The Stress-Tensor Noise Kernel

The noise kernel is the connected, symmetrized covariance of the renormalized stress tensor at two spacetime points. It contains information absent from the semiclassical source $\langle T_{\mu\nu}\rangle$, but it is still a quantum bi-distribution: positivity is asserted after smearing, and no classical random stress tensor or pointwise variance is assumed on this page.

Required background. Stress Bi-Tensors and Noise-Kernel Input supplies the separated-point composite operator and primed-index conventions; Quasifree States and Two-Point Functions supplies Wick factorization.

Helpful background. Wick Polynomials and Point Splitting explains local composite-field subtraction, while Connected Correlators and Cumulants fixes the connected subtraction.

Centered stress covariance

For a state $\omega$, define

$$\hat t_{\mu\nu}(x) \equiv \hat T_{\mu\nu}(x) -\omega\!\left(\hat T_{\mu\nu}(x)\right)\mathbf 1,$$

and, with the second tensor living at $y$,

$$N_{\mu\nu\rho'\sigma'}(x,y) \equiv \frac12\, \omega\!\left( \left\{\hat t_{\mu\nu}(x), \hat t_{\rho'\sigma'}(y)\right\} \right).$$

This is the chapter’s fixed normalization. It implies exchange symmetry

$$N_{\mu\nu\rho'\sigma'}(x,y) =N_{\rho'\sigma'\mu\nu}(y,x),$$

and symmetry within each index pair. If the renormalized stress tensor obeys its Ward identity, then $\nabla^\mu N_{\mu\nu\rho'\sigma'}=0$ and $\nabla^{\rho'}N_{\mu\nu\rho'\sigma'}=0$ as distributional identities, apart from explicitly retained contact terms. The anticommutator is essential: the stress commutator instead supplies causal response. These roles are separated in Hu and Verdaguer 2008, §§3.2 and 4.1, eqs. (11)–(12) and (22)–(27).

Let $f^{\mu\nu}\in C_0^\infty(S^2TM)$ be real and set

$$\hat t(f)=\int_M \mathrm d^4x\,\sqrt{-g}\, f^{\mu\nu}(x)\hat t_{\mu\nu}(x).$$

Then

$$N(f,f) =\int_{M^2}\!\mathrm dV_x\mathrm dV_y\, f^{\mu\nu}(x)N_{\mu\nu\rho'\sigma'}(x,y) f^{\rho'\sigma'}(y) =\omega\!\left(\hat t(f)^2\right)\ge 0.$$

This smeared inequality is the covariance-positivity condition needed for a real Gaussian representation. It says nothing about the undefined expression $N(x,x)$.

The structure map locates this object before both the response kernel and the stochastic metric equation. Inspect the fork: the symmetric covariance and antisymmetric response are not recoverable from one another in a generic state.

The noise kernel is a matter-state input to stochastic gravity, not yet a metric fluctuation and not a replacement for the retarded response kernel. The map is schematic and not to scale.

First application: a Gaussian scalar on an ultrastatic spacetime

Take $M=\mathbb R\times\Sigma$ with $\mathrm ds^2=\mathrm dt^2-h_{ij}\mathrm dx^i\mathrm dx^j$, compact $\Sigma$, and a free real scalar satisfying

$$P\phi=(\Box+m^2+\xi R)\phi=0, \qquad \xi_{\mathrm{conf}}=-\frac16.$$

Let $u_j(t,\mathbf x)=e^{-i\omega_jt}\psi_j(\mathbf x)/\sqrt{2\omega_j}$ be positive-frequency modes of a stationary quasifree state. Write the point-split stress tensor as a bidifferential operator $\mathcal D_{\mu\nu}^{xx'}$ acting on $\phi(x)\phi(x')$. Away from all partial diagonals, Wick’s theorem gives the connected stress covariance

$$N_{\mu\nu\rho'\sigma'}(x,y) =\operatorname{Re} \lim_{x'\to x\atop y'\to y} \mathcal D_{\mu\nu}^{xx'} \mathcal D_{\rho'\sigma'}^{yy'} \left[ G^+(x,y)G^+(x',y') +G^+(x,y')G^+(x',y) \right],$$

with any state-independent local subtraction already included in the definition of $\hat T_{\mu\nu}$. This is the general stress-bitensor construction specialized to a Gaussian scalar; explicit differential forms are given in Phillips and Hu 2001, §§II–III.

