THE U1 GAUGE FIELD IN A SUN GAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES

GU CHAO-HAO

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2024年10月30日

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

THE U₁ GAUGE FIELD IN A SU_N GAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES

GU CHAO-HAO

Futan Unvivercity

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Abstract：
Let F be a SU_N gauge field on the space-time manifold M₄, b_λ（x）（λ=0,1, 2, 3） the gauge potentials, the field strengths and Q（x） a Higgs field. All quantities b, f_λμ and Q（x） are SU_N'-valued, i.e. they are represented by N×N anti-hermitian traceless matrices.Let M₄' be the set of x such that Q（x）≠0 and define on M₄', 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_λ if and only if with Here s is an integer, 1≤s≤N-1.Suppose the conditions in theorem 1 are satisfied.Theorem 2. If s is a space-like two-dimensional surface, the value of dual charges contained in s defined by is equal to lq', where l is an integer and Theorem 3. The value of dual charges contained in S is 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'₄ into the AⅢ type symmetric space SU_N/S(U_s X U_N-s) and the integral is an extension of Kronecker index for N=2.

References

[1]

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 U₁ GAUGE FIELD IN A SU_N GAUGE FIELD AND THE QUANTIZED VALUES OF DUAL CHARGES[J]. Chinese Physics C, 1978, 2(4): 295-304.

GU CHAO-HAO. THE U₁ GAUGE FIELD IN A SU_N GAUGE 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

THE U₁ GAUGE FIELD IN A SU_N GAUGE 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: Let F be a SU_N gauge field on the space-time manifold M₄, b_λ（x）（λ=0,1, 2, 3） the gauge potentials, the field strengths and Q（x） a Higgs field. All quantities b, f_λμ and Q（x） are SU_N'-valued, i.e. they are represented by N×N anti-hermitian traceless matrices.Let M₄' be the set of x such that Q（x）≠0 and define on M₄', 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_λ if and only if with Here s is an integer, 1≤s≤N-1.Suppose the conditions in theorem 1 are satisfied.Theorem 2. If s is a space-like two-dimensional surface, the value of dual charges contained in s defined by is equal to lq', where l is an integer and Theorem 3. The value of dual charges contained in S is 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'₄ into the AⅢ type symmetric space SU_N/S(U_s X U_N-s) and the integral is an extension of Kronecker index for N=2.