Specified bosonic scalar and CAR Dirac model adoption

Fix four-dimensional affine Minkowski spacetime with metric $\eta =\mathrm{diag}(-1,1,1,1)$, an inertial length-coordinate chart $x=(x^0,\mathbf{x})$, $x^0=ct$, and coordinates measured in metres, as specified by Quantum field quantities and compatible SI normalizations. Write $\mathrm{\square }={\eta }^{ab}{\mathrm{\partial }}_a{\mathrm{\partial }}_b$. The mass-shell parameter $\mu \ge 0$ has unit ${\mathrm{m}}^{-1}$ and ${\omega }_{\mu }(\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2}$ has the same unit. For a massive species, $\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$; $\mu =0$ is the separately admitted massless branch of the mathematical free-field construction. Thus $\mathrm{\hslash }c{\omega }_{\mu }$ is physical energy and $\mathrm{\hslash }\mathbf{k}$ is physical momentum. The natural-unit action is written $S\mathrm{/}\mathrm{\hslash }$.

The scalar one-particle space is ${\mathcal{H}}_{\mu }=L^2(H_{\mu }^{+},d{\nu }_{\mu };\mathbb{C})$, where $H_{\mu }^{+}=\{({\omega }_{\mu }(\mathbf{k}),\mathbf{k}):\mathbf{k}\in {\mathbb{R}}^3,{\omega }_{\mu }(\mathbf{k})>0\}$ and $d{\nu }_{\mu }=d^3\mathbf{k}\mathrm{/}((2\pi {)}^{3}2{\omega }_{\mu }(\mathbf{k}))$; at $\mu =0$ the measure-zero apex $\mathbf{k}=0$ is omitted. These are the shell conventions of Smeared free scalar field and its Poincare representation.

For a smooth real classical scalar ${\varphi }_{\mathrm{n}}:{\mathbb{R}}^4\to \mathbb{R}$ with the canonical free-field length degree $-1$, adopt $\frac{S_{\varphi }}{\mathrm{\hslash }}=-\frac{1}{2}{\int }_{{\mathbb{R}}^4}({\eta }^{ab}{\mathrm{\partial }}_a{\varphi }_{\mathrm{n}}{\mathrm{\partial }}_b{\varphi }_{\mathrm{n}}+{\mu }^2{\varphi }_{\mathrm{n}}^2)d^{4}x\mathrm{.}$ For a smooth commuting classical Dirac spinor ${\psi }_{\mathrm{n}}:{\mathbb{R}}^4\to {\mathbb{C}}^4$ with canonical free-field length degree $-3\mathrm{/}2$, put ${\bar{\psi }}_{\mathrm{n}}={\psi }_{\mathrm{n}}^{†}\beta$ and use the Clifford convention $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}{I}_4$ from Quantum field quantities and compatible SI normalizations. Adopt $\frac{S_D}{\mathrm{\hslash }}={\int }_{{\mathbb{R}}^4}{\bar{\psi }}_{\mathrm{n}}(i{\gamma }^a{\mathrm{\partial }}_a-\mu ){\psi }_{\mathrm{n}}{d}^{4}x\mathrm{.}$ The smooth classical variables in these functionals commute; no Grassmann algebra or path-integral construction is being adopted. These formulas specify local action densities: for arbitrary smooth global fields the total integrals need not converge. Only compactly supported variations are used; the action difference under such a variation is an integral over a compact set and is finite for smooth fields. With these variations on the fixed background, the intended Euler–Lagrange equations are $(\mathrm{\square }-{\mu }^2){\varphi }_{\mathrm{n}}=0$ and $(i{\gamma }^a{\mathrm{\partial }}_a-\mu ){\psi }_{\mathrm{n}}=0$, conditional on the exact mathematical supplier recorded as [L1] below.

As the physical interpretation of this selected free model, assign the real scalar species to the bosonic symmetric Fock representation and assign the Dirac field to the particle/antiparticle CAR representation on ${\mathcal{H}}_{+}\oplus \overline{{\mathcal{H}}_{-}}$. Choose the zero-particle Fock vectors as the scalar and Dirac vacuum states. This is an adopted species and statistics assignment. The CCR/CAR and operator properties of the specified representations are mathematical model consequences; they do not prove a general spin-statistics theorem or establish that nature realizes this model.

