Proposition. Formal interaction under Gaussian elimination [fdip-0046]
AGENTDRAFTED
Eliminating a field with a quartic interaction generally creates
quadratic and sextic terms. The coefficients through second order
can be found by conditional Gaussian moments, while preserving a
supplied symmetry and composition of eliminations.
Use the finite orthogonal sum \(E=X\mathbin \perp Y\), positive
precision \(Q\), components \(A,B,C\), Schur precision \(S\) and scalar
\(c_C\) of Gaussian elimination. Let
\(T:E^4\to \mathbb R\) be a symmetric four-linear form and put
\[V(u)=\frac {T(u,u,u,u)}{4!}.\]
Write \(\mathcal P(X)\) for the polynomial function algebra.
For \(x\in X\), let \(\eta \) have centered Gaussian law with covariance
\(C^{-1}\) on \(Y\), and set
\[v_x=(x,-C^{-1}B^\dagger x+\eta ).\]
This is the conditional law of the eliminated field given the retained
value. It is the conditional kernel with its two named
summands exchanged; its covariance is \(C^{-1}\), whereas that
proposition's retained-given-eliminated kernel has covariance \(A^{-1}\).
For a formal variable \(\lambda \), use ordinary power-series
coefficients in \(\lambda \) and define the
fiber effective interaction by
\[\begin {aligned}
F_\lambda (x)&=-\log \mathbb E_\eta e^{-\lambda V(v_x)},\\
\mathbb E_\eta e^{-\lambda V(v_x)}
&=\sum _{n\geq 0}\frac {(-\lambda )^n}{n!}
\mathbb E_\eta V(v_x)^n.
\end {aligned}\]
These are formal series in \(\lambda \) with coefficients in
\(\mathcal P(X)\); the logarithm
is the formal logarithm of a series with constant coefficient one.
Every coefficient is finite by polynomial integrability.
The notation \(O(\lambda ^3)\) means a series divisible by \(\lambda ^3\),
not an analytic error estimate. With
\[\begin {aligned}
a(x)&=\mathbb E[V\mid x],\\
b(x)&=\mathbb E[V^2\mid x]-a(x)^2,
\end {aligned}\]
the first two coefficients are
\[F_\lambda =\lambda a-\frac {\lambda ^2}{2}b+O(\lambda ^3).\]
The degree-zero terms are retained: coefficientwise integration gives
\[\begin {aligned}
&\int _Y e^{-\frac 12\langle (x,y),Q(x,y)\rangle }
e^{-\lambda V(x,y)}\,dy\\
&\qquad =c_Ce^{-\frac 12\langle x,Sx\rangle }e^{-F_\lambda (x)}.
\end {aligned}\]
Here \(dy\) is the specified Euclidean volume. This fiber logarithm
integrates eliminated variables; it does not take the Legendre
transform of a source-generating function used for a one-particle
irreducible (1PI) effective action in
[etingof2024mathematical, v2, Proposition 3.23, p. 50].
For a specified orthogonal flag
\(E=X\mathbin \perp Y\mathbin \perp Z\), first eliminate \(Z\) and then
\(Y\). All conditional expectations below use the original Gaussian
law, with the \(Y\)-given-\(X\) kernel obtained after eliminating \(Z\).
Put
\[\begin {aligned}
a_1(x,y)&=\mathbb E[V\mid x,y],\\
b_1(x,y)&=\mathbb E[V^2\mid x,y]-a_1(x,y)^2.
\end {aligned}\]
Both stages together give
\[\begin {aligned}
F_\lambda (x)
={}&\lambda \mathbb E[a_1\mid x]\\
&-\frac {\lambda ^2}{2}\bigl (
\mathbb E[b_1\mid x]+\operatorname {Var}(a_1\mid x)\bigr )\\
&+O(\lambda ^3),
\end {aligned}\]
which agrees with direct elimination of \(Y\oplus Z\). The first
coefficient has degrees among \(4,2,0\); the second has degrees among
\(6,4,2,0\). Particular components may vanish.
Suppose a second such model is related by a linear isometric isomorphism
\(L=L_X\oplus L_Y:E\to E'\) satisfying
\(Q'L=LQ\) and \(V'\circ L=V\). Then
\[F'_\lambda (L_Xx)=F_\lambda (x)\pmod {\lambda ^3}.\]
Indeed the component identities give
\(C'L_Y=L_YC\) and \((B')^\dagger L_X=L_YB^\dagger \).
Thus \(L_Y\) transports the conditional mean and covariance, hence
its Gaussian law and both moments \(a,b\). For two-stage elimination
also require preservation of the supplied flag.
In particular, a supplied finite group of these isometries preserves
each induced coefficient. The result transports a given action;
it does not select that action.
Wick contractions are
[etingof2024mathematical, v2, Theorem 3.1 and Example 3.2, p. 29].
The perturbation convention in
[etingof2024mathematical, v2, Section 3.3, p. 33]
is \(B(u,u)/2-\sum _i g_iB_i(u,\ldots ,u)/i!\); here
\(g_4=-\lambda \) and \(B_4=T\), at \(\hbar =1\).
The conditional formulas and composition above are direct deductions
using these contractions and the specified Gaussian kernels. The
three-coordinate example exhibits every indicated degree.