Quantum albeit Continuous Physical Theories

Since the role that 𝓟𝓣-symmetric Hamiltonians could play in quantum mechanics theory was suggested in1998, much work has been conducted on the mathematical analysis of 𝓟𝓣-symmetry. Carl Benderchallenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing anyquantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant undercombined parity-time 𝓟𝓣-symmetry transformations likewise can exhibit real eigenvalue spectra.However, 𝓟𝓣-symmetric systems began as an analytic extension of classical and quantum mechanical tothe complex domain. These findings had a particularly profound impact in the field of photonics, where PTsymmetricpotential landscapes can be implemented by introducing dissipative terms appropriatelydistributing gain and loss in open or coupled systems. The refractive index profile plays the role of thecomplex potential. Similar research activity was transferred to other physical settings such as electronicscircuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems.Unconventional characteristics compared with Hermitian ones were found and allow to engineer newdevices.Likewise, topological phases of matter are another new field of physics which have had a rapid growth inthe last years, based on the concept of Geometric or Berry phase. Topological Insulators andSuperconductors, Topological photonics and nanophotonics, Topological Computers are some of theapplications widely described in scientific journals.In Nature all the processes have to be described by a property in a discrete or iterative way instead of acontinuous way which usually is time. Continuous processes, being a mathematical idealization, has beenthe best tool to model observable properties in physics and is the conventional tool used in Classical andQuantum Mechanics. The formal way to model discrete systems is through difference equations which aresimilar to differential equations but not equal.Identification of dynamical discrete system with recurrence equations, give us the possibility to consider ananalytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution ofthe system. Parametric solutions obtained by this procedure takes automatically complex values, avoidingto make any supposition about the possibility of their analytical extension to complex domain. Thisprocedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domainunknown in the real domain. This new mathematical tool is developed to enhance and provide a moresubtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than 𝓟𝓣-symmetric systems with real spectra, keeping also topological propertiesdue to a kind of geometric phase.