm{\hslash }:=h\mathrm{/}(2\pi )$. The magnetic constant ${\mu }_0>0$ has SI unit $\mathrm{N}{\mathrm{A}}^{-2}$ and is experimentally determined under the current SI; no exact numerical value or current estimate for it is adopted. Put ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ when that symbol is used. These are SI normalization inputs, not empirical claims about a selected QFT model.

For a species with rest mass $m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}>0$ in kilograms define its inverse-length mass parameter

$\mu :=\frac{m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}c}{\mathrm{\hslash }}[{\mathrm{m}}^{-1}]\mathrm{.}$

For a spatial wave vector $\mathbf{k}\in {\mathbb{R}}^3$, with each component in ${\mathrm{m}}^{-1}$ and the Euclidean norm, define ${\omega }_{\mu }(\mathbf{k}):=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2}\in {\mathbb{R}}_{>0}$, with unit ${\mathrm{m}}^{-1}$. Here ${\omega }_{\mu }$ is the temporal wave number, not the SI angular frequency; the corresponding SI angular frequency is $c{\omega }_{\mu }$. Energy and momentum are $E=\mathrm{\hslash }c{\omega }_{\mu }$ in joules and $\mathbf{p}=\mathrm{\hslash }\mathbf{k}$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$. With $p=({\omega }_{\mu }(\mathbf{k}),\mathbf{k})\in {\mathbb{R}}^4$ and $x=(x^0,\mathbf{x})\in {\mathbb{R}}^4$, the phase convention is

$p\circ x:={\omega }_{\mu }(\mathbf{k})x^0-\mathbf{k}\cdot \mathbf{x}=-\eta (p,x)\mathrm{.}$

For a selected local action represented on a chart region $\mathcal{U}\subseteq {\mathbb{R}}^4$ by $\frac{S}{\mathrm{\hslash }}={\int }_{\mathcal{U}}{\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}(x)d^{4}x,d^{4}x:=d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3,$ the natural Lagrangian density ${\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}$ has unit ${\mathrm{m}}^{-4}$. Since $x^0=ct$, the corresponding SI Lagrangian density is ${\mathcal{L}}_{\mathrm{S}\mathrm{I}}:=\mathrm{\hslash }c{\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}$ and has unit $\mathrm{J}{\mathrm{m}}^{-3}$. This unit conversion alone does not identify a Lagrangian density with the model's Hamiltonian energy density.

For canonical engineering dimensions in this four-dimensional normalization, use the formal field symbols ${\varphi }_{\mathrm{n}}$ and ${\psi }_{\mathrm{n}}$ in the selected scalar and Dirac kinetic terms, assigning them length degrees $-1$ and $-3\mathrm{/}2$, respectively. These are the dimensions fixed by those free kinetic terms, not exact scaling dimensions of interacting fields: anomalous dimensions, when present, are separate quantities. These symbols do not assert pointwise field maps or operator fields; a model supplies its field space and regularity. Let $A_{\mathrm{S}\mathrm{I}}^{\mu }=(\phi \mathrm{/}c,\mathbf{A})$ and $F_{\mathrm{S}\mathrm{I}}^{\mu \nu }$ use the component convention of The electromagnetic two-form, its observer decomposition and the four-current, namely $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={ϵ}_{ijk}{B}_k$, with $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$ only when a potential is supplied. Define the normalized photon potential and field strength by

$a^{\mu }:=\frac{A_{\mathrm{S}\mathrm{I}}^{\mu }}{\sqrt{{\mu }_0\mathrm{\hslash }c}},f^{\mu \nu }:=\frac{F_{\mathrm{S}\mathrm{I}}^{\mu \nu }}{\sqrt{{\mu }_0\mathrm{\hslash }c}}\mathrm{.}$

They have length degrees $-1$ and $-2$, respectively. The EM supplier is used only for SI units and tensor components: no classical Maxwell equation, Lorentz-force law, or classical-to-quantum identification is adopted here.

For any separately declared SI charge parameter $q_{\mathrm{S}\mathrm{I}}$ in coulombs put

$q_{\mathrm{n}}:=q_{\mathrm{S}\mathrm{I}}\sqrt{\frac{{\mu }_{0}c}{\mathrm{\hslash }}}\mathrm{.}$

It is dimensionless. For the positive exact SI elementary charge use $e_{\mathrm{n}}:=e_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}$ and define the rationalized dimensionless coupling $\alpha :=e_{\mathrm{n}}^2\mathrm{/}(4\pi )$. Since ${\mu }_0$ is measured, these formulas do not assign an exact numerical value to $e_{\mathrm{n}}$ or $\alpha$. If the model uses minimal coupling, its convention is $D_{\mu }={\mathrm{\partial }}_{\mu }-i{q}_{\mathrm{n}}{a}_{\mu }$; for an electron with $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}$ this reads $D_{\mu }={\mathrm{\partial }}_{\mu }+i{e}_{\mathrm{n}}{a}_{\mu }$. Bare, renormalized-at-scale, and measured effective charges are separately labelled model parameters; applying the conversion to one label does not identify it with either of the others or specify a running law.

