A Compact Formula for Rotations as Spin Matrix Polynomials (original) (raw)
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SU(2) GROUP and THEORETICAL ANALYSIS OF SPIN ANGULAR MOMENTUM
Page iii ACKNOWLEDGEMEN T All praises are due to Almighty Allah the Omnipotent, the Omnipresent, the most merciful and the most compassionate, who blessed us with sound health, sympathetic teachers and friends .We express our deepest gratitude to Almighty Allah for enabling us to complete this research work. We would also like to express the appreciation to our Professor Dr. Abdul-Aziz Bhatti, such a nice gentleman he continually and convincingly conveyed a spirit of adventure in regard to research, and an excitement in regard to teaching. His clear remarks, sense of great vision, salient attitude, and valuable advises and discussions were always a source of candle for us. Without his guidance and persistent help this dissertation would not have been possible. We also take this opportunity to express our profound gratitude to our all university teachers who work hard and give us proper guidelines & encouragement to work for the attainment of ideas. No acknowledgement would ever adequately express our obligation to our parents who have always wished to see us flying high above the skies of success. Without their prayers, sacrifice, encouragement, moral support, the present work would have been a merry dream.
The relationship between the spin transformations of the special linear group of order 2, SL (2, C) and the aggregate SO(3) of the three-dimensional pure rotations when considered as a group in itself (and not as a subgroup of the Lorentz group), is investigated. It is shown, by the spinor map CT AXA X → which is an action of SL(2.C) on the space of Hermitian matrices, that the one-parameter subgroup of rotations generated are precisely those of angles which are multiples of. 2π
Cayley transforms of su(2) representations
Cayley rational forms for rotations are given as explicit matrix polynomials for any quantized spin j. The results are compared to the Curtright-Fairlie-Zachos matrix polynomials for rotations represented as exponentials.
Linear versus spin: representation theory of the symmetric groups
Algebraic Combinatorics
We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition ξ with analogous normalized linear characters evaluated on the double partition D(ξ). We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups in terms of counting maps.
Lectures on spin representation theory of symmetric groups
Cornell University - arXiv, 2011
The representation theory of the symmetric groups is intimately related to geometry, algebraic combinatorics, and Lie theory. The spin representation theory of the symmetric groups was originally developed by Schur. In these lecture notes, we present a coherent account of the spin counterparts of several classical constructions such as the Frobenius characteristic map, Schur duality, the coinvariant algebra, Kostka polynomials, and Young's seminormal form. Contents 1. Introduction 1 2. Spin symmetric groups and Hecke-Clifford algebra 4 3. The (spin) characteristic map 10 4. The Schur-Sergeev duality 17 5. The coinvariant algebra and generalizations 24 6. Spin Kostka polynomials 34 7. The seminormal form construction 40 References 50
Journal of Russian Laser Research, 2009
Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S 2 coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.
Journal of Mathematical Physics, 1993
It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
Particle spin from representations of the diffeomorphism group
Communications in Mathematical Physics, 1983
The semidirect product if Λ Jf of Schwartz 5 space £f of functions on 1R 3 with the group Jf of diffeomorphisms of 1R 3 provides a model for quantum theory based on local currents. Certain unitary representations of tf are induced by representations of SL(3,R). From the local currents in these representations, we construct the generators of local rigid rotations, with respect to which the Hubert space decomposes into invariant subspaces of fixed spin carrying representations of local SU(2). The physical interpretation of this procedure is discussed.
Linear representations of SU(2) described by using Kravchuk polynomials
2016
We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of the linear representations of SU(2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing.
Polynomial algebras and higher spins
Physics Letters A, 1996
Polynomial relations for generators of su(2) Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modied Holstein-Primako transformations in nite dimensional Fock spaces. The connection between su(2) Lie algebra and q-oscillators with a root of unity q-parameter is considered. The meaning of the polynomial relations from the point of view of quantum mechanics on a sphere are discussed.