ar X iv : m at h - ph / 0 30 70 60 v 1 2 9 Ju l 2 00 3 Generally covariant Quantum Mechanics ∗
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چکیده
We present a complete theory, which is a generalization of Bargmann's theory of factors for ray representations. We apply the theory to the generally covariant formulation of the Quantum Mechanics. In the standard Quantum Mechanics (QM) and the Quantum Field Theory (QFT) the spacetime coordinates are pretty classical variables. Therefore the question about the general covariance of QM and QFT emerges naturally just like in the classical theory: what is the effect of a changing of the spacetime coordinates in QM and QFT when the changing does not form any symmetry transformation? It is a commonly accepted believe that there are no substantial difficulties if we refer the question to the wave equation. We simply treat the wave equation, and do not say why, in such a manner as if it was a classical equation. The only problem arising is to find the transformation rule ψ → T r ψ for the wave function ψ. This procedure, which on the other hand can be seriously objected, does not solve the above stated problem. The heart of the problem as well as of QM and QFT lies in the Hilbert space of states and just in finding the representation T r of the covariance group in question. The trouble gets its source in the fact that the covariance transformation changes the form of the wave equation such that ψ and T r ψ do not belong to the same Hilbert space, which means that T r does not act in the ordinary Hilbert space. This is not compatible with the paradigm worked out in dealing with symmetry groups.
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تاریخ انتشار 2008