Space-time geometry of relativistic particles

Noz²

2005

Found Phys

Self Cite

When Einstein formulated his special relativity, he developed his dynamics for point particles. Of course, many valiant efforts have been made to extend his relativity to rigid bodies, but this subject is forgotten in history. This is largely because of the emergence of quantum mechanics with wave-particle duality. Instead of Lorentz-boosting rigid bodies, we now boost waves and have to deal with Lorentz transformations of waves. We now have some understanding of plane waves or running waves in the covariant picture, but we do not yet have a clear picture of standing waves. In this report, we show that there is one set of standing waves which can be Lorentz-transformed while being consistent with all physical principle of quantum mechanics and relativity. It is possible to construct a representation of the Poincaré group using harmonic oscillator wave functions satisfying space-time boundary conditions. This set of wave functions is capable of explaining the quantum bound state for both slow and fast hadrons. In particular it can explain the quark model for hadrons at rest, and Feynman's parton model hadrons moving with a speed close to that of light.

Section: Einstein Einstein Wignermentioning

confidence: 99%

Standing Waves in the Lorentz-Covariant World

Noz²

2005

Found Phys

Self Cite

“…In 1987, Kim and Wigner [3] observed that it is also possible to make a cylindrical approximation of the spherical surface around the equatorial belt. While the correspondence between O(3) and the O(3)-like little group is transparent, the E(2)-like little group contains both the E(2) group and the cylindrical group [4]. We study this aspect in detail in this report.…”

Section: Introductionmentioning

confidence: 97%

“…The matrices Q 1 and Q 2 generate translations along the direction of z axis. The group generated by these three matrices is called the cylindrical group [3,4]. We can achieve the contractions to the Euclidean and cylindrical groups by taking the large-radius limits of…”

Section: Contraction Of O(3) To E(2)mentioning

confidence: 99%

Internal space-time symmetries of massive and massless particles and their unification

Nuclear Physics B - Proceedings Supplements

2001

It is noted that the internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted also that, while the rotational degree of freedom for a massless particle leads to its helicity, the two translational degrees of freedom correspond to its gauge degrees of freedom. It is shown that the E(2)-like symmetry of of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like symmetry of massive particles. This mechanism is illustrated in terms of a sphere elongating into a cylinder. In this way, the helicity degree of freedom remains invariant under the Lorentz boost, but the transverse rotational degrees of freedom become contracted into the gauge degree of freedom.

“…We are interested in the subgroups of the Lorentz group, whose transformations leave the four-momentum of a given particle invariant. This is an old problem and has been repeatedly discussed in the literature [6,7,9]. In this paper, we discuss this problem using the two-by-two formulation of the Lorentz group.…”

Section: Introductionmentioning

confidence: 99%

“…As for the problems in relativity, we shall discuss here Wigner's little groups dictating the internal space-time symmetries of relativistic particles [6]. In his original paper of 1939 [7], Wigner considered the subgroups of the Lorentz group, whose transformations leave the four-momentum of a given particle invariant.…”

Section: Introductionmentioning

confidence: 99%

Symmetries Shared by the Poincaré Group and the Poincaré Sphere

Noz

2013

Symmetry

Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner's little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics.