ULRICH H. GERLACH
Department of Mathematics, Ohio State University, Columbus, OH 43210, USAWe exhibit a purely quantum mechanical carrier of the imprints of gravitation by identifying for a relativistic charge a property which (i) is independent of its mass and (ii) expresses the Poincare invariance of spacetime in the absence of gravitation. This carrier is a Klein-Gordon-equation-determined vector field given by the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra.
Does there exist a purely quantum mechanical carrier of the imprints of gravitation? The motivation for considering this question arises from the following historical scenario: Suppose the time is 1907 when Einstein had the ``happiest thought of his life'', which launched him on the path toward his formulation of gravitation (general relativity). But suppose Einstein already knew relativistic quantum mechanics, and that, in fact, he accepted and appreciated it without any reservations before he started on his journey. How different would his theory of gravitation have been from what we have today? Put differently, how different would the course of history have been if Einstein had grafted relativistic quantum mechanics onto the roots of gravitation instead of its trunk or branches?
Nontrivial relativistic quantum mechanics starts with the Klein-Gordon
The objective of this brief report 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 particle mass m , the carrier and imprints must not depend on the particle species, i.e. the carrier must be independent of k2 . 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.
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 Dicke-Eotvos experiment, the motion of bodies in a gravitational field is independent of the composition of these bodies. 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. The superposition of different wave functions (states) of a relativistic particle yields interference fringes which do depend on the mass of a particle. 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 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, E2=m2 +p2z +p2y +p2x , of these waves depends on the internal mass m .
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 . 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 .
We shall now exhibit a carrier which fulfills the three fundamental
requirements. That carrier 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
This representation 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 usual Klein-Gordon inner product. Instead, we shall adopt a much more powerful viewpoint based on the vector bundle . Here C2 is the complex vector space of two-spinors, which is the fiber over the one-dimensional base manifold , the real line of Rindler frequencies in the mode integral, Eq.(3).
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).
The mode representation of Eq.(3) determines two parallel spinor
fields over R
, one corresponding to ``spin up'', the other to ``spin
down''. It is not difficult to verify that these two spinor fields are
(Klein-Gordon) orthonormal in each fiber over R
. The spinor field
We know that in the absence of gravitation each of the positive and
negative Minkowski plane wave solutions evolves independently of all
the others. This scenario does not change under Lorentz boosts and
spacetime translations. Will the proposed carriers comply with this
invariance, which is stipulated by fundamental requirement 3.? To
find out, consider a typical plane wave, which in the spinor
representation (3) is a state with a high degree of correlation
between the boost energy and the polarization (``spin'') degrees of
freedom. Suppose for each boost energy we determine the normalized
Stokes parameters of this polarization, i.e. the three Klein-Gordon
values of the ``spin'' operator
. This is a
three-dimensional vector field over the base manifold R
, and is given by
Its obvious but noteworthy feature is that its components coincide with the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra, in spite of the fact that we are only considering the quantum mechanics of a single charge.
 J.S.Anandan in B.L.Hu, M.P.Ryan, and C.V.Vishveshwara (eds.), Directions in General Relativity, Volume 1, (Cambridge University Press, 1993) p.10
 U.H.Gerlach in R.T.Jantzen and G.M.Keiser (eds.), The Seventh Marcel Grossmann Meeting, Part B, World Scientific Publishing Co. (1996), ibid International Jour. of Mod. Phys. 11, 3667 (1996) p.957