Irreducible Representations of the Five‐Dimensional Rotation Group. I

A system of interacting particles is treated by the methods of quantum mechanics. Among the variables the three Euler angles are introduced which give the orientation of the system as a whole in an inertial frame of reference. This allows an expansion of the Schrödinger eigenfunctions in terms of elements of irreducible representations of the rotation group. This method, originally suggested by Wigner, is extended to include particles with spin. Two and three particle systems, one particle having spin 1/2, the rest being spinless, are treated explicitly. If the two spinless particles are regarded as the core of a nucleus, the spin 1/2 particle as an extra nucleon, one can compare this “three particle nuclear model” with other nuclear models. It permits calculations of Nilsson diagrams and, compared with other models, a more straightforward adjustment of parameters to empirical data.

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On the unitary irreducible representations of the universal covering group of the 3 + 2 deSitter group

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100032321 ◽

1957 ◽

Vol 53 (2) ◽

pp. 290-303 ◽

Cited By ~ 26

Author(s):

Joachim B. Ehrman

Keyword(s):

Irreducible Representation ◽

Fractional Part ◽

Irreducible Unitary Representation ◽

Universal Covering ◽

Rotation Group ◽

Irreducible Representations ◽

Universal Covering Group ◽

Case Representation ◽

Covering Group ◽

Special Cases

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

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POVM construction: A simple recipe with applications to symmetric states

International Journal of Quantum Information ◽

10.1142/s0219749917500423 ◽

2017 ◽

Vol 15 (06) ◽

pp. 1750042

Author(s):

Swarnamala Sirsi ◽

Karthik Bharath ◽

S. P. Shilpashree ◽

H. S. Smitha Rao

Keyword(s):

Orthonormal Basis ◽

Dimensional Space ◽

Rotation Group ◽

Dimensional Subspace ◽

Irreducible Representations ◽

Projection Operators ◽

Simple Method ◽

Positive Operator Valued Measures ◽

Higher Dimensional ◽

Symmetric States

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

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