B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




           B EAUTY IN THE B REAKDOWN : S YMMETRY IN
                ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY ON THE SURFACE OF THE D EAD S EA




               ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY ON THE SURFACE OF THE D EAD S EA




          1   T...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY ON THE SURFACE OF THE D EAD S EA




          1   T...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M AKING R IPPLES – BREAKING SOME SYMMETRIES
      AND CREATING...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M AKING R IPPLES – BREAKING SOME SYMMETRIES
      AND CREATING...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M AKING R IPPLES – BREAKING SOME SYMMETRIES
      AND CREATING...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M AKING R IPPLES – BREAKING SOME SYMMETRIES
      AND CREATING...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M AKING R IPPLES – BREAKING SOME SYMMETRIES
      AND CREATING...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE S YMMETRIES OF E QUILATERAL T RIANGLES




              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R120 × RII = RI




                                         Z...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R120 × RII = RI




                                         Z...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      RII × R120 = RIII




                                        ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      RII × R120 = RIII




                                        ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE OPERATIONS DO NOT COMMUTE !




                         ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE OPERATIONS DO NOT COMMUTE !




       R120 × RII = RI


...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE OPERATIONS DO NOT COMMUTE !




       R120 × RII = RI


...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE OPERATIONS DO NOT COMMUTE !




       R120 × RII = RI   ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      M ULTIPLICATION TABLE FOR S3



                              ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY OF THE C IRCLE




                                 ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY OF THE C IRCLE




      Rotate by any arbitrary ang...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY O PERATIONS ON THE C IRCLE


              Define thi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY O PERATIONS ON THE C IRCLE


              Define thi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY O PERATIONS ON THE C IRCLE


              Define thi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY O PERATIONS ON THE C IRCLE


              Define thi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY O PERATIONS ON THE C IRCLE


              Define thi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G ROUPS



      A group G is essentially a set with a composi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G ROUPS



      A group G is essentially a set with a composi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G ROUPS



      A group G is essentially a set with a composi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G ROUPS



      A group G is essentially a set with a composi...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ISTINGUISHING F EATURE OF G ROUPS :
      C OMMUTATIVITY



...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ISTINGUISHING F EATURE OF G ROUPS :
      C OMMUTATIVITY



...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ISTINGUISHING F EATURE OF G ROUPS :
      C OMMUTATIVITY



...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS


            ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ISCRETE VS . C ONTINUOUS G ROUPS




              Discrete:...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ISCRETE VS . C ONTINUOUS G ROUPS




              Discrete:...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ESCRIBING C ONTINUOUS G ROUPS : U(1)




              U(1)
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ESCRIBING C ONTINUOUS G ROUPS : U(1)




              U(1)–...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ESCRIBING C ONTINUOUS G ROUPS : U(1)




              U(1)–...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ESCRIBING C ONTINUOUS G ROUPS : U(1)




              U(1)–...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      D ESCRIBING C ONTINUOUS G ROUPS : U(1)




              U(1)–...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R EPRESENTATIONS




              A representation of a group...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R EPRESENTATIONS




              A representation of a group...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R EPRESENTATIONS




              A representation of a group...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R EPRESENTATIONS




              A representation of a group...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                               ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R(θ): A REPRESENTATION OF U(1)
                       cos(θ) s...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      L IE G ROUPS
              Most of the continuous groups that ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      U NITARY G ENERATORS


              But we need a unitary gro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      U NITARY G ENERATORS


              But we need a unitary gro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      U NITARY G ENERATORS


              But we need a unitary gro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      U NITARY G ENERATORS


              But we need a unitary gro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      U NITARY G ENERATORS


              But we need a unitary gro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE G ENERATORS OF THE 2D ROTATION G ROUP



              Ou...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE G ENERATORS OF THE 2D ROTATION G ROUP



              Ou...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE G ENERATORS OF THE 2D ROTATION G ROUP



              Ou...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE G ENERATORS OF THE 2D ROTATION G ROUP



              Ou...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Ancient Greeks...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Ancient Greeks...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Ancient Greeks...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Ancient Greeks...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Copernicus: He...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Copernicus: He...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Copernicus: He...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Kepler: Believ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Kepler: Believ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS
              Kepler: Believ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS

                           ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS

                           ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS

                           ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS

                           ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS


              Mid-19th cen...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS


              Mid-19th cen...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS



              The idea of...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS



              The idea of...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS



              The idea of...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS



              The idea of...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS



              The idea of...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      N OETHER ’ S T HEOREM


              Using the Euler-Lagrange...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      N OETHER ’ S T HEOREM


              Using the Euler-Lagrange...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      N OETHER ’ S T HEOREM



              Connection between glob...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      N OETHER ’ S T HEOREM



              Connection between glob...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH
      C ENTURY
...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R EST OF 20 TH C ENTURY H ISTORY OF S YMMETRY:
      I T ’ S A...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE T HREE K EY D ISCRETE S YMMETRIES



              Global...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE T HREE K EY D ISCRETE S YMMETRIES



              Global...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE T HREE K EY D ISCRETE S YMMETRIES



              Global...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE T HREE K EY D ISCRETE S YMMETRIES



              Global...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE T HREE K EY D ISCRETE S YMMETRIES




                   ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE M AJOR I NTERNAL S YMMETRIES



         Gauge Transforma...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      G AUGE B OSONS AND L OCAL S YMMETRIES

              The QED L...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      W HAT U(1) INVARIANCE LOOKS LIKE

              Invariance und...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      I S PARITY CONSERVED ?


              All of the fundamental ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      I S PARITY CONSERVED ?


