Unitary representations of the (4 + 1)-de Sitter group on irreducible representation spaces of the Poincaré group

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Abstract

The construction of the principal continuous series of unitary representations of the simply-connected covering group of the (4 + 1)-de Sitter group on unitary irreducible representation spaces of the Poincaré group is presented. A unitary irreducible representation space of this covering group of the de Sitter group is realized as the direct sum of two irreducible representation spaces of the Poincaré group. Possible physical implications are indicated. In particular, an interpretation of the instantaneous velocity operator in the Dirac theory as the spin part of the de Sitter boosts is given. We obtain a simple method of computing the matrix elements of the generators of the de Sitter group in an SO(4) basis using the matrix elements of the generators of the four-dimensional Euclidean group. Also we obtain explicit expressions for certain matrix elements between the spinor and SO(4) basis of the representation space as functions on the coset space SO(4)/SO(3).

Original languageEnglish (US)
Pages (from-to)2706-2721
Number of pages16
JournalJournal of Mathematical Physics
Volume24
Issue number12
StatePublished - Dec 1 1982

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Unitary Representation
Irreducible Representation
coverings
Covering
generators
matrices
Generator
Coset
Spinor
acceleration (physics)
Direct Sum
Paul Adrien Maurice Dirac
Instantaneous
Euclidean
operators
Series
Computing
Operator

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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abstract = "The construction of the principal continuous series of unitary representations of the simply-connected covering group of the (4 + 1)-de Sitter group on unitary irreducible representation spaces of the Poincar{\'e} group is presented. A unitary irreducible representation space of this covering group of the de Sitter group is realized as the direct sum of two irreducible representation spaces of the Poincar{\'e} group. Possible physical implications are indicated. In particular, an interpretation of the instantaneous velocity operator in the Dirac theory as the spin part of the de Sitter boosts is given. We obtain a simple method of computing the matrix elements of the generators of the de Sitter group in an SO(4) basis using the matrix elements of the generators of the four-dimensional Euclidean group. Also we obtain explicit expressions for certain matrix elements between the spinor and SO(4) basis of the representation space as functions on the coset space SO(4)/SO(3).",
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AB - The construction of the principal continuous series of unitary representations of the simply-connected covering group of the (4 + 1)-de Sitter group on unitary irreducible representation spaces of the Poincaré group is presented. A unitary irreducible representation space of this covering group of the de Sitter group is realized as the direct sum of two irreducible representation spaces of the Poincaré group. Possible physical implications are indicated. In particular, an interpretation of the instantaneous velocity operator in the Dirac theory as the spin part of the de Sitter boosts is given. We obtain a simple method of computing the matrix elements of the generators of the de Sitter group in an SO(4) basis using the matrix elements of the generators of the four-dimensional Euclidean group. Also we obtain explicit expressions for certain matrix elements between the spinor and SO(4) basis of the representation space as functions on the coset space SO(4)/SO(3).

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