Geometric application of non-compact 𝑮 𝟐 group Ivane Javakhishvili Tbilisi State University Faculty of Exact and Natural Sciences Geometric application of non-compact 𝑮 𝟐 group Presenter: Alexandre Gurchumelia Master's supervisor: Merab Gogberashvili

Lie algebras in physics Conservation Laws Symmetries Groups Lie Algebras Noether's theorem 𝑞 𝛼 𝑒 𝑋∈𝔤 exp 𝛼𝑋 ∈𝐺

Classification of Lie algebras (with Dynkin diagrams) Infinite families Exceptional cases 𝐴 𝑛 𝐺 2 𝐵 𝑛 𝐹 4 𝐶 𝑛 𝐸 6 𝐷 𝑛 Notation 𝐴 𝑛 =𝑆𝑈 𝑛+1 , 𝐶 𝑛 =𝑆𝑝 2𝑛 , 𝐵 𝑛 =𝑆𝑂 2𝑛+1 , 𝐷 𝑛 =𝑆𝑂 2𝑛 , 𝒏 is rank 𝐸 7 𝐸 8

Something wrong with Poincare groups? Planck length isn’t supposed to Lorentz contract No division algebra in 3+1 D to describe transformations Why non-compact 𝑮 𝟐 ? 𝐺 2 is automorphism group for octonions Non-compact transformations give us boosts

Non-compact 𝑮 𝟐 group ( 𝑮 𝟐 𝑵𝑪 ) Generators 𝑋 𝑛𝑛 =− 𝑧 𝑛 𝜕 𝜕 𝑧 𝑛 + 𝑦 𝑛 𝜕 𝜕 𝑦 𝑛 + 1 3 𝑧 𝑚 𝜕 𝜕 𝑧 𝑚 − 𝑦 𝑚 𝜕 𝜕 𝑦 𝑚 𝑋 0𝑚 =−2𝑡 𝜕 𝜕 𝑦 𝑚 + 𝑧 𝑚 𝜕 𝜕𝑡 + 1 2 𝜖 𝑚𝑝𝑗 𝑦 𝑝 𝜕 𝜕 𝑧 𝑗 − 𝑦 𝑗 𝜕 𝜕 𝑧 𝑝 𝑋 𝑚𝑛 =− 𝑧 𝑛 𝜕 𝜕 𝑧 𝑚 + 𝑦 𝑚 𝜕 𝜕 𝑦 𝑛 𝑋 11 + 𝑋 22 + 𝑋 33 =0 (𝑚,𝑛=1,2,3) Metric: Space: 𝑝 𝑇 = 𝑦 𝑛 ,𝑡, 𝑧 𝑛 𝑔= 1 2 1 3×3 2 1 3×3 Quadratic form: 𝑝 𝑇 𝑔𝑝= 𝑡 2 + 𝑦 𝑛 𝑧 𝑛

7-dimensional equivalent representation Minkowski-like metric Cartan Ours Coordinates: 𝑝= 𝑦 𝑛 𝑡 𝑧 𝑛 𝓅= 𝜆 𝑛 𝑡 𝑥 𝑛 Metric: 𝑔= 1 2 1 3×3 2 1 3×3 ℊ= 1 3×3 1 − 1 3×3 Quadratic form: 𝑝 𝑇 𝑔𝑝= 𝑡 2 + 𝑦 𝑛 𝑧 𝑛 𝓅 𝑇 ℊ𝓅= 𝜆 𝑛 𝜆 𝑛 + 𝑡 2 − 𝑥 𝑛 𝑥 𝑛 𝑦 𝑛 = 𝜆 𝑛 + 𝑥 𝑛 𝑧 𝑛 = 𝜆 𝑛 − 𝑥 𝑛

