A Compact Formula for Rotations as Spin Matrix Polynomials

@article{Curtright2014ACF,
  title={A Compact Formula for Rotations as Spin Matrix Polynomials},
  author={Thomas L. Curtright and David B. Fairlie and C. Zachos},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2014}
}
Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed. 

Elementary results for the fundamental representation of SU(3)

More on Rotations as Spin Matrix Polynomials

Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of

Cayley transforms of su(2) representations

Cayley rational forms for rotations are given as explicit spin matrix polynomials for any j. The results are compared to the Curtright-Fairlie-Zachos matrix polynomials for rotations represented as

On rotations as spin matrix polynomials

Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the

Matrix exponentials, SU(N) group elements, and real polynomial roots

The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU(N) can be expressed

The Quantum Mechanical Rotation Operators of Spins 5/2 to 7/2

With the creation of logic gates and algorithms for quantum computers and entering our lives, it is predicted that great developments will take place in this area and important efforts are made. Spin

A New Relativistic Wave Equation for a Massive Boson and Fermion System

This article pretends to show some results from the study of The Relativistic Wave Equation in order to describe a massive system of boson and fermion. We defined the scattering matrix Vf i on the

Topology and entanglement in quench dynamics

  • P. Chang
  • Physics, Mathematics
    Physical Review B
  • 2018
We classify the topology of quench dynamics by homotopy groups. A relation between the topological invariant of a post-quench order parameter and the topological invariant of a static Hamiltonian is

Exact nonlinear dynamics of Spinor BECs applied to nematic quenches

In this thesis we study the nonlinear dynamics of spin-1 and spin-2 BoseEinstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order.

Spin-selective Aharonov-Casher caging in a topological quantum network

A periodic network of connected rhombii, mimicking a spintronic device, is shown to exhibit an intriguing spin selective extreme localization, when submerged in a uniform out of plane electric field.

References

SHOWING 1-10 OF 20 REFERENCES

On the direct calculations of the representations of the three-dimensional pure rotation group

  • Y. Lehrer-Ilamed
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1964
Abstract Explicit formulae are given for calculating the matrix elements of the irreducible representations of the three-dimensional pure rotation group by the direct method. In addition explicit

On rotations as spin matrix polynomials

Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the

Representations of the three‐dimensional rotation group by the direct method

The irreducible representations of the three‐dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in

Spin‐Matrix Polynomials and the Rotation Operator for Arbitrary Spin

A technique for the expansion of an arbitrary analytic function of a spin matrix is developed in terms of a complete set of polynomials based on the characteristic equations of the spin matrices. The

Spin-matrix polynomial development of the Hamiltonian for a free particle of arbitrary spin and mass.

Hamiltonian for free particle of arbitrary spin and mass formulated in terms of spin matrix polynomials, noting independence of expansion coefficients

Laplace-Runge-Lenz vector for arbitrary spin

A countable set of superintegrable quantum mechanical systems is presented which admit the dynamical symmetry with respect to algebra so(4). This algebra is generated by the Laplace-Runge-Lenz vector

Introduction to commutative algebra

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *

Central factorial numbers; their main properties and some applications.

The purpose of this paper is to present a systematic treatment of central factorial numbers (cfn), including their main properties, as well as to employ them in a variety of applications. The cfn are

Introduction to Fourier analysis and wavelets

This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary