The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown… (More)

It has been almost one hundred years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space-time symmetry is dictated by the Lorentz group. It is shown… (More)

According to Eugene Wigner, quantum mechanics is a physics of Fourier transformations, and special relativity is a physics of Lorentz transformations. Since two-by-two matrices with unit determinant… (More)

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the… (More)

The ABCD matrix is one of the essential mathematical instruments in optics. It is the two-by-two representation of the group Sp(2), which is applicable to many branches of physics, including squeezed… (More)

It has been almost 100 years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space–time symmetry is dictated by the Lorentz group. It is shown that this… (More)

Einstein had to learn the mathematics of Lorentz transformations in order to complete his covariant formulation of Maxwell’s equations. The mathematics of Lorentz transformations, called the Lorentz… (More)

The Stokes parameters form a Minkowskian four-vector under various optical transformations. As a consequence, the resulting two-by-two density matrix constitutes a representation of the Lorentz… (More)

Wigner’s little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of… (More)

It is possible to associate two angles with two successive non-collinear Lorentz boosts. If one boost is applied after the initial boost, the result is the final boost preceded by a rotation called… (More)