Symmetry group and group representations associated with the thermodynamic covariance principle

The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the ℝ-multipliers of the universal covering G* of G and the characters of G*. As an application, the Poincaré group and the Galilei group, both in 3+1 and 2+1 dimensions, are considered.

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On the coupling of non-compact group representations

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1966.0137 ◽

1966 ◽

Vol 292 (1430) ◽

pp. 314-321 ◽

Cited By ~ 2

Keyword(s):

Compact Group ◽

Symmetry Group ◽

Vertex Function ◽

Group Representations

The problem of coupling three ‘most-degenerate’ representations of an SL(v, C) group is solved. It is shown that if GL(v, C) represents a collinear subgroup of a non-compact rest-symmetry group U(v, v), the number of amplitudes to be expected for a vertex function on the basis of collinear symmetry is infinite.

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WAVELET TRANSFORMS ASSOCIATED TO GROUP REPRESENTATIONS AND FUNCTIONS INVARIANT UNDER SYMMETRY GROUPS

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691305000816 ◽

2005 ◽

Vol 03 (02) ◽

pp. 167-187 ◽

Cited By ~ 3

Author(s):

HOLGER RAUHUT

Keyword(s):

Wavelet Transform ◽

Wavelet Transforms ◽

Inversion Formula ◽

Symmetric Functions ◽

Group Representation ◽

Symmetry Groups ◽

Short Time Fourier Transform ◽

Group Representations ◽

Covariance Principle ◽

Short Time

We study the wavelet transform of functions invariant under a symmetry group, where the wavelet transform is associated to an irreducible unitary group representation. Among other results a new inversion formula and a new covariance principle are derived. As main examples we discuss the continuous wavelet transform and the short time Fourier transform of radially symmetric functions on ℝd.

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SYMMETRY, GROUP REPRESENTATIONS, AND PARTICLE MULTIPLETS

Unitary Symmetry and Elementary Particles ◽

10.1016/b978-0-12-448450-4.50007-x ◽

1970 ◽

pp. 22-35

Author(s):

D.B. LICHTENBERG

Keyword(s):

Symmetry Group ◽

Group Representations

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Group Representations

10.1016/s0304-0208(09)x7004-0 ◽

1980 ◽

Keyword(s):

Group Representations

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Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽

10.3390/e21030250 ◽

2019 ◽

Vol 21 (3) ◽

pp. 250

Author(s):

Frédéric Barbaresco ◽

Jean-Pierre Gazeau

Keyword(s):

Fourier Analysis ◽

Heat Equation ◽

Harmonic Analysis ◽

Lie Groups ◽

Coherent States ◽

Geometric Theory ◽

Plancherel Formula ◽

Group Representations ◽

Modern Development ◽

Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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Deformation classes in generalized Kähler geometry

Complex Manifolds ◽

10.1515/coma-2020-0101 ◽

2020 ◽

Vol 7 (1) ◽

pp. 241-256

Author(s):

Matthew Gibson ◽

Jeffrey Streets

Keyword(s):

Symmetry Group ◽

Ricci Flow ◽

Kahler Geometry ◽

Poisson Tensor ◽

Natural Extensions ◽

Generalized Kähler Geometry ◽

Kähler Structures

AbstractWe describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

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Block-diagonalization of the variational nodal response matrix using the symmetry group theory

Journal of Computational Physics ◽

10.1016/j.jcp.2017.09.029 ◽

2017 ◽

Vol 351 ◽

pp. 230-253 ◽

Cited By ~ 7

Author(s):

Zhipeng Li ◽

Hongchun Wu ◽

Yunzhao Li ◽

Liangzhi Cao

Keyword(s):

Group Theory ◽

Symmetry Group ◽

Response Matrix ◽

Block Diagonalization

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Local material symmetry group for first- and second-order strain gradient fluids

Mathematics and Mechanics of Solids ◽

10.1177/10812865211021640 ◽

2021 ◽

pp. 108128652110216

Author(s):

Victor A. Eremeyev

Keyword(s):

Symmetry Group ◽

Strain Gradient ◽

Constitutive Equations ◽

Strain Energy ◽

Mass Density ◽

Material Model ◽

Second Order ◽

Material Symmetry ◽

Local Material ◽

Current Mass

Using an unified approach based on the local material symmetry group introduced for general first- and second-order strain gradient elastic media, we analyze the constitutive equations of strain gradient fluids. For the strain gradient medium there exists a strain energy density dependent on first- and higher-order gradients of placement vector, whereas for fluids a strain energy depends on a current mass density and its gradients. Both models found applications to modeling of materials with complex inner structure such as beam-lattice metamaterials and fluids at small scales. The local material symmetry group is formed through such transformations of a reference placement which cannot be experimentally detected within the considered material model. We show that considering maximal symmetry group, i.e. material with strain energy that is independent of the choice of a reference placement, one comes to the constitutive equations of gradient fluids introduced independently on general strain gradient continua.

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