Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and conformal blocks.
The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicist Alexander Polyakov[2] and the Italian physicists Sergio Ferrara, Raoul Gatto and Aurelio Grillo.[3] Other early pioneers of this idea were Gerhard Mack and Ivan Todorov.
In higher dimensions, the conformal bootstrap started to develop following the 2008 paper by Riccardo Rattazzi, Slava Rychkov, Erik Tonni and Alessandro Vichi.[5] The method was since used to obtain many general results about conformal and superconformal field theories in three, four, five and six dimensions. Applied to the conformal field theory describing the critical point of the three-dimensional Ising model, it produced the most precise predictions for its critical exponents.[6][7][8]
Current research
The international Simons Collaboration on the Nonperturbative Bootstrap unites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory.[9]
History of the name
The modern usage of the term "conformal bootstrap" was introduced in 1984 by Belavin et al.[4] In the earlier literature, the name was sometimes used to denote a different approach to conformal field theories, nowadays referred to as the skeleton expansion or the "old bootstrap". This older method is perturbative in nature,[10][11] and is not directly related to the conformal bootstrap in the modern sense of the term.
External links
Open problems in conformal bootstrap
References
^"Using the 'Bootstrap,' Physicists Uncover Geometry of Theory Space | Quanta Magazine". Quanta Magazine. Retrieved 2018-01-03.
^ Polyakov, A. M. (1974). "Nonhamiltonian approach to conformal quantum field theory". Zh. Eksp. Teor. Fiz. 66: 23–42.
^Ferrara, S.; Grillo, A. F.; Gatto, R. (1973). "Tensor representations of conformal algebra and conformally covariant operator product expansion". Annals of Physics. 76 (1): 161–188. Bibcode:1973AnPhy..76..161F. doi:10.1016/0003-4916(73)90446-6.
^ abBelavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B. (1984). "Infinite conformal symmetry in two-dimensional quantum field theory". Nuclear Physics B. 241 (2): 333–380. Bibcode:1984NuPhB.241..333B. doi:10.1016/0550-3213(84)90052-X. ISSN 0550-3213.
^El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents". Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. Bibcode:2014JSP...157..869E. doi:10.1007/s10955-014-1042-7. S2CID 39692193.
^Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 174. arXiv:1502.02033. Bibcode:2015JHEP...06..174S. doi:10.1007/JHEP06(2015)174. ISSN 1029-8479. S2CID 35625559.
^Kadanoff, Leo P. (April 30, 2014). . Journal Club for Condensed Matter Physics. Archived from the original on July 22, 2015. Retrieved July 18, 2015.
^"Foundation Announces Simons Collaboration on the Non-Perturbative Bootstrap". 2016-08-25.
^Migdal, Alexander A. (1971). "Conformal invariance and bootstrap". Phys. Lett. B37 (4): 386–388. Bibcode:1971PhLB...37..386M. doi:10.1016/0370-2693(71)90211-5.
^Parisi, G. (1972). "On self-consistency conditions in conformal covariant field theory". Lettere al Nuovo Cimento. 4S2 (15): 777–780. doi:10.1007/BF02757039. S2CID 121431808.