The scalar construction is the Schwartz-smeared field ${\mathrm{\Phi }}_{\mu }(f)=a(K_{\mu }\bar{f})+a^{†}(K_{\mu }f)$ on the finite-particle domain, with the future-shell phase $p\circ x={\omega }_{\mu }{x}^0-\mathbf{k}\cdot \mathbf{x}$, exactly as in Smeared free scalar field and its Poincare representation. The Dirac construction is the bounded spacetime-smeared CAR field of Smeared free Dirac CAR fields and graded locality, using the positive and negative projections of the specified free Dirac matrix. In that carrier, ${\mathcal{H}}_D=L^2({\mathbb{R}}^3,d^3\mathbf{k}\mathrm{/}(2\pi {)}^3;{\mathbb{C}}^4)$, $h_D(\mathbf{k})=\mathit{\alpha }\cdot \mathbf{k}+\beta \mu$, and $P_{\pm }(\mathbf{k})=(I_4\pm h_D(\mathbf{k})\mathrm{/}{\omega }_{\mu }(\mathbf{k}))\mathrm{/}2$; the particle and antiparticle one-particle spaces are the closed ranges ${\mathcal{H}}_{+}$ and $\overline{{\mathcal{H}}_{-}}$. These carriers state their mathematical hypotheses and construction domains; their draft status is not an independent physics or proof approval.

Tong's lecture notes and Srednicki's book draft provide independent presentations of the scalar and Dirac free-field formulations. Their formal mode notation supplies context for this adoption, not a proof of the operator-domain claims and not evidence that the adopted model describes nature.

Use the fields, Fock spaces, vacua, coordinate conventions, and natural/SI normalizations adopted in Specified bosonic scalar and CAR Dirac model adoption. For an open spacetime region $O\subseteq {\mathbb{R}}^4$, a scalar test is a real $f\in C_c^{\mathrm{\infty }}(O;\mathbb{R})$ and a Dirac test is a $F\in C_c^{\mathrm{\infty }}(O;{\mathbb{C}}^4)$. These are compactly supported smooth test functions on the fixed inertial Minkowski chart.

For scalar tests use the bounded Weyl operators $W_{\mu }(f)=W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{K}_{\mu }f),$ where $K_{\mu }$ is the future-shell restriction and $W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}$ is the unitary generated by the closed Segal field, with the normalization and phase convention of Smeared free scalar field and its Poincare representation. Define the scalar bounded observable algebra ${\mathfrak{A}}_s(O)$ to be the norm closure of the unital $\ast$-algebra generated by these $W_{\mu }(f)$. The real smeared field is symmetric on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ and its self-adjoint closure has the exact multiplier domain supplied by Domain-qualified free observables and point-field obstruction and Self-adjoint Segal fields and explicit Weyl unitaries.

For Dirac tests, $\mathrm{\Psi }(F)$ is the bounded smeared CAR field of Smeared free Dirac CAR fields and graded locality. Define ${\mathfrak{A}}_D(O)$ as the norm closure of the unital algebra generated by products containing an even number of these field operators and their adjoints, with all test supports in $O$. “Even” here means even field parity, or equivalently an even number of odd CAR field generators in each product. The even algebra is the selected local Dirac observable algebra; an odd charged field is not included as a commuting local observable.

