1. Extension of the Poincare` Group and Non-Abelian Tensor Gauge Fields George Savvidy Demokritos National Research Center Athens Extension of the Poincare’ Group Non-Abelian Tensor Gauge Fields arXiv:1006.3005 PhysLett B625 (2005) 341 Int.J.Mod.Phys. A25 (2010) 6010 Int.J.Mod.Phys.A21(2006) 4931 Arm.J.Math.1 (2008) 1 Int.J.Mod.Phys.A21(2006) 4959 Miami 2010

2. 1.Space-time symmetry 2. Space-time and internal symmetries 3. Super-Extension of the Poincare' Group 4. Alternative Extension of the Poincare' Group 5. Gauge symmetry of the Extended Algebra 7. Representations of Extended Poincare' Algebra 8. Longitudinal and Transversal representations 9. Killing Form 10. High Spin Gauge Fields

3. Space-time symmetry - the Poincare Group 10 generators = 4-translations, 3-rotations and 3-boosts The Poincare’ algebra contains

4. The Coleman-Mandula theorem is the strongest no-go theorems, stating that the symmetry group of a consistent quantum field theory is the direct product of an internal symmetry group and the Poincare group. If G is a symmetry group of the S matrix and if the following five conditions holds: 1. G contains the Poincare’ group 2. Only finite number of particles with mass less than M 3. Occurrence of nontrivial two particle scatterings 4. Analyticity of amplitudes as the functions of s and t 5. The generators are integral operators in momentum space, then the group G is isomorphic to the direct product of an internal symmetry group and the Poincare’ group. Unification of Space-time and internal symmetries ?

5. Super-Extension of the Poincare Algebra Weakening the assumptions of the Coleman-Mandula theorem by allowing both commuting and anticommuting symmetry generators, allows a nontrivial extension of the Poincare algebra, namely the super-Poincare algebra. ……

6. Alternative Extension of the Poincare Group We shall add infinite many new tensor generators s= 0,1,2,… are the generators of the Lie algebra ….. are the new generators and

7. This symmetry group is a mixture of the space-time and internal symmetries because the new generators have internal and space-time indices and transform nontrivially under both groups. Space-time and internal symmetries The algebra incorporates the Poincare’ algebra and an internal algebra in a nontrivial way, which is different from direct product.

8. A) Both algebras have Poincare algebra as subalgebra. B) The commutators in the middle show that the extended generators are translationally invariant operators and carry a nonzero spin. C) The last commutators essentially different in both of the algebras, in super-Poincare algebra the generators anti-commute to the momentum operator, while in our case gauge generators commute to themselves forming an infinite dimensional current algebra (Similar to Faddeev or Kac-Moody algebras)

9. Gauge symmetry of the Extended Algebra Theorem. To any given representation of the generators of the extended algebra one can add the longitudinal generators, as it follows from the above transformation. All representations are defined therefore modulo longitudinal representation. (off-mass-shell invariance)

10. Representations s=1,2,… Example:- are in any representation and The new generators are therefore of “ the gauge field type ” then

11. Representations of the Poincare’ Algebra The little algebra contains the following generators with commutation relation The square of the Pauli-Lubanski pseudovector defines irreducible representations and ( )()

12. Representations of Extended Poincare ' Algebra Longitudinal Representations. Let us consider the representations, then and The longitudinal representations of the extended algebra can be characterized as representations in which the Poincare' generators are taken in the representation and the gauge generators are expressed as the direct products of the momentum operator.

13. Representations of Extended Poincare ' Algebra Transversal Representations. Let us consider the representations, then and where The transversal representation of the extended algebra can be characterized as representation in which the Poincare' generators are taken in the representation and the gauge generators are expressed as the direct products of the derivatives of Pauli-Lubanski vector over its length. S -> infinity

15. Killing Form Using the explicit matrix representation of the gauge generators we can compute the traces where In general

16. The gauge fields are defined as rank-(s+1) tensors and are totally symmetric with respect to the indices a priory the tensor fields have no symmetries with respect to the index 4x High Spin Gauge Fields free and interacting high spin fields …………

17. Extended Gauge Field The extended gauge field is a connection and is a algebra valued 1-form. The symmetry group acts simultaneously as a structure group on the fibers and as an isometry group of the base - space-time manifold.

18. The Lagrangian

19. Summary of the Particle Spectrum.? 2009

20. The generators are projecting out the components of the high spin gauge fields into the plane transversal to the momentum keeping only its positive definite space-like components. The helicity content of the field is: The Particle Spectrum

21. Interaction Vertices are Dimensionless The VVV vertex The VTT vertex

22. Interaction Vertices The VVVV and VVTT vertices

23. 1. Poincare’ invariant vertices. The problem is - do they propagate ghosts? 2. Brink light-front formulation. No ghosts – but are they Poincare’ invariant? General Properties of Interaction Vertices 3. Spinor formulation