The Algebra of Grand Unified Theories

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1、TheAlgebraofGrandUnifiedTheoriesTheAlgebraofGrandUnifiedTheoriesJohnHuertaDepartmentofMathematicsUCRiversideOralExamPresentationTheAlgebraofGrandUnifiedTheoriesIntroductionThistalkisanintroductiontotherepresentationtheoryusedinITheStandardModelofParticlePhysics(SM);ICertainext

2、ensionsoftheSM,calledGrandUnifiedTheories(GUTs).TheAlgebraofGrandUnifiedTheoriesIntroductionThere’salotIwon’ttalkabout:Iquantumfieldtheory;Ispontaneoussymmetrybreaking;Ianysortofdynamics.Thisstuffisessentialtoparticlephysics.WhatIdiscusshereisjustonesmallpiece.TheAlgebraofGran

3、dUnifiedTheoriesIntroductionThere’saloosecorrespondencebetweenparticlephysicsandrepresentationtheory:TheAlgebraofGrandUnifiedTheoriesIntroductionThere’saloosecorrespondencebetweenparticlephysicsandrepresentationtheory:IParticles→basisvectorsinarepresentationVofaLiegroupG.ICla

4、ssificationofparticles→decompositionintoirreps.IUnification→G,→H;particlesare“unified”intofewerirreps.IGrandUnification→asabove,butHissimple.ITheStandardModel→aparticularrepresentationVofSMaparticularLiegroupG.SMTheAlgebraofGrandUnifiedTheoriesTheStandardModelTheGroupTheStandard

5、ModelgroupisG=U(1)×SU(2)×SU(3)SMTheAlgebraofGrandUnifiedTheoriesTheStandardModelTheGroupTheStandardModelgroupisG=U(1)×SU(2)×SU(3)SMIThefactorU(1)×SU(2)correspondstotheelectroweakforce.Itrepresentsaunificationofelectromagnetismandtheweakforce.TheAlgebraofGrandUnifiedTheoriesThe

6、StandardModelTheGroupTheStandardModelgroupisG=U(1)×SU(2)×SU(3)SMIThefactorU(1)×SU(2)correspondstotheelectroweakforce.Itrepresentsaunificationofelectromagnetismandtheweakforce.ISpontaneoussymmetrybreakingmakestheelectromagneticandweakforceslookdifferent;athighenergies,they’re

7、thesame.TheAlgebraofGrandUnifiedTheoriesTheStandardModelTheGroupTheStandardModelgroupisG=U(1)×SU(2)×SU(3)SMIThefactorU(1)×SU(2)correspondstotheelectroweakforce.Itrepresentsaunificationofelectromagnetismandtheweakforce.ISpontaneoussymmetrybreakingmakestheelectromagneticandweak

8、forceslookdifferent;athighenergies,they’rethesame.ISU(3)correspondstothestrongforc