# On the Contraction of Groups and Their Representations.

@article{Inonu1953OnTC, title={On the Contraction of Groups and Their Representations.}, author={E Inonu and Eugene Paul Wigner}, journal={Proceedings of the National Academy of Sciences of the United States of America}, year={1953}, volume={39 6}, pages={ 510-24 } }

Classical mechanics is a limiting case of relativistic mechanics. Hence the group of the former, the Galilei group, must be in some sense a limiting case of the relativistic mechanics’ group, the representations of the former must be limiting cases of the latter’s representations. There are other examples for similar relations between groups. Thus, the inhomogeneous Lorentz group must be, in the same sense, a limiting case of the de Sitter groups. The purpose of the present note is to… Expand

#### 929 Citations

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

- Mathematics, Physics
- 2011

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these… Expand

The unitary representations of the Poincar\'e group in any spacetime dimension

- Physics, Mathematics
- 2006

An extensive group-theoretical treatment of linear relativistic wave equations on Minkowski spacetime of arbitrary dimension D>2 is presented in these lecture notes. To start with, the one-to-one… Expand

Dual pairing of symmetry and dynamical groups in physics

- Physics, Mathematics
- 2012

This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired… Expand

Limitable Dynamical Groups in Quantum Mechanics. I. General Theory and a Spinless Model

- Physics
- 1968

A pure group‐theoretical description of nonrelativistic interacting systems in terms of irreducible representations U(D) of a so‐called dynamical group D is investigated. The description is assumed… Expand

Contractions of representations of de Sitter groups

- Mathematics
- 1972

In order to construct the quantum field theory in a curved space with no “old” infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of… Expand

Covariant fields: Poincaré group representations and metric structure in the space of quantum states

- Physics
- 1974

The representations of the Poincaré group realized over the space of covariant fields transforming according to any irreducible representationD(m,n) of the Lorentz group are constructed explicitly… Expand

DISCRETE SERIES FOR THE UNIVERSAL COVERING GROUP OF THE 3 + 2 DE SITTER GROUP.

- Mathematics
- 1967

A classification is given of the irreducible unitary representations of the universal covering group of the 3 + 2 de Sitter group which contract to the usual physical representations of the Poincare… Expand

The contraction of the SU(1,1) discrete series of representations by means of coherent states

- Mathematics
- 1996

The group SU(1,1) is a deformation of the Poincare group. This relationship is studied both at the classical level (coadjoint orbits) and at the quantum level (unitary representations). The… Expand

General properties of the expansion methods of Lie algebras

- Mathematics, Physics
- 2013

The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new… Expand

Conformally Compactified Minkowski Space: Myths and Facts

- Mathematics
- 2012

Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a limiting case of the Poincare group,… Expand

#### References

SHOWING 1-3 OF 3 REFERENCES

Representations of the Galilei group

- Physics
- 1952

SummaryWhile the transition to a moving coordinate system (x→x+vt, t→t) commutes in classical mechanics with displacements (x→x+a, t→t), the corresponding operations in Schrödinger's nonrelativistic… Expand

Group Theoretical Discussion of Relativistic Wave Equations.

- Physics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1948

1 — The wave functions, ψ, describing the possible states of a quantum mechanical system form a linear vector space V which, in general,. is infinite dimensional and on which a positive definite… Expand