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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 ...
, the Callan–Symanzik equation is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
describing the evolution of the ''n''-point correlation functions under variation of the energy scale at which the theory is defined and involves the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...
of the theory and the anomalous dimensions. As an example, for a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
with one massless scalar field and one self-coupling term, denote the bare field strength by \phi_0 and the bare coupling constant by g_0. In the process of
renormalisation Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
, a mass scale ''M'' must be chosen. Depending on ''M'', the field strength is rescaled by a constant: \phi = Z\phi_0, and as a result the bare coupling constant g_0 is correspondingly shifted to the renormalised coupling constant ''g''. Of physical importance are the renormalised ''n''-point functions, computed from connected
Feynman diagrams 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 introduce ...
, schematically of the form :G^(x_1,x_2,\ldots,x_n;M,g) = \langle \phi(x_1)\phi(x_2)\cdots\phi(x_n)\rangle For a given choice of renormalisation scheme, the computation of this quantity depends on the choice of ''M'', which affects the shift in ''g'' and the rescaling of \phi. If the choice of M is slightly altered by \delta M, then the following shifts will occur: :M\to M + \delta M :g\to g + \delta g :\phi = Z\phi_0 \to Z'\phi_0 = (1+\delta\eta)\phi :G^ \to (1+n\,\delta\eta)G^ The Callan–Symanzik equation relates these shifts: :n\,\delta\eta G^ = \frac\delta M + \frac\delta g After the following definitions :\beta = \frac\delta g :\gamma = -\frac\delta\eta the Callan–Symanzik equation can be put in the conventional form: :\left \frac+\beta(g)\frac+n\gamma\rightG^(x_1,x_2,\ldots,x_n;M,g)=0 \beta(g) being the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...
. In
quantum electrodynamics 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 spec ...
this equation takes the form :\left \frac+\beta(e)\frac+n\gamma_2 +m\gamma_3\right^(x_1,x_2,\ldots,x_n;y_1,y_2,\ldots,y_m;M,e)=0 where ''n'' and ''m'' are the numbers of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
and
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 alwa ...
fields, respectively, for which the correlation function G^ is defined. The renormalised coupling constant is now the renormalised
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
''e''. The electron field and the photon field rescale differently under renormalisation, and thus lead to two separate functions, \gamma_2 and \gamma_3, respectively. The Callan–Symanzik equation was discovered independently by
Curtis Callan Curtis Gove Callan Jr. (born October 11, 1942) is an American theoretical physicist and the James S. McDonnell Distinguished University Professor of Physics at Princeton University. He has conducted research in gauge theory, string theory, instan ...
and
Kurt Symanzik Kurt Symanzik (November 23, 1923 – October 25, 1983) was a German physicist working in quantum field theory. Life Symanzik was born in Lyck (Ełk), East Prussia, and spent his childhood in Königsberg. He started studying physics in 1946 at ...
in 1970. Later it was used to understand
asymptotic freedom In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases. Asymptotic fre ...
. This equation arises in the framework of
renormalization group In theoretical physics, the term renormalization group (RG) refers to 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 t ...
. It is possible to treat the equation using
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.


See also

*
Exact renormalization group equation Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an Ameri ...
*
Beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^( ...


Notes


References

* Jean Zinn-Justin, ''Quantum Field Theory and Critical Phenomena '', Oxford University Press, 2003, * John Clements Collins, ''Renormalization'', Cambridge University Press, 1986, * Michael E. Peskin and Daniel V. Schroeder
''An Introduction to Quantum Field Theory''
Addison-Wesley, Reading, 1995. {{DEFAULTSORT:Callan-Symanzik equation Renormalization group Equations