Exact Dirac particle/antiparticle free model

Fix the oriented, time-oriented affine Minkowski spacetime and inertial length coordinates of Quantum field quantities and compatible SI normalizations, with $\eta =\mathrm{diag}(-1,1,1,1)$ and $x^0=ct$. The parameter $\mu \ge 0$ and wave vector $\mathbf{k}$ have unit ${\mathrm{m}}^{-1}$; $\omega (\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2}$ also has unit ${\mathrm{m}}^{-1}$. For $\mu >0$, $\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$; $\mu =0$ is the admitted massless branch. The corresponding SI energy and momentum are $\mathrm{\hslash }c\omega$ joules and $\mathrm{\hslash }\mathbf{k}$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$. The canonically normalized Dirac field has engineering length degree $-3\mathrm{/}2$; a spacetime spinor test has degree $-5\mathrm{/}2$, making its spacetime-smearing pairing dimensionless. These are the unit conventions of Quantum field quantities and compatible SI normalizations, not claims about interacting-field scaling.

Adopt the Dirac branch of Specified bosonic scalar and CAR Dirac model adoption: one free Dirac species, the selected CAR representation and zero-particle vacuum are physical model assumptions. Countable Choice is the mathematical choice strength used by the Fock construction The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Use ${\mathbb{C}}^4$ spinors and the matrices, mass-shell parameter, one-particle Hilbert space ${\mathcal{H}}_D$, multiplier $h_D(\mathbf{k})=\mathit{\alpha }\cdot \mathbf{k}+\beta \mu$, projections $P_{\pm }(\mathbf{k})$ and propagator $W_D$ defined in Dirac Clifford matrices and energy projections. The mathematical supplier Dirac matrix identities and orthogonal energy projections establishes that $P_{\pm }$ are complementary orthogonal projections of rank two, that ${\mathcal{H}}_{\pm }=\mathrm{ran}{P}_{\pm }$ are closed, and that $h_D$ is self-adjoint on its maximal multiplication domain. The one-particle space of the adopted field is ${\mathcal{K}}_D={\mathcal{H}}_{+}\oplus \overline{{\mathcal{H}}_{-}},$ where the bar denotes the conjugate Hilbert space. Its fermionic Fock space, vacuum and creation/annihilation maps use Schwartz/distribution spaces and Hilbert Fock conventions and Bosonic and fermionic Fock operators with exact domains.

For a Schwartz spinor test $F\in \mathcal{S}({\mathbb{R}}^4;{\mathbb{C}}^4)$, set $\widehat{F}(t,\mathbf{k})={\int }_{{\mathbb{R}}^3}{e}^{-i\mathbf{k}\cdot \mathbf{x}}F(t,\mathbf{x})d^3\mathbf{x},u_F(\mathbf{k})={\int }_{\mathbb{R}}{W}_D(t,\mathbf{k})\widehat{F}(t,\mathbf{k})dt\mathrm{.}$ The physical field is the bounded smeared map $\mathrm{\Psi }(F)=c(P_{+}{u}_F)+c^{†}(\overline{P_{-}{u}_F}),$ with the exact test-map and CAR construction of Smeared free Dirac CAR fields and graded locality. It is antilinear in $F$. Its positive-energy branch is the particle annihilation term; the conjugate negative-energy branch occurs as antiparticle creation. This is a model assignment: it does not identify negative-energy Fock excitations with physical states, invoke a filled Dirac sea, or prove the general spin-statistics theorem. No pointwise field operator is defined here.

Assume Countable Choice and the adopted Dirac branch of Specified bosonic scalar and CAR Dirac model adoption as specified by Exact Dirac particle/antiparticle free model. For all Schwartz spinor tests $F,G$, the bounded smeared field $\mathrm{\Psi }$ is antilinear in its test, satisfies $\{\mathrm{\Psi }(F),\mathrm{\Psi }(G)\}=0,\{\mathrm{\Psi }(F),\mathrm{\Psi }(G{)}^{†}\}=⟨u_F,u_G⟩I,$ and solves the tested free Dirac equation. If $F$ and $G$ have compact spacelike-separated supports, the field-field and field-adjoint anticommutators vanish; consequently, finite even polynomials in the bounded smeared fields commute across spacelike-separated regions.

Let ${\tilde{G}}_D$ be the translation and explicit spin-lift group defined in Dirac Fock covariance, vacuum cyclicity and spectrum. Its strongly continuous unitary Fock representation $U$ satisfies $U(\tilde{g})\mathrm{\Psi }(F)U(\tilde{g}{)}^{-1}=\mathrm{\Psi }({\tau }_{\tilde{g}}F),{\tau }_{\tilde{g}}F(x)=(S^{-1}{)}^{†}F({\mathrm{\Lambda }}^{-1}(x-a)),$ for $\tilde{g}=(a,\mathrm{\Lambda },S)$. For the explicit spatial rotation lift, $S_R(2\pi )=-I_4$ and $S_R(4\pi )=I_4$; hence the $2\pi$ test pullback is $F\mapsto -F$ and $U_{2\pi }\mathrm{\Psi }(F)U_{2\pi }^{-1}=-\mathrm{\Psi }(F)$, while the $4\pi$ test pullback and field action are identities. The vacuum is invariant and cyclic for the global graded field algebra. On the $n$-particle sector, $U_{2\pi }=(-1{)}^{n}I$ and $U_{4\pi }=I$.

The joint spectrum of translations lies in the closed future cone. If $K_D$ denotes the length-coordinate time-translation generator of Dirac Fock covariance, vacuum cyclicity and spectrum, its zero-energy eigenspace is $\ker K_D=\mathbb{C}\mathrm{\Omega }$, where $\mathrm{\Omega }$ is the normalized vacuum; thus normalized zero-energy vectors differ from $\mathrm{\Omega }$ only by a complex phase. For $\mu >0$, the nonvacuum spectrum has gap at least $\mu$ in inverse-length units, hence $m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}{c}^2$ in SI energy units. The physical energy is $H_D^{\mathrm{S}\mathrm{I}}=\mathrm{\hslash }c{K}_D\ge 0$ on the same domain and has the same kernel. This Fock generator acts on a different Hilbert space from the first-quantized multiplier $h_D$, which has both signs of energy. These are conditional deductions in the adopted model, not empirical claims.