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000076 | \begin{lemma}[Null-cone positive factorization]
000077 | For every nonzero null $k=(-r,p)$,
000078 | \begin{equation}
000079 |  K(k)=A(k)S P_\TT(p)S^TA(k)^T
000080 |        =L(k)L(k)^T,\qquad L(k)=A(k)S P_\TT(p).
000081 |  \label{eq:toe-r7-covariant-positive}
000082 | \end{equation}
000083 | Consequently $K(k)$ is positive semidefinite of rank exactly two.
000084 | It is nevertheless a polynomial of degree four in all four components of $k$.
000085 | \end{lemma}
000086 | \begin{proof}
000087 | Substitution in~\eqref{eq:toe-r7-covariant-A} gives $A(k)(k\xi^T+\xi k^T)=0$ for every
000088 | $\xi$. The identity~\eqref{eq:toe-r7-covariant-positive} can be checked with $p=(0,0,r)$:
000089 | the two surviving transverse orthonormal matrices are
000090 | \[
000091 |  h_+=\operatorname{diag}(1,-1,0)/\sqrt2,\qquad
000092 |  h_\times=(e_1e_2^T+e_2e_1^T)/\sqrt2,
000093 | \]
000094 | and direct contraction gives
000095 | $A\Pi A^T=(ASh_+)(ASh_+)^T+(ASh_\times)(ASh_\times)^T$.
000096 | Spatial rotational covariance and degree-four homogeneity give the result
000097 | for every nonzero $p$. The map is injective on the two-dimensional TT
000098 | space because $(A(k)Sh)_{0i0j}=r^2h_{ij}/2$. This proves rank two.
000099 | Polynomiality follows directly from~\eqref{eq:toe-r7-covariant-K}; no division by $r$
000100 | occurs there.
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000114 | 
000115 | The same map obeys both pair antisymmetries, pair interchange, the algebraic
000116 | Bianchi identity $R_{\mu[\nu\rho\sigma]}=0$, and the differential Bianchi
000117 | identity $k_{[\tau}R_{|\mu\nu|\rho\sigma]}=0$.
000118 | On the TT null range the Ricci contraction vanishes,
000119 | $\eta^{\mu\rho}R_{\mu\nu\rho\sigma}=0$. For example, insert $h_{0\mu}=0$,
000120 | $p_ih_{ij}=0$, $h_{ii}=0$, and $k^2=0$ into~\eqref{eq:toe-r7-covariant-A}; every contraction
000121 | contains one of those four zero factors. Thus the constructed vacuum
000122 | curvature is also its linearized Weyl tensor. These identities hold as
000123 | operator-valued distributional identities after the null quotient below.
000124 | 
000125 | \paragraph{Positive field, local commutator, and boosts.}
000126 | Index the independent pair-symmetric bivector entries by $I,J$.
000127 | Define the two-point distribution
000128 | \begin{equation}
000129 |  W_{IJ}(x-y)=\int_{\mathbb R^3}
000130 |  \frac{d^3p}{(2\pi)^3\,2r}\,
000131 |  e^{-ir(t_x-t_y)+ip\cdot(\mathbf x-\mathbf y)}K_{IJ}(-r,p).
000132 |  \label{eq:toe-r7-covariant-W}
000133 | \end{equation}
000134 | The Bianchi identities make one of the 21 algebraic components redundant;
000135 | retaining it with its null relation avoids an arbitrary choice of twenty
000136 | coordinates.
000137 | 
000138 | \begin{theorem}[Free covariant curvature field]
000139 | Equation~\eqref{eq:toe-r7-covariant-W}, with Wick's rule for higher vacuum correlations,
000140 | defines a positive free bosonic operator-valued tempered distribution on
000141 | a symmetric Fock space. Its commutators vanish at spacelike separation.
000142 | There is a strongly continuous unitary representation of the proper
000143 | orthochronous Poincare group, with invariant cyclic Fock vacuum, spectrum
000144 | in the closed forward cone, and covariant rank-four curvature transformation.
000145 | Its only one-particle helicities are $+2$ and $-2$.
000146 | \end{theorem}
000147 | \begin{proof}
000148 | For Schwartz test tensors $f$, factorization~\eqref{eq:toe-r7-covariant-positive} gives
000149 | \begin{equation}
000150 |  \|f\|_1^2=\int\frac{d^3p}{(2\pi)^3\,2r}
000151 |  \widehat f(k)^*K(k)\widehat f(k)
000152 |  =\int\frac{d^3p}{(2\pi)^3\,2r}
000153 |  \|L(k)^T\widehat f(k)\|^2\geq0.
000154 |  \label{eq:toe-r7-covariant-norm}
000155 | \end{equation}
000156 | Quotient the zero-norm tensors and complete to $\cH_1$.
000157 | Write $u_f$ for the one-particle image of the test tensor. With $a(u)$
000158 | antilinear in $u$, set $R(f)=a^\dagger(u_f)+a(u_{\bar f})$ on the
000159 | finite-particle domain of $\Gamma_s(\cH_1)$, using the Fourier convention
000160 | implicit in~\eqref{eq:toe-r7-covariant-W}. This is complex linear in $f$ and obeys
000161 | $R(f)^*=R(\bar f)$ on that domain. It gives~\eqref{eq:toe-r7-covariant-W}. The usual creation
000162 | estimate $\|a^\dagger(u)\psi_n\|\leq\sqrt{n+1}\|u\|\|\psi_n\|$
000163 | and continuity of~\eqref{eq:toe-r7-covariant-norm} make this an operator-valued tempered
000164 | distribution. Wick positivity follows from this explicit Fock
000165 | construction, not from assuming $\Pi$ is positive.
000166 | 
000167 | Infrared convergence and temperedness are immediate: $K=O(r^4)$,
000168 | so $K/(2r)=O(r^3)$ at zero and has only polynomial growth at infinity.
000169 | Together with the $d^3p$ measure the radial integrand is $O(r^5)$ near zero.
000170 | Schwartz test functions control every ultraviolet moment.