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Equation (13) is the center of a constellation consisting of the following three results: The main result is the recognition of the fact that a Klein-Gordon charge has a property which displays the Poincare invariance of Minkowski spacetime without involving its specific mass. This property is the set of expectation values given by Eq.(13). The presence of gravitation is expressed by distortions of this mass-independent vector field. This property is a quantum mechanical extension of the principle familiar from classical mechanics, and it is dictated by the Braginski-Dicke-Eotvos experiment, that the set of particle trajectories serve as the carrier of the imprints of gravitation.

The second result is the fact that Eq.(13) are (twice) the expectation values of the ``spin'' component operators

\begin{displaymath}{\bf L}:~L_1={\sigma _1\over 2},~L_2={\sigma _2\over 2},~
L_3={\sigma _3\over 2} \quad . 
\end{displaymath} 13

Their commutation relations

[L_3,L_1]&=&iL_2\quad ,

are those of the symmetry group SU(1,1) , which is precisely the invariance group of the fiber metric, Eq.(8). The fact that SU(1,1) is the invariance group of the single charge system extends into the classical regime: If one interprets the Klein-Gordon wave function as a classical field, then this SU(1,1) symmetry gives rise [13] to a conserved vectorial ``spin'', whose density is given by one half the expectation value of $\sigma$ , as in the numerator of Eq.(13).

The third result is as obvious as it is noteworthy: The components of the vector field coincide with the ``Planckian power'' and the ``r.m.s. thermal fluctuation'' spectra, in spite of the fact that we have not made any thermodynamic assumptions. In fact, we are considering only the quantum mechanics of a single charge, or the Klein-Gordon dynamics of a classical wave field. That these two spectra also arise within the framework of a strictly classical field theory, has already been observed in [14]. We would like to extend this observation by pointing out that in the absence of gravitation these two spectra express the invariance of spacetime under translations and Lorentz trasformations (requirement 3., Poincare invariance)

next up previous
Next: The Wider Perspective Up: SUMMARY Previous: SUMMARY
Ulrich Gerlach