In the following, we define   
   * Maxwell-Lorentz theory is the electrodynamic theory derived from the   
   Maxwell-Lorentz Lagrangian   
   * Yang-Mills theory is a gauge theory for a gauge field with a   
   semi-simple Lie group, derived from the corresponding Yang-Mills   
   Lagrangian   
   * for definiteness, the definitions and explicit forms of these items are laid   
   out below.   
      
   The constitutive law in electrodynamics is:   
      
   D = epsilon_0 E, B = mu_0 H   
      
   where D is the electric induction, E the electric field, B the magnetic   
   induction, H the magnetic field, epsilon_0 the permittivity in a vacuum,   
   and mu_0 the permeability in a vacuum. The two coefficients are related   
   in such a way that the speed 1/sqrt(epsilon_0 mu_0) is one and the same   
   as the invariant speed postulated by Special Relativity, which is   
   denoted "c".   
      
   A similar set of relations hold for NON-ABELIAN fields with the   
   following provisos:   
   (1) E, B are now Lie-valued vectors, but also still 3-space vectors,   
      
   (2) D, H are now dual Lie-valued covectors, but also still 3-space   
   vectors, using a,b,c,... for Lie indexes, they would be written D_a,   
   H_a, E^a, B^a, with a corresponding basis (Y_a) and dual basis (Y^a).   
      
   (3) for non-Abelian gauge fields, with a simple Lie group, epsilon_0 and   
   mu_0 are now Lie matrices; indexed respectively as (epsilon_0)_{ab},   
   (mu_0)^{ab} and are both diagonal; the matrices continue to satisfy the   
   relation (in matrix form)   
      
   (epsilon_0)_{ab} (mu_0)^{ba'} = 1/c^2 delta_a^a'   
      
   where delta is the Kronecker delta; and the diagonal elements are   
   related to the "coupling coeffcient" up to proportionality by   
      
   (epsilon_0)_{ab} c = delta_{ab}/g^2   
      
   (4) for non-Abelian gauge fields with a semi-simple Lie group, the   
   matrix epsilon_0 c = k, where k is an adjoint-invariant metric ... this   
   decomposes into a direct sum of forms given by (3) for each factor   
   simple Lie group making up the semi-simple Lie group, thereby giving one   
   coupling coefficient for each such subgroup. Abelian factors (e.g. the   
   U(1) in SU(3) x SU(2) x U(1)) have to be handled separately by   
   orthogonalization. This entails redefining the basis element for the   
   Abelian factor with suitable additions of basis elements from the   
   non-Abelian factors.   
      
   (5) The constitutive relations are the ones obtained from a Lagrangian   
   theory with Lagrangian density L by the definitions   
      
   D = @L/@E, H = -@L/@B (@ being used to denote "partial derivative")   
      
   in which   
      
   (5a) the Lagrangian is the Maxwell-Lorentz Lagrangian in the case of   
   electromagnetism (i.e. L = 1/2 epsilon_0 c (E^2 - B^2 c^2)   
      
   (5b) the Lagrangian is the Yang-Mills Lagrangian in the gauge theory   
   case; i.e. the Lagrangian expressed - in this notation - as L = 1/2   
   k_{ab} (E^a E^b - B^a B^b c^2).   
      
   Now the question is: are these Lagrangians ... or ANY Lagrangians   
   justified by the in vacuuo constitutive laws D = epsilon_0 E, B = mu_0   
   H.   
      
   And the answer is NO! It is "no" for the very simple reason that: these   
   relations follow from ALL LAGRANGIANS, subject to only very minor   
   restrictions as follows:   
      
   (6) The Lagrangian density L(E,B) as a function of E and B reduce to a   
   function L(I,J) of the relativistic invariants I = 1/2 (E^2 - B^2 c^2)   
   and J = E.B   
      
   (7) In the non-Abelian case, the invariants are I^{ab} = 1/2 (E^a E^b -   
   B^a B^b c^2) and J^{ab} = E^a . B^b; the Lagrangian reduces to a   
   function L(I^{ab}, J^{ab}) of these   
      
   (8) The derivative epsilon def= @L/@I be non-zero (while the derivative   
   theta def= @L/@J need not be subject to any restriction at all). In the   
   non-Abelian case, @L/@I is assumed to be a non-singular matrix.   
      
   (9) The derivative epsilon_{ab} = @L/@I^{ab} also be adjoint-invariant.   
      
   A consequence of this, and of the definitions for D and H, are that one   
   has the following constitutive law:   
      
   D = epsilon E + theta B, H = epsilon c^2 B - theta E   
      
   where epsilon(I,J) and theta(I,J) are functions of the invariants I, J   
   such that epsilon != 0 and   
      
   @epsilon/@J = @theta/@I.   
      
   When far-removed from matter, the field approaches a NULL-FIELD, which   
   also happens to be defined by the conditions: I = 0, J = 0.   
   Correspondingly, the constitutive law in regions remote from matter   
   reduce to the forms   
      
   D = epsilon_0 E + theta_0 B, H = epsilon_0 c^2 B - theta_0 E   
      
   where   
      
   epsilon_0 = epsilon(0, 0), theta_0 = theta(0, 0).   
      
   By a suitable redefinition of the (D,H) fields   
      
   D redefined as D - theta_0 B, H redefined as H + theta_0 E   
      
   (which is justified, since the modified fields continue to satisfy the   
   Maxwell equations div D = rho, curl H - @D/@t = J, if the original ones   
   do; a similar observation also applies in the non-Abelian case), this   
   reduces to   
      
   D = epsilon_0 E, B = mu_0 H   
      
   where epsilon_0 mu_0 = 1/c^2, mu_0 being defined as the 1/c^2 multiple   
   of the inverse of epsilon_0.   
      
   In the non-Abelian case, the null-field value gives us the adjoint   
   invariant metric   
      
   k_{ab} = epsilon(0,0)_{ab} c   
      
   which reduces to the forms described above in (3) and (4).   
      
   A corollary of this conclusion is that:   
      
   Neither the constitutive laws D = epsilon_0 E, B = mu_0 H, nor their   
   non-Abelian generalizations are justified by the in vacuuo constitutive   
   relations - not even as microphysical laws!   
      
   The only microphysical relations justified by the observations made as   
   those just laid out:   
      
   D = epsilon(I,J) E + theta(I,J) B, H = epsilon(I,J) c^2 B - theta(I,J) E   
      
   such that   
      
   epsilon(I,J) != 0, epsilon(0,0) = epsilon_0, theta(0,0) = 0.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)