The six authors of the 1964 PRL papers, who received the 2010 J. J. Sakurai Prize for their work. From left to right: Kibble, Guralnik, Hagen, Englert, Brout. Right: Higgs. つづく
>>17 つづき Particle physicists study matter made from fundamental particles whose interactions are mediated by exchange particles known as force carriers. At the beginning of the 1960s a number of these particles had been discovered or proposed, along with theories suggesting how they relate to each other; however, even accepted versions such as the Unified field theory were known to be incomplete. One omission was that they could not explain the origins of mass as a property of matter. Goldstone's theorem, relating to continuous symmetries within some theories, also appeared to rule out many obvious solutions.[12]
The Higgs mechanism is a process by which vector bosons can get rest mass[Note 2] without explicitly breaking gauge invariance. The proposal for such a spontaneous symmetry breaking mechanism originally was suggested in 1962 by Philip Warren Anderson[13] and developed into a full relativistic model, independently and almost simultaneously, by three groups of physicists: by Francois Englert and Robert Brout in August 1964;[6] by Peter Higgs in October 1964;[5] and by Gerald Guralnik, C. R. Hagen, and Tom Kibble (GHK) in November 1964.[7] Properties of the model were further considered by Guralnik in 1965 [14] and by Higgs in 1966.[15] The papers showed that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a finite mass. In 1967, Steven Weinberg and Abdus Salam were the first to apply the Higgs mechanism to the breaking of the electroweak symmetry, and showed how a Higgs mechanism could be incorporated into Sheldon Glashow's electroweak theory, [16][17][18] in what became the Standard Model of particle physics. つづく
>>18 つづき The three papers written in 1964 were each recognised as milestone papers during Physical Review Letters's 50th anniversary celebration.[19] Their six authors were also awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics for this work.[20] (A dispute also arose the same year; in the event of a Nobel Prize up to three scientists would be eligible, with six authors credited for the papers.[21] ) Two of the three PRL papers (by Higgs and by GHK) contained equations for the hypothetical field that eventually would become known as the Higgs field and its hypothetical quantum, the Higgs boson. Higgs's subsequent 1966 paper showed the decay mechanism of the boson; only a massive boson can decay and the decays can prove the mechanism.
In the paper by Higgs the boson is massive, and in a closing sentence Higgs writes that "an essential feature" of the theory "is the prediction of incomplete multiplets of scalar and vector bosons". In the paper by GHK the boson is massless and decoupled from the massive states. In reviews dated 2009 and 2011, Guralnik states that in the GHK model the boson is massless only in a lowest-order approximation, but it is not subject to any constraint and acquires mass at higher orders, and adds that the GHK paper was the only one to show that there are no massless Goldstone bosons in the model and to give a complete analysis of the general Higgs mechanism.[22][23]
In addition to explaining how mass is acquired by vector bosons, the Higgs mechanism also predicts the ratio between the W boson and Z boson masses as well as their couplings with each other and with the Standard Model quarks and leptons. 略 おわり
The mechanism was proposed in 1962 by Philip Warren Anderson,[4] who discussed its consequences for particle physics but did not work out an explicit relativistic model. The relativistic model was developed in 1964 by Peter Higgs,[5] and independently by Robert Brout and Francois Englert,[6] and Gerald Guralnik, C. R. Hagen, and Tom Kibble,[7] who worked out the results by the spring of 1963.[8] The mechanism is closely analogous to phenomena previously discovered by Yoichiro Nambu involving the "vacuum structure" of quantum fields in superconductivity.[9] A similar but distinct effect, known as the Stueckelberg mechanism, had previously been studied by Ernst Stueckelberg.