Change of basis Same quadratic form 𝓈 2 = 𝜆 𝑛 𝜆 𝑛 + 𝑡 2 − 𝑥 𝑛 𝑥 𝑛 𝓈 2 = 𝜆 𝑛 𝜆 𝑛 + 𝑡 2 − 𝑥 𝑛 𝑥 𝑛 Change of basis Generators 𝑇 𝑛 = 𝒳 0𝑛 − 𝒳 𝑛0 =−2 𝑥 𝑛 𝜕 𝜕𝑡 +𝑡 𝜕 𝜕 𝑥 𝑛 − 1 2 𝜖 𝑛𝑞𝑖 𝜆 𝑞 𝜕 𝜕 𝑥 𝑖 − 𝜆 𝑖 𝜕 𝜕 𝑥 𝑞 − 𝑥 𝑞 𝜕 𝜕 𝜆 𝑖 − 𝑥 𝑖 𝜕 𝜕 𝜆 𝑞 𝐹 𝑛 =− 𝒳 0𝑘 + 𝒳 𝑘0 =−2 𝜆 𝑛 𝜕 𝜕𝑡 −𝑡 𝜕 𝜕 𝜆 𝑛 − 1 2 𝜖 𝑛𝑞𝑖 𝜆 𝑞 𝜕 𝜕 𝜆 𝑖 − 𝜆 𝑖 𝜕 𝜕 𝜆 𝑞 − 𝑥 𝑞 𝜕 𝜕 𝑥 𝑖 − 𝑥 𝑖 𝜕 𝜕 𝑥 𝑞 𝐺 𝑘 = 𝜖 𝑘𝑚𝑛 𝒳 𝑚𝑛 = 1 2 𝜖 𝑘𝑛𝑚 𝑥 𝑛 𝜕 𝜕 𝜆 𝑚 + 𝑥 𝑚 𝜕 𝜕 𝜆 𝑛 + 𝜆 𝑛 𝜕 𝜕 𝑥 𝑚 + 𝜆 𝑚 𝜕 𝜕 𝑥 𝑛 𝑅 𝑘 = 𝜖 𝑘𝑚𝑛 𝒳 𝑚𝑛 = 1 2 𝜖 𝑘𝑛𝑚 𝜆 𝑛 𝜕 𝜕 𝜆 𝑚 − 𝜆 𝑚 𝜕 𝜕 𝜆 𝑛 + 𝑥 𝑛 𝜕 𝜕 𝑥 𝑚 − 𝑥 𝑚 𝜕 𝜕 𝑥 𝑛 Φ 𝑛 = 𝒳 𝑘𝑘 = 𝑥 𝑛 𝜕 𝜕 𝜆 𝑛 + 𝜆 𝑛 𝜕 𝜕 𝑥 𝑛 − 1 3 𝑥 𝑚 𝜕 𝜕 𝜆 𝑚 + 𝜆 𝑚 𝜕 𝜕 𝑥 𝑚

𝑮 𝟐 𝑵𝑪 group transformations in the new basis Infinitesimal transformations 𝜆 𝑘 ′ = 𝜆 𝑘 + 𝜖 𝑘𝑚𝑛 𝜙 𝑚 − 𝜌 𝑚 𝜆 𝑛 −2 𝜙 𝑘 𝑡− 𝜖 𝑘𝑚𝑛 𝜃 𝑚 + 𝜖 𝑘𝑚𝑛 𝛾 𝑚 𝑥 𝑛 − 𝛽 𝑘 𝑥 𝑘 𝑡 ′ =𝑡+2 𝜙 𝑚 𝜆 𝑚 + 𝜃 𝑚 𝑥 𝑚 𝑥′ 𝑘 = 𝑥 𝑘 + 𝜖 𝑘𝑚𝑛 𝜃 𝑚 − 𝜖 𝑘𝑚𝑛 𝛾 𝑚 𝜆 𝑛 +2 𝜃 𝑘 𝑡− 𝜖 𝑘𝑚𝑛 𝜙 𝑚 + 𝜌 𝑚 𝑥 𝑛 − 𝛽 𝑘 𝜆 𝑘 Finite transformation example exp 𝜌 3 𝑅 3 𝓅= cos 𝜌 3 sin 𝜌 3 0 0 0 0 0 − sin 𝜌 3 cos 𝜌 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 cos 𝜌 3 sin 𝜌 3 0 0 0 0 0 − sin 𝜌 3 cos 𝜌 3 0 0 0 0 0 0 0 1 𝜆 1 𝜆 2 𝜆 3 𝑡 𝑥 1 𝑥 2 𝑥 3