              All of the fundamental ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      I S PARITY CONSERVED ?


              All of the fundamental ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      I S PARITY CONSERVED ?


              All of the fundamental ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      I S PARITY CONSERVED ?


              All of the fundamental ...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      PARITY IS NOT CONSERVED



              1956—Yang and Lee pro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      PARITY IS NOT CONSERVED



              1956—Yang and Lee pro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      PARITY IS NOT CONSERVED



              1956—Yang and Lee pro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      PARITY IS NOT CONSERVED



              1956—Yang and Lee pro...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R ESULT OF G ARWIN , L EDERMAN , W EINRICH
      EXPERIMENT


...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      R ESULT OF G ARWIN , L EDERMAN , W EINRICH
      EXPERIMENT


...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T IME R EVERSAL S YMMETRY




              Laws of physics in...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T IME R EVERSAL S YMMETRY




              Laws of physics in...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T IME R EVERSAL S YMMETRY




              Laws of physics in...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C HARGE C ONJUGATION



              Dirac’s prediction and C...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C HARGE C ONJUGATION



              Dirac’s prediction and C...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C HARGE C ONJUGATION



              Dirac’s prediction and C...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C HARGE C ONJUGATION



              Dirac’s prediction and C...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE TEST OF C INVARIANCE



              The test of C invar...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE TEST OF C INVARIANCE



              The test of C invar...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE TEST OF C INVARIANCE



              The test of C invar...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      CP AND ITS VIOLATION




              1957—The µ+ found to ha...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      CP AND ITS VIOLATION




              1957—The µ+ found to ha...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      CP AND ITS VIOLATION




              1957—The µ+ found to ha...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      CP AND ITS VIOLATION




              1957—The µ+ found to ha...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      CP AND ITS VIOLATION




              1957—The µ+ found to ha...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT



              F...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT



              F...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT



              F...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT



              F...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      O UTLINE OF TALK



          1   Introduction: Examples of Sy...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY BREAKING




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B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      S YMMETRY BREAKING




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B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
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B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
      S YMMETR...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
      S YMMETR...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
      S YMMETR...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN
      S YMMETR...
B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS




