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This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, H n , together with the Euclidean, E n , and pseudo-Euclidean E p , q , groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, K p , q , that contain H p , q and E p , q as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of K p , q . We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type K p , q . By extending these Hilbert spaces, we obtain representations of K p , q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform. [ABSTRACT FROM AUTHOR]

A detailed description of the statistical properties of a system of bosons in a. harmonic trap at low temperature, which is expected to bear on the process of BE condensation, is given resorting only to the basic postulates of Gibbs and Bose, without assuming equipartition nor continuum statistics. Below Tc such discrete spectrum theory predicts for the thermo-dynamical variables a behavior different from the continuum case. In particular a new critical temperature Td emerges where the specific heat exhibits a λ-like spike. [ABSTRACT FROM AUTHOR]

A recipe is given to reproduce algebraically the Bose prescription in Quantum Statistics. Bosons are thus defined as coherent states of the D[sup +][sub 1/2] representation of the Lie-Hopf algebra su(1, 1), while h(1) is shown to be related to Boltzmann statistics. The technical power of group theory is used to show that black-body radiation formula as well as all other experimental results obtained in continuum limit do not require Bose distribution but are compatible with many other less symmetric ones. This may have dramatical effects on the predictions on Bose condensation. [ABSTRACT FROM AUTHOR]

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three-dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R-matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R-matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two-dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q. [ABSTRACT FROM AUTHOR]

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R-matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown. [ABSTRACT FROM AUTHOR]

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one-dimensional Heisenberg H(1)q and the two-dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q. [ABSTRACT FROM AUTHOR]

We introduce a multi-parameter family of bases in the Hilbert space L 2 (R) that are associated to a set of Hermite functions, which also serve as a basis for L 2 (R) . The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these "generalized Hermite functions". The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg–Weyl group and some of their extensions. [ABSTRACT FROM AUTHOR]

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle ( L 2 (C) ) and in l 2 (Z) , which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L 2 (C) and l 2 (Z) , so that all the mentioned operators are continuous. [ABSTRACT FROM AUTHOR]

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O (2) and functions on the unit circle, S U (2) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O (3 , 2) and spherical harmonics, s u (1 , 1) and Laguerre functions, s u (2 , 2) and algebraic Jacobi functions and, finally, s u (1 , 1) ⊕ s u (1 , 1) and Zernike functions on a circle. [ABSTRACT FROM AUTHOR]

In this paper, we present recent results in harmonic analysis in the real line R and in the half-line R + , which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based on the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. In addition, we introduce some new results concerning a discretized Fourier transform on the circle. We introduce new functions on the circle constructed with the use of Hermite functions with interesting properties under Fourier transformations. [ABSTRACT FROM AUTHOR]

Abstract: We show the presence of non-cyclic phases for oscillating neutrinos in the context of quantum field theory. Such phases carry information about the non-perturbative vacuum structure associated with the field mixing. By subtracting the condensate contribution of the flavor vacuum, the previously studied quantum mechanics geometric phase is recovered. [Copyright &y& Elsevier]