User:Sparkyscience/Yang-Mills theory
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Yang-Mills theory or Yang-Mills-Shaw theory is a mathematical generalization of Maxwell's equations. Yang-Mills equations can be used to describe the physics of the electromagnetic, weak and strong forces and forms the mathematical foundation of the Standard Model of particle physics. Yang-Mills theory primarily deals with symmetry breaking of gauge transformations. A gauge transformation is a symmetry transformation in physics that leads leads to the conservation of some local property known as a particle or quantum number (for instance the charge, spin, baryon number, lepton number etc.). A direct consequence of gauge symmetry conservation is that the relative phases between different states are unobservable .
One of the key ideas that helped Albert Einstein develop his theory of general relativity was the idea that the field equations should be the same in every co-ordinate system, Lorentz symmetry is preserved as a local property just like the symmetry principles of quantum numbers. From a modern point of view, this too is an example of a gauge symmetry. However, it is not known whether it is possible to cast general relativity as a Yang-Mills theory, which would allow the symmetry breaking of local Lorentz invariance. It is known how how to quantize Yang-Mills theories, but not general gauge theories. This unsolved issue is considered crucial for the progression of a theory of quantum gravity and is the essence the Yang–Mills existence and mass gap problem, one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 to the one who solves it.
The first observation of a broken gauge symmetry was seen in the Wu experiment, where P-symmetry (the symmetry between left and right) was violated by the weak interaction. Today broken gauge symmetries are able to explain a wide range of diverse phenomena including the Higgs mechanism, which describes how gauge bosons get their mass, to many electromagnetic phenomena such as the Aharonov–Bohm effect, Berry phase and Zeeman effect.
History
[edit]Shortly after Albert Einstein introduced his general theory of relativity in 1915, Hermann Weyl suggested an extension where the very notion of length becomes path dependant. Weyl was able to incorporate Maxwell's electromagnetic theory into a spacetime geometry.[1]
In 1954 Chen-Ning Yang and Robert Mills[2] published the first gauge theory of standing consequence and independently in the same year by Ronald Shaw[3], who was then a student of Abdus Salam’s, though he never published it except as a PhD thesis.[4] This occurred at about the same time that mathematicians were giving the final touches to the notion of fibre bundles.[5] It turns out fibre bundles stand in the background of both gauge theories and general relativity.[6]
Tsung-Dao Lee and Chen-Ning Yang proposed the idea of parity nonconservation in weak interactions. Experiments conducted in 1956 by Chien-Shiung Wu established that conservation of parity was violated (P-violation) by the weak interaction and both Lee and Yang received the 1957 Nobel Prize in physics for this result.
References
[edit]Notes
[edit]- ^ Penrose (2005), p. 451
- ^ Yang & Mills (1954)
- ^ Shaw (1955)
- ^ Kibble (2014), p. 4
- ^ For general contempory accounts see Steenrod (1951) and Lichnerowicz (1955)
- ^ See Trautman (1979); Daniel & Viallet (1980)
Bibliography
[edit]- Daniel, M.; Viallet, C. M. (1980). "The geometrical setting of gauge theories of the Yang-Mills type" (PDF). Reviews of Modern Physics. 52 (1): 175–197. Bibcode:1980RvMP...52..175D. doi:10.1103/RevModPhys.52.175. ISSN 0034-6861.
- Kibble, T. W. B. (2014). "Spontaneous symmetry breaking in gauge theories" (PDF). Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 373 (2032): 20140033. Bibcode:2014RSPTA.37340033K. doi:10.1098/rsta.2014.0033. ISSN 1364-503X.
- Lichnerowicz, A. (1955). Théorie Globale des Connexions et des Groupes d'Holonomie. Roma: Edizioni Cremonese. LCCN a56005667. OCLC 746575460.
- Penrose, R. (2005). The Road to Reality: A Complete Guide to the Laws of the Universe (1st American ed.). New York: A.A. Knopf. ISBN 0-679-45443-8. LCCN 2004061543. OCLC 214080870.
- Steenrod, N. (1951). The Topology of Fibre Bundles (1999 ed.). Princeton: Princeton University Press. ISBN 0-691-00548-6. LCCN 99017187. OCLC 40734875.
- 't Hooft, G. (1980). "Gauge Theories of the Force between Elementary Particles" (PDF). Scientific American. 242 (6): 104–138. doi:10.1038/scientificamerican0680-104. hdl:1874/4662.
- Trautman, A. (1979). "The Geometry of Gauge Fields". Czechoslovak Journal of Physics. 29 (1): 107–116. Bibcode:1979CzJPh..29..107T. doi:10.1007/BF01603811. ISSN 0011-4626.
- Weyl, H. (1919). "Eine neue Erweiterung der Relativitätstheorie". Annalen der Physik. 364 (10): 101–133. Bibcode:1919AnP...364..101W. doi:10.1002/andp.19193641002. ISSN 0003-3804.
- Weyl, H. (1929). "Elektron und Gravitation". I. Z. Phys. 56 (5–6): 330–352. Bibcode:1929ZPhy...56..330W. doi:10.1007/bf01339504.
- Yang, C. N.; Mills, R. L. (1954). "Conservation of Isotopic Spin and Isotopic Gauge Invariance" (PDF). Physical Review. 96 (1): 191–195. Bibcode:1954PhRv...96..191Y. doi:10.1103/PhysRev.96.191. ISSN 0031-899X.
- Shaw, R. (1955). "Invariance Under General Isotopic Gauge Transformations, Part II, chapter III". PHD Thesis, Uiversity of Cambridge, UK.
Books
[edit]- Baez, John C.; Muniain, Javier P. (1994). Gauge Fields, Knots and Gravity. Singapore: World Scientific. doi:10.1142/2324. ISBN 9810217293. LCCN 94003438. OCLC 611875857.
- 't Hooft;, Gerardus; Yang, Chen-Ning, eds. (2005). 50 Years of Yang-Mills Theory. Hackensack, New Jersey: World Scientific. ISBN 9789812389343. LCCN 2005043322. OCLC 249800059.
{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: editors list (link) - Chan, Hong-Mo; Tsou, Sheung Tsun (1993). Some Elementary Gauge Theory Concepts. SIngapore: World Scientific. doi:10.1142/1752. ISBN 9810210809. LCCN 94211637. OCLC 29514253.