Positive transverse free photon model

Use the oriented, time-oriented Minkowski model, length coordinates, metric, and SI/natural dictionary of Quantum field quantities and compatible SI normalizations, with $\eta =\mathrm{diag}(-1,1,1,1)$ and $x^0=ct$. Use the gamma matrices, Dirac adjoint, and effective Clifford metric of Charged gauge data and SI-natural conventions: ${\gamma }^0=\beta$, $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}{I}_4$, and $\bar{\psi }={\psi }^{†}\beta$. Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the mathematical QFT gauge-data supplier; the unit conversions below do not themselves spend a choice principle.

Identify the SI charge and mass symbols $q,m$ in Charged gauge data and SI-natural conventions with $q_{\mathrm{S}\mathrm{I}},m_{\mathrm{S}\mathrm{I}}$ here. Its natural coupling $g$ is $q_{\mathrm{n}}$ below, and its inverse-length mass $\mu$ is the same $m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$.

Let $q_{\mathrm{S}\mathrm{I}}\in \mathbb{R}$ be a selected charge parameter in coulombs and $m_{\mathrm{S}\mathrm{I}}>0$ a selected rest-mass parameter in kilograms. They are inputs of the chosen model; neither is derived from Maxwell's equations. Set $\mu =m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$ in ${\mathrm{m}}^{-1}$ and $q_{\mathrm{n}}=q_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}$, which is dimensionless under the framework units of Quantum field quantities and compatible SI normalizations. For an electron use $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}$ and $e_{\mathrm{n}}=e_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}>0$, so $q_{\mathrm{n}}=-e_{\mathrm{n}}$. Use ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ with the measured current-SI ${\mu }_0$ of Quantum field quantities and compatible SI normalizations. These definitions therefore do not assign an exact numerical value to $q_{\mathrm{n}}$, $e_{\mathrm{n}}$, or ${\epsilon }_0$.

In a named inertial chart, let $A_{\mathrm{S}\mathrm{I}}^{\mu }=(\phi \mathrm{/}c,\mathbf{A})$ be the electromagnetic potential when one is supplied, with components in $\mathrm{T}\mathrm{m}$ under Framework-local SI units, the vacuum constants and the exact/measured status of their values and The electromagnetic two-form, its observer decomposition and the four-current. Define the canonical potential and field-strength components 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 engineering dimensions ${\mathrm{m}}^{-1}$ and ${\mathrm{m}}^{-2}$, respectively. With the metric-lowered potential $A_{\mathrm{S}\mathrm{I},a}={\eta }_{ab}{A}_{\mathrm{S}\mathrm{I}}^b$, the potential component convention is $A_{\mathrm{S}\mathrm{I},0}=-\phi \mathrm{/}c$ and $A_{\mathrm{S}\mathrm{I},i}=A_i$. The covariant derivative in the selected QED model is

$D_a:={\mathrm{\partial }}_a-i{q}_{\mathrm{n}}{a}_a={\mathrm{\partial }}_a-\frac{i{q}_{\mathrm{S}\mathrm{I}}}{\mathrm{\hslash }}{A}_{\mathrm{S}\mathrm{I},a},$

since $q_{\mathrm{n}}{a}_a=q_{\mathrm{S}\mathrm{I}}{A}_{\mathrm{S}\mathrm{I},a}\mathrm{/}\mathrm{\hslash }$. Both terms in $D_a$ have unit ${\mathrm{m}}^{-1}$. If the SI gauge function $\chi$ has unit $\mathrm{W}\mathrm{b}=\mathrm{T}{\mathrm{m}}^2$, then $q_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }$ is dimensionless; the corresponding natural gauge function $\lambda =\chi \mathrm{/}\sqrt{{\mu }_0\mathrm{\hslash }c}$ is dimensionless and $q_{\mathrm{n}}\lambda =q_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }$. These conversions apply to smooth local gauge transformations; they assert no boundary condition, asymptotic charge sector, or global gauge-fixing result.

If a separately specified finite regulator introduces Hamiltonian eigenvalues, they have unit joule; this definition introduces no regulator. The field components and units refer to the selected inertial chart, not to a new observer congruence or a cross-framework identification.

