UV fixed point
Encyclopedia
In a quantum field theory
, 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
. 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 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
the Callan-Symanzik equation
.
The large length scale/small energy limit counterpart is the infrared fixed point
.
UV fixed point may not be an effective field theory
, 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
.
Noncommutative quantum field theories
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 freedom
.
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 dimension
s).
that gravity may satisfy asymptotic safety. http://arxiv.org/abs/gr-qc/0610018, http://relativity.livingreviews.org/Articles/lrr-2006-5/, http://arxiv.org/abs/0911.2727,
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics 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
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
. In
approximate calculations in several
quantum field theories, notably quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
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
Landau pole
In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...
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
"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-function
Beta-function
In theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....
in
the Callan-Symanzik equation
Callan-Symanzik equation
In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined and involves the beta-function of the theory and the anomalous dimensions...
.
The large length scale/small energy limit counterpart is the infrared fixed point
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 , to fixed stable values, usually predictable, at low energies...
.
Specific cases and details
Among other things, it means that a theory possessing aUV fixed point may not be an effective field theory
Effective field theory
In physics, an effective field theory is, as any effective theory, 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...
, 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
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
The converse statement, that any QFT
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
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
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....
.
Noncommutative quantum field theories
Noncommutative quantum field theory
In mathematical 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 coordinate functions are noncommutative...
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 freedom
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
.
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 dimension
Classical scaling dimension
In 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).
Asymptotic safety scenario in quantum gravity
Steven Weinberg has proposedthat gravity may satisfy asymptotic safety. http://arxiv.org/abs/gr-qc/0610018, http://relativity.livingreviews.org/Articles/lrr-2006-5/, http://arxiv.org/abs/0911.2727,
See also
- Ultraviolet divergenceUltraviolet divergenceIn 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. An infinite answer to a question that should have a...
- Landau poleLandau poleIn physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...
- Quantum trivialityQuantum trivialityIn a quantum field theory, charge screening can restrict 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