In the classical treatment, one merely considers the restricted Poincare Group which is formed with orthochronous components [L.sub.0].

One can then easily show that the complex Poincare group

This theory is based on the extremely important fact that inertial transformations form the

Poincare group.

We consider an algebra where the generators [P.sup.[mu]] and [M.sup.[mu]v] of the

Poincare group and n generators [S.sub.[alpha]] satisfy the relations:

The 10-parameter

Poincare group is the semi-direct product of the 6-parameter Lorentz group with the 4-parameter group of space-time translations.

The 11 papers in this collection review the role of nontrivial symmetries in equilibrium thermodynamics, the Lie derivative of spinor fields, Landen transformation formulas for Jacobi elliptic functions, and the quantum electrodynamics of the

Poincare group. Other topics include superselection rules induced by infrared divergence, Hermitian modifications of Toeplitz linear functionals and orthogonal polynomials, Dirac equations in cosmological backgrounds, and non-orthogonal signal representation.

Stick with the Lorentz group SO(1, 3) or introduce an affine structure in the fibres (to be sharply distinguished from an affine connection on the bundle), so the local symmetry group becomes the inhomogeneous Lorentz group, that is, the

Poincare group. Auyang follows some influential authors in claiming that the

Poincare group is needed if one wants to allow for torsion in the spacetime manifold.

They conjectured that the laws of physics are invariant under the symmetry group of de Sitter space (maximally symmetric space-time), rather than the

Poincare group of special relativity.

The variational principle and its relevant Lagrangian density; Wigner's analysis of the irreducible representations of the

Poincare group; the duality invariance of the homogeneous Maxwell equations.

Let us consider firstly the quantum

Poincare group with [omega] = 0.

The role of the conformal group in Gravity in these expressions (besides the holographic bulk/boundary AdS/CFT duality correspondence [13]) stems from the MacDowell-Mansouri-Chamseddine-West formulation of Gravity based on the conformal group SO(3,2) which has the same number of 10 generators as the 4D

Poincare group. The 4D vielbein [e.sup.a.sub.[mu]] which gauges the spacetime translations is identified with the SO(3,2) generator [A.sup.[a5].sub.[mu]], up to a crucial scale factor R, given by the size of the Anti de Sitter space (de Sitter space) throat.