Adopt the following specified free-photon model in the oriented, time-oriented affine Minkowski spacetime and length coordinates of Quantum field quantities and compatible SI normalizations, with metric $(-,+,+,+)$ and $x^0=ct$. The classical electromagnetic input is the smooth source-free branch of The microscopic vacuum Maxwell equations: smooth and weak branches; the electromagnetic two-form and its potential components are those of The electromagnetic two-form, its observer decomposition and the four-current, and a supplied potential is subject to Potential representations, gauge transformations and the Lorenz and Coulomb conditions. These EM suppliers are physical input and conventions: they do not by themselves construct a quantum representation.

For almost every $\mathbf{k}\in {\mathbb{R}}^3\setminus \{0\}$, set $\mathbf{n}=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ and $P_T(\mathbf{k})=I_3-\mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. The mathematical definition Positive transverse photon one-particle space specifies the positive one-photon Hilbert space

${\mathcal{H}}_T=\{u\in L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3):\mathbf{k}\cdot u(\mathbf{k})=0\text{ for almost every }\mathbf{k}\},$

with the inherited pairing $⟨u,v{⟩}_2=\int \overline{u(\mathbf{k})}\cdot v(\mathbf{k})d^3\mathbf{k}$, linear in the second variable as in Schwartz/distribution spaces and Hilbert Fock conventions. The value assigned to $P_T$ at $\mathbf{k}=0$ is immaterial because that point has Lebesgue measure zero; no continuous polarization frame over all momentum directions is chosen. Adopt the bosonic Fock representation ${\mathcal{F}}_s({\mathcal{H}}_T)$ and the free Hamiltonian

$H_{\gamma }=\mathrm{\hslash }cd\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid }),$

where on the $n$-photon sector $d\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ is multiplication by ${\sum }_{r=1}^n\mathrm{\mid }{\mathbf{k}}_r\mathrm{\mid }$ and is zero on the vacuum sector. Its exact self-adjoint domain and the unitary mass-cone action are those in the mathematical supplier Photon quotient Lorentz representation and positive dynamics. The actual smeared field-strength distributions and bounded Weyl operators are those constructed in Bounded local photon Weyl observable algebra; a pointwise potential or field-strength operator is not asserted to exist. These mathematical suppliers are draft carriers with mixed current Step 1 status: the transverse-space, null-quotient, covariance-spectrum, framework-units, and C01 measurement carriers have current ready records; the local-Weyl carrier is escalated through its field-strength prerequisite. Readiness records are not independent approval, and every use remains conditional on the exact statement and proof supplied.

For the physical interpretation of an actually constructed state and a supplied normalized measurement instrument, use Physical interpretation of a supplied constructed quantum model: Born probabilities are assigned only to that specified model and instrument. This postulate records the model's free Maxwell quanta as photons. It does not assert that the model has been empirically confirmed, and it introduces no interacting four-dimensional QED Hilbert space, scattering operator, LSZ limit, asymptotic completeness, or nonperturbative continuum limit. In calculations using an indefinite four-polarization auxiliary space, only the positive transverse model or its separately constructed null quotient is physical; four auxiliary components are not four positive photon states.

Adopt a smooth local classical Maxwell–Dirac model on an open region of the flat spacetime specified by Quantum field quantities and compatible SI normalizations. Let $A_{\mathrm{S}\mathrm{I},a}$ be a supplied real $C^{\mathrm{\infty }}$ electromagnetic covector potential and let $\psi :\text{\hspace{0.17em}⁣}U\to {\mathbb{C}}^4$ be a $C^{\mathrm{\infty }}$ commuting spinor field. The charge $q_{\mathrm{S}\mathrm{I}}$ and rest mass $m_{\mathrm{S}\mathrm{I}}>0$ are model inputs; for the electron take $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}$. Use the component, metric, Clifford, charge, field-normalization, and unit conventions of Charged gauge data and SI-natural conventions, Charge, canonical photon field and observer units, Quantum field quantities and compatible SI normalizations, and the electromagnetic suppliers of Positive transverse free photon model. In particular, $D_a={\mathrm{\partial }}_a-i{q}_{\mathrm{S}\mathrm{I}}{A}_{\mathrm{S}\mathrm{I},a}\mathrm{/}\mathrm{\hslash }={\mathrm{\partial }}_a-i{q}_{\mathrm{n}}{a}_a$ and $\mu =m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$.

