Proposition. Conditional propagators and induced vertices [fdip-0060]
AGENTDRAFTED
For an interacting selected background, put
\(n=\dim H\) and \(r=\operatorname {rank}P\), and choose
orthonormal basis maps \(B:\mathbb R^r\to X\) and
\(B_\perp :\mathbb R^{n-r}\to Y\). Thus a full field is
\(\phi =Bx+B_\perp y\); in this proposition \(x,y\) denote coordinates.
Set
\[\begin {aligned}
A&=B^TQB,&K&=B^TQB_\perp ,\\
D&=B_\perp ^TQB_\perp >0,&S&=A-KD^{-1}K^T.
\end {aligned}\]
The eliminated coordinates, conditionally on \(x\), have law
\[y=\mu (x)+\eta ,\qquad
\mu (x)=-D^{-1}K^Tx,\qquad \eta \sim N(0,D^{-1}).\]
In particular the full conditional mean and noise covariance are
\[h(x)=Bx+B_\perp \mu (x),\qquad
C=B_\perp D^{-1}B_\perp ^T.\]
The map \(h\) is the harmonic extension: \(B^Th(x)=x\) and
\(B_\perp ^TQh(x)=0\). It is the unique minimizer of the quadratic
energy among fields with these retained coordinates.
Write \(F_\lambda (x)=F_{b,\mathcal V}(\lambda ,Bx)\). Then
\[\begin {aligned}
a(x)&=\mathbb E\mathcal V(h(x)+B_\perp \eta ),\\
b(x)&=\operatorname {Var}\bigl (\mathcal V(h(x)+B_\perp \eta )\bigr ),\\
F_\lambda (x)&=\lambda a(x)-\frac {\lambda ^2}{2}b(x)
+O(\lambda ^3).
\end {aligned}\]
Thus external mean factors in the conditional moment expansion use
\(h(x)\), and each internal pairing of eliminated coordinates uses
\(D^{-1}\). These are the background-dependent Feynman rules for this
fiber integral. In full-field components the internal pairing is
\(C\); the eliminated mean \(\mu \) alone omits the retained part of
\(h\). For a coordinate selector retaining \(R\) and eliminating
\(U=V\setminus R\), the formulas become
\[\mu (x)=-Q_{UU}^{-1}Q_{UR}x,\qquad
\operatorname {Cov}(\eta )=Q_{UU}^{-1}.\]
For example, in full vertex coordinates with the symmetric
four-tensor \(T_{ijkl}\) and \(h=h(x)\), Wick's three pairings give
\[\begin {aligned}
24a(x)={}&T(h,h,h,h)
+6\sum _{i,j,k,l}T_{ijkl}h_i h_j C_{kl}\\
&+3\sum _{i,j,k,l}T_{ijkl}C_{ij}C_{kl}.
\end {aligned}\]
The covariance \(C\) can be singular on \(H\): it is supported on
\(Y\), and the formula follows by applying Wick to the nonsingular
\(D^{-1}\) in eliminated coordinates. No nondegeneracy on all of
\(H\) is needed for this expression.
TauCeti's
integral_mul_mul_mul_multivariateGaussian
proves the three-pairing identity for 4 centered Gaussian
coordinates, allowing repeated indices and any positive-semidefinite
covariance. Setting that covariance to \(C\) supplies the fourth-moment
step here. Contraction with \(T\), the formal logarithm and the
higher-moment pairing arguments are the additional deductions above.
The massive lattice analogue is
[friedli2017statistical, Theorem 8.26, p. 399]: on a finite
domain of \(\mathbb Z^d\), the conditional field has the massive
harmonic mean determined by its boundary values and a killed-walk
Green covariance. That theorem uses its normalized lattice Laplacian;
it does not supply the arbitrary-projector argument above. The
massless finite-domain analogue is Theorem 8.17, p. 391 of the same
source.
A formal conditional-distribution counterpart in TauCeti is
condDistrib_multivariateGaussian.
It assumes a positive-semidefinite joint covariance and a
positive-definite observed covariance block, and identifies the
regular conditional law with an explicit Gaussian kernel almost
everywhere in the observed marginal. To specialize, order the
coordinates as \(Y\oplus X\) and use \(G=Q^{-1}>0\). From \(QG=I\),
\[\begin {aligned}
DG_{YX}+K^TG_{XX}&=0,\\
DG_{YY}+K^TG_{XY}&=I.
\end {aligned}\]
Therefore \(G_{YX}G_{XX}^{-1}=-D^{-1}K^T\) and
\(G_{YY}-G_{YX}G_{XX}^{-1}G_{XY}=D^{-1}\), the theorem's mean
and covariance formulas. Its almost-everywhere identification is
distinct from the explicit density kernel used here at every \(x\).
The perturbative logarithm and vertex calculations are proved above.