7.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Asymptotic safety | 3/6 | https://en.wikipedia.org/wiki/Asymptotic_safety | reference | science, encyclopedia | 2026-05-05T13:41:25.065218+00:00 | kb-cron |
=== Renormalization group flow === The renormalization group (RG) describes the change of a physical system due to smoothing or averaging out microscopic details when going to a lower resolution. This brings into play a notion of scale dependence for the action functionals of interest. Infinitesimal RG transformations map actions to nearby ones, thus giving rise to a vector field on theory space. The scale dependence of an action is encoded in a "running" of the coupling constants parametrizing this action,
{
g
α
}
≡
{
g
α
(
k
)
}
{\displaystyle \{g_{\alpha }\}\equiv \{g_{\alpha }(k)\}}
, with the RG scale
k
{\displaystyle k}
. This gives rise to a trajectory in theory space (RG trajectory), describing the evolution of an action functional with respect to the scale. Which of all possible trajectories is realized in Nature has to be determined by measurements.
=== Taking the UV limit === The construction of a quantum field theory amounts to finding an RG trajectory which is infinitely extended in the sense that the action functional described by
{
g
α
(
k
)
}
{\displaystyle \{g_{\alpha }(k)\}}
is well-behaved for all values of the momentum scale parameter
k
{\displaystyle k}
, including the infrared limit
k
→
0
{\displaystyle k\rightarrow 0}
and the ultraviolet (UV) limit
k
→
∞
{\displaystyle k\rightarrow \infty }
. Asymptotic safety is a way of dealing with the latter limit. Its fundamental requirement is the existence of a fixed point of the RG flow. By definition this is a point
{
g
α
∗
}
{\displaystyle \{g_{\alpha }^{*}\}}
in the theory space where the running of all couplings stops, or, in other words, a zero of all beta-functions:
β
γ
(
{
g
α
∗
}
)
=
0
{\displaystyle \beta _{\gamma }(\{g_{\alpha }^{*}\})=0}
for all
γ
{\displaystyle \gamma }
. In addition that fixed point must have at least one UV-attractive direction. This ensures that there are one or more RG trajectories which run into the fixed point for increasing scale. The set of all points in the theory space that are "pulled" into the UV fixed point by going to larger scales is referred to as UV critical surface. Thus the UV critical surface consists of all those trajectories which are safe from UV divergences in the sense that all couplings approach finite fixed point values as
k
→
∞
{\displaystyle k\rightarrow \infty }
. The key hypothesis underlying asymptotic safety is that only trajectories running entirely within the UV critical surface of an appropriate fixed point can be infinitely extended and thus define a fundamental quantum field theory. It is obvious that such trajectories are well-behaved in the UV limit as the existence of a fixed point allows them to "stay at a point" for an infinitely long RG "time". With regard to the fixed point, UV-attractive directions are called relevant, UV-repulsive ones irrelevant, since the corresponding scaling fields increase and decrease, respectively, when the scale is lowered. Therefore, the dimensionality of the UV critical surface equals the number of relevant couplings. An asymptotically safe theory is thus the more predictive the smaller is the dimensionality of the corresponding UV critical surface. For instance, if the UV critical surface has the finite dimension
n
{\displaystyle n}
it is sufficient to perform only
n
{\displaystyle n}
measurements in order to uniquely identify Nature's RG trajectory. Once the
n
{\displaystyle n}
relevant couplings are measured, the requirement of asymptotic safety fixes all other couplings since the latter have to be adjusted in such a way that the RG trajectory lies within the UV critical surface. In this spirit the theory is highly predictive as infinitely many parameters are fixed by a finite number of measurements. In contrast to other approaches, a bare action which should be promoted to a quantum theory is not needed as an input here. It is the theory space and the RG flow equations that determine possible UV fixed points. Since such a fixed point, in turn, corresponds to a bare action, one can consider the bare action a prediction in the asymptotic safety program. This may be thought of as a systematic search strategy among theories that are already "quantum" which identifies the "islands" of physically acceptable theories in the "sea" of unacceptable ones plagued by short distance singularities.
=== Gaussian and non-Gaussian fixed points === A fixed point is called Gaussian if it corresponds to a free theory. Its critical exponents agree with the canonical mass dimensions of the corresponding operators which usually amounts to the trivial fixed point values
g
α
∗
=
0
{\displaystyle g_{\alpha }^{*}=0}
for all essential couplings
g
α
{\displaystyle g_{\alpha }}
. Thus standard perturbation theory is applicable only in the vicinity of a Gaussian fixed point. In this regard asymptotic safety at the Gaussian fixed point is equivalent to perturbative renormalizability plus asymptotic freedom. Due to the arguments presented in the introductory sections, however, this possibility is ruled out for gravity. In contrast, a nontrivial fixed point, that is, a fixed point whose critical exponents differ from the canonical ones, is referred to as non-Gaussian. Usually this requires
g
α
∗
≠
0
{\displaystyle g_{\alpha }^{*}\neq 0}
for at least one essential
g
α
{\displaystyle g_{\alpha }}
. It is such a non-Gaussian fixed point that provides a possible scenario for quantum gravity. As yet, studies on this subject thus mainly focused on establishing its existence.
=== Quantum Einstein gravity (QEG) ===