1. 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

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

3. 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

4. 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

5. Non-compact 𝑮 𝟐 group ( 𝑮 𝟐 𝑵𝑪 )
Generators
𝑋 𝑛𝑛 =− 𝑧 𝑛 𝜕 𝜕 𝑧 𝑛 + 𝑦 𝑛 𝜕 𝜕 𝑦 𝑛 𝑧 𝑚 𝜕 𝜕 𝑧 𝑚 − 𝑦 𝑚 𝜕 𝜕 𝑦 𝑚
𝑋 0𝑚 =−2𝑡 𝜕 𝜕 𝑦 𝑚 + 𝑧 𝑚 𝜕 𝜕𝑡 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= × ×3
Quadratic form: 𝑝 𝑇 𝑔𝑝= 𝑡 2 + 𝑦 𝑛 𝑧 𝑛

6. 7-dimensional equivalent representation Minkowski-like metric
Cartan
Ours
Coordinates:
𝑝= 𝑦 𝑛 𝑡 𝑧 𝑛
𝓅= 𝜆 𝑛 𝑡 𝑥 𝑛
Metric:
𝑔= × ×3
ℊ= 1 3× − 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

7. 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 𝑛 =−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 = 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 − 𝑥 𝑚 𝜕 𝜕 𝜆 𝑚 + 𝜆 𝑚 𝜕 𝜕 𝑥 𝑚

8. 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 − sin 𝜌 3 cos 𝜌 cos 𝜌 3 sin 𝜌 − sin 𝜌 3 cos 𝜌 𝜆 1 𝜆 2 𝜆 3 𝑡 𝑥 1 𝑥 2 𝑥 3

9. 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= ch 𝜃 sh 𝜃 ch 𝜃 − sh 𝜃 ch 2 𝜃 1 sh 2 𝜃 sh 2 𝜃 1 ch 2 𝜃 − sh 𝜃 ch 𝜃 sh 𝜃 ch 𝜃 𝜆 1 𝜆 2 𝜆 3 𝑡 𝑥 1 𝑥 2 𝑥 3

10. 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 − 𝑋 𝑘0 𝑋 0𝑘 + 𝑋 0𝑘 𝑋 𝑘0
𝐶 6 =…
Our basis:
𝒞 2 = 𝑘 𝑇 𝑘 2 − 1 3 𝐹 𝑘 2 + 𝐺 𝑘 2 − 𝑅 𝑘 2 +2 Φ 𝑘 2 …

11. 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 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 + 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 +2𝑡 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 + 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 𝜕 𝜕𝑡 + 𝑚 𝑛 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 𝑡 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 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 + 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 +2𝑡 𝑥 𝑘 𝜕 𝜕 𝑥 𝑘 + 𝜆 𝑘 𝜕 𝜕 𝜆 𝑘 𝜕 𝜕𝑡 + 𝑚 𝑛 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

12. 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

13. 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 ; doi: /2015/196708