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.

Jx yN = Jx − a y N = Jx yN − a J1 0N = Jx yN − a ∂xJx yN = exp 8−a ∂x < Jx yN

realization of 8Px< = ℜD HPxL = −i— 8∂x <

Jx yN = exp 9 i H−aL

— ℜD HPxL = Jx yN

x y x’ y’ a

J x yN =JCos HθL −Sin HθL Sin HθL Cos HθL N Jx yN= exp 9θ J0 −1 1 0 N= Jx yN

x’ y’ x y θ

realization of 8Mxy or Jz< = ℜD HMxyL = = ℜD HMxyL = −i— 8x ∂y− y ∂x <

x y = exp iθ

—

ℜD Mxy x y

Symmetry Generators

i k j j j j j j j j j ct x y z y { z z z z z z z z z = i k j j j j j j j j j j j j j j j j j

ct−vxêc " ################## 1−v2êc2 x−vt " ################## 1−v2êc2

y z y { z z z z z z z z z z z z z z z z z = i k j j j j j j j j j j j j j j j j j

1 "################## 1−v2êc2

−vêc "##################

1−v2êc2

−vêc "##################

1−v2êc2 1 "################## 1−v2êc2

1 1 y { z z z z z z z z z z z z z z z z z i k j j j j j j j j ct x y z y { z z z z z z z z =

exp i k j j j j j j j j j HvêcL i k j j j j j j j j j −1 −1 y { z z z z z z z z z y { z z z z z z z z z i k j j j j j j j j ct x y z y { z z z z z z z z = exp i k j j i Hv êcL — ℜM HM01Ly { z z i k j j j j j j j j ct x y z y { z z z z z z z z

x y z v x’ y’ z’

x y

• a

Px

Py↑ ⏐ ⏐ b

• −a

Px Py↓ ⏐ ⏐ −b

x’ y’ x y x’ y’ x y x’ y’ x y x’ y’ 81+ b ∂y<8

1+ a ∂x<8 1− b ∂y<8 1− a ∂x < H +ϑ a2 b2L H = 1+ ab ∂y∂x−∂x∂yL+ϑ Ha2 b2L =

1 − i H−aL H−bL — i k j j −i — y { z z@ℜD HPxL, ℜD HPyLD + ϑ Ha2 b2L = 1

@ℜD HPxL, ℜD HPyLD = 0 @Px, PzD = 0 @Py, PzD = 0 @Px, PyD = 0

Composition of Symmetry Transformations

Mxy, Px = i— Py

[0]3 +[0]1 =[0]2 It is an additive grading (additive quantum number), since the degree of a product is given by the addition of degrees

Underlying and Extending Grading

(0,0) (1,0) (0,1) (1,1)

(0,0) (1,0) (0,1) (1,1)

Extension I Z2XZ2

⊇

Poincaré Algebra Z2XZ2 I = Z2 X (Z4n X Z4n)

Clover Extensions:

a) Vector Scalar Clover Extension

2 2,

( 1) , , ( 1) , ( , ) T R s s R T R t t 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

Supersymmetry provides a Z2-graded extension, in which two spinor charges combine to produce a space-time translation:

x y z 2C 2C t

Summary:

The Clover extensions are Z4XZ4-graded extensions, in which three vector charges combine to produce a translation:

Z2X(Z4XZ4)- graded extensions:

Susy, Internal Symmetries, Dark Energy, Dim. confinement u(1)xu(1), su(2), su(3)?