Definition. Covariance and variance of Gaussian observables [fdip-002Z]
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  • October 25, 2026

Let \(\mu _Q\) be the Gaussian law on a finite real inner-product space \((E,h)\), and write \(\omega _Q\) for its expectation. For Borel \(X,Y:E\to \mathbb R\) with \(X\), \(Y\) and \(XY\) absolutely integrable, define their covariance by \[\begin {aligned} \operatorname {Cov}_Q(X,Y) &=\omega _Q(XY)-\omega _Q(X)\omega _Q(Y)\\ &=\omega _Q\bigl ((X-\omega _Q(X))(Y-\omega _Q(Y))\bigr ). \end {aligned}\] Expansion and normalization prove the second equality; all 4 terms are integrable by the stated hypotheses. This extends the finite-sum covariance to integrals.

When \(X\) and \(X^2\) are integrable, its variance is \[\begin {aligned} \operatorname {Var}_Q(X)&=\operatorname {Cov}_Q(X,X)\\ &=\omega _Q\bigl ((X-\omega _Q(X))^2\bigr )\\ &=\omega _Q(X^2)-\omega _Q(X)^2\geq 0. \end {aligned}\] The inequality is positivity of the integral. In particular all these expressions exist for polynomials by Proposition [fdip-002W], consistently with the polynomial state.

For \(\ell _j(x)=h(j,x)\), set \(K(j,k)=\operatorname {Cov}_Q(\ell _j,\ell _k)\). It is a finite symmetric bilinear form. There is a unique linear covariance operator \(C_Q:E\to E\), relative to the specified \(h\), such that \(h(j,C_Qk)=K(j,k)\).

Proof. 

  • October 25, 2026

In an orthonormal basis put \(C_Qk=\sum _i K(u_i,k)u_i\). Bilinearity of \(K\) follows from integral linearity and gives the linearity of \(C_Q\). Orthonormal expansion proves the required pairing. If \(C_Q'\) has the same pairings, then \(h(j,(C_Q-C_Q')k)=0\) for every \(j,k\); choosing \(j=(C_Q-C_Q')k\) proves equality. Thus the construction is independent of the basis. At \(E=0\) it is the unique map. This representation does not identify \(C_Q\) with inverse precision.

For the counting field space, the coordinate means vanish by Proposition [fdip-002Y], and the former coordinate notation is \[\begin {aligned} \operatorname {Cov}_Q(u,v) &=\left \langle (\phi _u-\langle \phi _u\rangle _Q) (\phi _v-\langle \phi _v\rangle _Q)\right \rangle _Q\\ &=\langle \phi _u\phi _v\rangle _Q. \end {aligned}\] The partition function, this two-point expectation, and the expectation of a different observable are different quantities. The separate inverse-precision theorem computes this covariance.

The definitions extend Notation [ftip-008H] directly; the integral identities use the preceding moment and state results. Positivity uses the ordinary real convention of [waldmann2004states, Definition 3.9, p. 11] through Definition [fdip-0024], not an additional inverse-covariance assumption.

The four-cycle covariance calculation evaluates these quantities for the running example.