Anomaly-Induced Nonlocal Actions

Integrating the trace anomaly determines how an effective action changes along a Weyl orbit, not the entire effective action. In four dimensions a convenient representative uses the inverse of a fourth-order conformally covariant operator. Its Weyl variation reproduces the Euler and Weyl-squared anomaly, but its Green function, boundary conditions, homogeneous solutions, local counterterms, and an arbitrary conformally invariant functional remain additional data.

Required background. Gravitational Actions and Boundary Terms provides metric variation and boundary completion; Curvature supplies Euler, Weyl, and conformal curvature identities. Helpful background. Flat-Spacetime Quantum Field Theory provides nonlocal form factors and auxiliary fields; Partial Differential Equations and Green Functions supplies inverse operators and homogeneous solutions; Causality and Global Structure establishes admissible Lorentzian Green functions; Quantum Fields in Curved Spacetime provides state dependence and renormalized stress–energy.

The integrability problem

Choose the finite $R^2$ counterterm so that the four-dimensional anomaly is represented as

$$\mathcal A[g] =\langle T^\mu{}_{\mu}\rangle =\frac{1}{(4\pi)^2} \left(cC^2-a\mathcal Q\right),$$

where

$$\mathcal Q=E_4-\frac23\Box R.$$

The coefficient of the remaining $\Box R$ term is scheme-dependent; selecting the combination $\mathcal Q$ is a useful representative, not a claim that the local $R^2$ freedom disappeared. We seek a functional satisfying

$$\delta_\sigma\Gamma_{\mathrm{anom}}[g] =\int d^4x\sqrt{-g}\,\sigma\mathcal A[g].$$

A local polynomial of the metric cannot generally solve this equation for the nontrivial type-A and type-B classes. Integrating the local Weyl variation therefore produces a nonlocal functional.

The Paneitz operator and the Riegert representative

For a scalar of Weyl weight zero, define the fourth-order operator

$$\Delta_4 =\Box^2+2R^{\mu\nu}\nabla_\mu\nabla_\nu -\frac23R\Box +\frac13(\nabla^\mu R)\nabla_\mu.$$

Its crucial property is

$$\delta_\sigma\!\left(\sqrt{-g}\,\mathcal Q\right) =4\sqrt{-g}\,\Delta_4\sigma,$$

while $\sqrt{-g}C^2$ is Weyl invariant in four dimensions. Let $G_4(x,y)$ be a chosen inverse,

$$\Delta_{4,x}G_4(x,y) =\frac{\delta^{(4)}(x,y)}{\sqrt{-g(x)}}.$$

Up to the qualifications below, one Riegert representative is

$$\begin{aligned} \Gamma_{\mathrm{anom}} =\frac{1}{8(4\pi)^2} \int d^4x\sqrt{-g_x}\int d^4y\sqrt{-g_y}\, \mathcal Q(x)G_4(x,y) \left[2cC^2(y)-a\mathcal Q(y)\right]. \end{aligned}$$

Varying the first $\mathcal Q$ in the cross term gives $cC^2$; varying both copies in the symmetric quadratic term gives $-a\mathcal Q$. This factor-of-two check is an efficient way to detect normalization errors. Riegert constructs the four-dimensional nonlocal action in Riegert 1984, complete article, pp. 56–60, while the Weyl-cohomological derivation and the explicit fourth-order Green function appear in Mazur and Mottola 2001, §§2–4, especially Eqs. (4.12)–(4.25).

For this variation, $G_4$ must be a symmetric two-sided inverse with a domain and boundary prescription preserved by the Weyl variations being considered. The density-valued operator $\sqrt{-g}\Delta_4$ is Weyl invariant on weight-zero scalars, so its inverse kernel can then be held fixed under those variations. These conditions are part of the calculation, not consequences of the one displayed inverse equation alone.

The formula is first controlled in Euclidean signature, where a self-adjoint elliptic realization of $\Delta_4$ can be specified. Zero modes must already have been removed or treated by additional global terms. Continuing it to the site’s Lorentzian convention requires a contour and state prescription. A symmetric Euclidean inverse continued with Feynman data yields an in-out object; a causal expectation equation requires the corresponding in-in construction. Substituting a retarded inverse into a single-field symmetric action is not by itself a consistent causal variational prescription.

What the anomaly does not determine

If two functionals have the same Weyl variation, their difference obeys

$$\delta_\sigma\Gamma_{\mathrm{inv}}[g]=0.$$

Therefore the conformally invariant functional $\Gamma_{\mathrm{inv}}$ is undetermined by the anomaly. It may be nonlocal and can carry physical information invisible to the trace, including transverse-traceless stress-tensor response. Local counterterms form another independent ambiguity, notably the finite $R^2$ term that shifts $\Box R$.

The Green-function choice is also not fixed. Two symmetric inverse prescriptions can differ by a symmetric bisolution $H$ obeying

$$\Delta_{4,x}H(x,y)=0, \qquad \Delta_{4,y}H(x,y)=0.$$

The homogeneous kernel changes the action without changing the local inverse equations. Its contribution has zero bulk Weyl variation when the kernel is itself held Weyl invariant and the same integration-by-parts conditions hold. An arbitrary metric-dependent $H[g]$ need not have this property: its own variation can contribute to the trace. Boundary conditions and zero modes decide whether an inverse exists and which homogeneous data are allowed. In Lorentzian spacetime these data are related to the quantum state and causal prescription, but the anomaly alone does not select a Hadamard state, an in-vacuum, or a thermal state.

