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.