Ultraviolet divergence
Encyclopedia
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
(approaching infinity), or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a finite answer is a potential problem. The ultraviolet (UV) divergences are often unphysical effects that can be removed by regularization
and renormalization
. If they cannot be removed, they imply that the theory is not perturbatively well-defined at very short distances.
The name comes from the earliest example of such a divergence, the "ultraviolet catastrophe
" in understanding blackbody radiation. According to then-current theory of radiation
, in this case, light
, the quantity of energy released at any specific wavelength
should increase with decreasing wavelength -- that is, there should be considerably more ultraviolet light released from a blackbody radiator than infrared light. Measurements showed the opposite; the most energy was released at wavelengths between the two extremes, which suggested classic mechanics simply didn't describe the phenomenon correctly. This problem led to the development of quantum mechanics
.
The success of the attack on the original ultraviolet catastrophe has led to the technique being widely applied in modern physics. A similar problem applying quantum field theory
to electromagnetism
was famously solved through the use of renormalization group
s and the successful creation of quantum electrodynamics
(QED). Similar techniques led to the modern standard model
of particle physics
. Ultraviolet divergences remain a key process in the exploration of new physical theories, like supersymmetry
.
A successful attack on an ultraviolet divergence is known as ultraviolet completion.
pointed out the following facts about such a procedure which are still as relevant today as in 1965: “The first is that we are led to a theory with differential wave propagation. The field functions are continuous functions of continuous parameters x and t, and the changes in the fields at a point x are determined by properties of the fields infinitesimally close to the point x. For most wave fields (for example, sound waves and the vibrations of strings and membranes) such a description is an idealization which is valid for distances larger than the characteristic length which measures the granularity of the medium. For smaller distances these theories are modified in a profound way. The electromagnetic field is a notable exception. Indeed, until the special theory of relativity obviated the necessity of a mechanistic interpretation, physicists made great efforts to discover evidence for such a mechanical description of the radiation field. After the requirement of an “ether” which propagates light waves had been abandoned, there was considerably less difficulty in accepting this same idea when the observed wave properties of the electron suggested the introduction of a new field. Indeed there is no evidence of an ether which underlies the electron wave. However, it is a gross and profound extrapolation of present experimental knowledge to assume that a wave description successful at “large” distances (that is, atomic lengths ≈10 -8 cm) may be extended to distances an indefinite number of orders of magnitude smaller (for example, to less than nuclear lengths ≈ 10 -13 cm). In the relativistic theory, we have seen that the assumption that the field description is correct in arbitrarily small space-time intervals has led—in perturbation theory—to divergent expressions for the electron self-energy and the bare charge. Renormalization theory has sidestepped these divergence difficulties, which may be indicative of the failure of the perturbation expansion. However, it is widely felt that the divergences are symptomatic of a chronic disorder in the small-distance behaviour of the theory. We might then ask why local field theories, that is, theories of fields which can be described by differential laws of wave propagation, have been so extensively used and accepted. There are several reasons, including the important one that with their aid a significant region of agreement with observations has been found. But the foremost reason is brutally simple: there exists no convincing form of a theory which avoids differential field equations”.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, an ultraviolet divergence is a situation in which an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, for example a Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
, diverges because of contributions of objects with very high energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
(approaching infinity), or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a finite answer is a potential problem. The ultraviolet (UV) divergences are often unphysical effects that can be removed by regularization
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
and renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. If they cannot be removed, they imply that the theory is not perturbatively well-defined at very short distances.
The name comes from the earliest example of such a divergence, the "ultraviolet catastrophe
Ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power....
" in understanding blackbody radiation. According to then-current theory of radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...
, in this case, light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
, the quantity of energy released at any specific wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
should increase with decreasing wavelength -- that is, there should be considerably more ultraviolet light released from a blackbody radiator than infrared light. Measurements showed the opposite; the most energy was released at wavelengths between the two extremes, which suggested classic mechanics simply didn't describe the phenomenon correctly. This problem led to the development of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
The success of the attack on the original ultraviolet catastrophe has led to the technique being widely applied in modern physics. A similar problem applying quantum field theory
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...
to electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
was famously solved through the use of renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
s and the successful creation of 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...
(QED). Similar techniques led to the modern standard model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
. Ultraviolet divergences remain a key process in the exploration of new physical theories, like supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
.
A successful attack on an ultraviolet divergence is known as ultraviolet completion.
Reason for ultraviolet divergence according to Bjorken and Drell
Commenting on the fact that contemporary theories about quantum scattering of fundamental particles grew out of application of the quantization procedure to classical fields that satisfy wave equations, Bjorken and DrellSidney Drell
Sidney David Drell is an American theoretical physicist and arms control expert. He is a professor emeritus at the Stanford Linear Accelerator Center and a senior fellow at Stanford University's Hoover Institution. Drell is a noted contributor in the field of quantum electrodynamics and particle...
pointed out the following facts about such a procedure which are still as relevant today as in 1965: “The first is that we are led to a theory with differential wave propagation. The field functions are continuous functions of continuous parameters x and t, and the changes in the fields at a point x are determined by properties of the fields infinitesimally close to the point x. For most wave fields (for example, sound waves and the vibrations of strings and membranes) such a description is an idealization which is valid for distances larger than the characteristic length which measures the granularity of the medium. For smaller distances these theories are modified in a profound way. The electromagnetic field is a notable exception. Indeed, until the special theory of relativity obviated the necessity of a mechanistic interpretation, physicists made great efforts to discover evidence for such a mechanical description of the radiation field. After the requirement of an “ether” which propagates light waves had been abandoned, there was considerably less difficulty in accepting this same idea when the observed wave properties of the electron suggested the introduction of a new field. Indeed there is no evidence of an ether which underlies the electron wave. However, it is a gross and profound extrapolation of present experimental knowledge to assume that a wave description successful at “large” distances (that is, atomic lengths ≈10 -8 cm) may be extended to distances an indefinite number of orders of magnitude smaller (for example, to less than nuclear lengths ≈ 10 -13 cm). In the relativistic theory, we have seen that the assumption that the field description is correct in arbitrarily small space-time intervals has led—in perturbation theory—to divergent expressions for the electron self-energy and the bare charge. Renormalization theory has sidestepped these divergence difficulties, which may be indicative of the failure of the perturbation expansion. However, it is widely felt that the divergences are symptomatic of a chronic disorder in the small-distance behaviour of the theory. We might then ask why local field theories, that is, theories of fields which can be described by differential laws of wave propagation, have been so extensively used and accepted. There are several reasons, including the important one that with their aid a significant region of agreement with observations has been found. But the foremost reason is brutally simple: there exists no convincing form of a theory which avoids differential field equations”.
See also
- infrared divergenceInfrared divergenceIn physics, an infrared divergence or infrared catastrophe is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances.The infrared ...
- cutoffCutoffIn theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...
- renormalization groupRenormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
- UV fixed pointUV fixed pointIn a quantum field theory, one may calculate an effectiveor running coupling constant that defines thecoupling of the theory measured at a given momentum scale.One example of such a coupling constantis the electric charge...
- Causal perturbation theoryCausal perturbation theoryCausal perturbation theory is a mathematically rigorous approach to renormalization theory, which makesit possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis....
- Zeta function regularizationZeta function regularizationIn mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators...