Noncommutative extensions of the space-time symmetries beyond - PowerPoint PPT Presentation

Noncommutative extensions of the space-time symmetries beyond supersymmetry L. A. Wills-Toro Department of Mathematics and Statistics, American University of Sharjah Abstract: Novel bosonic and fermionic graded extensions of the Poincaré algebra beyond supersymmetry are presented. Their nilpotent features and their combination with nonabelian symmetry give the possibility of going beyond Coleman & Mandula no-go theorems.

Symmetry Generators y y’ J x � y � N = J x − a N = J x y N − a J 1 0 N = J x y N − a ∂ x J x y N = exp 8 − a ∂ x < J x y N y realization of 8 P x < = ℜ D H P x L = − i — 8 ∂ x < i H − a L J x � y � N = exp 9 ℜ D H P x L = J x y N x’ x y � N = J Cos H θ L − Sin H θ L — J Cos H θ L N J x y N = exp 9 θ J 0 − 1 0 N= J x y N x � Sin H θ L a realization of 8 M xy or J z < = ℜ D H M xy L = 1 y’ = ℜ D H M xy L = − i — 8 x ∂ y − y ∂ x < y θ i θ x � x = exp ℜ D M xy x x’ y � y —

y’ y v z z’ x’ x i y − v ê c i ct − vx ê c y j z j z j z j z j z "################## "################## j z " ################## j z j z 1 i y i y 1 − v2 ê c2 j 1 − v2 ê c2 1 − v2 ê c2 z j z j z j z j z j z j z j z j z j ct � z j z j z j z j z j z − v ê c 0 0 j z j z ct j z j z j z j z j z j z j z j z x � j z j z "################## "################## " ################## j z x − vt j z j z j z j z 1 0 0 x j z j z 1 − v2 ê c2 1 − v2 ê c2 1 − v2 ê c2 j z = = = j z j z j z y � k { k { j z j z j z y j z j z z � k { k { z y 0 0 1 0 z 0 0 0 1 i i y y i y i y j j z z j z j z j j z z j z j z i H v ê c L j j z z j z j z − 1 = exp i ℜ M H M 01 Ly j j z z 0 0 0 j z j z j z ct ct j H v ê c L j z z j z j z j z j j z z j z j z j j z z j z j z − 1 k { j j z z j z j z 0 0 0 x x j j z z exp k { k { k k { { y y 0 0 0 0 — z z 0 0 0 0

Composition of Symmetry Transformations y’ Px � y’ y y − a x’ x x ⏐ x’ ⏐ − b P y ↑ ⏐ ⏐ b P y ↓ Px y’ y’ � y y a x x 8 1 + b ∂ y <8 1 + a ∂ x <8 1 − b ∂ y <8 1 − a ∂ x < +ϑ a 2 b 2 L H = 1 + ab ∂ y ∂ x −∂ x ∂ y L +ϑ H a 2 b 2 L = H x’ x’ i H − a L H − b L i y j z j z@ ℜ D H P x L , ℜ D H P y LD + ϑ H a 2 b 2 L = 1 − i k { 1 − — — @ ℜ D H P x L , ℜ D H P y LD = 0 @ P x , P y D = 0 @ P x , P z D = 0 @ P y , P z D = 0

Underlying and Extending Grading (0,0) (1,0) (0,1) (1,1) It is an additive grading (additive quantum = i — P y M xy , P x number), since the degree of a product [0]3 +[0]1 =[0]2 is given by the addition of degrees (0,0) (1,0) (0,1) (1,1)

Poincaré Algebra Z 2 XZ 2 Extension ⊇ I Z 2 XZ 2 I = Z 2 X (Z 4 n X Z 4 n)

Clover Extensions: a) Vector Scalar Clover Extension ⎡ ⎤ 2 ⎡ ⎤ = + = + 2 , T R s s ( 1) , R T , R t t ( 1) , R ( , ) s t ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

b) Minimal Vector Clover Extension

c) Full Clover Extension

d) VSC Extension with su(2) enhancement?

e)

f) The su(3) Full Clover Ext.

g)

g) su(3) Trefoil Extension

Summary: Supersymmetry provides a Z 2 -graded extension, in which two spinor charges combine to produce a space-time translation: y 2C 2C t z x The Clover extensions are Z 4 XZ 4 -graded extensions, in which three vector charges combine to produce a translation: Z 2 X(Z 4 XZ 4 )- graded extensions: Susy, Internal Symmetries, Dark Energy, Dim. confinement u(1)xu(1), su(2), su(3)?