Cylindrical group and massless particles

Kim, Y. S.; Wigner, E. P.

doi:10.1007/978-94-009-3051-3_35

Cited by 12 publications

(26 citation statements)

References 24 publications

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“…The boost generators along the y and z directions take similar forms. Let us then introduce the five-by-five contraction matrix [18,19]…”

Section: Contraction Of So(32) To Iso(31)mentioning

confidence: 99%

“…This mathematical method was extended to the contraction of the SO(3, 1) Lorentz group to the three-dimensional Euclidean group. More recently, Kim and Wigner considered a cylindrical surface tangent to the sphere [18,19] at its equatorial belt. This cylinder has one rotational degree of freedom and one up-down translational degree of freedom.…”

Section: Contraction Of So(32) To Iso(31)mentioning

confidence: 99%

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Poincaré Symmetry from Heisenberg’s Uncertainty Relations

2019

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It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the Sp(2) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the SO(2, 1) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group SO(3, 2), namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in SO(3, 2), it is possible, to construct the inhomogeneous Lorentz group ISO(3, 1) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz covariant world. This ISO(3, 1) group is commonly known as the Poincaré group.

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“…The boost generators along the y and z directions take similar forms. Let us then introduce the five-by-five contraction matrix [18,19]…”

Section: Contraction Of So(32) To Iso(31)mentioning

confidence: 99%

Section: Contraction Of So(32) To Iso(31)mentioning

confidence: 99%

Poincaré Symmetry from Heisenberg’s Uncertainty Relations

2019

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“…Besides the four-dimensional vector notation for spin operators (see (16), (17)), it is also convenient to use a three-dimensional notation:…”

Section: Discussionmentioning

confidence: 99%

“…Such an approach to the construction of wave functions, describing elementary particles, was suggested by Wigner in the article [15], where the discussion was restricted to particles of integer spin and to real-valued q µ with the constraints p 2 = m 2 , p µ q µ = 0, q 2 = −1. Different generalizations of the Wigner's approach were considered in [16,17,18,19,20].…”

Section: Spinmentioning

confidence: 99%

Classification of quantum relativistic orientable objects

Гитман¹,

Shelepin²

2011

Phys. Scr.

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Started from our work "Fields on the Poincaré Group and Quantum Description of Orientable Objects" (EPJC,2009), we consider here a classification of orientable relativistic quantum objects in 3 + 1 dimensions. In such a classification, one uses a maximal set of 10 commuting operators (generators of left and right transformations) in the space of functions on the Poincaré group. In addition to usual 6 quantum numbers related to external symmetries (given by left generators), there appear additional quantum numbers related to internal symmetries (given by right generators). We believe that the proposed approach can be useful for description of elementary spinning particles considering as orientable objects. In particular, their classification in the framework of the approach under consideration reproduces the usual classification but is more comprehensive. This allows one to give a group-theoretical interpretation to some facts of the existing phenomenological classification of known spinning particles. *

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“…These issues have been properly addressed since then [2,3,4,5]. The translationlike degrees of freedom for massless particles collapse into one gauge degree of freedom, and the E(2)-like little group can be obtained as the infinite-momentum limit of the O(3)-like little group.…”

Section: Introductionmentioning

confidence: 99%

Loop Representation of Wigner’s Little Groups

2017

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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.

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Cylindrical group and massless particles

Cited by 12 publications

References 24 publications

Poincaré Symmetry from Heisenberg’s Uncertainty Relations

Poincaré Symmetry from Heisenberg’s Uncertainty Relations

Classification of quantum relativistic orientable objects

Loop Representation of Wigner’s Little Groups

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