Theory of the Poincaré Group

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: Feynman Digramsmentioning

confidence: 77%

Section: Running Waves Standing Wavesmentioning

confidence: 99%

Section: Einstein Einstein Wignermentioning

confidence: 99%

Section: Einstein Einstein Wignermentioning

confidence: 99%

See 2 more Smart Citations

Standing Waves in the Lorentz-Covariant World

Noz²

2005

Found Phys

Self Cite

“…In 1973 [14], Kim and Noz constructed a ground-state harmonic oscillator wave function which can be Lorentz boosted. It was later found that this oscillator formalisms can be extended to represent the O(3)-like little group [15,16]. This oscillator formalism has a stormy history because it ultimately plays a pivotal role in combining quantum mechanics and special relativity [17,18].…”

Section: Further Considerationsmentioning

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.

Wigner's spins, Feynman's partons, and their common ground

2002

Czech J Phys

Self Cite

The connection between spin and symmetry was established by Wigner in his 1939 paper on the Poincaré group. For a massive particle at rest, the little group is O(3) from which the concept of spin emerges. The little group for a massless particle is isomorphic to the two-dimensional Euclidean group with one rotational and two translational degrees of freedom. The rotational degree corresponds to the helicity, and the translational degrees to the gauge degree of freedom. The question then is whether these two different symmetries can be united. Another hard-pressing problem is Feynman's parton picture which is valid only for hadrons moving with speed close to that of light. While the hadron at rest is believed to be a bound state of quarks, the question arises whether the parton picture is a Lorentz-boosted bound state of quarks. We study these problems within Einstein's framework in which the energy-momentum relations for slow particles and fast particles are two different manifestations one covariant entity.This note is based on the lectures delivered at the Advanced Study Institute on Symmetries and Spin (Praha, Czech Republic, July 2001).