Clifford traces and normalized spin completeness

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and use the energy-vector, metric, and unit conventions of QFT example units, metric, gauge, and pairing conventions. A rest-energy parameter $\mathsf{M}\ge 0$ and every energy-vector component are measured in joules. Set $D_{\mathsf{M}}={\mathbb{R}}^3$ when $\mathsf{M}>0$ and $D_0={\mathbb{R}}^3\setminus \{0\}$. Define ${\iota }_{\mathsf{M}}:D_{\mathsf{M}}\to {\mathbb{R}}^4$ by ${\iota }_{\mathsf{M}}(\mathbf{p})=(\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mathsf{M}}^2},\mathbf{p}),{\mathcal{H}}_{\mathsf{M}}^{+}:={\iota }_{\mathsf{M}}(D_{\mathsf{M}})\mathrm{.}$ Thus ${\mathcal{H}}_{\mathsf{M}}^{+}$ is the positive-energy shell, with its subspace Borel structure. Define the Borel shell measure ${\nu }_{\mathsf{M}}$ as the pushforward under ${\iota }_{\mathsf{M}}$ of the measure absolutely continuous with respect to three-dimensional Lebesgue measure on $D_{\mathsf{M}}$, with density $((2\pi {)}^{3}2\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mathsf{M}}^2}{)}^{-1}$. In particular, the massless apex is excluded.

Let ${\mathsf{M}}_1,{\mathsf{M}}_2\ge 0$ and let $s>({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2$. Define $k_{\ast }=\frac{\sqrt{[s-({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2][s-({\mathsf{M}}_1-{\mathsf{M}}_2{)}^2]}}{2\sqrt{s}},E_i^{\ast }=\sqrt{k_{\ast }^2+{\mathsf{M}}_i^2}\mathrm{.}$ Both factors under the square root are positive, so $k_{\ast }>0$. On the Euclidean unit sphere $S^2$, let $\mathrm{\Omega }$ be the chart surface Borel measure of Agreement with the existing polar sphere measure. Define the finite Borel measure ${\mathrm{\Phi }}_2$ on $S^2$ and the shell-pair map ${\mathrm{\Psi }}_2:S^2\to {\mathcal{H}}_{{\mathsf{M}}_1}^{+}\times {\mathcal{H}}_{{\mathsf{M}}_2}^{+}$ by $d{\mathrm{\Phi }}_2=\frac{k_{\ast }}{16{\pi }^2\sqrt{s}}d\mathrm{\Omega },{\mathrm{\Psi }}_2(\mathbf{n})=((E_1^{\ast },k_{\ast }\mathbf{n}),(E_2^{\ast },-k_{\ast }\mathbf{n}))\mathrm{.}$ The center-of-momentum measure on the shell-pair space is the pushforward $({\mathrm{\Psi }}_2{)}_{\mathrm{\#}}{\mathrm{\Phi }}_2$. The angular measure ${\mathrm{\Phi }}_2$ and its shell-pair pushforward are distinct typed measures.

For incoming energy four-vectors $p_i\in {\mathcal{H}}_{{\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}}^{+}$, where ${\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}\ge 0$ for $i=1,2$, define the invariant flux when its radicand is nonnegative by $F(p_1,p_2)=4\sqrt{(p_1\cdot p_2{)}^2-{\mathsf{M}}_{1,\mathrm{i}\mathrm{n}}^2{\mathsf{M}}_{2,\mathrm{i}\mathrm{n}}^2},p\cdot q=h(p,q)\mathrm{.}$ Let ${\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ and ${\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ be finite nonempty sets of initial and final spin labels, respectively, and let $w_i\ge 0$ for $i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ satisfy ${\sum }_i{w}_i=1$. A spin-resolved formal amplitude is a family of dimensionless Borel functions ${\mathcal{A}}_{if}:S^2\to \mathbb{C}$, indexed by $i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}$ and $f\in {\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Define the Borel function $\overline{\mathrm{\mid }\mathcal{A}(\mathbf{n}){\mathrm{\mid }}^2}:={\sum }_{i\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}}}{w}_i{\sum }_{f\in {\mathcal{S}}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\mathrm{\mid }{\mathcal{A}}_{if}(\mathbf{n}){\mathrm{\mid }}^2\mathrm{.}$ For a specified final two-particle channel, set $S_f=2$ when its two final particles are identical and $S_f=1$ when they are distinguishable. When the incoming flux is positive, $F>0$, define the angular cross-section coefficient measure ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ on $S^2$ by $d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{(\mathrm{\hslash }c{)}^2}{F{S}_f}\overline{\mathrm{\mid }\mathcal{A}(\mathbf{n}){\mathrm{\mid }}^2}d{\mathrm{\Phi }}_2\mathrm{.}$ For a parent at rest with energy ${\mathsf{M}}_0>0$ and a specified single-channel Borel formal decay amplitude $\mathcal{G}:S^2\to \mathbb{C}$ measured in joules, define the extended nonnegative coefficient ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{1}{2\mathrm{\hslash }{\mathsf{M}}_0{S}_f}{\int }_{S^2}\mathrm{\mid }\mathcal{G}(\mathbf{n}){\mathrm{\mid }}^{2}d{\mathrm{\Phi }}_2\mathrm{.}$ It is a finite rate coefficient when this integral is finite; otherwise the displayed integral has value $+\mathrm{\infty }$. These are mathematical coefficient maps for the explicitly specified formal amplitudes; their definitions assert no exact interacting scattering operator or decay law.

Use the finite Dirac matrices and adjoint of Dirac Clifford matrices and energy projections, and the separate geometric metric $\eta =\mathrm{diag}(-1,1,1,1)$ and algebraic contraction $h=-\eta =\mathrm{diag}(1,-1,-1,-1)$ of QFT example units, metric, gauge, and pairing conventions. Write $p=(E,\mathbf{p})$ for an energy four-vector with components in joules, $M\ge 0$ for a rest-energy parameter in joules, and $E=\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+M^2}>0$. In the natural-length variables of the Dirac definition, $\mathbf{k}=\mathbf{p}\mathrm{/}(\mathrm{\hslash }c)$, $\mu =M\mathrm{/}(\mathrm{\hslash }c)$, and $\omega =E\mathrm{/}(\mathrm{\hslash }c)$.

Let $\beta ={\gamma }^0$, ${\alpha }_j=\beta {\gamma }^j$, ${\mathrm{\Sigma }}_j=\mathrm{diag}({\sigma }_j,{\sigma }_j)$, and ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$. Define $H_{\pm }=\mathit{\alpha }\cdot \mathbf{p}\pm \beta M$ and $P_{\pm }=(I_4+H_{\pm }\mathrm{/}E)\mathrm{/}2$. Each $P_{\pm }$ is a Hermitian rank-two projection. Choose orthonormal bases ${\xi }_{\pm ,1},{\xi }_{\pm ,2}$ of their ranges and put $u_s=\sqrt{2E}{\xi }_{+,s}$ and $v_s=\sqrt{2E}{\xi }_{-,s}$ for $s=1,2$. With $\bar{z}=z^{†}\beta$ and $\mathrm{p̸}={\gamma }^{0}E-{\sum }_j{\gamma }^j{p}_j$, the following hold:

$\mathrm{Tr}{I}_4=4,\mathrm{Tr}{\gamma }^a=0,\mathrm{Tr}({\gamma }^a{\gamma }^b)=4h^{ab},\mathrm{Tr}({\gamma }^a{\gamma }^b{\gamma }^c)=0,$

$\mathrm{Tr}({\gamma }^a{\gamma }^b{\gamma }^c{\gamma }^d)=4(h^{ab}{h}^{cd}-h^{ac}{h}^{bd}+h^{ad}{h}^{bc}),{\sum }_s{u}_s{\bar{u}}_s=\mathrm{p̸}+M,{\sum }_s{v}_s{\bar{v}}_s=\mathrm{p̸}-M\mathrm{.}$

