On a new class of non-Abelian expanding waves

Article history: Received 21 January 2008 Received in revised form 23 April 2008 Accepted 4 May 2008 Available online 22 May 2008 Editor: J.-P. Blaizot PACS: 11.15.-q 13.10.+q

where μ, ν = 0, 1, 2, 3, a, b, c = 1, 2, . . . , N, A a,ν and F a,μν are potentials and strengths of a Yang-Mills field, respectively, j a,ν are field sources, f abc are the structure constants of an N-parameter gauge group and ∂ μ ≡ ∂/∂x μ , where x μ are orthogonal space-time coordinates of the Minkowski geometry. One of the important problems is a search for non-Abelian wave solutions to the Yang-Mills equations. In Refs. [3][4][5][6][7][8][9][10][11] non-Abelian plane waves and their generalizations were studied and a number of interesting results were obtained. The case of non-Abelian expanding waves was considered in our work [12] where a class of transverse wave solutions to the Yang-Mills equations (1)- (2) was found. In the present work we study non-Abelian expanding waves with not only transverse but also longitudinal components.
Let us consider an expanding wave radiated from a star and caused by movements of its electric charges. By convention, this wave is described by the Maxwell equations. However, such a big field source could generate not only photons but also Z 0 and W ± bosons. In this case the Maxwell equations may be inapplicable since they describe fields for which only photons are their carriers. On the other hand, the Yang-Mills equations with SU(2) symmetry, which present a reasonable nonlinear generalization of the Maxwell equations, play a leading role in various models of electroweak interactions caused by photons and Z 0 and W ± bosons [1,2,13,14].
That is why in our papers [15,16] a nonlinear electrodynamics based on the Yang-Mills equations with SU(2) symmetry was studied.
In these papers we considered the Yang-Mills equations (1)- (2) with N = 3 and classical sources j 1,ν = 0 and j 2,ν = j 3,ν = 0 in the spherically symmetric case and found a class of their non-trivial exact solutions. As follows from these solutions, the field strengths F 1,0ν (ν = 0) contain an arbitrary function and hence cannot be uniquely determined from the considered Yang-Mills equations. Besides, as shown in Refs. [17][18][19][20], the classical Yang-Mills equations (1)-(2) represent a chaotic system containing unstable solutions. This means that the nonlinear electrodynamics based only on the classical Yang-Mills equations cannot be a complete theory. That is why in Refs. [15,16] we posed the problem to find a new equation additional to Eqs. (1)-(2) in order to get rid of unnecessary solutions. Let us now consider the additional equation suggested in Refs. [15,16].
First, note that the Yang-Mills equation (1) can be represented as As easily follows from (3), the components J a,ν satisfy the differential equations of charge conservation ∂ ν J a,ν = 0, a = 1, 2, . . . , N.
From (4) and (5) we find that the components J a,ν can be interpreted as N four-dimensional vectors of full current densities which are the sum of the source components j a,ν and the field components generated by the source. Let us find a correlation between the current densities j a,ν and J a,ν . For this purpose, consider a small part of a field source and let q a be its intrinsic charges corresponding to the current densities j a,ν and Q a be its full charges corresponding to J a,ν and including, besides q a , the charges of field virtual particles created inside it.
As is well known, the classical intrinsic electric energy of a homogeneous body with charge q is proportional to q 2 . That is why, applying the law of energy conservation, let us require the invariance of the value N a=1 ( Q a ) 2 which is proportional to the energy of the source small part associated with its full charges Q a . Then, since the value N a=1 ( Q a ) 2 should be the same in both the trivial case when Q a = q a and the non-trivial case when Q a = q a , we come to the following correlation: which expresses the conservation of the energy in a small part of a field source.
Using the components J a,ν of full current densities and the source components j a,ν , this correlation for a small part of a field source can be represented as From (3) and (7) we derive the following correlation: Correlation (8) was suggested in our papers [15,16] as a new equation that should be added to the Yang-Mills equations (1)-(2) to get rid of unnecessary solutions.
Consider now expanding wave solutions to the Yang-Mills equations (1)-(2) and the additional equation (8) in the region outside field sources where j a,ν = 0.
Let us seek field potentials A a,ν satisfying the Yang-Mills equations (1)-(2) and Eqs. (8) and (9) in the following form which was proposed in Ref. [12]: where u a are some functions of the wave phase y 0 = x 0 − r and of the spatial coordinates y n = x n .
We will further consider gauge groups with compact semi-simple Lie algebras which have totally antisymmetric structure constants f abc [2,21]. Then substituting expressions (10) into formula (2) for the field strengths F a,μν , we readily find As will be shown below, these field strengths satisfy Eq. (8) in the considered case (9). Let us now substitute expressions (10) and (11) for A a,ν and F a,μν into the Yang-Mills equation (1).
Then when the index ν = 0 from (1) and (9) we obtain [12] 3 where y i = x i and y 0 = x 0 − r. From this point on we shall label x i by y i when i = 1, 2, 3.
When the index ν = n = 1, 2, 3 from Eqs. (1) and (9) we obtain after reductions [12] y n It should be noted that Eqs. (12) and (13) can be represented in the form From (14) we readily find that the field strengths F a,μν of the form (11) satisfy Eq. (8) in the considered case (9). Let us denote Then from (12) and (13) we find As follows from (11) and (15), in the case p a = 0, which was investigated in Ref. [12], the considered expanding waves are transverse. We will further study the case p a = 0 corresponding to expanding waves with longitudinal components.
As can be readily verified, these equations have the following solution: where s a are arbitrary differentiable functions of the argument y 0 . From (15), (16) and (19) we get As can be readily verified, Eq. (20) has the following solution: where g a are arbitrary differentiable functions. Actually, from (22) we derive Consider Eq. (21). For the functions g a (y 0 , ξ 1 , ξ 2 , ξ 3 ) we have [12] ∂ g a Let us substitute expression (22) for the function u a into Eq. (21) and take into account that the function 1/r is harmonic and the considered constants f abc are antisymmetric. Then using formulae (24) and the evident equality ξ 2 1 + ξ 2 2 + ξ 2 3 = 1, we obtain The arguments ξ i = y i /r of the functions g a are not independent, since ξ 2 1 + ξ 2 2 + ξ 2 3 = 1. That is why instead of ξ 1 , ξ 2 , ξ 3 , we can choose two independent arguments related to them.
As is shown in Ref. [12], it is convenient to choose the following two arguments θ and σ : Then we have [12] ∂ g a and as can be readily verified, the left-hand side of (25) acquires the form Since the variables ξ i = y i /r satisfy the equality ξ 2 2 + ξ 2 3 = 1 − ξ 2 1 , from (25), (26) and (28) we come to the following equation: Let us put where v a (y 0 , θ, σ ), κ(y 0 ) and d a (y 0 ) are some functions.
Then substituting (30) into (29) and taking into account that f abc are antisymmetric, we get Let us require that the N + 1 functions κ(y 0 ) and d a (y 0 ) should satisfy the following system of N algebraic equations which are linear with respect to them: Then from (31) we get , v a = v a (y 0 , θ, σ ).
where V a (y 0 , θ, ω) are some complex functions for which the integrals (35) are finite. Then substituting (35) into Eq. (33), we come to the following equation: which provides the fulfillment of (33).