Ultraviolet divergence

In physics, an ultraviolet divergence or UV divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with unbounded energy, or, equivalently, because of physical phenomena at infinitesimal distances.

Since an infinite result is unphysical, these divergences often require special treatment to remove unphysical effects inherent in the perturbative formalisms. In particular, UV divergences can often be removed by regularization and renormalization. Successful resolution of an ultraviolet divergence is known as ultraviolet completion. 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" first encountered in understanding blackbody radiation. According to then-current theory, the quantity of radiation in the form of light released at any specific wavelength should increase with decreasing wavelength—in particular, there should be considerably more ultraviolet light released from a blackbody radiator than infrared light. Measurements showed the opposite, with maximal energy released at intermediate wavelengths, suggesting a failure of classical mechanics. This problem eventually led to the development of quantum mechanics.

The successful resolution of the original ultraviolet catastrophe has prompted the pursuit of solutions to other problems of ultraviolet divergence. A similar problem in electromagnetism was famously solved by Richard Feynman by applying quantum field theory through the use of renormalization groups, leading to the successful creation of quantum electrodynamics (QED). Similar techniques led to the standard model of particle physics. Ultraviolet divergences remain a key feature in the exploration of new physical theories, like supersymmetry.

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