𝑮 𝟐 𝑵𝑪 group transformations in the new basis Infinitesimal transformations 𝜆 𝑘 ′ = 𝜆 𝑘 + 𝜖 𝑘𝑚𝑛 𝜙 𝑚 − 𝜌 𝑚 𝜆 𝑛 −2 𝜙 𝑘 𝑡− 𝜖 𝑘𝑚𝑛 𝜃 𝑚 + 𝜖 𝑘𝑚𝑛 𝛾 𝑚 𝑥 𝑛 − 𝛽 𝑘 𝑥 𝑘 𝑡 ′ =𝑡+2 𝜙 𝑚 𝜆 𝑚 + 𝜃 𝑚 𝑥 𝑚 𝑥′ 𝑘 = 𝑥 𝑘 + 𝜖 𝑘𝑚𝑛 𝜃 𝑚 − 𝜖 𝑘𝑚𝑛 𝛾 𝑚 𝜆 𝑛 +2 𝜃 𝑘 𝑡− 𝜖 𝑘𝑚𝑛 𝜙 𝑚 + 𝜌 𝑚 𝑥 𝑛 − 𝛽 𝑘 𝜆 𝑘 Finite transformation example exp 𝜃 1 𝑇 1 𝓅= 1 0 0 0 0 0 0 0 ch 𝜃 1 0 0 0 0 sh 𝜃 1 0 0 ch 𝜃 1 0 0 − sh 𝜃 1 0 0 0 0 ch 2 𝜃 1 sh 2 𝜃 1 0 0 0 0 0 sh 2 𝜃 1 ch 2 𝜃 1 0 0 0 0 − sh 𝜃 1 0 0 ch 𝜃 1 0 0 sh 𝜃 1 0 0 0 0 ch 𝜃 1 𝜆 1 𝜆 2 𝜆 3 𝑡 𝑥 1 𝑥 2 𝑥 3

Casimir Operators 𝐒𝐎 𝟑 𝐿 2 = 𝐿 𝑥 2 + 𝐿 𝑦 2 + 𝐿 𝑧 2 𝐿 2 𝜓= ℓ ℓ+1 𝜓 𝐿 2 𝜓= ℓ ℓ+1 𝜓 𝜓= 𝑌 ℓ𝑚 𝜃,𝜙 Poincaré group 𝑃 𝜇 𝑃 𝜇 = 𝜕 𝑡 2 − 𝛻 2 𝜕 𝑡 2 − 𝛻 2 𝜓= 𝑚 2 𝜓 𝑊 𝜇 = 1 2 𝜖 𝜇𝜈𝜌𝜎 𝑀 𝜈𝜌 𝑃 𝜎 𝑊 𝜇 𝑊 𝜇 𝜓=− 𝑚 2 𝑠 𝑠+1 𝜓 𝑮 𝟐 𝑵𝑪 2nd order casimir 6th order In Cartan’s basis: 𝐶 2 =2 𝑋 𝑚𝑛 𝑋 𝑛𝑚 − 2 3 𝑋 𝑘0 𝑋 0𝑘 + 𝑋 0𝑘 𝑋 𝑘0 𝐶 6 =… Our basis: 𝒞 2 = 𝑘 1 3 𝑇 𝑘 2 − 1 3 𝐹 𝑘 2 + 𝐺 𝑘 2 − 𝑅 𝑘 2 +2 Φ 𝑘 2 …