      H IGGS F IELD




                                         Z A...
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  1. 1. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS Zach McElrath Covenant College April 29, 2010 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  2. 2. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  3. 3. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  4. 4. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY ON THE SURFACE OF THE D EAD S EA Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  5. 5. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY ON THE SURFACE OF THE D EAD S EA 1 Translation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  6. 6. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY ON THE SURFACE OF THE D EAD S EA 1 Translation 2 Rotation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  7. 7. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M AKING R IPPLES – BREAKING SOME SYMMETRIES AND CREATING OTHERS Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  8. 8. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M AKING R IPPLES – BREAKING SOME SYMMETRIES AND CREATING OTHERS 1 Translational symmetry broken Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  9. 9. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M AKING R IPPLES – BREAKING SOME SYMMETRIES AND CREATING OTHERS 1 Translational symmetry broken 2 Rotational symmetry about the center of the ripple Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  10. 10. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M AKING R IPPLES – BREAKING SOME SYMMETRIES AND CREATING OTHERS 1 Translational symmetry broken 2 Rotational symmetry about the center of the ripple 3 How about 2 ripples? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  11. 11. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M AKING R IPPLES – BREAKING SOME SYMMETRIES AND CREATING OTHERS 1 Translational symmetry broken 2 Rotational symmetry about the center of the ripple 3 How about 2 ripples? 4 How about 3 ripples? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  12. 12. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  13. 13. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  14. 14. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Rotation by 120◦ (R120 ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  15. 15. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Rotation by 120◦ (R120 ) Rotation by 240◦ (R240 ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  16. 16. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Rotation by 120◦ (R120 ) Rotation by 240◦ (R240 ) Reflection axis I (RI ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  17. 17. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Rotation by 120◦ (R120 ) Rotation by 240◦ (R240 ) Reflection axis I (RI ) Reflection axis II (RII ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  18. 18. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE S YMMETRIES OF E QUILATERAL T RIANGLES Six Symmetry “Operations” “Identity” (1) Rotation by 120◦ (R120 ) Rotation by 240◦ (R240 ) Reflection axis I (RI ) Reflection axis II (RII ) Reflection axis III (RIII ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  19. 19. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R120 × RII = RI Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  20. 20. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R120 × RII = RI Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  21. 21. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS RII × R120 = RIII Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  22. 22. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS RII × R120 = RIII Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  23. 23. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE OPERATIONS DO NOT COMMUTE ! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  24. 24. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE OPERATIONS DO NOT COMMUTE ! R120 × RII = RI Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  25. 25. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE OPERATIONS DO NOT COMMUTE ! R120 × RII = RI Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  26. 26. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE OPERATIONS DO NOT COMMUTE ! R120 × RII = RI RII × R120 = RIII Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  27. 27. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R240 RI RIII 1 1 R120 R240 RI RII RIII R120 R240 1 RIII RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  28. 28. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R240 RI RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  29. 29. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R240 RI RII RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  30. 30. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R240 RI RII RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RI RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  31. 31. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R240 RI RII RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RI RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  32. 32. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R120 R240 RI RII RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RI RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  33. 33. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS M ULTIPLICATION TABLE FOR S3 1 R120 R240 RI RII RIII 1 1 R120 R240 RI RII RIII R120 R120 R240 1 RIII RI RII R240 R240 1 R120 RII RIII RI RI RI RII RIII 1 R120 R240 RII RII RIII RI R240 1 R120 RIII RIII RI RII R120 R240 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  34. 34. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY OF THE C IRCLE Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  35. 35. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY OF THE C IRCLE Rotate by any arbitrary angle θ ∈ R, and the circle still looks the same Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  36. 36. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY O PERATIONS ON THE C IRCLE Define this rotational symmetry operation as R(θ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  37. 37. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY O PERATIONS ON THE C IRCLE Define this rotational symmetry operation as R(θ) The set of possible symmetry operations on a circle forms a continuous symmetry group Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  38. 38. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY O PERATIONS ON THE C IRCLE Define this rotational symmetry operation as R(θ) The set of possible symmetry operations on a circle forms a continuous symmetry group Continuous symmetry groups: infinite number of elements, infinitesimal variation in parameters (i.e. θ). Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  39. 39. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY O PERATIONS ON THE C IRCLE Define this rotational symmetry operation as R(θ) The set of possible symmetry operations on a circle forms a continuous symmetry group Continuous symmetry groups: infinite number of elements, infinitesimal variation in parameters (i.e. θ). Impossible to construct a multiplication table as we did for the discrete group S3 , which involves discrete steps and has no infinitesimal operations. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  40. 40. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY O PERATIONS ON THE C IRCLE Define this rotational symmetry operation as R(θ) The set of possible symmetry operations on a circle forms a continuous symmetry group Continuous symmetry groups: infinite number of elements, infinitesimal variation in parameters (i.e. θ). Impossible to construct a multiplication table as we did for the discrete group S3 , which involves discrete steps and has no infinitesimal operations. Need a different way to describe all these operations Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  41. 41. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  42. 42. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G ROUPS A group G is essentially a set with a composition rule a · b, i.e. if a, b ∈ G, then a · b ∈ G. There are three other defining characteristics of groups: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  43. 43. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G ROUPS A group G is essentially a set with a composition rule a · b, i.e. if a, b ∈ G, then a · b ∈ G. There are three other defining characteristics of groups: Associativity: a(bc) = (ab)c Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  44. 44. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G ROUPS A group G is essentially a set with a composition rule a · b, i.e. if a, b ∈ G, then a · b ∈ G. There are three other defining characteristics of groups: Associativity: a(bc) = (ab)c The group has an identity element e s.t. ae = ea = a Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  45. 45. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G ROUPS A group G is essentially a set with a composition rule a · b, i.e. if a, b ∈ G, then a · b ∈ G. There are three other defining characteristics of groups: Associativity: a(bc) = (ab)c The group has an identity element e s.t. ae = ea = a The group has an inverse : ∀a ∈ G, ∃ a−1 ∈ G s.t. aa−1 = a−1 a = e Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  46. 46. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ISTINGUISHING F EATURE OF G ROUPS : C OMMUTATIVITY Key Distinguishing Feature: are the elements of the group (in our case, the symmetry operations) commutative? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  47. 47. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ISTINGUISHING F EATURE OF G ROUPS : C OMMUTATIVITY Key Distinguishing Feature: are the elements of the group (in our case, the symmetry operations) commutative? i.e. is a · b = b · a? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  48. 48. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ISTINGUISHING F EATURE OF G ROUPS : C OMMUTATIVITY Key Distinguishing Feature: are the elements of the group (in our case, the symmetry operations) commutative? i.e. is a · b = b · a? If the elements of the group are commutative, the group is called abelian. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  49. 49. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  50. 50. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  51. 51. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  52. 52. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer (i.e. 2 + 3 = 5) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  53. 53. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer (i.e. 2 + 3 = 5) Identity element: 0 (because 19 + 0 = 19) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  54. 54. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer (i.e. 2 + 3 = 5) Identity element: 0 (because 19 + 0 = 19) Inverse element: positive/negative (i.e. 23 + (−23) = 0) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  55. 55. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer (i.e. 2 + 3 = 5) Identity element: 0 (because 19 + 0 = 19) Inverse element: positive/negative (i.e. 23 + (−23) = 0) Associativity is satisfied, i.e. 3 + (5 + 7) = 15 = (3 + 5) + 7. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  56. 56. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS E XAMPLE OF AN A BELIAN G ROUP : T HE I NTEGERS The set of integers form a group under the operation of addition Take + as our composition rule Closure: For any elements a, b of the set of integers, a + b will yield another integer (i.e. 2 + 3 = 5) Identity element: 0 (because 19 + 0 = 19) Inverse element: positive/negative (i.e. 23 + (−23) = 0) Associativity is satisfied, i.e. 3 + (5 + 7) = 15 = (3 + 5) + 7. Addition is commutative (2 + 3 = 3 + 2 = 5), so the set of integers forms an abelian group under addition. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  57. 57. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ISCRETE VS . C ONTINUOUS G ROUPS Discrete: Equilateral Triangle (S3 )–finite number of distinguishable symmetry operations–6. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  58. 58. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ISCRETE VS . C ONTINUOUS G ROUPS Discrete: Equilateral Triangle (S3 )–finite number of distinguishable symmetry operations–6. Continous: Circle (Rotations in the 2D plane)–infinite number of possible operations Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  59. 59. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ESCRIBING C ONTINUOUS G ROUPS : U(1) U(1) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  60. 60. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ESCRIBING C ONTINUOUS G ROUPS : U(1) U(1)–(extremely important in physics) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  61. 61. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ESCRIBING C ONTINUOUS G ROUPS : U(1) U(1)–(extremely important in physics) The group containing all complex numbers with length 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  62. 62. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ESCRIBING C ONTINUOUS G ROUPS : U(1) U(1)–(extremely important in physics) The group containing all complex numbers with length 1 A subgroup of U(N), the group of N × N unitary matrices Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  63. 63. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS D ESCRIBING C ONTINUOUS G ROUPS : U(1) U(1)–(extremely important in physics) The group containing all complex numbers with length 1 A subgroup of U(N), the group of N × N unitary matrices Unitary Matrices: satisfy U † U = UU † = IN , where IN is the N × N identity matrix Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  64. 64. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R EPRESENTATIONS A representation of a group is a mapping which takes the elements a, b ∈ G into linear operators F that preserve the composition rule of the group Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  65. 65. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R EPRESENTATIONS A representation of a group is a mapping which takes the elements a, b ∈ G into linear operators F that preserve the composition rule of the group That is, F (a)F (b) = F (ab) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  66. 66. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R EPRESENTATIONS A representation of a group is a mapping which takes the elements a, b ∈ G into linear operators F that preserve the composition rule of the group That is, F (a)F (b) = F (ab) So, what’s the most basic representation of U(1)? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  67. 67. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R EPRESENTATIONS A representation of a group is a mapping which takes the elements a, b ∈ G into linear operators F that preserve the composition rule of the group That is, F (a)F (b) = F (ab) So, what’s the most basic representation of U(1)? eiθ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  68. 68. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  69. 69. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  70. 70. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  71. 71. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). For R(θ) to be a representation of U(1), it too must satisfy the properties of U(1) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  72. 72. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). For R(θ) to be a representation of U(1), it too must satisfy the properties of U(1) Unitarity: R(θ)† R(θ) = R(θ)R(−θ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  73. 73. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). For R(θ) to be a representation of U(1), it too must satisfy the properties of U(1) Unitarity: R(θ)† R(θ) = R(θ)R(−θ) = cos(θ) sin(θ) cos(θ) − sin(θ) − sin(θ) cos(θ) sin(θ) cos(θ) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  74. 74. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). For R(θ) to be a representation of U(1), it too must satisfy the properties of U(1) Unitarity: R(θ)† R(θ) = R(θ)R(−θ) = cos(θ) sin(θ) cos(θ) − sin(θ) = − sin(θ) cos(θ) sin(θ) cos(θ) cos2 θ + sin2 θ sin θ cos θ − sin θ cos θ sin θ cos θ − sin θ cos θ cos2 θ + sin2 θ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  75. 75. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R(θ): A REPRESENTATION OF U(1) cos(θ) sin(θ) R(θ) = − sin(θ) cos(θ) We can show that R(θ) is a group i.e. Composition rule satisfied: R(θ1 )R(θ2 ) = R(θ1 + θ2 ). For R(θ) to be a representation of U(1), it too must satisfy the properties of U(1) Unitarity: R(θ)† R(θ) = R(θ)R(−θ) = cos(θ) sin(θ) cos(θ) − sin(θ) = − sin(θ) cos(θ) sin(θ) cos(θ) cos2 θ + sin2 θ sin θ cos θ − sin θ cos θ = sin θ cos θ − sin θ cos θ cos2 θ + sin2 θ 1 0 0 1 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  76. 76. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  77. 77. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. (1) Elements of the group depend on a finite set of continuous parameters θ1 , θ2 , · · · , θn Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  78. 78. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. (1) Elements of the group depend on a finite set of continuous parameters θ1 , θ2 , · · · , θn (2) Derivatives of the group elements with respect to all of the group parameters exist. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  79. 79. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. (1) Elements of the group depend on a finite set of continuous parameters θ1 , θ2 , · · · , θn (2) Derivatives of the group elements with respect to all of the group parameters exist. Finding the Generators of a group representation–Take derivatives of the representation of the Lie group with respect to each of its parameters, and evaluate the derivative at zero Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  80. 80. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. (1) Elements of the group depend on a finite set of continuous parameters θ1 , θ2 , · · · , θn (2) Derivatives of the group elements with respect to all of the group parameters exist. Finding the Generators of a group representation–Take derivatives of the representation of the Lie group with respect to each of its parameters, and evaluate the derivative at zero If a group representation has n parameters, then there are n generators Xi of that representation, given by Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  81. 81. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS L IE G ROUPS Most of the continuous groups that turn up in studies of symmetries in physics, including U(1), are Lie groups. (1) Elements of the group depend on a finite set of continuous parameters θ1 , θ2 , · · · , θn (2) Derivatives of the group elements with respect to all of the group parameters exist. Finding the Generators of a group representation–Take derivatives of the representation of the Lie group with respect to each of its parameters, and evaluate the derivative at zero If a group representation has n parameters, then there are n generators Xi of that representation, given by ∂g Xi = ∂θ θi =0 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  82. 82. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS U NITARY G ENERATORS But we need a unitary group representation for our purposes, because U(1) is unitary. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  83. 83. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS U NITARY G ENERATORS But we need a unitary group representation for our purposes, because U(1) is unitary. To do this, we must have Hermitian generators Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  84. 84. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS U NITARY G ENERATORS But we need a unitary group representation for our purposes, because U(1) is unitary. To do this, we must have Hermitian generators How do you make a quantity involving a derivative Hermitian? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  85. 85. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS U NITARY G ENERATORS But we need a unitary group representation for our purposes, because U(1) is unitary. To do this, we must have Hermitian generators How do you make a quantity involving a derivative Hermitian? Multiply the generators by −i: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  86. 86. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS U NITARY G ENERATORS But we need a unitary group representation for our purposes, because U(1) is unitary. To do this, we must have Hermitian generators How do you make a quantity involving a derivative Hermitian? Multiply the generators by −i: ∂g Xi = −i ∂θ θi =0 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  87. 87. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE G ENERATORS OF THE 2D ROTATION G ROUP Our 2-D rotation group representation of U(1) has only one parameter, θ, so it has only one generator: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  88. 88. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE G ENERATORS OF THE 2D ROTATION G ROUP Our 2-D rotation group representation of U(1) has only one parameter, θ, so it has only one generator: ∂ cos(θ) sin(θ) X (θ) = −i ∂θ − sin(θ) cos(θ) θ=0 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  89. 89. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE G ENERATORS OF THE 2D ROTATION G ROUP Our 2-D rotation group representation of U(1) has only one parameter, θ, so it has only one generator: ∂ cos(θ) sin(θ) 0 −i X (θ) = −i = ∂θ − sin(θ) cos(θ) θ=0 i 0 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  90. 90. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE G ENERATORS OF THE 2D ROTATION G ROUP Our 2-D rotation group representation of U(1) has only one parameter, θ, so it has only one generator: ∂ cos(θ) sin(θ) 0 −i X (θ) = −i = ∂θ − sin(θ) cos(θ) θ=0 i 0 Lookie, lookie, its one of the Pauli Matrices! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  91. 91. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  92. 92. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Ancient Greeks: Regular polyhedra celebrated for their defining proportionality relationships, regular solids because they were recognized to be symmetric in the sense of their invariance under geometric operations Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  93. 93. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Ancient Greeks: Regular polyhedra celebrated for their defining proportionality relationships, regular solids because they were recognized to be symmetric in the sense of their invariance under geometric operations Aristotle: Heavenly bodies perfect spheres, orbits circular Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  94. 94. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Ancient Greeks: Regular polyhedra celebrated for their defining proportionality relationships, regular solids because they were recognized to be symmetric in the sense of their invariance under geometric operations Aristotle: Heavenly bodies perfect spheres, orbits circular Ptolemy: Theory of epicycles, dominant for 1500 years, grounded in supposition that the circle must characterize celestial orbits Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  95. 95. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Ancient Greeks: Regular polyhedra celebrated for their defining proportionality relationships, regular solids because they were recognized to be symmetric in the sense of their invariance under geometric operations Aristotle: Heavenly bodies perfect spheres, orbits circular Ptolemy: Theory of epicycles, dominant for 1500 years, grounded in supposition that the circle must characterize celestial orbits Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  96. 96. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Copernicus: Heliocentric model shattered Ptolemaic paradigm Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  97. 97. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Copernicus: Heliocentric model shattered Ptolemaic paradigm But it still preserved the cherished symmetry of the circular celestial orbits Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  98. 98. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Copernicus: Heliocentric model shattered Ptolemaic paradigm But it still preserved the cherished symmetry of the circular celestial orbits Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  99. 99. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Kepler: Believed that if he could find out where the predictions of Copernicus’ theory diverged from Tycho Brahe’s excellent data, he would “discover a magnificent new symmetry.” Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  100. 100. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Kepler: Believed that if he could find out where the predictions of Copernicus’ theory diverged from Tycho Brahe’s excellent data, he would “discover a magnificent new symmetry.” By abandoning the cherished theory of circular orbits, Kepler traded an approximate symmetry for a deeper symmetry associated with the conservation of angular momentum Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  101. 101. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Kepler: Believed that if he could find out where the predictions of Copernicus’ theory diverged from Tycho Brahe’s excellent data, he would “discover a magnificent new symmetry.” By abandoning the cherished theory of circular orbits, Kepler traded an approximate symmetry for a deeper symmetry associated with the conservation of angular momentum Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  102. 102. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS ´ Studies of Crystal Structure by Rene Hauy in 1801: the ¨ symmetry of a geometric figure redefined as its “invariance...when equal component parts are exchanged according to one of the specified operations.” Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  103. 103. