In a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, one may calculate an effective or
running coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the
electric charge.
In approximate calculations in several quantum field theories, notably
quantum electrodynamics and theories of the
Higgs particle, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the ''
Landau pole problem''.
It is not known whether the appearance of these inconsistencies is an artifact of the approximation, or a real fundamental problem in the theory. However, the problem can be avoided if an ultraviolet or UV fixed point appears in the theory. A quantum field theory has a UV fixed point if its
renormalization group flow
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...
approaches a
fixed point in the ultraviolet (i.e. short length scale/large energy) limit. This is related to zeroes of the
beta-function appearing in the
Callan–Symanzik equation.
The large length scale/small energy limit counterpart is the
infrared fixed point.
Specific cases and details
Among other things, it means that a theory possessing a UV fixed point may not be an
effective field theory
In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees ...
, because it is well-defined at arbitrarily small distance scales. At the UV fixed point itself, the theory can behave as a
conformal field theory.
The converse statement, that any
QFT which is valid at all distance scales (i.e. isn't an effective field theory) has a UV fixed point is false. See, for example,
cascading gauge theory
In theoretical physics, a cascading gauge theory is a gauge theory whose coupling rapidly changes with the scale in such a way that Seiberg duality must be applied many times.
Igor Klebanov and Matthew Strassler studied this kind of N=1 gauge th ...
.
Noncommutative quantum field theories have a UV cutoff even though they are not effective field theories.
Physicists distinguish between trivial and nontrivial fixed points. If a UV fixed point is
trivial (generally known as Gaussian fixed point), the theory is said to be
asymptotically free. On the other hand, a scenario, where a non-Gaussian (i.e. nontrivial) fixed point is approached in the UV limit, is referred to as
asymptotic safety
Asymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontr ...
.
Asymptotically safe theories may be well defined at all scales despite being ''
nonrenormalizable'' in perturbative sense (according to the
classical scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is scale ...
s).
Asymptotic safety scenario in quantum gravity
Steven Weinberg has proposed that the problematic
UV divergences appearing in
quantum theories of gravity may be cured by means of a nontrivial UV fixed point.
Such an
asymptotically safe theory is renormalizable in a nonperturbative sense, and due to the fixed point physical quantities are free from divergences. As yet, a general proof for the existence of the fixed point is still lacking, but there is mounting evidence for this scenario.
See also
*
Ultraviolet divergence
*
Landau pole
*
Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If
the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. ...
*
Asymptotic safety in quantum gravity
*
Asymptotic freedom
References
Statistical mechanics
Conformal field theory
Renormalization group
Fixed points (mathematics)
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