Loop Representation of Wigner’s Little Groups

Başkal, Sibel; Kim, Young S.; Noz, Marilyn E.

doi:10.3390/sym9070097

Cited by 3 publications

(8 citation statements)

References 42 publications

(78 reference statements)

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“…This aspect was observed first by Han et al in 1983 [19], and its geometry was given by Kim and Wigner in 1990 [10]. The most recent version of this geometry was given by the present authors in 2017 [9].…”

Section: Four-vectors Four-potential and The Gauge Transformationmentioning

confidence: 67%

“…Moreover, its Hermitian conjugate is not necessarily its inverse. This two-by-two representation has extensively been studied in the literature [4,5,6,7,8,9].…”

Section: Wigner's Little Groupsmentioning

confidence: 99%

“…When all types of the two-component spinors are combined in this regime, totally there are 16 states. The construction mechanism of those 16 states are similar to those that are constructed in the SU (2) regime, and its details were given in our earlier work [9]. For the purpose of this work it will suffice to give the results in Table 6.…”

Section: Massive and Massless Particlesmentioning

confidence: 99%

“…The matrices given in Eq. ( 49) were given in our earlier paper on this subject [9]. However, as we go deeper into the problem by starting from the transformation property of each spinor, it was inevitable to make a number of minus-sign adjustments.…”

Section: Four-vectors Four-potential and The Gauge Transformationmentioning

confidence: 99%

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Photons in the Quantum World

Başkal¹,

Kim²,

Noz³

2017

Preprint

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Einstein's photo-electric effect allows us to regard electromagnetic waves as massless particles. Then, how is the photon helicity translated into the electric and magnetic fields perpendicular to the direction of propagation? This is an issue of the internal space-time symmetries defined by Wigner's little group for massless particles. It is noted that there are three generators for the rotation group defining the spin of a particle at rest. The closed set of commutation relations is a direct consequence of Heisenberg's uncertainty relations. The rotation group can be generated by three two-by-two Pauli matrices for spin-half particles. This group of two-bytwo matrices is called SU (2), with two-component spinors. The direct product of two spinors leads to four states leading to one spin-0 state and one spin-1 state with three sub-states. The SU (2) group can be expanded to another group of two-by-two matrices called SL(2, c), which serves as the covering group for the group of Lorentz transformations. In this Lorentz-covariant regime, it is possible to Lorentz-boost the particle at rest to its infinite-momentum or massless state. Also in this SL(2, c) regime, there are four spin states for each particle, as in the case of the Dirac equation. The direct product of two SL(2, c) spinors thus leads sixteen states. Among them, four of them can be used for the electromagnetic four-potential, and six for the Maxwell tensor. The gauge degree of freedom is shown to be a Lorentz-boosted rotation. The polarization of massless neutrinos is interpreted as a consequence of the gauge invariance.

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Section: Four-vectors Four-potential and The Gauge Transformationmentioning

confidence: 67%

“…Moreover, its Hermitian conjugate is not necessarily its inverse. This two-by-two representation has extensively been studied in the literature [4,5,6,7,8,9].…”

Section: Wigner's Little Groupsmentioning

confidence: 99%

Section: Massive and Massless Particlesmentioning

confidence: 99%

Section: Four-vectors Four-potential and The Gauge Transformationmentioning

confidence: 99%

See 2 more Smart Citations

Photons in the Quantum World

Başkal¹,

Kim²,

Noz³

2017

Preprint

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“…We can use the Lagrangian F µνi F µνi to describe the interaction of left-handed fermions with the connection A i µ . We choose a representation of SL(2, C) in which we have K i = iJ i , where K i are boost generators and J i are rotation generators [24]. Our connection A i is a one-form complex given by the self-dual projection A i µ = P i IJ ω IJ µ of the spin connection ω IJ µ according to the decomposition so(3, 1 : C) = so(3 : C) ⊕ so(3 : C) [25].…”

Section: Beta Functionmentioning

confidence: 99%

Yang–Mills Theory of Gravity

Matwi¹

2019

Physics

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The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

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Probing Wigner rotations for any group

Oblak

2018

Journal of Geometry and Physics

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Wigner rotations are transformations that affect spinning particles and cause the observable phenomenon of Thomas precession. Here we study these rotations for arbitrary symmetry groups with a semi-direct product structure. In particular we establish a general link between Wigner rotations and Thomas precession by relating the latter to the holonomies of a certain Berry connection on a momentum orbit. Along the way we derive a formula for infinitesimal, Liealgebraic transformations of one-particle states. *

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Loop Representation of Wigner’s Little Groups

Cited by 3 publications

References 42 publications

Photons in the Quantum World

Photons in the Quantum World

Yang–Mills Theory of Gravity

Probing Wigner rotations for any group

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