In this work, we present 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 ��, a space-shift operator ��, and a mass operator ��, we derive operator equations for massless particles which can be classified into two types of topological structures: the symmetric type-I with commutative �� and ��, representing a boson, and anti-symmetric type-II with {��,��} = 0, representing the fermion. We illustrate their topological differences and show that the fermion wave exhibits a twist during propagation like a Möbius strip. In contrast, the type-I boson behaves like a simple loop strip without a twist. The massless boson in our model resembles a 2D photon or a Higgs boson before symmetry breaking, while the fermion resembles a massless 2D Majorana particle. Unlike conventional string theories, we use a Möbius strip and a simple loop as the most fundamental topological structures of the quantum field excitations in 2D spacetime, representing fermionic and bosons. As an alternative to the string and loop quantum gravity theories, our approach could potentially serve as potential building blocks to construct elementary particles in the Standard Model, meriting an investigation into their topological properties in 4D spacetime.