It is crucial to look for a suitable

gauge transformation of a matrix spectral problem; it can transform the matrix spectral problem into another spectral problem of the same form [5, 6, 8, 23].

Gauge transformations have been found among them [9,10].

In the next Section we will consider this particular case implying accelerations of all charged particles as negligible and will show that the corresponding EMEM tensor (8) can be subjected to an appropriate

gauge transformation, which allows elimination of the divergent terms in its structure and logically brings into the absence of bound EM field retardation within the near zone.

At this juncture, we lay emphasis on the fact that the quantity (z - [??]) remains invariant under the

gauge transformations (2).

We usually describe the set by a

gauge transformation that joins each subclass.

To also be gauge invariant off-shell one must endow the matter fields with a

gauge transformation rule, such as [[phi].sub.i] [right arrow] [[phi].sub.i] + [kappa][[alpha].sub.[mu]][[partial derivative].sup.[mu]][[phi].sub.i] for scalars [4].

In particular, using the inverse scattering transform, they have shown global existence and uniqueness up to a

gauge transformation for small initial data in [W.sup.2,1]([R.sup.2]).

The existence of the gradient of the function [phi](z) = (iv/2)[[absolute value of z].sup.2], in the

gauge transformation [[theta].sub.v,[mu]] = [[theta].sup.s.sub.[mu]] + (iv/2)d[[absolute value of z].sup.2], is equivalent to multiplying the eigenstates of the Landau Hamiltonian [H.sub.[mu]]; v = 0, by the phase factor [mathematical expression not reproducible].

A

gauge transformation of the form (65) multiplies the Polyakov loop by the center element z, that is,

One can easily show that the

gauge transformation of a section [psi] = {[[psi].sub.i]} [member of] Sec(F) is an element of Sec(F).

The first one which is called the positive representation has a

gauge transformation [[psi]'.sub.+] = U(x) * [[psi].sub.+], while the second one which is called the negative representation is [[psi]'.sub.-] = [[psi].sub.-] * [U.sup.-1] (x), where the *-product is a realization of algebra (1).

An alternative modification of U(I) gauge invariance explored in ([21], where we demand that the Lagrangian density be invariant under a time-local (rather than space-local) U(1)

gauge transformation [psi] [right arrow] [psi]' = U[psi] with U being time-dependent (rather than space-dependent), generated a scalar spin0 field (rather than a 3-vector spin1 field) which we identify as the gravitational field (instead of the magnetic field).