Topology and Minimal Group Representation of the Classical Dual Sector of Quantum Theory

Abstract

In this paper we compute and analyze the geometrical and topological changes in quantum physics within the new framework of our recent work [APL Quantum 2, 016104 (2025)] on Classicalization. Thus, the results of our paper here are twofold: Topology, geometry and Classicalization and the relationship between them. With this end ,we consider various types of states on the simplest polygon: states with triangle topology as solution of the Schrodinger equation, eigenstates of a lowering operator and coset coherent states. The results of this paper are: (1) From the point of view of the metaplectic group and the classicalization, the application of the Minimal Group Representation (MGR) on the states on the triangle and the circle can be performed only if the ladder operators [Formula: see text] and [Formula: see text] are related to those of the oscillator [Formula: see text] and [Formula: see text]. (2) For the coherent state case, the application of the MGR can be viewed in the same way as (1) but considering an eigenstate of the operator [Formula: see text], or a more general case, as an eigenstate of [Formula: see text]. (3) These 2D states are edge states geometrically dual to the quantum states within the boundary of the triangle: there is interplay between states at the edge and the states inside the triangular boundary. (4) The coherent states in the triangle are generated by applying the group [Formula: see text] to the coset coherent states of the circle as a fiducial vector (e.g., a polygon with infinity sides). (5) The justification for the proposal in item (4) is easy to see by applying the cyclic subgroup [Formula: see text] of [Formula: see text] to the fiducial (coset circle coherent states): as the number of sides tends to infinity, the elements of [Formula: see text] tend to the identity. (6) The application of MGR in the case of [Formula: see text] ([Formula: see text] coherent state, reveals to us the quantum structure underlying each of the sides of the triangle (polygon). (7) The norm (probability) found shows a clear interaction/interference between the sides of the triangle and between the even and odd sectors of the Hibert space. (8) The application of MGR allows topological isolation between the projected sectors (even and odd). It can be seen that the topological effect is of a quantum nature and vice versa. (9) These topological isolation effects do not disappear at the macroscopic level (e.g. they are independent of the space-time parameters of the physical system). In general, the results here show too that quantum states in geometries with angles and vertices classicalize more than in geometries without them. (10) Within this context, a generalization of the Bargmann representation, not previously introduced in the literature, is presented. (11) These theoretical results can have impact and applications in : Classical and quantum information processing, Quantum computation, classical-quantum interaction and interpretation measurements, classical-quantum duality, classical or quantum optimization.