The selected vacuum states on bounded operators are the vector states ${\omega }_s(A)=⟨{\mathrm{\Omega }}_s,A{\mathrm{\Omega }}_s⟩$ and ${\omega }_D(B)=⟨{\mathrm{\Omega }}_D,B{\mathrm{\Omega }}_D⟩$. For the scalar Fock space, define the total number, Hamiltonian, and physical momentum components by $N_s=d{\mathrm{\Gamma }}_s(1),H_s=\mathrm{\hslash }cd{\mathrm{\Gamma }}_s({\omega }_{\mu }),P_{s,j}=\mathrm{\hslash }d{\mathrm{\Gamma }}_s(k_j)\mathrm{.}$ Here $d{\mathrm{\Gamma }}_s$ acts on the $n$-particle sector by summing the displayed one-particle multiplier over its $n$ arguments. Their maximal domains are the square-integrability domains of those sector sums; in particular, $D(N_s)=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n):{\sum }_{n\ge 1}{n}^2\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ These physical components have one-particle momentum $+\mathrm{\hslash }{k}_j$, as in Quantum field quantities and compatible SI normalizations. The mathematical spatial generators in Smeared free scalar field and its Poincare representation and Domain-qualified free observables and point-field obstruction use $G_{s,j}=d{\mathrm{\Gamma }}_s(-k_j)$ with the positive-exponent convention; thus $P_{s,j}=-\mathrm{\hslash }{G}_{s,j}$. For a translation increment $a=(a^0,\mathbf{a})$ in length coordinates, the scalar translation unitary is the sectorwise phase $\exp (i(a^0{H}_s\mathrm{/}(\mathrm{\hslash }c)-{\sum }_j{a}^j{P}_{s,j}\mathrm{/}\mathrm{\hslash }))$. Changing the sign does not change the maximal domain, since the squared sector multiplier is unchanged. This also follows directly from Second-quantized multipliers, joint spectral measure and exact domains with multiplier $k_j$. For the Dirac Fock space, let $H$ and $P_j$ be the mathematical translation generators of Fermionic Dirac translation generators and positive spectrum. Their one-particle multipliers are respectively $({\omega }_{\mu },\mathbf{k})$ for particles and $({\omega }_{\mu },-\mathbf{k})$ for antiparticles, and the sector multiplier is their sum. Define the physical Dirac Hamiltonian and momentum by $H_D=\mathrm{\hslash }cH$ and $P_{D,j}=\mathrm{\hslash }{P}_j$, using the positive SI constants from Quantum field quantities and compatible SI normalizations. Their exact domains are unchanged by these nonzero scale factors; their multipliers have units joules and momentum, respectively.

For any of these self-adjoint operators $A$, a normalized state $\mathrm{\Psi }\in D(A)$ has model mean $⟨\mathrm{\Psi },A\mathrm{\Psi }⟩$ and variance ${\mathrm{Var}}_{\mathrm{\Psi }}(A)=\mathrm{\parallel }(A-⟨\mathrm{\Psi },A\mathrm{\Psi }⟩)\mathrm{\Psi }{\mathrm{\parallel }}^2$. This requires the second spectral moment but does not require $\mathrm{\Psi }\in D(A^2)$. Products of unbounded operators use their composition domains.

If a signed charge parameter $q_{\mathrm{S}\mathrm{I}}\in \mathbb{R}$ measured in coulombs is supplied, the intended Dirac charge-number operator is $Q_{\mathrm{S}\mathrm{I}}=q_{\mathrm{S}\mathrm{I}}(N_p-N_{\bar{p}})$. Here ${\mathrm{\Pi }}_p$ and ${\mathrm{\Pi }}_{\bar{p}}$ are the orthogonal projections onto ${\mathcal{H}}_{+}$ and $\overline{{\mathcal{H}}_{-}}$; on the $n$-particle sector, $N_p$ and $N_{\bar{p}}$ are the sums of the corresponding projections over the $n$ tensor factors. Its candidate maximal direct-sum domain is $D(Q_{\mathrm{S}\mathrm{I}})=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n):{\sum }_{n\ge 0}{\parallel q_{\mathrm{S}\mathrm{I}}{\sum }_{r=1}^n({\mathrm{\Pi }}_p-{\mathrm{\Pi }}_{\bar{p}}{)}^{(r)}{\mathrm{\Psi }}_n\parallel }^2<\mathrm{\infty }\}\mathrm{.}$ This is a candidate definition only; no self-adjointness or proved-domain claim is asserted here.