2nd order Casimir of 𝔤 𝟐 algebra Neglecting change in 𝝀 𝑑𝜆 𝑘 =0 Quadratic form 𝓈 2 = 𝜆 𝑛 𝜆 𝑛 + 𝑡 2 − 𝑥 𝑛 𝑥 𝑛 𝒞 2 =6𝑡 𝜕 𝜕𝑡 − 𝓈 2 − 𝑡 2 𝜕 2 𝜕 𝑡 2 + 𝑘 𝓈 2 + 𝑥 𝑘 2 𝜕 2 𝜕 𝑥 𝑘 2 − 𝓈 2 − 𝜆 𝑘 2 𝜕 2 𝜕 𝜆 𝑘 2 +6 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 + 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 +2𝑡 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 + 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 𝜕 𝜕𝑡 + 𝑚 𝑛 1+ 𝛿 𝑚𝑛 𝜆 𝑚 𝑥 𝑛 𝜕 2 𝜕 𝜆 𝑚 𝜕 𝑥 𝑛 + 𝜖 𝑘𝑚𝑛 𝑥 𝑚 𝑥 𝑛 𝜕 2 𝜕 𝑥 𝑚 𝜕 𝑥 𝑛 + 𝜆 𝑚 𝜆 𝑛 𝜕 2 𝜕 𝜆 𝑚 𝜕 𝜆 𝑛 +2𝑡 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 + 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 𝜕 𝜕𝑡 𝑘 𝓈 2 + 𝑥 𝑘 2 𝜕 2 𝜕 𝑥 𝑘 2 − 𝓈 2 − 𝜆 𝑘 2 𝜕 2 𝜕 𝜆 𝑘 2 +6 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 + 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 +2𝑡 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 + 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 𝜕 𝜕𝑡 + 𝑚 𝑛 1+ 𝛿 𝑚𝑛 𝜆 𝑚 𝑥 𝑛 𝜕 2 𝜕 𝜆 𝑚 𝜕 𝑥 𝑛 + 𝜖 𝑘𝑚𝑛 𝑥 𝑚 𝑥 𝑛 𝜕 2 𝜕 𝑥 𝑚 𝜕 𝑥 𝑛 + 𝜆 𝑚 𝜆 𝑛 𝜕 2 𝜕 𝜆 𝑚 𝜕 𝜆 𝑛

Comparing Casimir operators 2nd order Casimir of 𝔤 𝟐 in 4-vector notation 𝒞 2 =− 𝝀 𝒏 𝝀 𝒏 𝜂 𝜇𝜈 𝜕 𝜇 𝜕 𝜈 − 𝑥 𝜇 𝑥 𝜇 𝜕 𝜈 𝜕 𝜈 + 𝑥 𝜇 𝑥 𝜇 𝜕 𝜇 𝜕 𝜇 +2 𝑥 𝜇 𝜕 𝜇 + 𝜎 𝜖 𝜎𝜇𝜈 𝑥 𝜇 𝑥 𝜈 𝜕 𝜇 𝜕 𝜈 𝒞 2 = 𝝀 𝟐 𝑃 𝜇 𝑃 𝜇 − 1 𝑚 2 𝑊 𝜇 𝑊 𝜇 Poincaré group 𝑃 𝜇 𝑃 𝜇 = 𝜕 𝑡 2 − 𝛻 2 𝑊 𝜇 𝑊 𝜇 𝑊 𝜇 = 1 2 𝜖 𝜇𝜈𝜌𝜎 𝑀 𝜈𝜌 𝑃 𝜎

Summary & further research Thank you! Summary & further research 𝐺 2 𝑁𝐶 contains Lorentz transformtions with some corrections[1] Casimir found and expressed in terms of Poincare group casimirs Next step: construct field theory with 𝐺 2 𝑁𝐶 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