where

The objective is to deduce from this equation a carrier of the imprints of gravitation with the following three fundamental requirements:

1. The imprints must be carried by the evolving dynamics of a quantum mechanical wavefunction.

2. Even though the dynamical system is characterized by its mass
*m*
, the carrier and the imprints must *not* depend on this mass, i.e.
the carrier
must be *independent* of *k ^{2}*
. This requirement is analogous to the
classical one in which the world line of a particle is independent of
its mass.

3. In the absence of gravitation the carrier should yield measurable results (expectation values) which are invariant under Lorentz boosts and spacetime translations.

We shall now expand on these three requirements. In quantum mechanics the wave function plays the role which in Newtonian mechanics is played by a particle trajectory or in relativistic mechanics by a particle world line. That the wave function should also assume the task of carrying the imprints of gravitation is, therefore, a reasonable requirement.

Because of the Braginski-Dicke-Eotvos experiment, the motion of bodies
in a gravitational field is independent of the composition of these
bodies, in particular their mass. Consequently, the motion of free
particles in spacetime traces out particle histories whose details
depend only on the gravitational environment of these particles, not
on their internal constitution (uniqueness of free fall, ``weak
equivalence principle''). Recall the superposition of different wave
functions (states) of a relativistic particle yields interference
fringes which do depend on the mass of a particle (``incompatibility
between quantum and equivalence principle''
[10]).
If the task of
these wave functions is to serve as carriers of the imprints of
gravitation, then, unlike in classical mechanics, these interfering
wave functions would do a poor job at their task: They would respond
to the presence (or absence) of gravitation in a way which depends on
the details of the internal composition (mass) of a particle. This
would violate the simplicity implied by the Braginski-Dicke-Eotvos
experiment. Thus we shall not consider such carriers. This eliminates
any quantum mechanical framework based on energy and momentum
eigenfunctions because the dispersion relation, *E ^{2}*=

Recall that momentum and energy are constants of motion which imply
the existence of a locally inertial reference frame. Consequently,
requirement 2. rules out *inertial* frames as a viable spacetime
framework to accomodate any quantum mechanical carrier of the imprints
of gravitation. Requirement 2. also rules out a proposal to use the
interference fringes of the gravitational Bohm-Aharanov effect to
carry the imprints of gravitation
[11]. This is because the fringe
spacing depends on the rest mass of the quantum mechanical particle.

Requirement 3. expresses the fact that the quantum mechanical carrier must remain unchanged under the symmetry transformations which characterize a two-dimensinal spacetime. By overtly suppressing the remaining two spatial dimensions we are ignoring the requisite rotational symmetry. Steps towards remedying this neglect have been taken elsewhere [13].

We shall now exhibit a carrier which fulfills the three fundamental
requirements. It resides in the space of Klein-Gordon solutions whose
spacetime domain is that of a *pair* of frames accelerating into
opposite directions (``Rindler frames''). These frames partition
spacetime into a pair of isometric and achronally related Rindler
Sectors *I*
and *II*
,

Suppose we represent an arbitrary solution to the K-G equation in the form of a complex two-component vector normal mode expansion

This is a

4 |

solutions to the Rindler wave equation

5 |

which is the equation obtained by applying the coordinate transformation Eq.(2) to the Klein-Gordon Eq.(1).