Adopt the local action density in natural normalization

${\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}=-\frac{1}{4}{f}_{ab}{f}^{ab}+\frac{i}{2}(\bar{\psi }{\gamma }^a{D}_a\psi -(D_a\bar{\psi }){\gamma }^a\psi )-\mu \bar{\psi }\psi ,$

where $f_{ab}={\mathrm{\partial }}_a{a}_b-{\mathrm{\partial }}_b{a}_a$ on the region where the potential is supplied, $\bar{\psi }={\psi }^{†}\beta$, and $D_a\bar{\psi }={\mathrm{\partial }}_a\bar{\psi }+i{q}_{\mathrm{n}}{a}_a\bar{\psi }$. The physical action is $S_{\mathrm{c}\mathrm{l}}=\mathrm{\hslash }{\int }_U{\mathcal{L}}_{\mathrm{n}\mathrm{a}\mathrm{t}}{d}^{4}x$ with $x^0=ct$. For compactly supported local variations this is the adopted classical Maxwell–Dirac formulation; its Euler–Lagrange consequences are not postulated as a proof in this item.

For a smooth gauge function $\chi$ in webers, the adopted local transformation is $A_{\mathrm{S}\mathrm{I},a}\mapsto A_{\mathrm{S}\mathrm{I},a}+{\mathrm{\partial }}_a\chi$ and $\psi \mapsto e^{i{q}_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }}\psi$. It leaves $F=dA$ invariant and makes $D_a\psi$ transform by the same phase. This is the covector form of the SI potential rule in Potential representations, gauge transformations and the Lorenz and Coulomb conditions. Gauge functions in the local variation argument are compactly supported; no assertion is made about large gauge transformations, nontrivial bundles, boundary charges, or asymptotic sectors.

The model adopts the separately specified free Dirac representation of Specified bosonic scalar and CAR Dirac model adoption and the positive free photon representation of Positive transverse free photon model as the free-field correspondence used to describe the selected species. This extra correspondence does not quantize the interacting classical action. In particular, this postulate supplies no interacting continuum four-dimensional QED Hilbert space, no globally well-posed coupled PDE, no pointwise operator-valued composite current, no scattering or LSZ construction, and no regulator-removal or nonperturbative-existence result.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the selected Fock-space construction, with the positive constants and physical energy units of Quantum field quantities and compatible SI normalizations. In the adopted model Positive transverse free photon model, for $\mathbf{k}\mathrm{\ne }0$ set $\mathbf{n}=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ and $P_T(\mathbf{k})=I_3-\mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. Then for almost every nonzero momentum, $P_T(\mathbf{k})$ is a positive orthogonal projection of rank two, and the positive one-photon Hilbert space is its closed range ${\mathcal{H}}_T\subset L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3)$. This description chooses no global continuous polarization frame.

If an auxiliary four-polarization space $\mathcal{K}=L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^4)$ with indefinite form $[ϵ,\delta ]=\int {ϵ}^{†}\eta \delta d^3\mathbf{k}$ is used, the physical subsidiary space ${\mathcal{K}}_0=\{ϵ:-{ϵ}^0+\mathbf{n}\cdot \mathit{ϵ}=0\text{ a.e.}\}$ has null radical $\mathcal{N}=\{b(\mathbf{k})(1,\mathbf{n}):b\in L^2\}$. The quotient ${\mathcal{K}}_0\mathrm{/}\mathcal{N}$ is isometric to ${\mathcal{H}}_T$ with positive inner product. The physical Fock Hamiltonian $H_{\gamma }=\mathrm{\hslash }cd\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ is self-adjoint, nonnegative, and has the exact domain and strongly continuous Lorentz-covariant mass-cone action in Photon quotient Lorentz representation and positive dynamics. The vector-valued auxiliary space is not a positive state space, and none of its negative directions is assigned a negative Born probability.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the constructed Fock representation. In the adopted model Positive transverse free photon model, let $h^{ab}\in C_c^{\mathrm{\infty }}({\mathbb{R}}^{1,3};\mathbb{R})$ be a real antisymmetric two-tensor test. By the natural field-strength normalization of Quantum field quantities and compatible SI normalizations, choose each component of $h^{ab}$ with unit ${\mathrm{m}}^{-2}$. Since $f_{ab}$ has unit ${\mathrm{m}}^{-2}$ and $d^{4}x$ has unit ${\mathrm{m}}^4$, $f(h)$ below is dimensionless. The corresponding one-particle vector $g_h$ has profile unit ${\mathrm{m}}^{3\mathrm{/}2}$ under the coefficient formula of Smeared Coulomb photon operator distribution: its effective test $j^i=-{\mathrm{\partial }}_a{h}^{ai}$ has unit ${\mathrm{m}}^{-3}$, the Fourier integral contributes ${\mathrm{m}}^1$, and the factor $(2\omega {)}^{-1\mathrm{/}2}$ contributes ${\mathrm{m}}^{1\mathrm{/}2}$. Thus its Hilbert norm under $d^3\mathbf{k}$ of unit ${\mathrm{m}}^{-3}$ is dimensionless, as is the Weyl parameter $\sqrt{2}{g}_h$. This is the test normalization adopted here, not a universal test-function unit. Here $f_{ab}$ is the natural-normalized field-strength distribution of the selected free model, corresponding to $F_{ab}$ in the mathematical free-photon suppliers and distinct from the SI tensor $F_{\mathrm{S}\mathrm{I},ab}$. Its smearing $f(h)=\frac{1}{2}\int f_{ab}(x)h^{ab}(x)d^{4}x$ is a well-defined operator-valued distribution on the common finite-particle domain ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, not a pointwise field operator. It is gauge invariant and covariant under the selected free mass-cone unitary action. For two such tests with spacelike-separated supports, $[f(h),f(\mathrm{\ell })]=0$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. Let $g_h$ be the associated one-particle test vector and define the Weyl observable $W_h:=W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{g}_h)$ as in Bounded local photon Weyl observable algebra. These are bounded unitaries of norm one; their real and imaginary parts are bounded Hermitian operators and commute for spacelike-separated supports. No locality claim is made for a Coulomb-gauge potential.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the selected free-field correspondence in Explicit charged Dirac/Maxwell local coupling model. Let $U$ be an open subset of flat Minkowski spacetime with $\eta =\mathrm{diag}(-1,1,1,1)$ and length coordinates $x^0=ct$. Let $A_{\mathrm{S}\mathrm{I},a}:U\to \mathbb{R}$ be a supplied $C^{\mathrm{\infty }}$ covector potential, let $\psi :U\to {\mathbb{C}}^4$ be a $C^{\mathrm{\infty }}$ commuting spinor, and take a selected real charge $q_{\mathrm{S}\mathrm{I}}$ and mass $m_{\mathrm{S}\mathrm{I}}>0$ with the conventions of Charge, canonical photon field and observer units and Charged gauge data and SI-natural conventions. Use independent compactly supported smooth variations of $A_{\mathrm{S}\mathrm{I}}$ and of both real and imaginary components of $\psi$. If $(A_{\mathrm{S}\mathrm{I}},\psi )$ is stationary for the adopted classical action, then

$(i{\gamma }^a{D}_a-\mu )\psi =0,{\mathrm{\partial }}_a{F}_{\mathrm{S}\mathrm{I}}^{ab}=-{\mu }_0{j}^b,{\mathrm{\partial }}_a{j}^a=0,$

