The Yang-Mills fields on the Minkowski space期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

The Yang-Mills fields on the Minkowski space

Authors:

Qikeng Lu

Affiliation:

(1) Institute of Mathematics, Shantou University, 515063 Shantou, China;(2) Institute of Mathematics, Chinese Academy of Sciences, 100080 Beijing, China

Abstract:

Let the coordinatex=(x ⁰,x ¹,x ²,x ³) of the Minkowski spaceM ⁴ be arranged into a matrix

$H_x = left( {begin{array}{*{20}c} {x^0 + x^1 x^2 + ix^3 } {x^2 - ix^3 x^0 - x^1 } end{array} } right).$

Then the Minkowski metric can be written as

$ds^2 = eta _{jk} dx^j dx^k = det dH_x$

. Imbed the space of 2 × 2 Hermitian matrices into the complex Grassmann manifoldF(2,2), the space of complex 4-planes passing through the origin ofC ^2×4. The closure $$bar M^4 $$

ofM ⁴ inF(2,2) is the compactification ofM ⁴. It is known that the conformal group acts on $$bar M^4 $$

. It has already been proved that onF(2,2) there is anSu(2)-connection

$B(Z, dZ) = Gamma (Z, dZ) - Gamma (Z, dZ)^ + - frac{{tr[Gamma (Z, dZ) - Gamma (Z, dZ^ + ]}}{2}I.$

whereZ is a 2 × 2 complex matrix andZ ^†the complex conjugate and transposed matrix ofZ. Restrict this connection to $$bar M^4 $$

$C(H_x ,dH_x ) = [B(Z, dZ)]_{z = H_x } = C_j (x)dx^j ,$

which is anSu(2)-connection on $$bar M^4 $$

. It is proved that its curvature form

$F: = dC + C Lambda C = frac{1}{2}left[ {frac{{partial C_k }}{{partial x^j }} - frac{{partial C_j }}{{partial x^k }} + C_j C_k - C_k C_j } right]dx^j Lambda dx^k = :F_{jk} dx^j Lambda dx^k$

satisfies the Yang-Mills equation

$eta ^mu left[ {frac{{partial F_{jk} }}{{partial x^l }} + C_l F_{jk} - F_{jk} C_l } right] = 0.$

. Project partially supported by the National Natural Science Foundation of China (Grant No. 19131010) and Fundamental Research Bureau of CAS.

Keywords:

Yang-Mills fields Minkowski space Lorentz manifold

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