The columns satisfy $(\mathrm{p̸}-M)u_s=0$ and $(\mathrm{p̸}+M)v_s=0$. For all $x,y\in {\mathbb{C}}^4$, complex conjugation of a Dirac current is $\overline{\stackrel{ˉ}{x}{\gamma }^{a}y}=\bar{y}{\gamma }^{a}x$. The Countable Choice hypothesis written in the current draft Dirac Clifford matrices and energy projections is retained as a supplier assumption; this finite-dimensional proof makes no additional choice use.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), and use QFT example units, metric, gauge, and pairing conventions and Two-body phase-space and formal coefficient data. Write ${\mathcal{H}}_{\mathsf{M}}^{+}=\{(E,\mathbf{q})\in (0,\mathrm{\infty })\times {\mathbb{R}}^3:E=\sqrt{\mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^2+{\mathsf{M}}^2}\}$ for the positive shell in that Definition, and $C_{+}=\{(E,\mathbf{q})\in \mathbb{R}\times {\mathbb{R}}^3:E\ge \mathrm{\mid }\mathbf{q}\mathrm{\mid }\}$ for the closed future cone of $h=(+,-,-,-)$. A proper orthochronous linear $h$-Lorentz map is a real linear $\mathrm{\Lambda }$ with ${\mathrm{\Lambda }}^{T}h\mathrm{\Lambda }=h$, $\det \mathrm{\Lambda }=1$, and $\mathrm{\Lambda }(C_{+})=C_{+}$. For energy four-vectors on these positive shells, the shell measure ${\nu }_{\mathsf{M}}$ defined there is invariant under every such map. Let ${\mathsf{M}}_1,{\mathsf{M}}_2\ge 0$, let $P$ be future timelike, meaning $P^0>0$ and $h(P,P)>0$, set $s=h(P,P)>({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2$, and write $E_i(r)=\sqrt{r^2+{\mathsf{M}}_i^2}$. Let $f$ be a nonnegative Borel function on ${\mathcal{H}}_{{\mathsf{M}}_1}^{+}\times {\mathcal{H}}_{{\mathsf{M}}_2}^{+}$. For a proper orthochronous $h$-Lorentz map ${\mathrm{\Lambda }}_P$ with ${\mathrm{\Lambda }}_{P}P=(\sqrt{s},\mathbf{0})$, define ${\mathrm{\Phi }}_{2,P}$ as the pushforward of the angular measure ${\mathrm{\Phi }}_2$ from that Definition under $\mathbf{n}⟼({\mathrm{\Lambda }}_P^{-1}(E_1^{\ast },k_{\ast }\mathbf{n}),{\mathrm{\Lambda }}_P^{-1}(E_2^{\ast },-k_{\ast }\mathbf{n})),$ where $E_i^{\ast }$ and $k_{\ast }$ are from Two-body phase-space and formal coefficient data. The proof below shows that ${\mathrm{\Phi }}_{2,P}$ is independent of the choice of ${\mathrm{\Lambda }}_P$; for $P=(\sqrt{s},\mathbf{0})$ it is the pushforward $({\mathrm{\Psi }}_2{)}_{\mathrm{\#}}{\mathrm{\Phi }}_2$ under the center-of-momentum map ${\mathrm{\Psi }}_2$ defined there. The standard two-body on-shell integral $(2\pi {)}^4\int d{\nu }_{{\mathsf{M}}_1}(k_1)d{\nu }_{{\mathsf{M}}_2}(k_2){\delta }^{(4)}(P-k_1-k_2)f(k_1,k_2)$ is interpreted by first evaluating the spatial delta in $k_2$, then the one-dimensional energy delta by the increasing radial change of variable $r\mapsto E_1(r)+E_2(r)$. Explicitly, for each fixed angular direction and nonnegative Borel radial function $g:(0,\mathrm{\infty })\to [0,\mathrm{\infty }]$, the radial action after spatial evaluation is ${\int }_0^{\mathrm{\infty }}g(r)\delta (\sqrt{s}-E_1(r)-E_2(r))dr=g(k_{\ast })\mathrm{/}(d(E_1+E_2)\mathrm{/}dr{\mathrm{\mid }}_{k_{\ast }})$; this is the pushforward evaluation under that $C^1$ diffeomorphism, not a product of distributions. It equals $\int fd{\mathrm{\Phi }}_{2,P}$. The threshold root is $k_{\ast }$ in that Definition, and ${\mathrm{\Phi }}_{2,P}({\mathcal{H}}_{{\mathsf{M}}_1}^{+}\times {\mathcal{H}}_{{\mathsf{M}}_2}^{+})={\mathrm{\Phi }}_2(S^2)=\frac{k_{\ast }}{4\pi \sqrt{s}}\mathrm{.}$ For equal rest energies ${\mathsf{M}}_1={\mathsf{M}}_2=\mathsf{M}$, set $\beta =\sqrt{1-4{\mathsf{M}}^2\mathrm{/}s}$; then $k_{\ast }=\sqrt{s}\beta \mathrm{/}2$ and ${\mathrm{\Phi }}_{2,P}({\mathcal{H}}_{\mathsf{M}}^{+}\times {\mathcal{H}}_{\mathsf{M}}^{+})={\mathrm{\Phi }}_2(S^2)=\beta \mathrm{/}(8\pi )$. For the scattering flux identities, take incoming $p_i=(E_i,{\mathbf{p}}_i)\in {\mathcal{H}}_{{\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}}^{+}$, where ${\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}\ge 0$, and require $p_1+p_2=P$. A center-of-momentum map is a proper orthochronous $h$-Lorentz map $\mathrm{\Lambda }$ with $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$; write $\mathrm{\Lambda }{p}_1=(E_1^{\ast },{\mathbf{p}}_1^{\ast })$ and define $p_{\ast }=\mathrm{\mid }{\mathbf{p}}_1^{\ast }\mathrm{\mid }$. The value is independent of the choice of $\mathrm{\Lambda }$, since two such maps differ by a spatial rotation. Their invariant flux is $F=4p_{\ast }\sqrt{s}$. For equal incoming rest energies ${\mathsf{M}}_{\mathrm{i}\mathrm{n}}\ge 0$ with $s\ge 4{\mathsf{M}}_{\mathrm{i}\mathrm{n}}^2$, define ${\beta }_{\mathrm{i}\mathrm{n}}=\sqrt{1-4{\mathsf{M}}_{\mathrm{i}\mathrm{n}}^2\mathrm{/}s}$; then $p_{\ast }=\sqrt{s}{\beta }_{\mathrm{i}\mathrm{n}}\mathrm{/}2$ and $F=2s{\beta }_{\mathrm{i}\mathrm{n}}$. For finite nonempty incoming and outgoing label sets, normalized initial weights, and the Borel amplitude family of the Definition, whenever $F>0$ the cross-section coefficient map gives $\frac{d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}}{d\mathrm{\Omega }}=\frac{(\mathrm{\hslash }c{)}^2}{64{\pi }^{2}s}\frac{k_{\ast }}{p_{\ast }}\frac{\overline{\mathrm{\mid }\mathcal{A}{\mathrm{\mid }}^2}}{S_f}\mathrm{.}$ For a parent at rest, represented by $P_0=({\mathsf{M}}_0,\mathbf{0})$ with ${\mathsf{M}}_0>2\mathsf{M}$, and two equal-rest-energy daughters on ${\mathcal{H}}_{\mathsf{M}}^{+}$ with total four-vector $P_0$ and constant amplitude $G_E$ in joules, the rate coefficient is $\mathrm{\mid }{G}_E{\mathrm{\mid }}^2\sqrt{1-4{\mathsf{M}}^2\mathrm{/}{\mathsf{M}}_0^2}\mathrm{/}(32\pi \mathrm{\hslash }{\mathsf{M}}_0)$ when the daughters are identical. This is a coefficient calculation, not a claim of exponential survival.

