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.