where $D_a={\mathrm{\partial }}_a-i{q}_{\mathrm{S}\mathrm{I}}{A}_{\mathrm{S}\mathrm{I},a}\mathrm{/}\mathrm{\hslash }$, $\mu =m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$, and $j^a=q_{\mathrm{S}\mathrm{I}}c\bar{\psi }{\gamma }^a\psi$. In laboratory components this current is ${\rho }_q=q_{\mathrm{S}\mathrm{I}}{\psi }^{†}\psi$ and ${\mathbf{J}}_q=q_{\mathrm{S}\mathrm{I}}c{\psi }^{†}\mathit{\alpha }\psi$, so $j=(c{\rho }_q,{\mathbf{J}}_q)$. With $[\psi ]={\mathrm{m}}^{-3\mathrm{/}2}$, $[q_{\mathrm{S}\mathrm{I}}]=\mathrm{C}$, and $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, the charge density ${\rho }_q$ has unit $\mathrm{C}{\mathrm{m}}^{-3}$, while ${\mathbf{J}}_q$ and every component of $j^a$ have unit $\mathrm{A}{\mathrm{m}}^{-2}$; here $x^0=ct$, so $j^0=c{\rho }_q$ has the same component unit as $J_q^i$. Together with $F=dA$ on the supplied potential domain, the field equation and Bianchi identity have exactly the smooth SI Maxwell signs in The microscopic vacuum Maxwell equations: smooth and weak branches. The current and spinor here are classical smooth commuting fields; this statement is not a construction of a pointwise quantum composite operator.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the torus Fourier construction. Let $L>0$, let $I\subseteq \mathbb{R}$ be an open time interval, and let the spatial domain be the flat torus ${\mathbb{T}}_L^3=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$, where $L$ is a positive length in metres. Coordinates use $x^0=ct$ with $t$ in seconds under Quantum field quantities and compatible SI normalizations. Use the smooth classical branch of Explicit charged Dirac/Maxwell local coupling model with smooth periodic electromagnetic potentials and spinor. Here $\psi$ has the canonical length dimension ${\mathrm{m}}^{-3\mathrm{/}2}$ of Quantum field quantities and compatible SI normalizations, so the scalar-potential source ${\rho }_q:=q_{\mathrm{S}\mathrm{I}}{\psi }^{†}\psi$ has unit $\mathrm{C},{\mathrm{m}}^{-3}$. On stationary solutions it agrees with the current density in Conditional classical charged gauge equations and current. Use the gamma-matrix and $\alpha ,\beta$ conventions of Charged gauge data and SI-natural conventions. In Coulomb gauge $\mathrm{div}\mathbf{A}=0$, the multiplier equation for any smooth periodic configuration is $\mathrm{div}{\mathit{\pi }}_A+{\rho }_q=0$. On the explicitly neutral branch ${\int }_{{\mathbb{T}}_L^3}{\rho }_q(\mathbf{x},t)d^3\mathbf{x}=0$ for every $t\in I$, the zero-mean scalar potential is the unique smooth periodic solution of $-\mathrm{\Delta }\phi ={\rho }_q\mathrm{/}{\epsilon }_0$. With ${\widehat{\rho }}_q(\mathbf{n},t)=L^{-3}{\int }_{{\mathbb{T}}_L^3}{\rho }_q(\mathbf{x},t)e^{-2\pi i\mathbf{n}\cdot \mathbf{x}\mathrm{/}L}{d}^3\mathbf{x}$ and ${\mathit{\kappa }}_{\mathbf{n}}=2\pi \mathbf{n}\mathrm{/}L$, the longitudinal energy is

$\frac{{\epsilon }_0}{2}{\int }_{{\mathbb{T}}_L^3}\mathrm{\mid }\mathrm{\nabla }\phi {\mathrm{\mid }}^2{d}^3\mathbf{x}=\frac{1}{2}{\int }_{{\mathbb{T}}_L^3}{\rho }_q\phi d^3\mathbf{x}=\frac{L^3}{2{\epsilon }_0}{\sum }_{\mathbf{n}\in {\mathbb{Z}}^3\setminus \{0\}}\frac{\mathrm{\mid }{\widehat{\rho }}_q(\mathbf{n},t){\mathrm{\mid }}^2}{\mathrm{\mid }{\mathit{\kappa }}_{\mathbf{n}}{\mathrm{\mid }}^2}\ge 0.$

The corresponding classical Hamiltonian, before imposing the Gauss constraint, contains

$H={\int }_{{\mathbb{T}}_L^3}[\frac{\mathrm{\mid }{\mathit{\pi }}_A{\mathrm{\mid }}^2}{2{\epsilon }_0}+\frac{\mathrm{\mid }\mathbf{B}{\mathrm{\mid }}^2}{2{\mu }_0}+{\psi }^{†}\text{\hspace{0.17em}⁣}(c\mathit{\alpha }\cdot (-i\mathrm{\hslash }\mathrm{\nabla }-q_{\mathrm{S}\mathrm{I}}\mathbf{A})+\beta m_{\mathrm{S}\mathrm{I}}{c}^2)\text{\hspace{0.17em}⁣}\psi +\phi (\mathrm{div}{\mathit{\pi }}_A+{\rho }_q)]d^3\mathbf{x},$