Let $m,M\ge 0$ be the electron and muon rest energies in joules. Use the positive shells of Two-body phase-space and formal coefficient data: $p_i=(E_i,{\mathbf{p}}_i)$ with $E_i=\sqrt{\mathrm{\mid }{\mathbf{p}}_i{\mathrm{\mid }}^2+m^2}>0$ for $i=1,2$, and $k_i=({\epsilon }_i,{\mathbf{k}}_i)$ with ${\epsilon }_i=\sqrt{\mathrm{\mid }{\mathbf{k}}_i{\mathrm{\mid }}^2+M^2}>0$ for $i=1,2$. With $p_1+p_2=k_1+k_2=P$, set $s=h(P,P)>4\max (m^2,M^2)$, using the algebraic metric $h=(+,-,-,-)$ of QFT example units, metric, gauge, and pairing conventions. Since $P^0=E_1+E_2>0$ and $h(P,P)=s>0$, $P$ is future timelike. A center-of-momentum map is a proper orthochronous linear $h$-Lorentz map $\mathrm{\Lambda }$ with ${\mathrm{\Lambda }}^{T}h\mathrm{\Lambda }=h$, $\det \mathrm{\Lambda }=1$, $\mathrm{\Lambda }(C_{+})=C_{+}$ for $C_{+}=\{(E,\mathbf{q})\in \mathbb{R}\times {\mathbb{R}}^3:E\ge \mathrm{\mid }\mathbf{q}\mathrm{\mid }\}$, and $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$. Write ${\mathbf{p}}_1^{\ast }$ and ${\mathbf{k}}_1^{\ast }$ for the spatial components of $\mathrm{\Lambda }{p}_1$ and $\mathrm{\Lambda }{k}_1$, respectively. The strict threshold makes both nonzero; define $\theta \in [0,\pi ]$ by $\cos \theta =({\mathbf{p}}_1^{\ast }{\cdot }_{{\mathbb{R}}^3}{\mathbf{k}}_1^{\ast })\mathrm{/}(\mathrm{\mid }{\mathbf{p}}_1^{\ast }\mathrm{\mid }\mathrm{\mid }{\mathbf{k}}_1^{\ast }\mathrm{\mid })$. This angle is independent of the chosen center-of-momentum map because any two choices differ by a spatial rotation. For $a,b\in \{0,1,2,3\}$, set $h_{ab}=h^{ab}$ equal to the diagonal entries of $h=\mathrm{diag}(1,-1,-1,-1)$, sum repeated Lorentz indices over this range, and define ${\gamma }_a:={\sum }_{b=0}^3{h}_{ab}{\gamma }^b$ and $P_a:={\sum }_{b=0}^3{h}_{ab}{P}^b$. Use the dimensionless coupling $g=\sqrt{4\pi \alpha }$ from QFT example units, metric, gauge, and pairing conventions. Adopt only the formal single-photon tree coefficient $\mathcal{M}=\frac{g^2}{s}[\bar{v}(p_2){\gamma }^{a}u(p_1)][\bar{u}(k_1){\gamma }_{a}v(k_2)],$ with equal weight $1\mathrm{/}4$ on the four initial spin-label pairs and a sum over final spin labels. The outgoing labels denote the distinct ${\mu }^{-}$ and ${\mu }^{+}$ species, so $S_f=1$ in the coefficient map. With the spinor normalization of Clifford traces and normalized spin completeness, $\mathcal{M}$ is dimensionless. No interacting continuum $S$-matrix or exact QED cross section is assumed.

