Infrared divergence
Encyclopedia
In 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 (IR) divergence only appears in theories with massless particles (such as photon
s). They represent a legitimate effect that a complete theory often implies. One way to deal with it is to impose an infrared cutoff and take the limit as the cutoff approaches zero and/or refine the question. Another way is to assign the massless particle a fictitious mass, and then take the limit as the fictitious mass vanishes.
The divergence is usually in terms of particle number and not empirically troubling, in that all measurable quantities remain finite. (Unlike in the case of the UV catastrophe where the energies involve diverge.)
is accelerated (or de-accelerated) it emits Bremsstrahlung radiation. Semiclassical electromagnetic theory, or the full quantum electrodynamic analysis, shows that an infinite number of soft photons
are created. But only a finite number are detectable, the remainder, due to their low energy, falling below any finite energy detection threshold, which must necessarily exist.
However even though most of the photons are not detectable they can't be ignored in the theory; Quantum electrodynamic calculations show that the transition amplitude between any states with a finite number of photons vanishes. Finite transition amplitudes are obtained only by summing over states with an infinite number of soft photons.
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 infrared divergence or infrared catastrophe 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 small 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 zero, or, equivalently, because of physical phenomena at very long distances.
The infrared (IR) divergence only appears in theories with massless particles (such as photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s). They represent a legitimate effect that a complete theory often implies. One way to deal with it is to impose an infrared cutoff and take the limit as the cutoff approaches zero and/or refine the question. Another way is to assign the massless particle a fictitious mass, and then take the limit as the fictitious mass vanishes.
The divergence is usually in terms of particle number and not empirically troubling, in that all measurable quantities remain finite. (Unlike in the case of the UV catastrophe where the energies involve diverge.)
Bremsstrahlung example
When an electric chargeElectric 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...
is accelerated (or de-accelerated) it emits Bremsstrahlung radiation. Semiclassical electromagnetic theory, or the full quantum electrodynamic analysis, shows that an infinite number of soft photons
Soft photons
In particle physics, soft photons are photons with energies much smaller than the energies of the particles participating in a particular scattering process, and they are not energetic enough to be detected. Such photons can be emitted from the external lines of charged particles of the Feynman...
are created. But only a finite number are detectable, the remainder, due to their low energy, falling below any finite energy detection threshold, which must necessarily exist.
However even though most of the photons are not detectable they can't be ignored in the theory; Quantum electrodynamic calculations show that the transition amplitude between any states with a finite number of photons vanishes. Finite transition amplitudes are obtained only by summing over states with an infinite number of soft photons.