In a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, 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 chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
. In
approximate calculations in several
quantum field theories, notably
quantum electrodynamicsQuantum electrodynamics is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons...
and theories of the Higgs particle, the running coupling
appears to become infinite at a finite momentum scale.
This is
sometimes called the
Landau poleIn physics, the Landau pole is the energy scale at which a coupling constant of a quantum field theory appears to become infinite. Such a possibility was pointed out by the physicist Lev Davidovich Landau...
problem.
In a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, 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 chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
. In
approximate calculations in several
quantum field theories, notably
quantum electrodynamicsQuantum electrodynamics is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons...
and theories of the Higgs particle, the running coupling
appears to become infinite at a finite momentum scale.
This is
sometimes called the
Landau poleIn physics, the Landau pole is the energy scale at which a coupling constant of a quantum field theory appears to become infinite. Such a possibility was pointed out by the physicist Lev Davidovich Landau...
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 approaches a
fixed point"Fixed point" has many meanings in science, most of them mathematical.* Fixed point * Fixed point combinator* Fixed-point arithmetic, a manner of doing arithmetic on computers* Benchmark , fixed points used by geodesists...
in the ultraviolet (i.e. short length scale/large energy) limit. This is related to zeroes of the
beta-functionIn theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process....
in
the
Callan-Symanzik equationIn physics, the Callan–Symanzik equation is the renormalization relation that tells how the coupling constant changes with momentum in a quantum field theory.It was discovered independently by Curtis Callan and Kurt Symanzik in 1970...
.
Specific cases and details
Among other things, it means that a theory possessing a
UV fixed point may not be an
effective field theoryIn physics, an effective field theory is an approximate theory that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization group:Presently, effective...
, because it is well-defined at arbitrarily small distance scales. At the UV fixed point itself, the theory can behave as a
conformal field theoryA conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
The converse statement, that any
QFTQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
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 theoryIn 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....
.
Noncommutative quantum field theoriesIn mathematical physics, a branch of mathematics and physics, noncommutative quantum field theory is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the spatial coordinates are...
have a UV cutoff even though they are not effective field theories.
If the UV fixed point is trivial (aka Gaussian), we say that we have
asymptotic freedomIn physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles, such as quarks, to become arbitrarily weak at shorter distances, i.e...
.
If the UV fixed point is nontrivial, we say that we have "asymptotic safety". Theories with asymptotic safety may be well defined at all scales despite being nonrenormalizable in perturbative sense (according to the
classical scaling dimensionIn theoretical physics, namely quantum field theory, the classical scaling dimension of an operator O is the power of mass of an operator determined by dimensional analysis from the Lagrangian...
s).
See also
- Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances...
- Landau pole
In physics, the Landau pole is the energy scale at which a coupling constant of a quantum field theory appears to become infinite. Such a possibility was pointed out by the physicist Lev Davidovich Landau...
- Quantum triviality
In a quantum field theory, charge screening canrestrict the value of the observable"renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero,the theory is said to be "trivial" or noninteracting....
- Asymptotic Safety gravity