11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Asymptotic safety | 5/6 | https://en.wikipedia.org/wiki/Asymptotic_safety | reference | science, encyclopedia | 2026-05-05T13:41:25.065218+00:00 | kb-cron |
Γ
k
[
Φ
,
Φ
¯
]
=
∑
α
=
1
N
g
α
(
k
)
P
α
[
Φ
,
Φ
¯
]
,
{\displaystyle \Gamma _{k}[\Phi ,{\bar {\Phi }}]=\sum \limits _{\alpha =1}^{N}g_{\alpha }(k)P_{\alpha }[\Phi ,{\bar {\Phi }}],}
leading to a system of finitely many coupled differential equations,
k
∂
k
g
α
(
k
)
=
β
α
(
g
1
,
⋯
,
g
N
)
{\displaystyle k\partial _{k}g_{\alpha }(k)=\beta _{\alpha }(g_{1},\cdots ,g_{N})}
, which can now be solved employing analytical or numerical techniques. Clearly a truncation should be chosen such that it incorporates as many features of the exact flow as possible. Although it is an approximation, the truncated flow still exhibits the nonperturbative character of the FRGE, and the
β
{\displaystyle \beta }
-functions can contain contributions from all powers of the couplings.
== Evidence from truncated flow equations ==
=== Einstein–Hilbert truncation === As described in the previous section, the FRGE lends itself to a systematic construction of nonperturbative approximations to the gravitational beta-functions by projecting the exact RG flow onto subspaces spanned by a suitable ansatz for
Γ
k
{\displaystyle \Gamma _{k}}
. In its simplest form, such an ansatz is given by the Einstein–Hilbert action where Newton's constant
G
k
{\displaystyle G_{k}}
and the cosmological constant
Λ
k
{\displaystyle \Lambda _{k}}
depend on the RG scale
k
{\displaystyle k}
. Let
g
μ
ν
{\displaystyle g_{\mu \nu }}
and
g
¯
μ
ν
{\displaystyle {\bar {g}}_{\mu \nu }}
denote the dynamical and the background metric, respectively. Then
Γ
k
{\displaystyle \Gamma _{k}}
reads, for arbitrary spacetime dimension
d
{\displaystyle d}
,
Γ
k
[
g
,
g
¯
,
ξ
,
ξ
¯
]
=
1
16
π
G
k
∫
d
d
x
g
(
−
R
(
g
)
+
2
Λ
k
)
+
Γ
k
gf
[
g
,
g
¯
]
+
Γ
k
gh
[
g
,
g
¯
,
ξ
,
ξ
¯
]
.
{\displaystyle \Gamma _{k}[g,{\bar {g}},\xi ,{\bar {\xi }}]={\frac {1}{16\pi G_{k}}}\int {\text{d}}^{d}x\,{\sqrt {g}}\,{\big (}-R(g)+2\Lambda _{k}{\big )}+\Gamma _{k}^{\text{gf}}[g,{\bar {g}}]+\Gamma _{k}^{\text{gh}}[g,{\bar {g}},\xi ,{\bar {\xi }}].}
Here
R
(
g
)
{\displaystyle R(g)}
is the scalar curvature constructed from the metric
g
μ
ν
{\displaystyle g_{\mu \nu }}
. Furthermore,
Γ
k
gf
{\displaystyle \Gamma _{k}^{\text{gf}}}
denotes the gauge fixing action, and
Γ
k
gh
{\displaystyle \Gamma _{k}^{\text{gh}}}
the ghost action with the ghost fields
ξ
{\displaystyle \xi }
and
ξ
¯
{\displaystyle {\bar {\xi }}}
. The corresponding
β
{\displaystyle \beta }
-functions, describing the evolution of the dimensionless Newton constant
g
k
=
k
d
−
2
G
k
{\displaystyle g_{k}=k^{d-2}G_{k}}
and the dimensionless cosmological constant
λ
k
=
k
−
2
Λ
k
{\displaystyle \lambda _{k}=k^{-2}\Lambda _{k}}
, have been derived for the first time in reference for any value of the spacetime dimensionality, including the cases of
d
{\displaystyle d}
below and above
4
{\displaystyle 4}
dimensions. In particular, in
d
=
4
{\displaystyle d=4}
dimensions they give rise to the RG flow diagram shown on the left-hand side. The most important result is the existence of a non-Gaussian fixed point suitable for asymptotic safety. It is UV-attractive both in
g
{\displaystyle g}
-
and in
λ{\displaystyle \lambda }
-direction. This fixed point is related to the one found in
d
=
2
+
ϵ
{\displaystyle d=2+\epsilon }
dimensions by perturbative methods in the sense that it is recovered in the nonperturbative approach presented here by inserting
d
=
2
+
ϵ
{\displaystyle d=2+\epsilon }
into the
β
{\displaystyle \beta }
-functions and expanding in powers of
ϵ
{\displaystyle \epsilon }
. Since the
β
{\displaystyle \beta }
-functions were shown to exist and explicitly computed for any real, i.e., not necessarily integer value of
d
{\displaystyle d}
, no analytic continuation is involved here. The fixed point in
d
=
4
{\displaystyle d=4}
dimensions, too, is a direct result of the nonperturbative flow equations, and, in contrast to the earlier attempts, no extrapolation in
ϵ
{\displaystyle \epsilon }
is required.
=== Extended truncations === Subsequently, the existence of the fixed point found within the Einstein–Hilbert truncation has been confirmed in subspaces of successively increasing complexity. The next step in this development was the inclusion of an
R
2
{\displaystyle R^{2}}
-term in the truncation ansatz. This has been extended further by taking into account polynomials of the scalar curvature
R
{\displaystyle R}
(so-called
f
(
R
)
{\displaystyle f(R)}
-truncations), and the square of the Weyl curvature tensor. Also, f(R) theories have been investigated in the Local Potential Approximation finding nonperturbative fixed points in support of the Asymptotic Safety scenario, leading to the so-called Benedetti–Caravelli (BC) fixed point. In such BC formulation, the differential equation for the Ricci scalar R is overconstrained, but some of these constraints can be removed via the resolution of movable singularities. Moreover, the impact of various kinds of matter fields has been investigated. Also computations based on a field reparametrization invariant effective average action seem to recover the crucial fixed point. In combination these results constitute strong evidence that gravity in four dimensions is a nonperturbatively renormalizable quantum field theory, indeed with a UV critical surface of reduced dimensionality, coordinatized by only a few relevant couplings.
== Microscopic structure of spacetime == Results of asymptotic safety related investigations indicate that the effective spacetimes of QEG have fractal-like properties on microscopic scales. It is possible to determine, for instance, their spectral dimension and argue that they undergo a dimensional reduction from 4 dimensions at macroscopic distances to 2 dimensions microscopically. In this context it might be possible to draw the connection to other approaches to quantum gravity, e.g. to causal dynamical triangulations, and compare the results.
== Physics applications ==