Quantum Noise from Massive Lorentz--Violating Gravitons: Interferometer Signatures in Very Special Linear Gravity

What carries the argument

The calculation is carried by four linked objects. First, the VSR graviton action, which in the natural gauge $\partial_\mu h^{\mu\nu}=n^\mu h_{\mu\nu}=0$, $h=0$ reduces to a Klein-Gordon field $(\partial^2-m_g^2)h_{\mu\nu}=0$ with only two physical polarizations. Second, the gauge-invariant detector coupling $\Lambda^{(s)}_{ij}=\varepsilon^{(s)}_{00}k_i k_j+\varepsilon^{(s)}_{ij}\omega^2+(\varepsilon^{(s)}_{0i} k_j+\varepsilon^{(s)}_{0j} k_i)\omega$, built from the polarization tensor; the physics is carried by the sum of $\Lambda^{(s)}_{ij}\Lambda^{(s)}_{lm}$ over the two modes. Third, the reduction of that sum to a double kernel contraction $\hat b^{\mu i}\hat b^{\nu j}\hat b^{\rho l}\hat b^{\sigma m}S_{\mu\nu\rho\sigma}$ with $\hat b^{\mu i}=\omega\delta^{\mu i}+k^i\delta^{\mu 0}$ (using $k_0=-\omega$), whose angular integral gives $4\pi\omega^4[\widetilde T^{(0)}_{ijlm}+(m_g^2/\omega^2)\Lambda_{ijlm}+(m_g^2/\omega^2)\ln(4\omega^2/m_g^2)\Xi_{ijlm}]$. Fourth, the influence functional with Gaussian initial density matrices that interpolate between vacuum, thermal, and squeezed states in one expression; the squeezed-state frequency integrals $I_0$, $I_1$, $I_2$ place the $\sin\phi$ term into $I_2$ as $\tfrac{\pi}{4}\sin\phi\,\sinh(2r)\ln(m_g|t+t'|)$, while in the massless $I_0$ the same phase appears only inside $\delta'(t+t')$. The direction-dependent tensor $\Xi_{ijlm}$, built from $\delta_{ij}$ and the preferred null direction $n^i$, carries the new channel, and the logarithm is what makes it a persistent memory rather than a boundary effect.