States of two-level systems (systems with two eigenstates, such as optical propagation of polarized light or spin 1/2) are conventionally represented by 2 $\times$ 1 complex vectors or real 3 $\times$ 1 geometric vectors. A limitation of geometrical representations is an inability to represent overall phase, important in optical and quantum interference applications. We propose an extension of the usual geometrical representation for such systems, representing the overall phase as an additional, internal, spin of the geometric state vector. We generalize the earlier representations to include this phase, establish rules for its representation and calculation, and illustrate our model by analyzing an optical interferometer.
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