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.