Measurements in this model are assigned only when a state and a normalized finite bounded Kraus instrument have actually been constructed. For such an instrument, the adopted outcome probabilities and positive-probability conditional states are exactly those of Physical interpretation of a supplied constructed quantum model. They are model probabilities, not observed outcomes. The field definitions are test-smeared: neither ${\mathrm{\Phi }}_{\mu }(x)$ nor $\mathrm{\Psi }(x)$ is a pointwise operator supplied here, and the smearing results do not define coincident products or a diagonal restriction.

The current Domain-qualified free observables and point-field obstruction Statement and Proof establish the scalar and Dirac field and translation-generator claims above, but do not state or prove the planned charged-number operator or its domain. The research module research/extended-frameworks-2026-10-03/quantum-field-theory/workers/free-fields/proof-modules.md#FF7 contains a species-count domain argument, but that argument is not a current MF04 item carrier or readiness input. The displayed charge expression is therefore a candidate definition conditional on that missing mathematical interface; no self-adjointness or proved-domain claim is made here.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. In the selected model of Specified bosonic scalar and CAR Dirac model adoption, let $H_{\mu }$ be the scalar one-particle Hilbert space and let $\mathcal{K}={\mathcal{H}}_{+}\oplus \overline{{\mathcal{H}}_{-}}$ be the Dirac particle/antiparticle one-particle space. Let ${\mathrm{\Omega }}_s$ and ${\mathrm{\Omega }}_D$ be their zero-particle Fock vectors. The assignments of $H_{\mu }$ to bosonic Fock space and $\mathcal{K}$ to CAR Fock space are the physical assumptions of that postulate.

For every unit scalar wave-packet mode $h\in H_{\mu }$, define $\mathrm{\mid }n{⟩}_h=\frac{(a^{†}(h){)}^n}{\sqrt{n!}}{\mathrm{\Omega }}_s,n\ge 0.$ These vectors are normalized and mutually orthogonal, belong to the finite-particle domain, and $a(h)\mathrm{\mid }n{⟩}_h=\sqrt{n}\mathrm{\mid }n-1{⟩}_h,a^{†}(h)\mathrm{\mid }n{⟩}_h=\sqrt{n+1}\mathrm{\mid }n+1{⟩}_h,$ where $a(h)\mathrm{\mid }0{⟩}_h=0$. The maximal operator domains supplied by Bosonic and fermionic Fock operators with exact domains are $D(a(h))=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n)\in {\mathcal{F}}_s(H_{\mu }):{\sum }_{n\ge 1}\mathrm{\parallel }{a}_n(h){\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\},D(a^{†}(h))=\{\mathrm{\Psi }=({\mathrm{\Psi }}_n)\in {\mathcal{F}}_s(H_{\mu }):{\sum }_{n\ge 0}\mathrm{\parallel }{a}_n^{†}(h){\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ Define the one-mode number operator as the composition $N_h:=a^{†}(h)a(h),D(N_h):=\{\mathrm{\Psi }\in D(a(h)):a(h)\mathrm{\Psi }\in D(a^{†}(h))\},N_h\mathrm{\Psi }:=a^{†}(h)(a(h)\mathrm{\Psi })\mathrm{.}$ The displayed finite-particle vectors $\mathrm{\mid }n{⟩}_h$ belong to $D(N_h)$ and satisfy $N_h\mathrm{\mid }n{⟩}_h=n\mathrm{\mid }n{⟩}_h$.

For every unit Dirac particle mode $u\in {\mathcal{H}}_{+}$ or antiparticle mode $u\in \overline{{\mathcal{H}}_{-}}$, regarded as a vector of $\mathcal{K}$, the bounded operator $N_u=c^{†}(u)c(u)$ is an orthogonal projection. The one-mode subspace generated by $u$ is the closed span of $\{(c^{†}(u){)}^n{\mathrm{\Omega }}_D:n\ge 0\}$. It is spanned by ${\mathrm{\Omega }}_D$ and $c^{†}(u){\mathrm{\Omega }}_D$; the latter is normalized and $c^{†}(u{)}^2=0$. Thus a single Dirac CAR mode has occupation $0$ or $1$. All statements are conditional on the adopted representations and ${\mathrm{A}\mathrm{C}}_{\omega }$. They do not derive which statistics nature assigns to all fields and do not assert an experimental observation.