This is the central nonuniqueness: the trace fixes one functional derivative—the derivative along conformal directions. It does not fix variations transverse to the Weyl orbit.

Auxiliary-field localization

The type-A quadratic piece can be represented locally by introducing a scalar $\varphi$:

$$\Gamma_A[g,\varphi] =-\frac{a}{4(4\pi)^2} \int d^4x\sqrt{-g} \left[-\frac12\varphi\Delta_4\varphi +\varphi\mathcal Q\right].$$

The auxiliary equation is $\Delta_4\varphi=\mathcal Q$. Eliminating $\varphi$ with a specified inverse gives

$$\Gamma_A[g] =-\frac{a}{8(4\pi)^2} \int\sqrt{-g_x}\sqrt{-g_y}\, \mathcal Q(x)G_4(x,y)\mathcal Q(y).$$

The mixed $C^2G_4\mathcal Q$ structure can likewise be localized with an enlarged auxiliary system. Localization does not remove nonlocal physics: the fourth-order auxiliary equations require boundary or initial data, and their homogeneous solutions store exactly the information carried by the chosen Green function. Treating every auxiliary solution as a new fundamental particle would overcount what began as a representation of a nonlocal quantum functional.

Two-dimensional normalization check

In two dimensions the corresponding conformal operator is the scalar Laplacian and the integrated anomaly gives the Polyakov action. For one conformal scalar in Vassilevich’s Euclidean convention,

$$W[g_E] =\frac{1}{96\pi} \int d^2x\sqrt{g_E}\, R_E\frac{1}{\Delta_E}R_E,$$

where $\Delta_E=\nabla_E^2$ has nonpositive spectrum, and zero modes are treated separately. On a conformal patch $g_{E\mu\nu}=e^{2\rho}\delta_{\mu\nu}$ with suitable decay or boundary data, $R_E=-2e^{-2\rho}\partial^2\rho$ and $\Delta_E^{-1}R_E=-2\rho$ up to a homogeneous solution. Thus

$$W[\rho]=\frac{1}{24\pi}\int d^2x\,\rho\partial^2\rho =-\frac{1}{24\pi}\int d^2x\,(\partial\rho)^2.$$

Its variation is $\delta W=-(24\pi)^{-1}\int\sqrt{g_E}\,\delta\rho R_E$. Vassilevich defines the Euclidean trace by $\delta W=-\int\sqrt{g_E}\,\delta\rho\,T^\mu{}_{\mu}$, so this is $T^\mu{}_{\mu}=R_E/(24\pi)$ in that convention. The derivation and zero-mode caveat are Vassilevich 2003, §9.1, Eqs. (9.1)–(9.5), PDF. Defining a positive Laplacian $-\nabla_E^2$ instead would reverse the displayed sign in front of its inverse. The example checks that locality in the conformal factor can coexist with covariance and nonlocality in the full metric.

Boundaries and integrated identities

On a manifold with boundary, integrations by parts used in proving the Weyl identity generate surface terms. The Euler invariant needs a boundary completion, $\Delta_4$ needs a boundary-value problem, and the anomaly itself has boundary invariants. Consequently, the bulk Riegert expression is not a complete boundary effective action.

Likewise, on a compact manifold zero modes obstruct a literal inverse. One must project them out and add the corresponding global term or collective coordinate. These are not small technical corrections: they change the integrated response to a constant Weyl rescaling while leaving the local bulk differential identity intact.

Diagnostic: vary the Green-function data

Let $G_4' = G_4+H$, where $H$ is a symmetric bisolution, both inverse prescriptions have compatible boundary data, and $\delta_\sigma H=0$ for the compactly supported Weyl variations under consideration. Show why the two actions have the same bulk trace anomaly. Explain why $\Delta_{4,x}H=0$ alone would not prove this.

Solution

Replace $G_4$ by $H$ in the difference of the two bilinear expressions. Varying either $\sqrt{-g}\mathcal Q$ gives $4\sqrt{-g}\Delta_4\sigma$; integration by parts transfers this operator to the corresponding argument of $H$, where it vanishes. The Weyl-squared density and $H$ have zero variation by hypothesis, and the assumed boundary terms vanish. Therefore the added term is Weyl invariant and both actions reproduce the same bulk trace. A one-sided equation alone controls neither the second argument nor $\delta_\sigma H[g]$, and it says nothing about boundary terms. The allowed difference can still change traceless response and encode different boundary or state data; zero-mode changes additionally require a consistent treatment of the global terms. An anomaly determines a Weyl variation, not a unique effective action.

References

• Mazur, Pawel O., and Emil Mottola. “Weyl Cohomology and the Effective Action for Conformal Anomalies.” Physical Review D 64 (2001): 104022. DOI. Open PDF, arXiv:hep-th/0106151.
• Riegert, Robert J. “A Nonlocal Action for the Trace Anomaly.” Physics Letters B 134 (1984): 56–60. DOI.
• Vassilevich, Dmitri V. “Heat Kernel Expansion: User’s Manual.” Physics Reports 388 (2003): 279–360. DOI. Open PDF, arXiv:hep-th/0306138.