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS ´ Studies of Crystal Structure by Rene Hauy in 1801: the ¨ symmetry of a geometric figure redefined as its “invariance...when equal component parts are exchanged according to one of the specified operations.” 1st turning point in history of physics–19th century: The idea of symmetry as invariance generalized to algebraic structure of groups Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  104. 104. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS ´ Studies of Crystal Structure by Rene Hauy in 1801: the ¨ symmetry of a geometric figure redefined as its “invariance...when equal component parts are exchanged according to one of the specified operations.” 1st turning point in history of physics–19th century: The idea of symmetry as invariance generalized to algebraic structure of groups The set of symmetry operations applicable to a given geometric figure satisfies the conditions for a group Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  105. 105. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS ´ Studies of Crystal Structure by Rene Hauy in 1801: the ¨ symmetry of a geometric figure redefined as its “invariance...when equal component parts are exchanged according to one of the specified operations.” 1st turning point in history of physics–19th century: The idea of symmetry as invariance generalized to algebraic structure of groups The set of symmetry operations applicable to a given geometric figure satisfies the conditions for a group In both its ancient and modern formulations, then, symmetry is associated with a unity of parts which are equal “with respect to the whole in the sense of their interchangeability.” Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  106. 106. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Mid-19th century–Jacobi develops a method of solving dynamical equations formulated using Hamilton’s canonical variables by applying transformations of these variables which leave the equations invariant Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  107. 107. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS Mid-19th century–Jacobi develops a method of solving dynamical equations formulated using Hamilton’s canonical variables by applying transformations of these variables which leave the equations invariant This spawned a slew of research on how transformations affect physical theories, culminating in the study of the connection between the invariance of observable physical quantities, such as momentum, with the algebraic and geometric theory of invariants. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  108. 108. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS The idea of the classical “action” S Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  109. 109. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS The idea of the classical “action” S = ˙ L(q, q, t)dt Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  110. 110. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS The idea of the classical “action” S = ˙ L(q, q, t)dt Hamilton’s Principle: The path that a particle actually takes is the one that results in δS = 0 (at least to first-order) Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  111. 111. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS The idea of the classical “action” S = ˙ L(q, q, t)dt Hamilton’s Principle: The path that a particle actually takes is the one that results in δS = 0 (at least to first-order) This results in the Euler-Lagrange Equations: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  112. 112. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS The idea of the classical “action” S = ˙ L(q, q, t)dt Hamilton’s Principle: The path that a particle actually takes is the one that results in δS = 0 (at least to first-order) This results in the Euler-Lagrange Equations: d ∂L ∂L = . dt ˙ ∂q ∂q Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  113. 113. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS N OETHER ’ S T HEOREM Using the Euler-Lagrange Equations, it can be shown that IF a small transformation qk → qk + δqk leaves S invariant, then there is an associated conserved quantity X , such that dX =0 dt Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  114. 114. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS N OETHER ’ S T HEOREM Using the Euler-Lagrange Equations, it can be shown that IF a small transformation qk → qk + δqk leaves S invariant, then there is an associated conserved quantity X , such that dX =0 dt Emmy Noether rigorously generalized this idea–if a transformation of one of the quantities in the action leaves its form invariant, then there is a conserved quantity associated with that symmetry in the physical system Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  115. 115. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS N OETHER ’ S T HEOREM Connection between global symmetries and conservation principles Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  116. 116. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS N OETHER ’ S T HEOREM Connection between global symmetries and conservation principles Invariance Conserved Quantity Translation in time Energy Translation in space Linear Momentum Rotation in space Angular Momentum Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  117. 117. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 1st 20th century turning point: Einstein’s paper on special relativity in 1905 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  118. 118. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 1st 20th century turning point: Einstein’s paper on special relativity in 1905 Up until Einstein, the global spatiotemporal symmetries had been derived from the laws of classical electrodynamics Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  119. 119. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 1st 20th century turning point: Einstein’s paper on special relativity in 1905 Up until Einstein, the global spatiotemporal symmetries had been derived from the laws of classical electrodynamics He reversed this order: start with the universality of the global symmetries, and then use as the test of the validity of the laws of special relativistic dynamics Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  120. 120. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 2nd turning point: Heisenberg’s description of the indistinguishability of quantum particles in terms of a “permutation symmetry”’ in 1926 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  121. 121. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 2nd turning point: Heisenberg’s description of the indistinguishability of quantum particles in terms of a “permutation symmetry”’ in 1926 First time that a symmetry observed in quantum mechanical phenomena dealt with using the techniques of group theory Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  122. 122. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 2nd turning point: Heisenberg’s description of the indistinguishability of quantum particles in terms of a “permutation symmetry”’ in 1926 First time that a symmetry observed in quantum mechanical phenomena dealt with using the techniques of group theory Shout-out to fellow quantumers: the exchange operator! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  123. 123. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 2nd turning point: Heisenberg’s description of the indistinguishability of quantum particles in terms of a “permutation symmetry”’ in 1926 First time that a symmetry observed in quantum mechanical phenomena dealt with using the techniques of group theory Shout-out to fellow quantumers: the exchange operator! Hollah! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  124. 124. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H ISTORY OF S YMMETRY IN P HYSICS : T HE 20 TH C ENTURY 2nd turning point: Heisenberg’s description of the indistinguishability of quantum particles in terms of a “permutation symmetry”’ in 1926 First time that a symmetry observed in quantum mechanical phenomena dealt with using the techniques of group theory Shout-out to fellow quantumers: the exchange operator! Hollah!Pˆ |a, b >= |b, a > 12 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  125. 125. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R EST OF 20 TH C ENTURY H ISTORY OF S YMMETRY: I T ’ S ALL SYMMETRY, BABY “The history of the application of symmetry principles in quantum mechanics and the quantum field theory coincides with the history of the developments of 20th century theoretical physics.” Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  126. 126. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  127. 127. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE T HREE K EY D ISCRETE S YMMETRIES Global symmetries–apply to all points in space. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  128. 128. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE T HREE K EY D ISCRETE S YMMETRIES Global symmetries–apply to all points in space. Local symmetries apply only to individual space-time locations Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  129. 129. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE T HREE K EY D ISCRETE S YMMETRIES Global symmetries–apply to all points in space. Local symmetries apply only to individual space-time locations Each quantum field theory possesses various global and local symmetries Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  130. 130. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE T HREE K EY D ISCRETE S YMMETRIES Global symmetries–apply to all points in space. Local symmetries apply only to individual space-time locations Each quantum field theory possesses various global and local symmetries Each field theory has a global gauge symmetry associated with its bosons Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  131. 131. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE T HREE K EY D ISCRETE S YMMETRIES Invariance Conserved Quantity Charge conjugation (C) Charge Parity Coordinate inversion (P) Spatial Parity Time reversal (T) Time Parity Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  132. 132. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE M AJOR I NTERNAL S YMMETRIES Gauge Transformation Invariance Conserved Quantity U(1) Electric Charge U(1) Hypercharge U(1)Y Weak Hypercharge U(2) [U(1) × SU(2)] Electroweak force SU(2) Isospin SU(3) Quark Color SU(3) (approximate) Quark Flavor S(U(2) × U(3)) [U(1) × SU(2) × SU(3)] Standard Model Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  133. 133. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  134. 134. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Associated gauge boson–the photon. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  135. 135. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Associated gauge boson–the photon. Weinberg-Salam Electroweak theory–invariant under an SU(2) transformation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  136. 136. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Associated gauge boson–the photon. Weinberg-Salam Electroweak theory–invariant under an SU(2) transformation Associated gauge bosons–the W+, W-, and Z intermediate vector bosons Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  137. 137. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Associated gauge boson–the photon. Weinberg-Salam Electroweak theory–invariant under an SU(2) transformation Associated gauge bosons–the W+, W-, and Z intermediate vector bosons QCD Lagrangian–invariant under global SU(3) transformation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  138. 138. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS G AUGE B OSONS AND L OCAL S YMMETRIES The QED Lagrangian–is invariant under a global U(1) transformation Associated gauge boson–the photon. Weinberg-Salam Electroweak theory–invariant under an SU(2) transformation Associated gauge bosons–the W+, W-, and Z intermediate vector bosons QCD Lagrangian–invariant under global SU(3) transformation (N 2 − 1 = 9 − 1 = 8) Generators, corresponding to 8 Gauge Bosons–the 8 gluons Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  139. 139. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  140. 140. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  141. 141. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: L = ∂µ φ∗ ∂ µ φ − m 2 φ∗ φ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  142. 142. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1) transformation φ → e−iθ φ: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  143. 143. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1) transformation φ → e−iθ φ: ∗ ∗ L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  144. 144. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1) transformation φ → e−iθ φ: ∗ ∗ L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ = e+iθ e−iθ ∂µ φ∗ ∂ µ φ − e+iθ e−iθ m2 φ∗ φ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  145. 145. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS W HAT U(1) INVARIANCE LOOKS LIKE Invariance under U(1) transformation Example: the Lagrangian for a complex scalar field: L = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ is invariant under the U(1) transformation φ → e−iθ φ: ∗ ∗ L → ∂µ e−iθ φ ∂ µ e−iθ φ − m2 e−iθ φ e−iθ φ = e+iθ e−iθ ∂µ φ∗ ∂ µ φ − e+iθ e−iθ m2 φ∗ φ = ∂µ φ∗ ∂ µ φ − m 2 φ∗ φ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  146. 146. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS I S PARITY CONSERVED ? All of the fundamental forces are invariant under a local coordinate inversion symmetry...except for the weak force Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  147. 147. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS I S PARITY CONSERVED ? All of the fundamental forces are invariant under a local coordinate inversion symmetry...except for the weak force The τ − θ problem Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  148. 148. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS I S PARITY CONSERVED ? All of the fundamental forces are invariant under a local coordinate inversion symmetry...except for the weak force The τ − θ problem θ+ → π+ π0 τ + → π+π−π+ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  149. 149. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS I S PARITY CONSERVED ? All of the fundamental forces are invariant under a local coordinate inversion symmetry...except for the weak force The τ − θ problem θ+ → π+ π0 τ + → π+π−π+ They appear to be the exact same particle Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  150. 150. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS I S PARITY CONSERVED ? All of the fundamental forces are invariant under a local coordinate inversion symmetry...except for the weak force The τ − θ problem θ+ → π+ π0 τ + → π+π−π+ They appear to be the exact same particle But...if the weak force is invariant under parity exchange, the same particle can NOT decay in 2 different ways! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  151. 151. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS PARITY IS NOT CONSERVED 1956—Yang and Lee propose that parity might be violated for weak interactions Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  152. 152. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS PARITY IS NOT CONSERVED 1956—Yang and Lee propose that parity might be violated for weak interactions 1957—Chien-Shung Wu’s group observes parity violation in β-decay of cobalt-60 Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  153. 153. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS PARITY IS NOT CONSERVED 1956—Yang and Lee propose that parity might be violated for weak interactions 1957—Chien-Shung Wu’s group observes parity violation in β-decay of cobalt-60 1957—Garwin, Lederman, Weinrich and the decay of the pion (π − ): π − → µ− + νµ ¯ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  154. 154. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS PARITY IS NOT CONSERVED 1956—Yang and Lee propose that parity might be violated for weak interactions 1957—Chien-Shung Wu’s group observes parity violation in β-decay of cobalt-60 1957—Garwin, Lederman, Weinrich and the decay of the pion (π − ): π − → µ− + νµ ¯ They expected to find approximately equal distribution of positive and negative helicity muons Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  155. 155. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R ESULT OF G ARWIN , L EDERMAN , W EINRICH EXPERIMENT n (a), the helicity is positive–both resultant particles’ velocities are aligned with their spins; in (b), the helicity is negative. Only (b) is ever actually observed. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  156. 156. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS R ESULT OF G ARWIN , L EDERMAN , W EINRICH EXPERIMENT In (a), the helicity is positive–both resultant particles’ velocities are aligned with their spins; in (b), the helicity is negative. Only (b) is ever actually observed. Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  157. 157. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T IME R EVERSAL S YMMETRY Laws of physics invariant under time reversal transformation T Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  158. 158. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T IME R EVERSAL S YMMETRY Laws of physics invariant under time reversal transformation T Doesn’t appear to hold at macroscopic level–poking hole in a balloon example Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  159. 159. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T IME R EVERSAL S YMMETRY Laws of physics invariant under time reversal transformation T Doesn’t appear to hold at macroscopic level–poking hole in a balloon example This is only a manifestation of the laws of statistical mechanics Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  160. 160. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C HARGE C ONJUGATION Dirac’s prediction and Carl Anderson’s 1932 discovery of antimatter provides: Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  161. 161. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C HARGE C ONJUGATION Dirac’s prediction and Carl Anderson’s 1932 discovery of antimatter provides: Another discrete symmetry: the invariance of the laws of physics after replacing all particles with their antiparticles Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  162. 162. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C HARGE C ONJUGATION Dirac’s prediction and Carl Anderson’s 1932 discovery of antimatter provides: Another discrete symmetry: the invariance of the laws of physics after replacing all particles with their antiparticles “Charge conjugation,” or C Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  163. 163. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C HARGE C ONJUGATION Dirac’s prediction and Carl Anderson’s 1932 discovery of antimatter provides: Another discrete symmetry: the invariance of the laws of physics after replacing all particles with their antiparticles “Charge conjugation,” or C If C is not violated, then atomic antimatter should be exactly like atomic matter–antihydrogen Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  164. 164. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE TEST OF C INVARIANCE The test of C invariance came shortly after the discovery that P is violated Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  165. 165. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE TEST OF C INVARIANCE The test of C invariance came shortly after the discovery that P is violated If we exchange particles for antiparticles in the π − decay equation π − → µ− + νµ , we get the π + decay process, ¯ π + → µ+ + ν µ Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  166. 166. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE TEST OF C INVARIANCE The test of C invariance came shortly after the discovery that P is violated If we exchange particles for antiparticles in the π − decay equation π − → µ− + νµ , we get the π + decay process, ¯ π + → µ+ + ν µ If C is a symmetry of this reaction, both µ+ and µ− should have the same helicity Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  167. 167. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS CP AND ITS VIOLATION 1957—The µ+ found to have positive helicity! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  168. 168. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS CP AND ITS VIOLATION 1957—The µ+ found to have positive helicity! C is violated! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  169. 169. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS CP AND ITS VIOLATION 1957—The µ+ found to have positive helicity! C is violated! BUT...if C and P are considered as a joint CP operation, then the reactions are invariant! Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  170. 170. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS CP AND ITS VIOLATION 1957—The µ+ found to have positive helicity! C is violated! BUT...if C and P are considered as a joint CP operation, then the reactions are invariant! Invert the parity and charge, and everything is alright Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  171. 171. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS CP AND ITS VIOLATION 1957—The µ+ found to have positive helicity! C is violated! BUT...if C and P are considered as a joint CP operation, then the reactions are invariant! Invert the parity and charge, and everything is alright This IS what we observe Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  172. 172. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT For 7 years, CP appeared to be inviolable Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  173. 173. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT For 7 years, CP appeared to be inviolable Cronon-Fitch experiment in 1964—CP is violated very slightly Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  174. 174. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT For 7 years, CP appeared to be inviolable Cronon-Fitch experiment in 1964—CP is violated very slightly In every physical process we have yet observed, the combination CPT IS conserved Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  175. 175. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS C RAP —CP IS VIOLATED TOO ! B UT CPT IS NOT For 7 years, CP appeared to be inviolable Cronon-Fitch experiment in 1964—CP is violated very slightly In every physical process we have yet observed, the combination CPT IS conserved This combined CPT symmetry is a necessary condition for probability to be conserved in quantum mechanics Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  176. 176. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS O UTLINE OF TALK 1 Introduction: Examples of Symmetries in Nature 2 Groups 3 Brief History of Use of Symmetry in Physics 4 Symmetry and Quantum Field Theory 5 Symmetry Breaking 6 Conclusion: Why Symmetry? Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  177. 177. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY BREAKING Pencil standing on its point Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  178. 178. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS S YMMETRY BREAKING Pencil standing on its point Ferromagnetism—total rotational symmetry above Curie point, but below it, the spins spontaneously line up in a fixed direction Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  179. 179. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN S YMMETRY OF BCS T HEORY Yoichiro Nambu noticed that the BCS ground state violated the gauge symmetry of QED Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  180. 180. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN S YMMETRY OF BCS T HEORY Yoichiro Nambu noticed that the BCS ground state violated the gauge symmetry of QED He recast BCS theory into perturbative quantum field theory Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  181. 181. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN S YMMETRY OF BCS T HEORY Yoichiro Nambu noticed that the BCS ground state violated the gauge symmetry of QED He recast BCS theory into perturbative quantum field theory Showed that all of the phenomenon peculiar to superconductivity are the result of the spontaneous breaking of the underlying QED gauge symmetry Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  182. 182. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN S YMMETRY OF BCS T HEORY Yoichiro Nambu noticed that the BCS ground state violated the gauge symmetry of QED He recast BCS theory into perturbative quantum field theory Showed that all of the phenomenon peculiar to superconductivity are the result of the spontaneous breaking of the underlying QED gauge symmetry Suggested that any theory in which a continuous symmetry is spontaneously broken will give rise to massless, spinless bosonic particles (Nambu-Goldstone bosons) like those observed in BCS theory Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  183. 183. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS T HE 2008 N OBEL P RIZE : NAMBU AND THE B ROKEN S YMMETRY OF BCS T HEORY Yoichiro Nambu noticed that the BCS ground state violated the gauge symmetry of QED He recast BCS theory into perturbative quantum field theory Showed that all of the phenomenon peculiar to superconductivity are the result of the spontaneous breaking of the underlying QED gauge symmetry Suggested that any theory in which a continuous symmetry is spontaneously broken will give rise to massless, spinless bosonic particles (Nambu-Goldstone bosons) like those observed in BCS theory Showed that the same mechanism of spontaneous symmetry breaking in BCS theory gives rise to the mechanism required to support the existence of a nucleon Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS
  184. 184. B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS H IGGS F IELD Z ACH M C E LRATH B EAUTY IN THE B REAKDOWN : S YMMETRY IN P HYSICS

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