Propagation of the Energy of Yang-Mills Fields

Abstract

Let us begin with the familiar Maxwell equations. We write a spatial point as x = (x₁,x₂,x₃) and a space-time point as (t,x) = (t,x₁,x₂,x₃); also ∂₀=∂/∂^t, ∂_k =∂/∂x (k=1,2,3), and ▽x = curl, ▽. = divergence. In terms of the electric field E and the magnetic field H, the Maxwell’s equations are

$$\left\{ {_{_{\nabla .H = \nabla \bullet E = 0}^{{\partial _{0E = \nabla xH}}}}^{{\partial _0}H = - \nabla xE}} \right. (1)$$

((1))

They may alternatively be written in a Lagrangian formulation with the Lagrangian

$$L = 1/2{\int {(\left| E \right|} ^2} - {\left| H \right|^2})dxdt (2)$$

((2))

subject to the constraints

$$\left\{ {_{E = {\partial _0}a + \nabla {A_0}}^{H = - \nabla xA}} \right. (3)$$

((3))

for some 3-vector A = (A₁,A₂,A₃) and some scalar A ₀ . This 4-vector (A₀,A₁,A₂,A₃) is the potential. It is not unique: one can add to ita 4-dimensional gradient (-∂₀,∂₁,∂₂,∂₃) without affecting E and H. The process of adding such a gradient is called a gauge transformation. (Equation (2) is to be understood formally; the integral is not assumed to converge).