Möbius and simple loop strips as 2D topological spacetime structures of preonic fermions and bosons

This work presents a model to treat the relativistic quantum dynamics of particles in a 2D Minkowski spacetime. Using independent 2x2 real-value matrices to represent a time-shift operator E, a space-shift operator P, and a mass operator M, we first derive and show these exist only two types of operator equations, representing a bosonic preon for the symmetric type-I case with commutative E and P, and a fermionic preon for the anti-symmetric type-II case with an anti-commutative relation. We illustrate their topological differences and show that the wave during propagation of the type-II preon as a Weyl-fermion exhibits a twist like a Mobius strip. In contrast, the type-I bosonic preon behaves like a simple loop strip without a twist. We have also examined the case with a rest mass for a 2D particle and a Dirac particle in 4D. Unlike the conventional string theories, our model consists of two fundamental structures, a Mobius-strip fermionic preon, and a simple-loop bosonic preon. These two topological preonic structures can be used as the most fundamental building blocks for constructing elementary particles of higher dimensions.