Is Maxwell-Lorentz theory (and Yang-Mills theory) justified by the in
From
rockbrentwood@gmail.com@21:1/5 to
All on Thu Jun 11 08:39:39 2020
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.
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