RIGHT TRANSLATION INVARIANT METRICS AND VARIATIONAL PRINCIPLES ON A PRINCIPAL BUNDLE——TREATED AS THE UNION OF SPACE-TIME AND AN INTERNAL SPACE

Wu Yong-shi; Lu Qi-keng

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2024年10月30日

Chinese Physics C> 1980, Vol. 4> Issue(3) : 322-336

RIGHT TRANSLATION INVARIANT METRICS AND VARIATIONAL PRINCIPLES ON A PRINCIPAL BUNDLE——TREATED AS THE UNION OF SPACE-TIME AND AN INTERNAL SPACE

Wu Yong-shi ,
Lu Qi-keng

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Abstract：
In this paper, we discuss how to assign a metric on a principal bundle and howto rewrite the variational principles for a particle and for matter fields in an inva-riant from on the bundle in the principal-bundle formulation of gauge theories. Weshow that the right-translation invariant metric on the bundle must contain quantitieswhich transform exactly as gauge potentials, thus providing a new formalism for gaugefields. And we formulate the variational principle for a particle moving in the gaugefield as follows: The particle moves along a horizontal geodesic on the principalbundle. Starting from this we derive the Wong's equations of motion. Moreover, we elucidate the physical view-point which treats the bundle space asthe union of space-time and the internal space. Advantages of this viewpoint for un-derstanding the essentialities of gauge transformations and gauge invariance and forestablishing unified theories of gravitation and gauge fields are also discussed.

References

[1]

吴泳时、陈时、郭汉英,中国科学,1918年279页;Scientia Sinica, 21(1978),193.[2] R. Jackiw and C. Rebbi, Phys. Rev., D14(1976), 517.[3] 吴泳时、吴建时,科学通报,23(1978),601.[4] R .Jackiw and C. Rebbi. Phys. Rev. Lett., 36(1976), 1116.[5] 有关纤维丛的数学知识,可参阅陆启铿“纤维丛与规范场讲义”,(1973);或S. Kabayashi and B. Nomizu, "Foundations of Differential Geometry," Vol.1,1963.[6] Y. M. Cho, J. Math. Phys., 16(1975), 2029.[7] L. N. Chang, Ii. I.Macme and F. lsansouriy Pays. Rev., D13(1976), 235[8] 吴泳时、郭汉英,中国科学,1977年308页;Sctentia S4nica, 20(1977),186,[9] S. K. along. Nuovo Cvmento, 65A(1970), 689.[10] 陆启铿,物理学报,23(1974),249.[11] W. Greub, S. Halperin and R. Vanstein. "Connections, CurvatureCohomology", Vol.II,1973.[12] H. Whitney, ann. Math., 37(1935), 645.[13] C. N. Fang, Phys. rev. Lett., 33(1974), 445.[14] 郭汉英,科学通报,23(1978),407.[15] 吴泳时,Preprint IHES/P/79/271.,物理学报,29(1980),395.

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Wu Yong-shi and Lu Qi-keng. RIGHT TRANSLATION INVARIANT METRICS AND VARIATIONAL PRINCIPLES ON A PRINCIPAL BUNDLE——TREATED AS THE UNION OF SPACE-TIME AND AN INTERNAL SPACE[J]. Chinese Physics C, 1980, 4(3): 322-336.

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Received: 1979-03-13
Revised: 1900-01-01

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1.
沈阳化工大学材料科学与工程学院沈阳 110142

RIGHT TRANSLATION INVARIANT METRICS AND VARIATIONAL PRINCIPLES ON A PRINCIPAL BUNDLE——TREATED AS THE UNION OF SPACE-TIME AND AN INTERNAL SPACE

Wu Yong-shi
Lu Qi-keng

Received Date: 1979-03-13
Accepted Date: 1900-01-01
Available Online: 1980-06-05

Abstract: In this paper, we discuss how to assign a metric on a principal bundle and howto rewrite the variational principles for a particle and for matter fields in an inva-riant from on the bundle in the principal-bundle formulation of gauge theories. Weshow that the right-translation invariant metric on the bundle must contain quantitieswhich transform exactly as gauge potentials, thus providing a new formalism for gaugefields. And we formulate the variational principle for a particle moving in the gaugefield as follows: The particle moves along a horizontal geodesic on the principalbundle. Starting from this we derive the Wong's equations of motion. Moreover, we elucidate the physical view-point which treats the bundle space asthe union of space-time and the internal space. Advantages of this viewpoint for un-derstanding the essentialities of gauge transformations and gauge invariance and forestablishing unified theories of gravitation and gauge fields are also discussed.