These physicists discovered that when a gauge theory is combined with an additional field breaking spontaneously the symmetry group, the gauge bosons can consistently acquire a finite mass. In spite of the large values involved (see below) this permits a gauge theory description of the weak force, which was independently developed by Steven Weinberg and Abdus Salam in 1967. Higgs's original article presenting the model was rejected by Physics Letters. When revising the article before resubmitting it to Physical Review Letters, he added a sentence at the end,[10] mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete representations of the symmetry group; these are the Higgs bosons. つづく
>>24 つづき The three papers by Brout and Englert; Higgs; and Guralnik, Hagen, and Kibble were each recognized as "milestone letters" by Physical Review Letters in 2008.[11] While each of these seminal papers took similar approaches, the contributions and differences among the 1964 PRL symmetry breaking papers are noteworthy. All six physicists were jointly awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics for this work.[12]
Benjamin W. Lee is often credited with first naming the "Higgs-like" mechanism, although there is debate around when this first occurred.[13][14][15] One of the first times the Higgs name appeared in print was in 1972 when Gerardus 't Hooft and Martinus J. G. Veltman referred to it as the "Higgs-Kibble mechanism" in their Nobel winning paper.[16][17]
On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to three scientists for their work in subatomic physics symmetry breaking. Yoichiro Nambu, 87, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry in the context of the strong interactions, specifically chiral symmetry breaking. Physicists Makoto Kobayashi and Toshihide Maskawa shared the other half of the prize for discovering the origin of the explicit breaking of CP symmetry in the weak interactions.[6] This origin is ultimately reliant on the Higgs mechanism, but, so far understood as a "just so" feature of Higgs couplings, not a spontaneously broken symmetry phenomenon.
Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey lanl.arxiv.org/abs/hep-th/9802142 Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey R. Brout, F. Englert (Submitted on 20 Feb 1998 (v1), last revised 18 May 1998 (this version, v2)) Talks presented at the award ceremony of the 1997 High Energy and Particle Physics Prize of the European Physical Society (Jerusalem, 24 August 1997)
1.Renormalization of massless Yang-Mills fields. Nucl. Phys. B33 (1971) 173 - 199. www.phys.uu.nl/~thooft/gthpub/massless.pdf 2.Renormalizable Lagrangians for massive Yang-Mills fields. Nucl. Phys. B35 (1971) 167-188. www.phys.uu.nl/~thooft/gthpub/massive.pdf ・ ・ ・ 240.Quantum Mechanics from Classical Logic, Proceedings EmerQuM 11: Emergent Quantum Mechanics 2011 (Heinz von Foerster Congress) 10?13 November 2011, Vienna, Austria, J. Phys.: Conf. Ser. 361 012024 doi:10.1088/1742-6596/361/1/012024 The Pursuit of Quantum Gravity, Memoirs of Bryce De Witt from 1946 to 2004, by Cecile DeWitt-Morette. Book Review. Found. Phys. (2012)42:685-687. DOI 10.1007/s10701-012-9638-8 241.Relating the quantum mechanics of discrete systems to standard canonical quantum mechanics, ITP-UU-12/14; SPIN-12/12, arXiv:1204.4926 242Duality between a deterministic cellular automaton and a bosonic quantum field theory in 1+1 dimensions, ITP-UU-12/18; SPIN-12/16, arxiv.org/abs/1205.4107.
N.B. The papers with a gr-qc archive number can be obtained from the Gravity and Quantum Cosmology archive. Last revised date: May 21, 2012.
mathsoc.jp/meeting/kikaku/2009haru/2009_haru_ito.pdf 2009 年3 月26 日(木) 13:00?14:00 佐藤‐テイト予想の解決と展望 ? 非可換類体論の進展? Solution of the Sato-Tate Conjecture and Beyond ? Recent Developments in Non-abelian Class Field theory ? 伊藤哲史1 Tetsushi Ito 京都大学大学院理学研究科数学教室
局所指数定理 ジョン-ローの「指数定理」の本の解説．（結構詳しく書きました）．１２４ページもある． 測地線座標 geodesic.pdf リーマン多様体の測地線座標（正規座標）に関しての詳しい解説（学部向け） ４次元自己双対ケーラー多様体とアインシュタイン多様体 A. Derdzinskiの「self-dual K\"ahler manifolds and einstein manifolds of dimension four」の解説．計算もちゃんとしたので長くなってしまいました．
シンプレクティック幾何入門 シンプレクティック幾何の勉強ノートです．群作用がある場合の話しはかなり詳しく書いてあります． もとになってる本はAna Cannas da Silvaの「Lectures on symplectic geometry」とGuillemin Sternbergの「super symmteryy and equivariant de Rham theory」（３５０ページぐらいあります．重い）． これも専門家ではないので責任もたないけど，一応幾何学者ではあるんでね．
ameblo.jp/koto-tokyo-kodomomamoru/ 石川あや子の活動記録 〜Tomorrow is another day〜 Twitter 2012.6.28「都議有志の尖閣諸島視察について」 2012年06月30日(土)koto-tokyo-kodomomamoruの投稿
都知事定例会見「センセンとカクカク」発言。NHK（首都圏）も朝日新聞も東京新聞も、どうでもよいこの皮肉発言だけを真面目な顔で報道する。都庁記者クラブはレベル低すぎ。 丹羽中国大使の発言を巡ってメディアが「しっかりしろ」と喝を入れられたことなど、報道するわけもないか･･･。 posted at 21:34:50
なるほど〜。いかついパンダになりそう。 RT @shitakeo_cs137 : ネタにマジレスすると「尖」はJi?nと読み、「閣」はgeと読むので、「ジエンジエン」と「ガガ」ですかね。 posted at 22:25:39
例えば、佐藤‐テイト予想 英語版 en.wikipedia.org/wiki/Sato-Tate_conjecture Proofs and claims in progress On March 18, 2006, Richard Taylor of Harvard University announced on his web page the final step of a proof, joint with Laurent Clozel, Michael Harris, and Nicholas Shepherd-Barron, of the Sato?Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime.[4] Two of the three articles have since been published.[5] Further results are conditional on improved forms of the Arthur?Selberg trace formula. Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula. As of 8 July 2008, Richard Taylor has posted on his website an article (joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris) which claims to prove a generalized version of the Sato?Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris' "Book project"[8] and work of Sug Woo Shin.[9][10]
Generalisation There are generalisations, involving the distribution of Frobenius elements in Galois groups involved in the Galois representations on etale cohomology. In particular there is a conjectural theory for curves of genus n > 1. （略） More precise questions
英語版の方が情報量多いね en.wikipedia.org/wiki/Richard_Taylor_(mathematician) Personal life Taylor is married to Christine Taylor (a mathematical biologist). They have two children: Jeremy and Chloe. He is also the son of British physicist, John C. Taylor.
論文集のサイトがあるね www.math.ias.edu/~rtaylor/ Here are some recent papers. They are available either as dvi or as pdf files. They may be slightly different from the published versions, e.g. they may not include corrections made to the proofs. This work was partially supported by the NSF.
References [Cha] Ngo Bao Chau. Le lemme fondamental pour les algebres de Lie, arXiv:0801.0446 [GW] Sergei Gukov and Edward Witten. Gauge Theory, Rami?cation, And The Geometric Langlands Program, arXiv:hep-th/0612073 . [KW] Anton Kapustin and Edward Witten. Electric-Magnetic Duality And The Geometric Langlands Program, arXiv:hep-th/0604151 .
Kapustin と Witten の 論 文 によると , 2004 年 の Institute for Advanced Study の conference での Ben-Zvi の 講 演 が 鍵 にな っ たようである 。
References [Frea] Edward Frenkel. Gauge Theory and Langlands Duality, arXiv:0906.2747 . [Freb] Edward Frenkel. Lectures on the Langlands Program and Conformal Field Theory, arXiv:hep-th/0512172 . [GNO77] P. Goddard, J. Nuyts, and D. Olive. Gauge theories and magnetic charge. Nuclear Phys. B , 125(1):1?28, 1977. [KW] Anton Kapustin and Edward Witten. Electric-Magnetic Duality And The Geometric Langlands Program, arXiv:hep-th/0604151 . [MO77] C. Montonen and D. Olive. Magnetic monopoles as gauge particles? Physics Letters B , 72(1):117?120, December 1977. Updated on: Thu Nov 19 06:21:40 +0900 2009
Edward Witten Mirror Symmetry, Hitchin's Equations, And Langlands Duality "2. Mirror Symmetry And Hitchin’s Equations" "3. The Hitchin Fibration" "3.1. A Few Hints."・・・か これがどれだけNgo は Fundamental Lemmaに寄与したか不明だが Langlands DualityもMirror Symmetryに呑み込まれようとしている・・
arxiv.org/abs/0802.0999 Mirror Symmetry, Hitchin's Equations, And Langlands Duality Edward Witten (Submitted on 7 Feb 2008) Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding these statements is extremely difficult without picking a complex structure on $C$ and using Hitchin's equations. We sketch the essential statements both for the ``unramified'' case that $C$ is a compact oriented two-manifold without boundary, and the ``ramified'' case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.
