Definition. An interaction on a selected background [fdip-005Y]

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An interacting selected background consists of a selected Gaussian background \(b=(\Pi ,w,m,P)\), a real polynomial \(\mathcal V:H\to \mathbb R\), and a formal variable \(\lambda \). Thus the precision \(Q>0\), metric and selector are supplied along with the interaction. Write \(X=\operatorname {im}P\) and \(Y=\ker P\), so \(H=X\mathbin \perp Y\).

For every \(x\in X\), define a probability law on \(Y\) by \[\begin {aligned} Z_b(x)&=\int _Y e^{-\frac 12\langle x+y,Q(x+y)\rangle }\,dy,\\ \kappa _{b,x}(dy) &=Z_b(x)^{-1}e^{-\frac 12\langle x+y,Q(x+y)\rangle }\,dy. \end {aligned}\] The volume \(dy\) is the induced Euclidean volume. Positive precision makes \(Z_b(x)\) positive and finite. The square-completion formula shows that this is a Gaussian law whose mean depends linearly on \(x\). It is the conditional law of the eliminated field given the retained value, using the canonical density formula at every \(x\); general regular conditional laws are unique only almost everywhere.

The fiber effective interaction is the formal series \[\begin {aligned} M_{b,\mathcal V}(\lambda ,x) &=\sum _{n\ge 0}\frac {(-\lambda )^n}{n!} \int _Y\mathcal V(x+y)^n\,\kappa _{b,x}(dy),\\ F_{b,\mathcal V}(\lambda ,x)&=-\log M_{b,\mathcal V}(\lambda ,x). \end {aligned}\] Each coefficient is a polynomial on \(X\): expand the polynomial about the conditional mean and use the fixed centered Gaussian moments. These moments are finite by polynomial integrability. The series \(M\) has constant coefficient one, so its logarithm is a formal power series in \(\lambda \) with coefficients in the real polynomial algebra \(\mathcal P(X)\) on \(X\). This extends the quartic construction to any polynomial interaction and any orthogonal selector. Constants in \(x\) remain part of \(F\).

This definition is coefficientwise. It imposes no positivity on \(\mathcal V\), asserts no convergence for nonzero \(\lambda \), and uses no interchange of an infinite sum with an integral. The notation \(O(\lambda ^k)\) means divisibility by \(\lambda ^k\). A fiber logarithm is different from the Legendre transform that defines a one-particle irreducible effective action in [etingof2024mathematical, v2, Proposition 3.23, p. 50]. Here the retained field is fixed while its orthogonal complement is integrated.