Kim and Noz introduce the mathematical tool that Eugene Paul Wigner presented in his 1939 article "On Unitary Representations of the Inhomogeneous Lorentz Group
," in the Annals of Mathematics and show how to use it to extend Einstein's special relativity to extended objects like the hydrogen atom or the proton in the quark model.
L is here the element of the Lorentz group
and N is the boost 4-vector.
As is well known, the massless spin 2 unitary representation of the Lorentz group
involves two helicities in 3+1 dimensions.
The Lorentz group
is the group of all isometries of Minkovski spacetime.
Proceeding from the Jacobi identities, it is easy to demonstrate that [B.sup.[mu]v.sub.[alpha][beta]] are the matrix elements of the generators [B.sup.[mu]v] of some n-dimensional representation (reducible or irreducible) of the Lorentz group
and [A.sub.[mu]],[alpha][beta] are matrix elements of matrices [A.sub.[mu]] which are connected with the [beta]-matrices of an invariant first-order equation
He then develops the theory of generalized Gelfand pairs and deals with examples related to the generalized Lorentz group
. The text is aimed at advanced undergraduates or beginning graduate students and the prerequisites are elementary real, complex, and functional analysis; familiarity with the spectral theory of (unbounded) self-adjoint operators on a Hilbert space; and knowledge of distribution theory and Lie theory.
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.
But it is frequently overlooked that physical kinematics (the behaviour of real rods and clocks in motion) cannot be derived directly from the symmetry group (the Lorentz group
) of Minkowski space-time without the further assumption that the covariance group of all the physical laws governing the behaviour of all conceivable rods and clocks coincides with the said symmetry group.
Another possible DSR scenario [3, 4] in which the speed of light is constant is also heavily studied with the help of nonlinear realizations of the Lorentz group
(this is referred to as model 1 in the rest of the paper).
But any bijective linear operator in the Minkowski space-time, preserving the Lorentz-Minkowski pseudo-metric, belongs to the general Lorentz group
, and it can not be coordinate transform for superluminal reference frame.
Called space forms in the literature on differential geometry, the manifolds are classified and consist of the Euclidean spaces, the hyperboloids and the spheres with the corresponding orthonormal frame bundles equal to the Euclidean group of motions, the rotation group and the Lorentz group
. Jurdjevic covers Cartan decomposition and the generalized elastic problems, the maximum principle and the Hamiltonians, the left-invariant symplectic form, symmetries and the conservation laws, complex Lie groups and complex Hamiltons, complexified elastic problems, complex elasticae of Euler and its n-dimensional extensions, Cartan algebras, root spaces and extra integrals of motion, and elastic curves in the cases of Lagrange and Kowalewski.