The representation (3) puts us at an important mathematical juncture: We shall forego the usual picture of viewing this solution as an element of Hilbert space ( ) with the Klein-Gordon inner product,

Instead, we shall adopt a qualitatively new and superior viewpoint.

We know that one can add vectors in the *same* vector (fiber)
space. However, one may not, in general, add vectors belonging to
different vector spaces at different
's. The exception is when
vectors in different vector spaces are *parallel*. In that case
one may add these vectors. The superposition of modes, Eq.(3),
demands that one do precisely that in order to obtain the two
respective total amplitudes of Eq.(3). In
brief, we are about to show that that *the linear superposition principle
determines a unique law of parallel transport*.

The mode representation of Eq.(3) determines two parallel basis spinor
fields over *R*
,

one corresponding to ``spin up'' ( has zero support in Rindler

One can see from Eq.(6) that, relative to this
basis, the fiber metric over each
is given by

It is evident that the law of parallel transport defined by
Eq.(7) is *compatible* with this
(Klein-Gordon induced) fiber metric. This is because the two parallel
vector fields, which are represented by

relative to the basis (7), are orthonormal with respect to the the fiber metric Eq.(8) at each point of the base space

9 |

is a section of the fiber bundle and it represents a linear combination of the two parallel vector fields. It is clear that there is a