In a supplied four-component spinor representation, let ${\gamma }^a\in {\mathrm{End}}_{\mathbb{C}}({\mathbb{C}}^4)$, $a=0,1,2,3$, be dimensionless linear maps satisfying $\{{\gamma }^a,{\gamma }^b\}:={\gamma }^a{\gamma }^b+{\gamma }^b{\gamma }^a=-2{\eta }^{ab}{I}_4$, where $I_4$ is the identity map. Set $\beta :={\gamma }^0$ and ${\alpha }^i:=\beta {\gamma }^i$; then ${\beta }^2=I_4$ and ${\gamma }^i=\beta {\alpha }^i$. For a supplied four-component spinor $\psi$ write $\bar{\psi }:={\psi }^{†}\beta$. The effective Clifford metric is $-\eta$. A supplied free Dirac model may use $(i{\gamma }^a{\mathrm{\partial }}_a-\mu )\psi =0$ for its model-supplied spinor field $\psi$; that model supplies its field domain and regularity. The phase and Clifford choices are conventions; this item proves no representation or field equation.

For the autonomous evolution convention, assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): every sequence of nonempty sets has a choice function. Let a selected model supply a complex Hilbert space $\mathcal{H}$ as in Hilbert space and a time-independent complex-linear Hamiltonian $H:D(H)\to \mathcal{H}$ with a specified dense linear domain $D(H)\subseteq \mathcal{H}$ and SI energy unit joules. Require self-adjointness as in Symmetric, self-adjoint and essentially self-adjoint operators: $H=H^{\ast }$, including equality of domains and values. Here $y\in D(H^{\ast })$ precisely when there is a $z\in \mathcal{H}$ such that $⟨Hx,y⟩=⟨x,z⟩$ for every $x\in D(H)$, and $H^{\ast }y=z$; density makes $z$ unique.

The published mathematical supplier CC spectral calculus on an arbitrary complex Hilbert space by compatible vector-local Cayley subspaces then supplies the spectral measure $E_H$ on the Borel subsets of $\mathbb{R}$. This is a normalized projection-valued measure as in Projection valued measure: its values are bounded orthogonal projections, $E_H(\mathbb{R})=I$, $E_H(B\cap C)=E_H(B)E_H(C)$, and it is countably additive in norm on every vector. For $\xi \in \mathcal{H}$ put ${\nu }_{\xi }(B):=⟨E_H(B)\xi ,\xi ⟩$, a finite positive measure. The spectral integral of $\lambda$ represents $H$, on exactly the domain $D(H)=\{\xi \in \mathcal{H}:{\int }_{\mathbb{R}}{\lambda }^{2}d{\nu }_{\xi }(\lambda )<\mathrm{\infty }\}$. For real $t$ in seconds define the autonomous evolution by bounded Borel functional calculus, $U(t):=\exp (-itH\mathrm{/}\mathrm{\hslash }):={\int }_{\mathbb{R}}{e}^{-it\lambda \mathrm{/}\mathrm{\hslash }}d{E}_H(\lambda )\mathrm{.}$ The spectral variable $\lambda$ carries unit joule. This exponential is a bounded operator on all of $\mathcal{H}$, even when $H$ is unbounded; it does not mean an operator power series on arbitrary vectors. The supplier gives a strongly continuous unitary group: $U(0)=I$, $U(t+s)=U(t)U(s)$ and $\mathrm{\parallel }U(t)\xi -U(t_0)\xi \mathrm{\parallel }\to 0$ as $t\to t_0$ for each $\xi \in \mathcal{H}$. A model may restrict it to a declared $I\subseteq \mathbb{R}$. Each $U(t)$ is a surjective complex-linear map satisfying $⟨U(t)\xi ,U(t)\zeta ⟩=⟨\xi ,\zeta ⟩$ for all $\xi ,\zeta \in \mathcal{H}$; write $U(t{)}^{\ast }:=U(t{)}^{-1}$. The exponent is dimensionless because $\mathrm{\hslash }$ has unit $\mathrm{J}\mathrm{s}$. For any model-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 the Hamiltonian and operator domains; the spectral supplier justifies this evolution only for those data and hypotheses. This dictionary constructs no model-specific Hamiltonian or field representation and does not adopt a physical dynamics law. Where a free-field module uses an inner product linear in its second variable, convert to the repository's first-variable-linear pairing by $⟨u,v{⟩}_{\mathrm{F}\mathrm{F}}=(v,u{)}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$.

For precision about domains, let $\mathcal{H}$ be a complex Hilbert space with inner product linear in its first variable and let $\mathcal{D}\subseteq \mathcal{H}$ be a dense complex linear subspace. A weak operator-valued distribution on the common domain $\mathcal{D}$ means a complex-linear assignment $g\mapsto \mathrm{\Phi }(g)$ for $g\in \mathcal{D}({\mathbb{R}}^4)$, where each $\mathrm{\Phi }(g):\mathcal{D}\to \mathcal{H}$ is linear and, for every $u,v\in \mathcal{D}$, the scalar map $g\mapsto ⟨\mathrm{\Phi }(g)u,v⟩$ is a continuous complex-linear functional on the test-function space with the topology of Test function topology, hence a distribution as in Distribution. This definition gives no pointwise field $\mathrm{\Phi }(x)$ and no product $\mathrm{\Phi }(x)\mathrm{\Phi }(x)$; a product or extension needs a separate construction. It asserts only matrix-element continuity and the stated linear maps; all further operator properties, locality, local observable algebras, graded locality, and gauge-specific potential behavior must be supplied by a constructed model. Test-function units can be chosen so a specified smeared observable is dimensionless; there is no universal test-function unit asserted here.

Primitive data for a selected model include its species and rest masses, declared charge parameters, regulator geometry, couplings, state and any constructed measurement data. Vacuum-subtracted Hamiltonians, regulator eigenvalues, distributions, correlations and outcome probabilities are derived only when their constructions supply the required domains and well-definedness. This dictionary neither constructs an interacting four-dimensional QED Hilbert space nor asserts pointwise field products, scattering, an LSZ limit, asymptotic completeness, or removal of a regulator.

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.