[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-43509-en":3,"doc-seo-43509-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},43509,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Quantum Yang Mills Theory - Arthur Jaffe and Edward Witten","Explores how quantum mechanics leads to field-based descriptions where classical particle trajectories fail and observables become noncommuting operators. Develops gauge theory from Maxwell’s abelian U(1) electromagnetism to non-abelian Yang–Mills theory with curvature modified by A∧A and nonlinear equations. Traces historical motivations to incorporate quantum fields, antimatter predictions, and explains how electroweak unification uses the Higgs mechanism to give short-range weak interactions while preserving long-range electromagnetism.","QUANTUM YANG{MILLS THEORY  \nARTHUR JAFFE AND EDWARD WITTEN  \n1. The Physics of Gauge Theory  \nSince the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics. In quantum mechanics, the position and velocity of a particle are noncommuting operators acting on a Hilbert space, and classical notions such as \\the trajectory of a particle\" do not apply.  \nBut quantum mechanics of particles is not the whole story. In 19th and early 20th century physics, many aspects of nature were described in terms of 􀀌elds|the electric and magnetic 􀀌elds that enter in Maxwell's equations, and the gravitational 􀀌eld governed by Einstein's equations. Since 􀀌elds interact with particles, it became clear by the late 1920s that an internally coherent account of nature must incorporate quantum concepts for 􀀌elds as well as for particles.  \nAfter doing this, quantities such as the components of the electric 􀀌eld at di􀀋erent points in space-time become non-commuting operators. When one attempts to construct a Hilbert space on which these operators act, one 􀀌nds many surprises. The distinction between 􀀌elds and particles breaks down, since the Hilbert space of a quantum 􀀌eld is constructed in terms of particle-like excitations. Conventional particles, such as electrons, are reinterpreted as states of the quantized 􀀌eld. In the process, one 􀀌nds the prediction of \\antimatter\"; for every particle, there must be a corresponding antiparticle, with the same mass and opposite electric charge. Soon after P.A.M. Dirac predicted this on the basis of quantum 􀀌eld theory, the \\positron\" or oppositely charged antiparticle of the electron was discovered in cosmic rays.  \nThe most important Quantum Field Theories (QFTs) for describing elementary particle physics are gauge theories. The classical example of a gauge theory is Maxwell's theory of electromagnetism. For electromagnetism the gauge symmetry group is the abelian group U (1) . If A denotes the U (1) gauge connection, locally a one-form on space-time, then the curvature or electromagnetic 􀀌eld tensor is the two-form F = dA, and Maxwell's equations in the absence of charges and currents read 0 = dF = d 􀀃 F. Here 􀀃 denotes the Hodge duality operator; indeed, Hodge introduced his celebrated theory of harmonic forms as a generalization of the solutions to Maxwell's equations. Maxwell's equations describe large-scale electric and magnetic 􀀌elds and also|as Maxwell discovered|the propagation of light waves, at a characteristic velocity, the speed of light.  \nThe idea of a gauge theory evolved from the work of Hermann Weyl. One can 􀀌nd in [34] an interesting discussion of the history of gauge symmetry and the discovery of Yang{Mills theory [50], also known as \\non-abelian gauge theory.\"At the classical level one replaces the gauge group U (1) of electromagnetism by a compact gauge group G. The de􀀌nition of the curvature arising from the connection  \n2 ARTHUR JAFFE AND EDWARD WITTEN  \nmust be modi􀀌ed to F = dA + A ^ A, and Maxwell's equations are replaced by the Yang{Mills equations, 0 = dA F = dA 􀀃 F , where dA is the gauge-covariant extension of the exterior derivative.  \nThese classical equations can be derived as variational equations from the Yang{ Mills Lagrangian  \n(1) L = ~~4~~1g2 Z Tr F ^ 􀀃F;  \nwhere Tr denotes an invariant quadratic form on the Lie algebra of G. The Yang{ Mills equations are nonlinear|in contrast to the Maxwell equations. Like the Einstein equations for the gravitational 􀀌eld, only a few exact solutions of the classical equation are known. But the Yang{Mills equations have certain properties in common with the Maxwell equations: In particular they provide the classical description of massless waves that travel at the speed of light.  \nBy the 1950s, when Yang{Mills theory was discovered, it was already known that the quantum version of Maxwell theory|known as Quantum Electrodynamics or QED|gives an extremely accurat","cbCaifRi2eivx3To","https://ap.wps.com/l/cbCaifRi2eivx3To","pdf",195976,5,1,14,"English","en",105,"# The Physics of Gauge Theory\n## From quantum mechanics to quantum fields\n## Yang–Mills theory as a non-abelian gauge theory\n## From classical waves to quantum interactions\n## Electroweak unification and the Higgs mechanism","[{\"question\":\"Why does classical particle language break down in quantum mechanics and quantum field theory?\",\"answer\":\"In quantum mechanics, position and velocity become noncommuting operators rather than well-defined trajectories. In quantum field theory, states are built from particle-like excitations of the fields, blurring the particle/field distinction.\"},{\"question\":\"How does Yang–Mills theory generalize Maxwell’s gauge theory?\",\"answer\":\"Maxwell’s theory uses an abelian U(1) gauge symmetry with curvature F=dA. Yang–Mills theory replaces U(1) with a compact gauge group G, modifies the curvature to F=dA + A∧A, and yields nonlinear Yang–Mills equations.\"},{\"question\":\"How does the Higgs mechanism address the issue of massless gauge bosons in the weak force?\",\"answer\":\"By adding a Higgs field, the vacuum expectation value reduces the gauge structure group H=SU(2)×U(1) down to a U(1) subgroup. This avoids long-range massless Yang–Mills behavior for the weak interaction while leaving long-range electromagnetism.\"}]",1783381925,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"quantum-yang-mills-theory-arthur-jaffe-and-edward-witten","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quantum-yang-mills-theory-arthur-jaffe-and-edward-witten/43509/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-20","2026-07-06",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why does classical particle language break down in quantum mechanics and quantum field theory?","Question",{"text":76,"@type":77},"In quantum mechanics, position and velocity become noncommuting operators rather than well-defined trajectories. In quantum field theory, states are built from particle-like excitations of the fields, blurring the particle/field distinction.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does Yang–Mills theory generalize Maxwell’s gauge theory?",{"text":81,"@type":77},"Maxwell’s theory uses an abelian U(1) gauge symmetry with curvature F=dA. Yang–Mills theory replaces U(1) with a compact gauge group G, modifies the curvature to F=dA + A∧A, and yields nonlinear Yang–Mills equations.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the Higgs mechanism address the issue of massless gauge bosons in the weak force?",{"text":85,"@type":77},"By adding a Higgs field, the vacuum expectation value reduces the gauge structure group H=SU(2)×U(1) down to a U(1) subgroup. This avoids long-range massless Yang–Mills behavior for the weak interaction while leaving long-range electromagnetism.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]