Next we come to the special case of symmetries, i.e.,
invariance under under coordinate transformations.
Symmetry transformations change the arguments of a function .
For example for the translation of a function
.
Therefore it is useful to see how
acts on the arguments of a function.
Denoting the
(possibly improper) eigenvectors
of the coordinate operator with
eigenvalue by = , i.e.,
,
function values can be expressed as scalar products, e.g.
= for a function in ,
or, in two variables,
=
.
(Note that in this `eigenvalue' notation,
frequently used by physicists,
for example .)
Thus, we see that the action of
on some function
is equivalent to the action of
( = if orthogonal)
on
(214) |
(215) |
(216) |
Notice that, for a general operator ,
the transformed argument
does not have to be an eigenvector
of the coordinate operator again.
In the general case can map a specific
to arbitrary vectors being linear combinations
of all , i.e.,
=
.
A general orthogonal
maps an orthonormal basis
to another orthonormal basis.
Coordinate transformations, however, are represented by operators ,
which map coordinate eigenvectors to other
coordinate eigenvectors
.
Hence, such coordinate transformations
just changes
the argument of a function into , i.e.,
(217) |
A special case are linear coordinate transformations for which we can write , with (in contrast to ) acting in the space of . An example of such are coordinate rotations which preserve the norm in -space, analogously to Eq. (211) for , and form a Lie group acting on coordinates, analogously to Eq. (212).