**Authors:** M. Pitkänen

Few years has passed from the latest formulation for the hierarchy of Planck constant. The original hypothesis seven years ago was that the hierarchy is real. In this formulation the imbedding space was replaced with its covering space assumed to decompose to a Cartesian product of singular finite-sheeted coverings of M^{4} and CP_{2}.

Few years ago came the realization that the hierarchy could be only effective but have same practical implications. The basic observation was that the effective hierarchy need not be postulated separately but follows as a prediction from the vacuum degeneracy of Kähler action. In this formulation Planck constant at fundamental level has its standard value and its effective values come as its integer multiples so that one should write hbar_{eff}=n×hbar rather than hbar= nhbar_{0} as I have done. For most practical purposes the states in question would behave as if Planck constant were an integer multiple of the ordinary one. It was no more necessary to assume that the covering reduces to a Cartesian product of singular coverings of M^{4} and CP_{2} but for some reason I kept this assumption.

In the recent formulation this assumption is made and the emphasis is on the interpretation of the multi-sheetedness (in the sense of Riemann surfaces) resulting as a multi-furcation for a preferred extremal taking place at the partonic 2-surfaces. This gives a connection with complexity theory (say in living systems), with transition to chaos, and with general ideas about fractality. Second quantization of the multi-furcation means accepting not only superpositions of branches as single particle states but also the analogs of many-particle states obtained by allowing several branches up to the maximum number. This revives the ideas of N-atom, N-molecule etc.. already given up as too adventurous.

The question whether gravitational Planck constant h_{gr} having gigantic values results as an effective Planck constant has remained open. A simple argument suggests that gravitational four-momentum could be identified as a projection of the inertial four-momentum to the space-time surface and that the square of the gravitational four-momentum obtained using the effective metric defined by the anti-commutators of the modified gamma matrices appearing in the modified Dirac equation naturally leads to the emergence of h_{gr}.

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[v1] 2012-10-11 23:39:54

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