In the auxiliary finite-matrix setting of [L1], assume $d\ge 1$, an invertible $K_0$, and the stated covariance of $K_0$ and $V$ under the finite-graph phase action; then the coefficientwise inverse and covariance identities in [L1] hold. Separately stipulate the displayed formal pair-production amplitude above. Its on-shell spin average is $\overline{\mathrm{\mid }\mathcal{M}{\mathrm{\mid }}^2}=g^4[3-{\beta }_e^2-{\beta }_{\mu }^2+{\beta }_e^2{\beta }_{\mu }^2{\cos }^2\theta ],{\beta }_e=\sqrt{1-\frac{4m^2}{s}},{\beta }_{\mu }=\sqrt{1-\frac{4M^2}{s}},$ where the angle $\theta$ is defined above. The resulting formal differential and total coefficients are $\frac{d{\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}}{d\mathrm{\Omega }}=\frac{{\alpha }^2(\mathrm{\hslash }c{)}^2{\beta }_{\mu }}{4s{\beta }_e}[3-{\beta }_e^2-{\beta }_{\mu }^2+{\beta }_e^2{\beta }_{\mu }^2{\cos }^2\theta ],$ ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\pi {\alpha }^2(\mathrm{\hslash }c{)}^2}{s}\frac{{\beta }_{\mu }}{{\beta }_e}[3-{\beta }_e^2-{\beta }_{\mu }^2+\frac{1}{3}{\beta }_e^2{\beta }_{\mu }^2]\mathrm{.}$ If $\delta =4m^2\mathrm{/}s\le 1\mathrm{/}2$, the full differential coefficient divided by its $m=0$ value is between $1$ and $1+4\delta$. This bound concerns only omission of the incoming electron mass in this formal tree coefficient, not loop, soft-radiation, threshold, or empirical errors.

The same on-shell equations give $P_a\bar{v}(p_2){\gamma }^{a}u(p_1)=0$ and $P_a\bar{u}(k_1){\gamma }^{a}v(k_2)=0$. Thus a propagator term proportional to $P_a{P}_b$ does not change this specified tree coefficient. These current cancellations are proved locally for all $m,M\ge 0$; the actual QFT-MG03 matrix-Ward carrier supplies only its finite-matrix inverse and covariance interface, not the broader tree-current clauses in the historical Q7 research module.