This work presents a model to treat the relativistic quantum dynamics of massless and massive particles in a 2D Minkowski spacetime. Using a set of three independent 2x2 real-value matrices to represent a time-shift operator E, a space-shift operator P, and a mass operator M, we derive operator equations for massless particles which can be classified into two types of topological preons, the symmetric type-I with commutative E and P, representing a bosonic preon, and anti-symmetric type-II with {E,P}=0, representing the fermionic preon. We illustrate their topological differences and show that the wave of the fermionic preon exhibits a twist during propagation like a Möbius strip. In contrast, the type-I bosonic preon behaves like a simple loop strip without a twist. The massless bosonic preon in our model resembles a 2D photon or a Higgs boson before symmetry-breaking, while the fermionic preon resembles a massless 2D Majorana particle. Unlike conventional string theories involving 1D strings, in this work, we use a Möbius strip and a simple loop in 2D, representing fermionic and bosonic preons, respectively. The extension of this model to 4D spacetime as potential building blocks to construct elementary particles deserves further studies.