Lie Algebras in Physics: Geometric Applications and Non-Compact Groups
Explore the fascinating world of Lie algebras in physics, delving into geometric and non-compact groups, Conservation Laws, Noether's theorem, and the symmetries they reveal. Discover the classification of Lie algebras with Dynkin diagrams, Infinite families, and Exceptional cases. Dive into the complexities of Poincare groups, non-compact transformations, and the significance of automorphism groups for octonions. Uncover the nuances of non-compact groups and their generators, along with the role of quadratic forms in these structures.
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Ivane Javakhishvili Tbilisi State University Faculty of Exact and Natural Sciences compact ?? group Geometric application of non Geometric application of non- -compact group Presenter: Master's supervisor: Alexandre Gurchumelia Merab Gogberashvili
Lie algebras in physics Conservation Laws Noether's theorem Symmetries ? ? ? Groups ? ? exp ?? ? Lie Algebras
Classification of Lie algebras (with Dynkin diagrams) Infinite families Exceptional cases ?2 ?? ?? ?4 ?? ?6 ?? ?7 Notation ??= ?? ? + 1 , ??= ?? 2? , ??= ?? 2? + 1 , ??= ?? 2? , ?8 ?is rank
Something wrong with Poincare groups? Something wrong with Poincare groups? Planck length isn t supposed to Lorentz contract No division algebra in 3+1 D to describe transformations compact ??? ? Why non Why non- -compact ?2 is automorphism group for octonions Non-compact transformations give us boosts
??) ) Non-compact ?? group ( (?? Generators ? ???+1 ? ???+1 ? ??+1 ? ??? ? ? ? ???= ?? ???+ ?? ? ???+ ?? ? ???+ ?? ? ???+ ?? ?? ?? 3 ??? ? ??? ??? ? ??? ??? ???= ?? ?? ?? 3 2???? ??? ?0?= 2? ??? ??? ???= ?? ?11+ ?22+ ?33= 0 (?,? = 1,2,3) Metric: Space: ??= ??,?,?? 13 3 ? =1 2 2 13 3 Quadratic form: ???? = ?2+ ????
7-dimensional equivalent representation Minkowski-like metric Representation: Cartan Ours ?? ? ?? ?? ? ?? ? = ? = Coordinates: 13 3 13 3 ? =1 2 1 Metric: : = 2 13 3 13 3 ???? = ?2+ ???? ?? ? = ????+ ?2 ???? Quadratic form: : ??= ??+ ?? ??= ?? ??
Same quadratic form Same quadratic form Change of basis ?2= ????+ ?2 ???? Generators ? ??+ ? ? 1 ??? ? ??? ??? ??? ?? ??= ?0? ??0 = 2 ?? 2???? ??? ??? ??? ??? ? ?? ? ? 1 ??? ??? ??? ? ??? ?? ??= ?0?+ ??0 = 2 ?? 2???? ??? ??? ??? ??? =1 ? ? ? ? ??= ??????? 2???? ?? ???+ ?? + ?? ???+ ?? ??? ??? =1 ? ? ? ? ??= ??????? 2???? ?? ??? ?? + ?? ??? ?? ??? ??? ? ??? 1 ? ? ? ?= ??? = ?? ???+ ?? ?? + ?? 3 ??? ???
?? group transformations in the new basis ?? Infinitesimal transformations = ??+ ?????? ???? 2??? ??????+ ???????? ???? ?? ? = ? + 2 ????+ ???? ? ?= ??+ ?????? ????????+ 2??? ??????+ ???? ???? Finite transformation example Finite transformation example cos?3 sin?3 sin?3 cos?3 0 0 0 0 0 0 0 0 0 0 ?1 ?2 ?3 ? ?1 ?2 ?3 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 exp ?3?3? = cos?3 sin?3 0 sin?3 cos?3 0
?? group transformations in the new basis ?? Infinitesimal transformations = ??+ ?????? ???? 2??? ??????+ ???????? ???? ?? ? = ? + 2 ????+ ???? ? ?= ??+ ?????? ????????+ 2??? ??????+ ???? ???? Finite transformation example Finite transformation example 0 ch?1 0 0 ch?1 0 0 0 0 0 sh?1 sh?1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ?1 ?2 ?3 ? ?1 ?2 ?3 sh?1 0 0 0 0 ch?1 sh?1 0 0 ch?1 0 ch2?1 sh2?1 0 0 sh2?1 ch2?1 0 0 exp ?1?1? =
Casimir Operators Poincar group ?? ? ?2= ?? 2+ ?? 2+ ?? 2 ??=1 2?????????? ????= ??2 ?2 ?2? = + 1 ? ??2 ?2? = ?2? ????? = ?2? ? + 1 ? ? = ? ??,? ?? 2 2nd nd order casimir order casimir 6 6th th order order ?? ?2= 2?????? 2 ??0?0?+ ?0???0 ?6= In Cartan s basis: 3 1 3?? 2 1 2+ ?? 2 ?? 2+ 2 ? 2 ?2= 3?? Our basis: : ?
2nd order Casimir of ?? algebra Quadratic form Quadratic form Neglecting change in ? ?2= ????+ ?2 ???? ???= 0 ?2 ??2 ?2= 6?? ?? ?2 ?2 ?2 ??? ?2 ??? ? ? 2 2 ?2+ ?? 2 ?2 ?? + 2+ 6 ?? + ?? ??? ??? ? ? ? ? ?? + 2? ?? + ?? ??? ??? ?2 ?2 ?2 + 1 + ??????? + ???? ???? + ???? ?????? ?????? ?????? ? ?
Comparing Casimir operators 2nd order Casimir of ??in 4-vector notation ?2= ??????????? ???? ????+ ????????+ 2????+ ???????????? ? 1 ?2= ?????? ?2???? Poincar group ????= ??2 ?2 ???? ??=1 2??????????
Thank you! Summary & further research Summary & further research ?? contains Lorentz transformtions with some corrections[1] ?2 Casimir found and expressed in terms of Poincare group casimirs Next step: construct field theory with ?2 ?? symmetry Reference [1] Gogberashvili M, Sakhelashvili O, (2015). Geometrical Applications of Split Octonions; Hindawi Publishing Corporation: Adv. Math. Phys. p.196708; doi: 10.1155/2015/196708
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