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’ AlgebraThe 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' AlgebraTransversal 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
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