### Abstract

It is argued that the familiar algebra of non-commutative space-time with c-number θ^{μν} is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting θ^{μν} to an anti-symmetric tensor operator θ̂^{μν}. The simplest among them is the Doplicher-Fredenhagen-Roberts (DFR) algebra, in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional θ-space of the hermitian operators, θ̂^{μν} . It is shown that we then recover the Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need for the compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to the division of the θ-space into two disjoint spaces not connected by any Lorentz transformation, so that the CCZ covariant moment formula holds in each space separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on M_{4} × Z_{2} obtained in the commutative limit.

Original language | English |
---|---|

Pages (from-to) | 663-685 |

Number of pages | 23 |

Journal | Progress of Theoretical Physics |

Volume | 109 |

Issue number | 4 |

DOIs | |

Publication status | Published - 01-04-2003 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

### Cite this

*Progress of Theoretical Physics*,

*109*(4), 663-685. https://doi.org/10.1143/PTP.109.663

}

*Progress of Theoretical Physics*, vol. 109, no. 4, pp. 663-685. https://doi.org/10.1143/PTP.109.663

**Lorentz-invariant non-commutative space-time based on DFR algebra.** / Kase, Hiromi; Morita, Katsusada; Okumura, Yoshitaka; Umezawa, Eizo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Lorentz-invariant non-commutative space-time based on DFR algebra

AU - Kase, Hiromi

AU - Morita, Katsusada

AU - Okumura, Yoshitaka

AU - Umezawa, Eizo

PY - 2003/4/1

Y1 - 2003/4/1

N2 - It is argued that the familiar algebra of non-commutative space-time with c-number θμν is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting θμν to an anti-symmetric tensor operator θ̂μν. The simplest among them is the Doplicher-Fredenhagen-Roberts (DFR) algebra, in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional θ-space of the hermitian operators, θ̂μν . It is shown that we then recover the Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need for the compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to the division of the θ-space into two disjoint spaces not connected by any Lorentz transformation, so that the CCZ covariant moment formula holds in each space separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on M4 × Z2 obtained in the commutative limit.

AB - It is argued that the familiar algebra of non-commutative space-time with c-number θμν is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting θμν to an anti-symmetric tensor operator θ̂μν. The simplest among them is the Doplicher-Fredenhagen-Roberts (DFR) algebra, in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional θ-space of the hermitian operators, θ̂μν . It is shown that we then recover the Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need for the compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to the division of the θ-space into two disjoint spaces not connected by any Lorentz transformation, so that the CCZ covariant moment formula holds in each space separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on M4 × Z2 obtained in the commutative limit.

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U2 - 10.1143/PTP.109.663

DO - 10.1143/PTP.109.663

M3 - Article

VL - 109

SP - 663

EP - 685

JO - Progress of Theoretical Physics

JF - Progress of Theoretical Physics

SN - 0033-068X

IS - 4

ER -