Misplaced Pages

Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman Axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Schwinger-Dyson equation Renormalization group equation
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Cardy Coleman Connes Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gell-Mann Glimm Goldstone Gribov Gross Gupta Guralnik Haag Hagen Han Heisenberg Hepp Higgs 't Hooft Iliopoulos Ivanenko Jackiw Jaffe Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Kim Klebanov Kontsevich Kreimer Kuraev Landau Lee Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Matsubara Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osborn Osterwalder Parisi Pauli Peccei Peskin Plefka Polchinski Polyakov Pomeranchuk Popov Proca Quinn Rouet Rubakov Ruelle Sakurai Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Takahashi Thirring Tomonaga Tyutin Vainshtein Veltman Veneziano Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
v t e

In physics, the Callan–Symanzik equation is a differential equation 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 of the theory and the anomalous dimensions.

As an example, for a quantum field theory with one massless scalar field and one self-coupling term, denote the bare field strength by ${\displaystyle \phi _{0}}$ and the bare coupling constant by ${\displaystyle g_{0}}$. In the process of renormalisation, a mass scale M must be chosen. Depending on M, the field strength is rescaled by a constant: ${\displaystyle \phi =Z\phi _{0}}$, and as a result the bare coupling constant ${\displaystyle g_{0}}$ is correspondingly shifted to the renormalised coupling constant g.

Of physical importance are the renormalised n-point functions, computed from connected Feynman diagrams, schematically of the form

${\displaystyle G^{(n)}(x_{1},x_{2},\ldots ,x_{n};M,g)=\langle \phi (x_{1})\phi (x_{2})\cdots \phi (x_{n})\rangle }$

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 ${\displaystyle \phi }$. If the choice of ${\displaystyle M}$ is slightly altered by ${\displaystyle \delta M}$, then the following shifts will occur:

${\displaystyle M\to M+\delta M}$

${\displaystyle g\to g+\delta g}$

${\displaystyle \phi =Z\phi _{0}\to Z'\phi _{0}=(1+\delta \eta )\phi }$

${\displaystyle G^{(n)}\to (1+n\,\delta \eta )G^{(n)}}$

The Callan–Symanzik equation relates these shifts:

${\displaystyle n\,\delta \eta G^{(n)}={\frac {\partial G^{(n)}}{\partial M}}\delta M+{\frac {\partial G^{(n)}}{\partial g}}\delta g}$

After the following definitions

${\displaystyle \beta ={\frac {M}{\delta M}}\delta g}$

${\displaystyle \gamma =-{\frac {M}{\delta M}}\delta \eta }$

the Callan–Symanzik equation can be put in the conventional form:

${\displaystyle \leftG^{(n)}(x_{1},x_{2},\ldots ,x_{n};M,g)=0}$

${\displaystyle \beta (g)}$ being the beta function.

In quantum electrodynamics this equation takes the form

${\displaystyle \leftG^{(n,m)}(x_{1},x_{2},\ldots ,x_{n};y_{1},y_{2},\ldots ,y_{m};M,e)=0}$

where n and m are the numbers of electron and photon fields, respectively, for which the correlation function ${\displaystyle G^{(n,m)}}$ is defined. The renormalised coupling constant is now the renormalised elementary charge e. The electron field and the photon field rescale differently under renormalisation, and thus lead to two separate functions, ${\displaystyle \gamma _{2}}$ and ${\displaystyle \gamma _{3}}$, respectively.

The Callan–Symanzik equation was discovered independently by Curtis Callan and Kurt Symanzik in 1970. Later it was used to understand asymptotic freedom.

This equation arises in the framework of renormalization group. It is possible to treat the equation using perturbation theory.

See also

• Renormalization group
• Beta function

Notes

1. Callan, Curtis G. (1970-10-15). "Broken Scale Invariance in Scalar Field Theory". Physical Review D. 2 (8). American Physical Society (APS): 1541–1547. Bibcode:1970PhRvD...2.1541C. doi:10.1103/physrevd.2.1541. ISSN 0556-2821.
2. Symanzik, K. (1970). "Small distance behaviour in field theory and power counting". Communications in Mathematical Physics. 18 (3). Springer Science and Business Media LLC: 227–246. Bibcode:1970CMaPh..18..227S. doi:10.1007/bf01649434. ISSN 0010-3616. S2CID 76654566.
3. Symanzik, K. (1971). "Small-distance-behaviour analysis and Wilson expansions". Communications in Mathematical Physics. 23 (1). Springer Science and Business Media LLC: 49–86. Bibcode:1971CMaPh..23...49S. doi:10.1007/bf01877596. ISSN 0010-3616. S2CID 119431863.

References

• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena , Oxford University Press, 2003, ISBN 0-19-850923-5
• John Clements Collins, Renormalization, Cambridge University Press, 1986, ISBN 0-521-31177-2
• Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995. 2nd edition, pbk. Westview Press. 2015.

1. Berg, Michael (10 February 2016). "Review of An introduction to quantum field theory by Peskin & Schroeder". MAA Reviews, maa.org.

• Eponymous equations of physics
• Renormalization group