Discretization and Quantum Group Description as Different Aspects of Finite Measurement Resolution

Abstract

The identification of discretization and quantum group approach as classical and quantal aspects for the description of finite measurement resolution is one of the key ideas of quantum TGD. The article of Etera Livine motivated by loop quantum gravity provides a stimulus leading to a more detailed vision how the two views relate to each other a group description as its quantum counterpart. As an outcome the hypothesis can be now formulated more precisely and makes a lot of very general predictions. In particular, the representations of quantum group should be obtained as one decomposes the representations of group with respect to discrete algebraic subgroup.  This insight would explain and generalize some key observations about quantum group representations (finite number of spins for SU(2)_q). R-matrix defining the action of braid group defines quantum group. A connection with p-adic physics emerges: in p-adic sectors the discretization is always necessary since only discrete phases (rather than continuous angles) definable as roots of unity and their hyperbolic counterparts exist in the extensions of p-adic numbers. An infinite hierarchy of quantum groups associated with the algebraic extensions of rationals emerges if the interpretation is correct.