Single-Particle Electrodynamics - Assassination Science

as a mathematically G-covariant quantity. Any mathematical quantity that
does not transform as a representation of the G group, and which, even if
considered together with an arbitrary number of other quantities that do not
transform as a representation of the G group, can still never be made to transform
as a representation of the G group, is referred to as a mathematically
non-G-covariant quantity.
Any symbolic representation of a mathematically covariant quantity is
referred to as a manifestly G-covariant quantity, or, simply, a G-covariant
quantity, where the adjective “manifest” is used for emphasis or disambiguation.
Any symbolic representation of a subpart of a G-covariant quantity, that
is not itself mathematically G-covariant, is referred to as a non-G-covariant
quantity.
Note that, according to these definitions, for any given non-G-covariant
quantity there always exist other non-G-covariant quantities such that, when
considered together, the resultant structure is, as a whole, a G-covariant
quantity. Thus, non-G-covariant quantities are never mathematically non-Gcovariant;
conversely, mathematically non-G-covariant quantities are never
non-G-covariant quantities. All quantities are either G-covariant, non-Gcovariant,
or mathematically non-G-covariant quantities.
These definitions appear to be counterintuitive, but their application will
reveal their usefulness. In particular, the “mathematically” forms of the
above definitions are rarely used in this thesis.
When unambiguous, the “G-” prefix in the above may be omitted from
a discussion where the group under consideration is understood. In such
circumstances, the “G-” may be reïnserted at any point for emphasis or
disambiguation.
The adjective “covariant” is also used in its traditional form as a conjugate
to “contravariant”, when applied to indices. This meaning of the word
“covariant” has nothing to do with the definitions above.
336