For a real compactly supported sampler $f^{\mu\nu}$, insert the mode expansion. In the ground state the smeared operator has a pair-creation part

$$\hat t(f)=C_f+ \sum_{jk}\left(A_{jk}a_j^\dagger a_k +\frac12B_{jk}a_j a_k +\frac12B_{jk}^*a_j^\dagger a_k^\dagger\right),$$

where $C_f$ cancels in the centered operator. Direct contraction yields

$$N(f,f)=\frac12\sum_{jk}\lvert B_{jk}\rvert^2\ge0.$$

The exact coefficient depends on the convention used to symmetrize $B_{jk}$, but the norm-square form and its sign do not. Conservation follows by integrating the Ward identity against $f$; for a pure-gauge sampler $f^{\mu\nu}=\nabla^{(\mu}v^{\nu)}$ with compactly supported $v^\nu$, integration by parts gives $N(f,f)=0$. Exchange symmetry is immediate from the real part. These are three independent checks: algebraic symmetry, the Ward identity, and covariance positivity.

Same mean stress, different fluctuations

One oscillator mode already defeats the idea that the mean fixes the noise. Compare

$$\rho_A=|1\rangle\langle1|, \qquad \rho_B=\frac12|0\rangle\langle0| +\frac12|2\rangle\langle2|.$$

Both have $\langle n\rangle=1$ and no anomalous two-point function, so their contribution to the stationary mean stress is identical. Yet

$$\langle n^2\rangle_A=1, \qquad \langle n^2\rangle_B=2.$$

A stress sampler sensitive to that mode therefore has different connected four-point data and a different noise kernel. Mean semiclassical backreaction cannot distinguish the states; stochastic gravity can at second order. For non-quasifree states, the connected four-point function must be supplied rather than reconstructed from $G^+$.

Domain and failure conditions

Use the chapter’s comparison table to locate this page at the “Gaussian stochastic” input layer. The result requires a state on the stress algebra, a renormalized stress prescription, and admissible test tensors. Boundaries, non-Hadamard singularities, or partial-diagonal restrictions may change the distributional domain. Coincidence extensions belong to the next page, metric propagation belongs to Einstein–Langevin dynamics, and the commutator cannot be inferred from $N$.

The failure map highlights the first decisive mistake: replacing the smeared quadratic form by an unsmeared value at $x=y$.

Covariance positivity licenses a Gaussian representation only on the declared test-function space; it does not license a pointwise random stress tensor. The map is schematic and not to scale.

Exercises

Show that $N(f,f)$ is unchanged if a constant multiple of the identity is added to $\hat T_{\mu\nu}$.

Solution

The same constant is added to $\langle\hat T_{\mu\nu}\rangle$, so it cancels in $\hat t_{\mu\nu}=\hat T_{\mu\nu}-\langle\hat T_{\mu\nu}\rangle\mathbf1$. Hence neither $\hat t(f)$ nor its variance changes. This does not remove derivative contact ambiguities in a renormalized product of two stress tensors.

References

• Hu, B. L., and E. Verdaguer. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3 (2008). doi:10.12942/lrr-2008-3. Open PDF
• Phillips, N. G., and B. L. Hu. “Noise Kernel in Stochastic Gravity and Stress Energy Bitensor of Quantum Fields in Curved Spacetimes.” Physical Review D 63, 104001 (2001). doi:10.1103/PhysRevD.63.104001. Open PDF