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conformal, bootstrap, conformal, bootstrap, perturbative, mathematical, method, constrain, solve, conformal, field, theories, models, particle, physics, statistical, physics, that, exhibit, similar, properties, different, levels, resolution, contents, overview. The conformal bootstrap is a non perturbative mathematical method to constrain and solve conformal field theories i e models of particle physics or statistical physics that exhibit similar properties at different levels of resolution 1 Contents 1 Overview 2 Current research 3 History of the name 4 External links 5 ReferencesOverview EditUnlike more traditional techniques of quantum field theory conformal bootstrap does not use the Lagrangian of the theory Instead it operates with the general axiomatic parameters such as the scaling dimensions of the local operators and their operator product expansion coefficients A key axiom is that the product of local operators must be expressible as a sum over local operators thus turning the product into an algebra the sum must have a non zero radius of convergence This leads to decompositions of correlation functions into structure constants and conformal blocks The main ideas of the conformal bootstrap were formulated in the 1970s by the Soviet physicist Alexander Polyakov 2 and the Italian physicists Sergio Ferrara Raoul Gatto and Aurelio Grillo 3 Other early pioneers of this idea were Gerhard Mack and Ivan Todorov In two dimensions the conformal bootstrap was demonstrated to work in 1983 by Alexander Belavin Alexander Polyakov and Alexander Zamolodchikov 4 Many two dimensional conformal field theories were solved using this method notably the minimal models and the Liouville field theory In higher dimensions the conformal bootstrap started to develop following the 2008 paper by Riccardo Rattazzi Slava Rychkov Erik Tonni and Alessandro Vichi 5 The method was since used to obtain many general results about conformal and superconformal field theories in three four five and six dimensions Applied to the conformal field theory describing the critical point of the three dimensional Ising model it produced the most precise predictions for its critical exponents 6 7 8 Current research EditThe international Simons Collaboration on the Nonperturbative Bootstrap unites researchers devoted to developing and applying the conformal bootstrap and other related techniques in quantum field theory 9 History of the name EditThe modern usage of the term conformal bootstrap was introduced in 1984 by Belavin et al 4 In the earlier literature the name was sometimes used to denote a different approach to conformal field theories nowadays referred to as the skeleton expansion or the old bootstrap This older method is perturbative in nature 10 11 and is not directly related to the conformal bootstrap in the modern sense of the term External links EditOpen problems in conformal bootstrapReferences Edit Using the Bootstrap Physicists Uncover Geometry of Theory Space Quanta Magazine Quanta Magazine Retrieved 2018 01 03 Polyakov A M 1974 Nonhamiltonian approach to conformal quantum field theory Zh Eksp Teor Fiz 66 23 42 Ferrara S Grillo A F Gatto R 1973 Tensor representations of conformal algebra and conformally covariant operator product expansion Annals of Physics 76 1 161 188 Bibcode 1973AnPhy 76 161F doi 10 1016 0003 4916 73 90446 6 a b Belavin A A Polyakov A M Zamolodchikov A B 1984 Infinite conformal symmetry in two dimensional quantum field theory Nuclear Physics B 241 2 333 380 Bibcode 1984NuPhB 241 333B doi 10 1016 0550 3213 84 90052 X ISSN 0550 3213 Rattazzi Riccardo Rychkov Vyacheslav S Tonni Erik Vichi Alessandro 2008 Bounding scalar operator dimensions in 4D CFT JHEP 2008 12 031 arXiv 0807 0004 Bibcode 2008JHEP 12 031R doi 10 1088 1126 6708 2008 12 031 S2CID 8954304 El Showk Sheer Paulos Miguel F Poland David Rychkov Slava Simmons Duffin David Vichi Alessandro 2014 Solving the 3d Ising Model with the Conformal Bootstrap II c Minimization and Precise Critical Exponents Journal of Statistical Physics 157 4 5 869 914 arXiv 1403 4545 Bibcode 2014JSP 157 869E doi 10 1007 s10955 014 1042 7 S2CID 39692193 Simmons Duffin David 2015 A semidefinite program solver for the conformal bootstrap Journal of High Energy Physics 2015 6 174 arXiv 1502 02033 Bibcode 2015JHEP 06 174S doi 10 1007 JHEP06 2015 174 ISSN 1029 8479 S2CID 35625559 Kadanoff Leo P April 30 2014 Deep Understanding Achieved on the 3d Ising Model Journal Club for Condensed Matter Physics Archived from the original on July 22 2015 Retrieved July 18 2015 Foundation Announces Simons Collaboration on the Non Perturbative Bootstrap 2016 08 25 Migdal Alexander A 1971 Conformal invariance and bootstrap Phys Lett B37 4 386 388 Bibcode 1971PhLB 37 386M doi 10 1016 0370 2693 71 90211 5 Parisi G 1972 On self consistency conditions in conformal covariant field theory Lettere al Nuovo Cimento 4S2 15 777 780 doi 10 1007 BF02757039 S2CID 121431808 This quantum mechanics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Conformal bootstrap amp oldid 1114261427, wikipedia, wiki, book, books, library,
