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) 4931Arm.J.Math.1 (2008) 1 Int.J.Mod.Phys.A21(2006) 4959

Miami 2010

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

Space-time symmetry - the Poincare Group

10 generators = 4-translations, 3-rotations and 3-boosts

The Poincare’ algebra contains

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 t5. 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 ?

Super-Extension of the Poincare AlgebraWeakening 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.

……

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 generatorsand

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.

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)

Gauge symmetry of the Extended Algebra

Theorem. To any given representation of the generatorsof the extended algebra one can add the longitudinal generators, as it follows from the above transformation. All representationsare defined therefore modulo longitudinal representation.

(off-mass-shell invariance)

Representations

s=1,2,…

Example: - are in any representation and

The new generators are therefore of “ the gauge field type ”

then

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 ( ) ( )

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 asthe direct products of the momentum operator.

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

Killing Form

Using the explicit matrix representation of the gauge generators we cancompute the traces

where

In general

The gauge fields are defined as rank-(s+1) tensors

and are totally symmetric with respect to the indicesa priory the tensor fields have no symmetries with respect to the index

4x

High Spin Gauge Fields

free and interacting high spin fields…………

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 fibersand as an isometry group of the base - space-time manifold.

The Lagrangian

Summary of the Particle Spectrum

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2009

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

Interaction Vertices are Dimensionless

The VVV vertex

The VTT vertex

Interaction Vertices

The VVVV and VVTT vertices

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