Elimination of a source sector: Schur-complement effective operators

Remark 41 (Nonlocality of the effective operator).

The correction to the visible operator is $$\Delta:=-\,BC^{-1}B^{*} ,\tag{9.5}$$ and it is derived, not postulated. What the theorem establishes is exactly the form of $\Delta$; whether $\Delta$ is nonlocal is a further question about the composed operator, and must not be read off from $C^{-1}$ alone. In field theory, when $\mathcal{H}_{\chi}$ is a space of functions on a spacetime region and $C$ is (the bounded realization of) a differential operator, $C^{-1}$ is an integral operator with a Green kernel $G(y,y')$, and $$(\Delta\varphi)(x)=-\int\!\!\int\!\!\int B(x,y)\,G(y,y')\,B^{*}(y',x')\,\varphi(x')\,dy\,dy'\,dx' .$$ This composition can be nonlocal: the effective equation for $\varphi$ at a point then involves $\varphi$ elsewhere, weighted by the Green function of the eliminated sector, even though the coupled source equations for $(\varphi,\chi)$ were strictly local.

The qualification is not pedantic, because nonlocality of $C^{-1}$ is not sufficient. The correction depends on the coupling as well: if $B=0$ there is no correction at all, and more generally $B$ may fail to couple to the nonlocal directions of $G$, or the two compositions $B$ and $B^{*}$ may conspire so that $BC^{-1}B^{*}$ collapses to a local (e.g. multiplication or differential) operator. A concrete instance: if $B$ is a rank-one coupling $B=\lvert f\rangle\langle h\rvert$, then $BC^{-1}B^{*}=\langle h , C^{-1}h \rangle\,\lvert f\rangle\langle f\rvert$, whose kernel is the product $\overline{f(x')}f(x)$ times a single scalar, the whole Green function $G$ has entered only through the number $\langle h , C^{-1}h \rangle$. We stress what this example does and does not show. The rank-one operator $\lvert f\rangle\langle f\rvert$ is itself generally nonlocal (its kernel $\overline{f(x')}f(x)$ does not concentrate on the diagonal), so the example does not exhibit a local effective operator; what it exhibits is that the detailed structure of $G$, the very thing one might expect to be imprinted on the correction, can collapse to a single number, leaving a correction whose nonlocal structure is dictated entirely by $B$ and not by the eliminated sector’s propagator. Nonlocality of the effective operator is therefore a property of $BC^{-1}B^{*}$, requiring that $B$ couple to the relevant Green-operator directions without cancellation; it is an additional verification, not a corollary of Theorem 9.2.

Subject to that verification, elimination of a sector converts locality-with-extra-fields into nonlocality-with-fewer-fields, and there is an exact structural analogy with the memory kernel of Theorem 6.2: there, eliminating $\mathcal{H}_{Q}$ converted a first-order ODE into an equation nonlocal in time, and there too the kernel $\mathcal{K}=PLQ\,e^{\tau QLQ}\,QLP$ can degenerate, it vanishes identically when $PLQ=0$ or $QLP=0$, so that nonlocality in time likewise depends on the couplings and not on the eliminated sector’s propagator alone. In both settings nonlocality is a possible price of the reduction, not an automatic one, and never a feature of the source.

Remark 42 (Unbounded operators, and what would have to be supplied).

Theorem 9.2 is stated for bounded $A$, $B$, $C$ with $C$ boundedly invertible, and the proof uses boundedness at every step (in the differentiation, in the existence of $C^{-1}$ as a bounded operator, in the adjoint manipulations). It does not automatically cover the case where $A$ and $C$ are unbounded differential operators, which is the case of primary interest in field theory. To extend it one must supply, and verify: a common dense domain, or a formulation in terms of closed, densely defined, semibounded quadratic forms with form domains [7, Ch. VI]; boundary conditions making $C$ self-adjoint and rendering the integration by parts in the variation legitimate; invertibility of $C$ (equivalently, that $0\notin\sigma(C)$, which fails whenever the eliminated sector has zero modes, e.g. a massless field on a compact region with the wrong boundary conditions); and the boundedness of $C^{-1}$ from the form dual back to the form domain, so that $BC^{-1}B^{*}$ is defined on the form domain of $A$. None of this follows from the bounded proof, and we do not assert it here. Where the physics literature performs such eliminations with unbounded operators, the Green operator $C^{-1}=G$ is being used, and the validity of the manipulation is exactly the validity of the associated elliptic/hyperbolic solvability theory, which must be supplied case by case.

Remark 43 (Tree level, not the quantum effective action).

Theorem 9.2 is a statement of classical stationary elimination: $\chi$ is replaced by the solution of its classical field equation (9.2). Equivalently, in a functional-integral language, (9.4) is the tree-level (saddle-point) contribution. It is not the quantum effective action for $\varphi$. Integrating $\chi$ out of a Gaussian functional integral with weight $e^{iS[\varphi,\chi]}$ produces, in addition to (9.4), a $\varphi$-independent determinant factor $(\det C)^{-1/2}$; if $C$ depends on $\varphi$ (a nonquadratic coupling, absent from Assumption 9.1) the determinant becomes $\varphi$-dependent and contributes a genuine one-loop term $\tfrac{i}{2}\operatorname{Tr}\log C[\varphi]$; and beyond one loop there are further corrections [12, 17]. Nothing in Theorem 9.2 computes or bounds these. The theorem is exactly what it says: substitution of a stationary point into a quadratic functional.