In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
,
momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
, or
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
, used in order that objects with larger or smaller values than these
physical quantities
A physical quantity (or simply quantity) is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a '' numerical value'' and a '' ...
are ignored in some calculation. It is usually represented within a particular
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
or
length scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot af ...
, such as
Planck units
In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: ''Speed of light, c'', ''Gravitational constant, G'', ''Reduced Planck constant, ħ ...
.
When used in this context, the traditional terms "
infrared
Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
" and "
ultraviolet
Ultraviolet radiation, also known as simply UV, is electromagnetic radiation of wavelengths of 10–400 nanometers, shorter than that of visible light, but longer than X-rays. UV radiation is present in sunlight and constitutes about 10% of ...
" are not literal references to specific regions of the spectrum, but rather refer by analogy to portions of a calculation for low energies (infrared) and high energies (ultraviolet), respectively.
Infrared and ultraviolet cutoff
An infrared cutoff (long-distance cutoff) is the minimal value of energy – or, equivalently, the maximal
wavelength
In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats.
In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
(usually a very large distance) – that will be taken into account in a calculation, typically an integral.
At the opposite end of the energy scale, an ultraviolet cutoff is the maximal allowed energy or the shortest allowed distance (usually a very short
length scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot af ...
). An example of this is "the maximum energy the classically driven photoelectron can convert into a photon energy."
This "cutoff formula", most importantly, can be experimentally verified.
Effect on calculation
A typical use of cutoffs is to prevent
singularities from appearing during calculation. If some quantities are computed as
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s over energy or another physical quantity, these cutoffs determine the limits of integration. The exact physics is reproduced when the appropriate cutoffs are sent to zero or infinity. However, these integrals are often divergent – see
IR divergence and
UV divergence – and a cutoff is needed. The dependence of physical quantities on the chosen cutoffs (especially the ultraviolet cutoffs) is the main focus of the theory of the
renormalization group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
.
See also
*
Infrared fixed point
In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from arbitrary initial values at very high energies (short distance) to fixed, stable values, usually predictable, at low energies (large distance ...
*
Ultraviolet fixed point
References
Bibliography
* J.C. Collins, "Renormalization", Cambridge University Press, Cambridge, 1984.
* {{cite web, url=http://documents.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=1973-009, title=G. 't Hooft, M.Veltman, "Diagrammar", CERN Report 73-9, 1973.
* M.J. Veltman, "Diagrammatica", The path to Feynman diagrams, Cambridge University Press, Cambridge, 1995.
* L.S. Brown, "Quantum field theory", Cambridge University Press, Cambridge, 1992.
* M.E. Peskin, D.V. Schroeder, "An introduction to quantum field theory", Westview Press, 1995.
* C. Itzykson and J.B. Zuber, "Quantum field theory", Mcgraw-hill, New York, 1980.
* S. Weinberg, "The quantum theory of fields", Cambridge University Press, Cambridge, 2000.
Quantum field theory
Statistical mechanics
Renormalization group