On Some Unitary Representations of the Galilei Group I. Irreducible Representations

1965 ◽  
Vol 6 (10) ◽  
pp. 1519-1529 ◽  
Author(s):  
J. Voisin
Keyword(s):  
Unitary Representations ◽  
Irreducible Representations ◽  
Galilei Group ◽  
Group I

scholarly journals Certain Unitary Representations of the Infinite Symmetric Group, I

1987 ◽  
Vol 105 ◽  
pp. 121-128 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Type I ◽  
Discrete Topology ◽  
Type Ii ◽  
Infinite Symmetric Group ◽  
Von Neumann ◽  
Group I

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

2009 ◽  
Vol 324 (12) ◽  
pp. 2490-2505 ◽  
Author(s):  
V. Colussi ◽  
H. Reich ◽  
S. Wickramasekara
Keyword(s):  
Unitary Representations ◽  
Galilei Group

Representations of the Bondi—Metzner—Sachs group I. Determination of the representations

1972 ◽  
Vol 330 (1583) ◽  
pp. 517-535 ◽  
Keyword(s):  
Elementary Particles ◽  
Wave Functions ◽  
Unitary Representations ◽  
Continuous Spin ◽  
Gravitational Fields ◽  
Group I ◽  
Irreducible Unitary Representations ◽  
Infinite Component ◽  
Striking Contrast

Following a historical introduction, it is suggested that irreducible unitary representations of the Bondi-Metzner-Sachs group may be used to classify elementary particles in a quantum theory which takes ‘asymptotically flat5 gravitational fields into account. The unitary representations of the group induced from irreducible unitary representations of the connected little groups are all determined. It is shown that the connected little groups are all compact, so that the ‘spins’ of the corresponding particles are necessarily discrete, and the wave functions have a finite number of components. Furthermore, the spins are of precisely the observed type. This is in striking contrast to the situation for the Poincare group, for which the spins may be discrete or continuous. (The continuous spin wave functions are infinite-component.) It is concluded that the B.M.S. group may provide an explanation for the observed discreteness of the spins of elementary particles.

Irreducible Representations of the Five‐Dimensional Rotation Group. I

1968 ◽  
Vol 9 (8) ◽  
pp. 1224-1230 ◽  
Author(s):  
N. Kemmer ◽  
D. L. Pursey ◽  
S. A. Williams
Keyword(s):  
Rotation Group ◽  
Irreducible Representations ◽  
Group I

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

2013 ◽  
Vol 28 (05n06) ◽  
pp. 1350013 ◽  
Author(s):  
R. G. G. AMORIM ◽  
M. C. B. FERNANDES ◽  
F. C. KHANNA ◽  
A. E. SANTANA ◽  
J. D. M. VIANNA
Keyword(s):  
Quantum Mechanics ◽  
Group Theory ◽  
Phase Space ◽  
Wigner Function ◽  
Physical Theory ◽  
Star Product ◽  
Unitary Representations ◽  
Theory Approach ◽  
Galilei Group ◽  
Symplectic Representations

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

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