Chinese Physics C > 1978, Vol. 2 > Issue(4): 295-304

THEU₁GAUGE FIELD IN ASU_NGAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES

GU CHAO-HAO

Futan Unvivercity

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Abstract：
Let Fbe a SU_Ngauge field on the space-time manifold M ₄, b_λ（ x）（λ=0,1, 2, 3） the gauge potentials, the field strengths and Q（ x） a Higgs field. All quantities b, f _λμand Q（ x） are SU_N'-valued, i.e. they are represented by N× Nanti-hermitian traceless matrices.Let M ₄' be the set of xsuch that Q（ x）≠0 and define on M ₄', where The following results are obtained:Theorem 1. The 1st set of Maxwell equations F _λμ,v+ F _μv,λ+ F_v _λ,μ=0 are satisfied for arbitrary b _λif and only if with Here s is an integer, 1≤ s≤ N-1.Suppose the conditions in theorem 1 are satisfied.Theorem 2. If sis a space-like two-dimensional surface, the value of dual charges contained in sdefined by is equal to lq', where lis an integer and Theorem 3. The value of dual charges contained in Sis equal to the integral which is independent of the gauge potentials.Theorem 4. The least positive value q' of dual charge can be attained by some Higgs fields.Remarks（a） When N=2, the results obtained are consistent with those of t Hooft, Arafune and Hou etc.（b） For N=3, we give an answer to the question of quantized values of dual charges which was discussed by Marciano and Pagels.（c） The Higgs field ø（ x） is a mapping from M' ₄into the AⅢ type symmetric space SU_N/ S( U_sX U _N-s) and the integral is an extension of Kronecker index for N=2.

References

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G. 't Hooft, Nucl. Phys., B79(1974), 276.[2] 侯伯宇、段一士、葛墨林，兰州大学学报(自然科学版)，1975, 2, 26.[3] J. Arafune, P. G. O. Freund and C. J. Goebel, Jour. Math. Phys., 16(1975), 433.[4] T. T. Wu, C. N. Yang, Phys. Rev., D12(1975), 3845;C. N. Yang, "Gauge Fields", (Proc. 1975 Hawaii Conf.).[5] 谷超豪、杨振宁，中国科学，1976, 6, 610; Scientia Sinica, 20 (1977), 177.[6] Z. F. Ezawa and H. C. Tze, Nucl. Phys., B100(1975), 1.[7] W. J. Marciano and H. Pagels, Phys. Rev., D12(1975), 1093.[8] 谷超豪，复旦学报(自然科学版)，1975, 4, 82;中国科学，1976, 3, 320.[9] И. М. Mxanoc, H 3. R., "TIpeArrabneHxA rpyrm BpaiuexHH“rppm}TIopeHUa, , (1958, Mocx日a), 33-34.[10] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press (1962), 354.

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GU CHAO-HAO. THE U ₁GAUGE FIELD IN A SU_NGAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES[J]. Chinese Physics C, 1978, 2(4): 295-304.

GU CHAO-HAO. THE U ₁GAUGE FIELD IN A SU_NGAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES[J]. Chinese Physics C, 1978, 2(4): 295-304. shu

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Received: 1976-09-07
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

THEU₁GAUGE FIELD IN ASU_NGAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES

GU CHAO-HAO

Futan Unvivercity

Received Date:1976-09-07
Accepted Date:1900-01-01
Available Online:1978-08-05

Abstract:LetFbe aSU_Ngauge field on the space-time manifoldM₄,b_λ（x）（λ=0,1, 2, 3） the gauge potentials, the field strengths andQ（x） a Higgs field. All quantities b,f_λμandQ（x） areSU_N'-valued, i.e. they are represented byN×Nanti-hermitian traceless matrices.LetM₄' be the set ofxsuch thatQ（x）≠0 and define onM₄', where The following results are obtained:Theorem 1. The 1st set of Maxwell equationsF_λμ,v+F_μv,λ+F_v_λ,μ=0 are satisfied for arbitraryb_λif and only if with Here s is an integer, 1≤s≤N-1.Suppose the conditions in theorem 1 are satisfied.Theorem 2. Ifsis a space-like two-dimensional surface, the value of dual charges contained insdefined by is equal tolq', wherelis an integer and Theorem 3. The value of dual charges contained inSis equal to the integral which is independent of the gauge potentials.Theorem 4. The least positive valueq' of dual charge can be attained by some Higgs fields.Remarks（a） WhenN=2, the results obtained are consistent with those of t Hooft, Arafune and Hou etc.（b） ForN=3, we give an answer to the question of quantized values of dual charges which was discussed by Marciano and Pagels.（c） The Higgs field ø（x） is a mapping fromM'₄into the AⅢ type symmetric spaceSU_N/S(U_sXU_N-s) and the integral is an extension of Kronecker index forN=2.