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 that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

. In approximate calculations in several quantum field theories, notably

quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...

and theories of the

Higgs particle The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stand ...

, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the ''

Landau pole In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the ph ...

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 the ...

approaches a fixed point in the ultraviolet (i.e. short length scale/large energy) limit. This is related to zeroes of the

beta-function In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is ...

appearing in the

Callan–Symanzik equation In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the Correlation function (quantum field theory), ''n''-point correlation functions under variation of the energy scale at which the theory is define ...

. The large length scale/small energy limit counterpart is the

infrared fixed point In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from initial values at very high energies (short distance) to fixed stable values, usually predictable, at low energies (large distance). This usu ...

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 A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...

. 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. 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 Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

(generally known as Gaussian fixed point), the theory is said to be

asymptotically free In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases. Asymptotic fr ...

. 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 Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...

'' 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 Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interactio ...

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.

References

Statistical mechanics Conformal field theory Renormalization group Fixed points (mathematics) {{Quantum-stub

Specific cases and details

Asymptotic safety scenario in quantum gravity

See also

References