Comments: 15 pp Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph) Cite as: arXiv:0802.0999v1 [math.RT] Submission history From: Edward Witten [view email] [v1] Thu, 7 Feb 2008 16:11:53 GMT (18kb)
>>83 訂正：これがどれだけNgo は Fundamental Lemmaに寄与したか→Ngoの Fundamental Lemmaの証明に
参考文献を見ると、Wittenのはないね。自分の2006年ころのFibration de Hitchin・・という論文がある
arxiv.org/abs/0801.0446 Le lemme fondamental pour les algebres de Lie Ngo Bao Chau (Submitted on 3 Jan 2008 (v1), last revised 2 May 2008 (this version, v3)) We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.
R´ef´erences [52] Laumon, G. et Ng?o B.C. : Le lemme fondamental pour les groupes unitaires,`a para??tre aux Annals of Math. [53] Matsumura, H. : Commutative ring theory.Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge,1989. [54] Mumford, D. Abelian varieties. Oxford University Press. [55] Ng?o B.C. : Fibration de Hitchin et endoscopie. Inv. Math. 164 (2006)399?453. [56] Ng?o B.C. : Fibration de Hitchin et structure endoscopique de la formule des traces. International Congress of Mathematicians Vol. II, 1213?1225, Eur. Math. Soc., Z¨urich, 2006. [57] Ng?o B.C. : Fibrations de Hitchin pour les groupes classiques. En pr´eparation. [58] Nitsure : Moduli space of semistable pairs on a curve. Pr
en.wikipedia.org/wiki/Nigel_Hitchin Nigel Hitchin (b. 2 August 1946 in Holbrook, Derbyshire) is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.
Academic career Hitchin attended Ecclesbourne School, Duffield, and earned his BA in mathematics from Jesus College, Oxford in 1968.[2] After moving to Wolfson College, he received his D.Phil. in 1972. In 1997 he was appointed to the Savilian Chair of Geometry at Oxford University, a position previously held by his doctoral supervisor (and later research collaborator) Sir Michael Atiyah.
Amongst his notable discoveries are the Hitchin integrable system, the Hitchin?Thorpe inequality, Hitchin's projectively flat connection over Teichmuller space, Hitchin's self-duality equations, the Atiyah?Hitchin monopole metric, the ADHM construction of instantons (of Atiyah, Drinfeld, Hitchin, and Manin), and the Hyperkahler quotient (of Hitchin, Karlhede, Lindstrom and Rocek).
In his article [3] on generalized Calabi?Yau manifolds, he introduced the notion of generalized complex manifolds, providing a single structure that incorporates, as examples, Poisson manifolds, symplectic manifolds and complex manifolds. These have found wide applications as the geometries of flux compactifications in string theory and also in topological string theory.
en.wikipedia.org/wiki/Fundamental_lemma_of_Langlands_and_Shelstad Fundamental lemma (Langlands program) In the theory of automorphic forms, an area of mathematics, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gerard Laumon and Ngo B?o Chau in the case of unitary groups and then by Ngo for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. Time magazine placed Ngo's proof on the list of the "Top 10 scientific discoveries of 2009".[1] In 2010 Ngo was awarded the Fields medal for this proof.
Motivation and history Robert Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur?Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way. This relationship takes the form of identities between orbital integrals on reductive groups G and H over a nonarchimedean local field F, where the group H, called an endoscopic group of G, is constructed from G and some additional data.
The first case considered was G = SL2 (Labesse & Langlands 1979). Langlands and Shelstad (1987) then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures. However, during the next two decades only partial progress was made towards proving the fundamental lemma.[2][3] Harris called it a "bottleneck limiting progress on a host of arithmetic questions".[4]
Dear community, In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands correspondence. In particular, in Teruyoshi Yoshida's thesis, one can see a glimpse of this for , 略 Sincerely, Moshe Adrian edited May 7 2011 at 3:13
Teruyoshi Yoshida responded to my question by e-mail and he is ok with my posting his response on mathoverflow : "Dear Moshe, thanks for your interest - yes it would be very interesting to do this with more general Rapoport-Zink spaces, but i) I haven't been successful in finding an intrinsic moduli interpretation of my model for tame Lubin-Tate space, hence the difficulty in generalizing to other groups ii) the so-called Drinfeld level structures do not seem to give nice integral models for the Rapoport-Zink spaces with deeper levels. In spite of these obstacles in arithmetic-geometry, it would be interesting to investigate the cohomology for other RZ spaces (there are works by Ito-Mieda, Shin, Strauch etc). Feel free to quote my email in mathoverflow.
