physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, an infrared divergence (also IR divergence or infrared catastrophe) is a situation in which an

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...

, for example a

Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...

, diverges because of contributions of objects with very small

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

approaching zero, or equivalently, because of physical phenomena at very long distances.

Overview

The infrared divergence only appears in theories with

massless particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, g ...

s (such as

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...

s). They represent a legitimate effect that a complete theory often implies. In fact, in the case of photons, the energy is given by

E=h\nu

, where

\nu

is the frequency associated to the particle and as it goes to zero, like in the case of soft photons, there will be an infinite number of particles in order to have a finite amount of energy. One way to deal with it is to impose an

infrared cutoff In theoretical physics, cutoff (AE: cutoff, BE: cut-off) 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 calculat ...

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 involved diverge.)

Bremsstrahlung example

When an

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 ...

is accelerated (or decelerated) it emits

Bremsstrahlung radiation ''Bremsstrahlung'' (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typicall ...

. Semiclassical

electromagnetic theory In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

, or the full

quantum electrodynamic 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 speci ...

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., pages 177-184 and appendix A6 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. The zero-energy photons become important in analyzing the

in the coaccelerated frame in which the charge experiences a thermal bath due to the

Unruh effect The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...

. In this case, the static charge will only interact with these zero-energy (Rindler) photons in a sense similar to virtual photons in the coulomb interaction.

References

Quantum field theory Renormalization group {{quantum-stub

Overview

Bremsstrahlung example

See also

References