Asymptotic Safety and UV Fixed-Point Claims

Asymptotic safety proposes that dimensionless gravitational couplings approach an interacting ultraviolet fixed point with finitely many relevant directions. A candidate fixed point in a truncated Euclidean flow is evidence for the scenario; ultraviolet completion additionally requires truncation stability, a physical state space, regulator-independent observables, and a causal unitary Lorentzian interpretation.

Required background. Ultraviolet and Infrared Fixed Points: Criteria and Evidence supplies fixed-point definitions. Validity, Unitarity, and Breakdown fixes the gravity-EFT boundary.

Helpful background. Critical Surfaces, Crossover, and Corrections to Scaling supplies predictivity criteria. Quantum-Gravity Consistency Claims and Comparison Contract supplies the cross-program standard.

Fixed points and critical surfaces

For running couplings $\bar G_k$ and $\bar\Lambda_k$ in four dimensions, define $g=k^2\bar G_k$ and $\lambda=\bar\Lambda_k/k^2$. A fixed point $u_*$ satisfies $\beta_i(u_*)=0$. Linearizing,

$$k\frac{d}{dk}\delta u_i=B_{ij}\delta u_j,\qquad B_{ij}=\left.\frac{\partial\beta_i}{\partial u_j}\right|_* .$$

If $B V^{(a)}=-\theta_aV^{(a)}$, directions with $\operatorname{Re}\theta_a>0$ are ultraviolet-attractive and span the critical surface. Finitely many such directions imply finitely many measurements select a trajectory.

Application: a two-coupling flow

Consider the illustrative normalized flow

$$\beta_g=2g-bg^2,\qquad \beta_\lambda=-2\lambda+ag,\qquad a,b>0.$$

Besides the Gaussian point, $g_*=2/b$ and $\lambda_*=a/b$. Its stability matrix is

$$B_*=\begin{pmatrix}-2&0\\a&-2\end{pmatrix},$$

so both critical exponents are $\theta=2$ in this model. This translates a flow into fixed-point coordinates and a candidate two-dimensional critical surface, but the values depend on normalization and beta functions.

Functional-RG calculations beginning with Reuter find a non-Gaussian point in Einstein–Hilbert and many enlarged curvature truncations Reuter 1998. High-order curvature studies find a small number of relevant directions and stability within selected bases Falls et al. 2018. These are substantive but correlated truncation evidence.

Adversarial fixed-point tests

Vary regulator shape, background, gauge, field parametrization, and operator basis. Universal critical exponents and physical observables should stabilize even though coordinates $g_*$ and $\lambda_*$ need not. Check modified Ward identities and distinguish background from fluctuation couplings. Continue with a declared Lorentzian contour and test spectral positivity. A Lorentzian Einstein–Hilbert flow has yielded a candidate fixed point from universal terms D’Angelo 2023, but not yet a complete Lorentzian theory.

Controlled conclusion

Current evidence supports a persistent fixed-point scenario across approximations. It does not yet give a regulator-independent construction of all observables or prove unitarity. Projection errors are developed on Functional Renormalization Group and Truncation Control.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• D’Angelo, E. “Asymptotic Safety in Lorentzian Quantum Gravity.” (2023). arXiv:2310.20603.
• Falls, K. G., C. R. King, D. F. Litim, K. Nikolakopoulos, and C. Rahmede. “Asymptotic Safety of Quantum Gravity beyond Ricci Scalars.” Physical Review D 97 (2018): 086006. DOI.
• Reuter, M. “Nonperturbative Evolution Equation for Quantum Gravity.” Physical Review D 57 (1998): 971–985. DOI.