12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Asymptotic safety | 4/6 | https://en.wikipedia.org/wiki/Asymptotic_safety | reference | science, encyclopedia | 2026-05-05T13:41:25.065218+00:00 | kb-cron |
The name quantum Einstein gravity (QEG) describes any quantum field theory of gravity that (regardless of its bare action) takes the spacetime metric as the dynamical field variable and whose symmetry is given by diffeomorphism invariance. This fixes the theory space and an RG flow of the effective average action defined over it, but it does not single out a priori any specific action functional. However, the flow equation determines a vector field on that theory space which can be investigated. If it displays a non-Gaussian fixed point by means of which the UV limit can be taken in the "asymptotically safe" way, this point acquires the status of the bare action.
=== Quantum quadratic gravity (QQG) ===
A specific realisation of QEG is quantum quadratic gravity (QQG). This a quantum extension of general relativity obtained by adding all local quadratic-in-curvature terms to the Einstein-Hilbert Lagrangian. QQG, besides being renormalizable, has also been shown to feature a UV fixed point (even in the presence of realistic matter sectors). It can, therefore, be regarded as a concrete realisation of asymptotic safety.
== Implementation via the effective average action ==
=== Exact functional renormalization group equation ===
The primary tool for investigating the gravitational RG flow with respect to the energy scale
k
{\displaystyle k}
at the nonperturbative level is the effective average action
Γ
k
{\displaystyle \Gamma _{k}}
for gravity. It is the scale dependent version of the effective action where in the underlying functional integral field modes with covariant momenta below
k
{\displaystyle k}
are suppressed while only the remaining are integrated out. For a given theory space, let
Φ
{\displaystyle \Phi }
and
Φ
¯
{\displaystyle {\bar {\Phi }}}
denote the set of dynamical and background fields, respectively. Then
Γ
k
{\displaystyle \Gamma _{k}}
satisfies the following Wetterich–Morris-type functional RG equation (FRGE):
k
∂
k
Γ
k
[
Φ
,
Φ
¯
]
=
1
2
STr
[
(
Γ
k
(
2
)
[
Φ
,
Φ
¯
]
+
R
k
[
Φ
¯
]
)
−
1
k
∂
k
R
k
[
Φ
¯
]
]
.
{\displaystyle k\partial _{k}\Gamma _{k}{\big [}\Phi ,{\bar {\Phi }}{\big ]}={\frac {1}{2}}\,{\mbox{STr}}{\Big [}{\big (}\Gamma _{k}^{(2)}{\big [}\Phi ,{\bar {\Phi }}{\big ]}+{\mathcal {R}}_{k}[{\bar {\Phi }}]{\big )}^{-1}k\partial _{k}{\mathcal {R}}_{k}[{\bar {\Phi }}]{\Big ]}.}
Here
Γ
k
(
2
)
{\displaystyle \Gamma _{k}^{(2)}}
is the second functional derivative of
Γ
k
{\displaystyle \Gamma _{k}}
with respect to the quantum fields
Φ
{\displaystyle \Phi }
at fixed
Φ
¯
{\displaystyle {\bar {\Phi }}}
. The mode suppression operator
R
k
[
Φ
¯
]
{\displaystyle {\mathcal {R}}_{k}[{\bar {\Phi }}]}
provides a
k
{\displaystyle k}
-dependent mass-term for fluctuations with covariant momenta
p
2
≪
k
2
{\displaystyle p^{2}\ll k^{2}}
and vanishes for
p
2
≫
k
2
{\displaystyle p^{2}\gg k^{2}}
. Its appearance in the numerator and denominator renders the supertrace
(
STr
)
{\displaystyle ({\mbox{STr}})}
both infrared and UV finite, peaking at momenta
p
2
≈
k
2
{\displaystyle p^{2}\approx k^{2}}
. The FRGE is an exact equation without any perturbative approximations. Given an initial condition it determines
Γ
k
{\displaystyle \Gamma _{k}}
for all scales uniquely. The solutions
Γ
k
{\displaystyle \Gamma _{k}}
of the FRGE interpolate between the bare (microscopic) action at
k
→
∞
{\displaystyle k\rightarrow \infty }
and the effective action
Γ
[
Φ
]
=
Γ
k
=
0
[
Φ
,
Φ
¯
=
Φ
]
{\displaystyle \Gamma [\Phi ]=\Gamma _{k=0}{\big [}\Phi ,{\bar {\Phi }}=\Phi {\big ]}}
at
k
→
0
{\displaystyle k\rightarrow 0}
. They can be visualized as trajectories in the underlying theory space. Note that the FRGE itself is independent of the bare action. In the case of an asymptotically safe theory, the bare action is determined by the fixed point functional
Γ
∗
=
Γ
k
→
∞
{\displaystyle \Gamma _{*}=\Gamma _{k\rightarrow \infty }}
.
=== Truncations of the theory space === Let us assume there is a set of basis functionals
{
P
α
[
⋅
]
}
{\displaystyle \{P_{\alpha }[\,\cdot \,]\}}
spanning the theory space under consideration so that any action functional, i.e. any point of this theory space, can be written as a linear combination of the
P
α
{\displaystyle P_{\alpha }}
's. Then solutions
Γ
k
{\displaystyle \Gamma _{k}}
of the FRGE have expansions of the form
Γ
k
[
Φ
,
Φ
¯
]
=
∑
α
=
1
∞
g
α
(
k
)
P
α
[
Φ
,
Φ
¯
]
.
{\displaystyle \Gamma _{k}[\Phi ,{\bar {\Phi }}]=\sum \limits _{\alpha =1}^{\infty }g_{\alpha }(k)P_{\alpha }[\Phi ,{\bar {\Phi }}].}
Inserting this expansion into the FRGE and expanding the trace on its right-hand side in order to extract the beta-functions, one obtains the exact RG equation in component form:
k
∂
k
g
α
(
k
)
=
β
α
(
g
1
,
g
2
,
⋯
)
{\displaystyle k\partial _{k}g_{\alpha }(k)=\beta _{\alpha }(g_{1},g_{2},\cdots )}
. Together with the corresponding initial conditions these equations fix the evolution of the running couplings
g
α
(
k
)
{\displaystyle g_{\alpha }(k)}
, and thus determine
Γ
k
{\displaystyle \Gamma _{k}}
completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since there are infinitely many couplings, and the
β
{\displaystyle \beta }
-functions can depend on all of them. This makes it very hard to solve the system in general. A possible way out is to restrict the analysis on a finite-dimensional subspace as an approximation of the full theory space. In other words, such a truncation of the theory space sets all but a finite number of couplings to zero, considering only the reduced basis
{
P
α
[
⋅
]
}
{\displaystyle \{P_{\alpha }[\,\cdot \,]\}}
with
α
=
1
,
⋯
,
N
{\displaystyle \alpha =1,\cdots ,N}
. This amounts to the ansatz