### Abstract

A proof of the existence of an essentially self-adjoint extension of a symmetric SO_{0}(4,1) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincaré group, is presented. Our analysis of SO_{0}(4,1) and its Lie algebra provides us with an example of an observation of Harish-Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for SO_{0}(n,1). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO_{0}(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n-dimensional Minkowski space, which has one timelike dimension.

Original language | English (US) |
---|---|

Pages (from-to) | 365-374 |

Number of pages | 10 |

Journal | Journal of Mathematical Physics |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1985 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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_{0}(4,1)',

*Journal of Mathematical Physics*, vol. 26, no. 3, pp. 365-374. https://doi.org/10.1063/1.526669

**On the integrabillty of certain symmetric representations of the Lie algebra of SO _{0}(4,1).** / Bohm, A.; Moylan, Patrick J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the integrabillty of certain symmetric representations of the Lie algebra of SO0(4,1)

AU - Bohm, A.

AU - Moylan, Patrick J.

PY - 1985/1/1

Y1 - 1985/1/1

N2 - A proof of the existence of an essentially self-adjoint extension of a symmetric SO0(4,1) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincaré group, is presented. Our analysis of SO0(4,1) and its Lie algebra provides us with an example of an observation of Harish-Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for SO0(n,1). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO0(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n-dimensional Minkowski space, which has one timelike dimension.

AB - A proof of the existence of an essentially self-adjoint extension of a symmetric SO0(4,1) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincaré group, is presented. Our analysis of SO0(4,1) and its Lie algebra provides us with an example of an observation of Harish-Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for SO0(n,1). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO0(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n-dimensional Minkowski space, which has one timelike dimension.

UR - http://www.scopus.com/inward/record.url?scp=36549096016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36549096016&partnerID=8YFLogxK

U2 - 10.1063/1.526669

DO - 10.1063/1.526669

M3 - Article

VL - 26

SP - 365

EP - 374

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -