This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the calculation of Wigner's angle for arbitrary Lorentz transformations. Our derivation highlights the geometrical content of the sequence of little group transformations leading to Wigner's matrix and links it to classical theorems in spherical trigonometry.
This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the calculation of Wigner's angle for arbitrary Lorentz transformations. Our derivation highlights the geometrical content of the sequence of little group transformations leading to Wigner's matrix and links it to classical theorems in spherical trigonometry.
Lorentz transformations in time and two space dimensions consist of boosts and rotations, and combinations thereof. In general, the combination of two boosts is not another boost: It is a boost followed by a rotation. The rotation angle is called the Wigner angle. Although it is straightforward to determine the energy and direction of the combined boost, it is difficult to determine the Wigner angle. In this article, the vector, matrix and spinor derivations of formulas for the Wigner angle are reviewed, and the underlying mathematics and physics are discussed briefly. Although the derivations are different, the results they produce are equivalent, as they should be. Like many physics problems, if one looks at the problem in the right way, it is not difficult to solve.
This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincare sphere on polarization optics.
The concept of the ‘Wigner rotation’, familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset representatives. Examples of Abelian and non-Abelian Wigner rotations are given. The Lorentz group Wigner rotation, occurring in the coset space SL (2, R )/ SO (2) ≃ SO (2, 1)/ SO (2), is shown to be an analytic continuation of a Wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations, belonging to the coset space SU (2)/ U (1) ≃ SO (3)/ SO (2). The possibility of interpreting these two Wigner rotations as geometric phases is shown in detail. Essential background material on geometric phases, Bargmann invariants and null phase curves, all of which are needed for this purpose, is provided.
This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincare sphere on polarization optics.
This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincaré sphere on polarization optics.
It is known that entangled particles involve Lorentz symmetry violation. Hence, we pay attention to Lorentz transformations of signature (m,n) for all positive integers m and n. We show that these form the symmetry groups by which systems of m entangled n-dimensional particles can be understood, just as the common Lorentz group of signature (1,3) forms the symmetry group by which Einstein's special theory of relativity is understood. A novel, unified parametric realization of the Lorentz transformations of any signature (m,n) shakes down the underlying matrix algebra into elegant and transparent results.