### Abstract

The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HP^{n} (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang's SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov's spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess-Zumino term unlike the one in the complex case.

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

Pages (from-to) | 2347-2357 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 32 |

Issue number | 9 |

Publication status | Published - 1991 |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*32*(9), 2347-2357.

**Quaternionic gauge fields and the geometric phase.** / Lévay, P.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 32, no. 9, pp. 2347-2357.

}

TY - JOUR

T1 - Quaternionic gauge fields and the geometric phase

AU - Lévay, P.

PY - 1991

Y1 - 1991

N2 - The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang's SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov's spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess-Zumino term unlike the one in the complex case.

AB - The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang's SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov's spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess-Zumino term unlike the one in the complex case.

UR - http://www.scopus.com/inward/record.url?scp=4243930568&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4243930568&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4243930568

VL - 32

SP - 2347

EP - 2357

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

ER -