Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two-component spinors

Han, D.; Kim, Y. S.; Son, D.

doi:10.1007/978-94-009-3051-3_34

Cited by 4 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…According to the Euler decomposition, the rotation around the y axis, in addition, will accommodate rotations along all three directions. For this reason, it is enough to study what happens in transformations within the xz plane [14].…”

Section: Loop Representation Of Wigner's Little Groupsmentioning

confidence: 99%

Loop Representation of Wigner’s Little Groups

2017

View full text Add to dashboard Cite

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 relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner's little group.

show abstract

Section: Loop Representation Of Wigner's Little Groupsmentioning

confidence: 99%

Loop Representation of Wigner’s Little Groups

2017

View full text Add to dashboard Cite

show abstract

“…However, for many physical problems, it is more convenient to study the problem in the two-dimensional (x, z) plane first and generalize it to three-dimensional space by rotating the system around the z axis. This process can be called Euler decomposition and Euler generalization [2].…”

Section: Groups Of Two-by-two Matricesmentioning

confidence: 99%

“…The matrix G is not a unitary matrix, because its Hermitian conjugate is not always its inverse. This group has a unitary subgroup called SU (2) and another consisting only of real matrices called Sp (2). For this later subgroup, it is sufficient to work with the three matrices R(θ), S(λ), and B(η) given in Eqs.…”

Section: Two-by-two Formulation Of Lorentz Transformationsmentioning

confidence: 99%

“…Let the direction of the momentum be along the z axis, and let the perpendicular direction be along the x axis. We therefore study the kinematics of the group within the zx plane, then see what happens when we rotate the system around the z axis without changing the momentum [2].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the Poincaré sphere extends Wigner's picture into the three space-like and one time-like coordinates. Specifically, this extension adds rotations around the given momentum which leaves the four-momentum invariant [2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

2014

View full text Add to dashboard Cite

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 that this oscillator system contains the essential features of Wigner's little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail.

show abstract

Eugene Paul Wigner's Nobel Prize

Kim¹

2016

Preprint

Self Cite

View full text Add to dashboard Cite

In 1963, Eugene Paul Wigner was awarded the Nobel Prize in Physics for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles. There are no disputes about this statement. On the other hand, there still is a question of why the statement did not mention Wigner's 1939 paper on the Lorentz group, which was regarded by Wigner and many others as his most important contribution in physics. By many physicists, this paper was regarded as a mathematical exposition having nothing to do with physics. However, it has been more than one half century since 1963, and it is of interest to see what progress has been made toward understanding physical implications of this paper and its historical role in physics. Wigner in his 1963 paper defined the subgroups of the Lorentz group whose transformations do not change the four-momentum of a given particle, and he called them the little groups. Thus, Wigner's little groups are for internal space-time symmetries of particles in the Lorentz-covariant world. Indeed, this subgroup can explain the electron spin and spins of other massive particles. However, for massless particles, there was a gap between his little group and electromagnetic waves derivable Maxwell's equations. This gap was not completely removed until 1990. The purpose of this report is to review the stormy historical process in which this gap is cleared. It is concluded that Wigner's little groups indeed can be combined into one Lorentz-covariant formula which can dictate the symmetry of the internal space-time time symmetries of massive and massless particles in the Lorentz covariant world, just like Einstein's energy-momentum relation applicable to both slow and massless particles.

show abstract

Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two-component spinors

Cited by 4 publications

References 31 publications

Loop Representation of Wigner’s Little Groups

Loop Representation of Wigner’s Little Groups

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

Eugene Paul Wigner's Nobel Prize

Contact Info

Product

Resources

About