Functional Renormalization Group Equation

The functional renormalization-group equation evolves an effective action in which fluctuations with covariant momentum below a scale $k$ are suppressed. Its right-hand side resembles a one-loop trace, but the inverse propagator is the full, scale-dependent Hessian. This page derives that identity, then identifies the additional background, gauge, ghost, and projection choices needed for gravity.

Required background. Flat-Spacetime Quantum Field Theory supplies generating functionals, Legendre transforms, and functional traces; Gravitational Actions and Boundary Terms supplies the gravitational Hessian and boundary assumptions.

Helpful background. Einstein Equations supplies the classical limit; Quantum Information helps distinguish gauge from physical fluctuations; and Quantum Gravity supplies the comparison criteria for continuum methods.

Regulated generating functional

Begin with a Euclidean field multiplet $\chi$ and add a quadratic infrared regulator,

$$\Delta S_k[\chi]=\frac12\chi\cdot R_k\cdot\chi, \qquad Z_k[J]=\int\mathcal D\chi\, \exp\!\left[-S[\chi]-\Delta S_k[\chi]+J\cdot\chi\right].$$

The kernel should behave as $R_k(p^2)\sim k^2$ for $p^2\ll k^2$ and vanish for $p^2\gg k^2$. Thus low-momentum modes acquire a masslike suppression while high-momentum modes are essentially unchanged. Let $W_k=\ln Z_k$ and $\phi=\delta W_k/\delta J$. The effective average action is a modified Legendre transform,

$$\Gamma_k[\phi]=J\cdot\phi-W_k[J]-\Delta S_k[\phi].$$

Subtracting $\Delta S_k[\phi]$ is essential: it makes $\Gamma_{k\to0}$ the ordinary one-particle-irreducible effective action rather than an action retaining the artificial mass. Wetterich derived the general exact evolution equation in this form Wetterich 1993, pp. 91–94.

Differentiating the Legendre transform

Take $t=\ln(k/k_0)$, with $k_0$ an arbitrary reference scale, and differentiate at fixed mean field. Terms containing $\partial_tJ$ cancel because $\delta W_k/\delta J=\phi$. Only the explicit regulator dependence remains:

$$\partial_t\Gamma_k =-\partial_tW_k-\frac12\phi\cdot\partial_tR_k\cdot\phi.$$

Differentiating $W_k$ under the path integral gives

$$\partial_tW_k =-\frac12\left\langle\chi\cdot\partial_tR_k\cdot\chi\right\rangle.$$

Split the two-point function into connected and disconnected pieces, $\langle\chi\chi\rangle=G_k+\phi\phi$. The disconnected term cancels the derivative of $\Delta S_k[\phi]$, leaving $\partial_t\Gamma_k=\tfrac12\operatorname{Tr}(G_k\partial_tR_k)$. A second derivative of the Legendre transform yields the inverse relation

$$G_k=\left(\Gamma_k^{(2)}+R_k\right)^{-1}.$$

Combining the two steps gives

$$\boxed{\displaystyle \partial_t\Gamma_k =\frac12\operatorname{STr}\!\left[ \left(\Gamma_k^{(2)}+R_k\right)^{-1}\partial_tR_k \right]}.$$

The supertrace includes spacetime or momentum integration and internal indices, with a minus sign for Grassmann-odd fields. The equation is formally exact because no loop expansion was used. Its “one-loop” appearance reflects the differentiated quadratic regulator; nonperturbative information enters through the full $\Gamma_k^{(2)}$.

Gravitational fields, backgrounds, and ghosts

For gravity one normally introduces a background metric $\bar g_{\mu\nu}$ and a fluctuation $h_{\mu\nu}$ through a specified split, adds background-covariant gauge fixing, and integrates over $h_{\mu\nu}$ plus ghosts. The regulator is built from a background Laplacian so that traces can be evaluated. The resulting functional $\Gamma_k[h,\bar g,\bar c,c]$ depends separately on fluctuation and background fields. Reuter’s construction adapts the exact flow to this setting Reuter 1998, §§ II–III.

Background covariance is not the same as independence from the arbitrary split. The regulator and gauge fixing break split symmetry at finite $k$, producing modified Ward or Nielsen identities. A consistent approximation should solve or monitor those identities and recover the appropriate symmetry as $k\to0$. Ghosts contribute with the fermionic sign in the supertrace; omitting them gives the wrong gauge-mode balance.

Projection onto running couplings

To obtain beta functions, choose an ansatz $\Gamma_k=\sum_i\bar u_i(k)\mathcal O_i$, evaluate the trace on backgrounds capable of separating the operators, and match coefficients:

$$\partial_t\Gamma_k =\sum_i\partial_t\bar u_i\,\mathcal O_i \quad\Longrightarrow\quad \partial_tu_i=\beta_i(u).$$

Heat-kernel expansions are efficient for curvature invariants, while vertex expansions retain external momenta and distinguish fluctuation from background couplings. Neither projection is neutral. A background with constant curvature cannot separate every invariant, and a finite momentum sample can alias tensor structures. Regulator and gauge variation, enlarged operator bases, and symmetry identities provide independent checks.

The limiting checks are direct. As $k\to0$, $R_k\to0$ and the flow stops at the ordinary effective action. At a large ultraviolet initialization scale, a sufficiently strong regulator suppresses fluctuations so that $\Gamma_k$ approaches the specified microscopic action under suitable assumptions. Expanding the full inverse Hessian about the classical Hessian reproduces ordinary loop perturbation theory and its signs.

The trace also carries a useful normalization check. A real bosonic degree of freedom contributes the prefactor $1/2$ because its regulator term is quadratic, whereas a complex ghost pair contributes without that factor and with an overall minus sign. Gauge modes, Jacobians from field decompositions, and zero modes must be treated consistently before a coefficient is interpreted. Dimensionless beta functions contain both the trace contribution and canonical scaling: if $\bar u_i$ has dimension $d_i$, then $\partial_tu_i=-d_i u_i+k^{-d_i}\partial_t\bar u_i$. Forgetting the first term can manufacture or erase a fixed point.

Finally, the equation does not choose a Lorentzian contour. Most gravitational computations evaluate an elliptic Euclidean trace and later ask whether the result admits a physical continuation. A Euclidean fixed point is therefore evidence about a regulated functional integral, not by itself a proof of a unitary Lorentzian theory. That boundary should accompany every projected flow.

Exercise

For one bosonic mode with $\Gamma_k^{(2)}=p^2+m_k^2$ and regulator $R_k(p^2)$, write its contribution to $\partial_t\Gamma_k$. Explain why modes with $p^2\gg k^2$ and $p^2\ll k^2$ do not dominate for a well-shaped regulator.

Solution

The contribution is

$$\frac12\int\frac{d^dp}{(2\pi)^d}\, \frac{\partial_tR_k(p^2)} {p^2+m_k^2+R_k(p^2)}.$$

For $p^2\gg k^2$, a valid regulator and its scale derivative vanish, so ultraviolet modes are absent from this infinitesimal shell. For $p^2\ll k^2$, the denominator contains the masslike $R_k\sim k^2$, preventing an infrared divergence, while the phase-space volume is small. The dominant contribution comes from momenta of order $k$. This shell localization is a check on the regulator, not proof that a finite truncation is regulator independent.

References

• Reuter, Martin (1998), “Nonperturbative Evolution Equation for Quantum Gravity,” Physical Review D 57, 971–985, arXiv:hep-th/9605030.
• Wetterich, Christof (1993), “Exact Evolution Equation for the Effective Potential,” Physics Letters B 301, 90–94, DOI: 10.1016/0370-2693(93)90726-X.