### Abstract

The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group, S L(2, R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group of SL(2, R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.

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

Pages (from-to) | 169-204 |

Number of pages | 36 |

Journal | Communications in Mathematical Physics |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1972 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Ultralocal quantum field theory in terms of currents - I. The free theory.** / Newman, Charles.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 26, no. 3, pp. 169-204. https://doi.org/10.1007/BF01645089

}

TY - JOUR

T1 - Ultralocal quantum field theory in terms of currents - I. The free theory

AU - Newman, Charles

PY - 1972/9

Y1 - 1972/9

N2 - The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group, S L(2, R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group of SL(2, R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.

AB - The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group, S L(2, R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group of SL(2, R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.

UR - http://www.scopus.com/inward/record.url?scp=0042105529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042105529&partnerID=8YFLogxK

U2 - 10.1007/BF01645089

DO - 10.1007/BF01645089

M3 - Article

VL - 26

SP - 169

EP - 204

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -