Dirac Clifford matrices and energy projections

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the mathematical scalar shell conventions of Scalar future mass shell, invariant measure and test restriction, and the second-variable-linear Hilbert pairing of Schwartz/distribution spaces and Hilbert Fock conventions. In the fixed inertial coordinates $x=(x^0,\mathbf{x})$ of Inertial frames and inertial charts of the affine Minkowski model on the affine Minkowski model Oriented time-oriented affine Minkowski space (the spacetime primitives), $\eta =\mathrm{diag}(-1,1,1,1)$; these coordinates have length units under Scalar future mass shell, invariant measure and test restriction. The parameter $\mu \ge 0$ and momentum coordinate $\mathbf{k}$ have inverse-length units, and $\omega (\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2}$ has the same units. The parameter is not identified here with a physical rest mass. Matrix $h_D$ has inverse-length units; the dimensionless spin matrices have no physical unit assignment in this mathematical item.

Use the Pauli matrices ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. On ${\mathbb{C}}^4={\mathbb{C}}^2\oplus {\mathbb{C}}^2$, define $\beta =\left(\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right),{\alpha }_j=\left(\begin{array}{cc}0& {\sigma }_j\\ {\sigma }_j& 0\end{array}\right),{\gamma }^0=\beta ,{\gamma }^j=\beta {\alpha }_j\mathrm{.}$ For a spinor column $z\in {\mathbb{C}}^4$, its Dirac adjoint is $\bar{z}=z^{†}\beta$. The Clifford contraction is $\mathrm{\Gamma }(p):={\gamma }^0{p}^0-{\sum }_{j=1}^3{\gamma }^j{p}_j$; this contraction uses the effective matrix $-\eta$ in the relation $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}{I}_4$.

Let ${\mathcal{H}}_D$ be the space of measurable functions $u:{\mathbb{R}}^3\to {\mathbb{C}}^4$, modulo equality almost everywhere, with $\mathrm{\parallel }u{\mathrm{\parallel }}^2={\int }_{{\mathbb{R}}^3}u(\mathbf{k}{)}^{†}u(\mathbf{k})\frac{d^3\mathbf{k}}{(2\pi {)}^3}<\mathrm{\infty },$ and pairing $⟨u,v⟩=\int u(\mathbf{k}{)}^{†}v(\mathbf{k})d^3\mathbf{k}\mathrm{/}(2\pi {)}^3$. Define the measurable matrix $h_D(\mathbf{k})={\sum }_j{\alpha }_j{k}_j+\beta \mu$ and its maximal multiplication operator with domain $D(h_D)=\{u\in {\mathcal{H}}_D:h_{D}u\in {\mathcal{H}}_D\}$. For $\omega (\mathbf{k})>0$, define the matrix fields $P_{\pm }(\mathbf{k})=\frac{1}{2}(I_4\pm \frac{h_D(\mathbf{k})}{\omega (\mathbf{k})})\mathrm{.}$ When $\mu =0$ at $\mathbf{k}=0$, set $P_{+}(0)=\mathrm{diag}(I_2,0)$ and $P_{-}(0)=\mathrm{diag}(0,I_2)$. These are complementary rank-two orthogonal projections; the value at this Lebesgue-null point does not change their multiplication operators. Write $M_{P_{\pm }}$ for their maximal multiplication operators and ${\mathcal{H}}_{\pm }=\mathrm{ran}{M}_{P_{\pm }}$. The companion matrix lemma on this page proves these are bounded orthogonal projections with closed ranges. The conjugate Hilbert space $\overline{{\mathcal{H}}_{-}}$ has vectors $\bar{u}$, scalar multiplication $z\bar{u}=\overline{\stackrel{ˉ}{z}u}$, and pairing $⟨\bar{u},\bar{v}⟩=⟨v,u⟩$.

Use the sine and cosine definitions of Sine and cosine defined by their real power series. For a length-coordinate increment $t\in \mathbb{R}$ and $\omega (\mathbf{k})>0$, define the matrix multiplier $W_D(t,\mathbf{k})=\cos (\omega t)I_4+i\frac{\sin (\omega t)}{\omega }{h}_D(\mathbf{k})\mathrm{.}$ At the massless apex set $W_D(t,0)=I_4$. Its unitarity, group law and generator identity are proved by the companion matrix lemma on this page.

The symbols $P_{+}$ and $P_{-}$ designate the positive- and negative-energy matrix formulas for this fixed free Dirac operator. This is purely mathematical data; no physical spin-statistics postulate or empirical assertion is included.