with ${\pi }_{\phi }=0$ the primary multiplier momentum. In a separate finite mode truncation, fix $N\ge 1$ time-independent modes $u_1,\dots ,u_N\in C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{C}}^4)$ satisfying ${\int }_{{\mathbb{T}}_L^3}{u}_j(\mathbf{x}{)}^{†}{u}_k(\mathbf{x})d^3\mathbf{x}={\delta }_{jk}\mathrm{.}$ Restrict the commuting spinor field to ${\psi }_N(\mathbf{x},t)={\sum }_{j=1}^N{z}_j(t)u_j(\mathbf{x})$, where $z_j\in C^{\mathrm{\infty }}(I;\mathbb{C})$ and ${\bar{z}}_j=\overline{z_j}$. Under the canonical field normalization $[\psi ]={\mathrm{m}}^{-3\mathrm{/}2}$, these modes have unit ${\mathrm{m}}^{-3\mathrm{/}2}$ and the amplitudes $z_j$ are dimensionless. This is only a kinematic restriction of field configurations; the mode subspace is not asserted invariant under the field equations or to contain a solution on the neutral branch.

For its finite-dimensional canonical phase space ${\mathcal{P}}_N={\mathbb{R}}^{4N}$, use real coordinates $(x_j,y_j,P_j^x,P_j^y{)}_{j=1}^N$ and, for $F,G\in C^1({\mathcal{P}}_N;\mathbb{C})$, the Poisson bracket $\{F,G\}={\sum }_{j=1}^N(\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_j}\frac{\mathrm{\partial }G}{\mathrm{\partial }{P}_j^x}-\frac{\mathrm{\partial }F}{\mathrm{\partial }{P}_j^x}\frac{\mathrm{\partial }G}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }F}{\mathrm{\partial }{y}_j}\frac{\mathrm{\partial }G}{\mathrm{\partial }{P}_j^y}-\frac{\mathrm{\partial }F}{\mathrm{\partial }{P}_j^y}\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_j}),$ extended complex-bilinearly. Define $z_j=x_j+i{y}_j$, ${\bar{z}}_j=x_j-i{y}_j$, $p_j=(P_j^x-i{P}_j^y)\mathrm{/}2$, and ${\bar{p}}_j=(P_j^x+i{P}_j^y)\mathrm{/}2$. These definitions give $\{z_j,p_k\}={\delta }_{jk}$, $\{{\bar{z}}_j,{\bar{p}}_k\}={\delta }_{jk}$, and all other elementary brackets zero. Substitution into the adopted action gives the velocity-dependent part of the finite-mode physical Lagrangian $L_{\mathrm{k}\mathrm{i}\mathrm{n}}=\frac{i\mathrm{\hslash }}{2}{\sum }_{j=1}^N({\bar{z}}_j{\dot{z}}_j-{\dot{\bar{z}}}_j{z}_j)\mathrm{.}$ Its primary constraints are $C_j=p_j-i\mathrm{\hslash }{\bar{z}}_j\mathrm{/}2$ and ${\bar{C}}_j={\bar{p}}_j+i\mathrm{\hslash }{z}_j\mathrm{/}2$. For $\mathrm{\hslash }>0$ their constraint matrix is $M=\left(\begin{array}{cc}0& -i\mathrm{\hslash }{I}_N\\ i\mathrm{\hslash }{I}_N& 0\end{array}\right),$ which is invertible; the reduced bracket satisfies $\{z_j,{\bar{z}}_k{\}}_D=-i{\delta }_{jk}\mathrm{/}\mathrm{\hslash }$ and $\{z_j,z_k{\}}_D=\{{\bar{z}}_j,{\bar{z}}_k{\}}_D=0$. This finite-dimensional classical calculation does not derive fermionic CAR. The neutral torus branch is an explicit hypothesis. For the selected electron, $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}\mathrm{\ne }0$, so a neutral single-species commuting-spinor configuration has $\psi =0$; no nontrivial neutral charged classical Dirac configuration is asserted. The finite-mode constraint calculation is separate and does not claim to be a solution in that branch.