Quantum Mechanics on Periodic and Non-Periodic Lattices and Almost Unitary Schwinger Operators

In this work we uncover the mathematical structure of the Schwinger algebra and introduce an almost unitary Schwinger operators which are derived by considering translation operators on a finite lattice. We calculate mathematical relations between these algebras and show that the almost unitary Schwinger operators are equivalent to the Schwinger algebra. We introduce new representations for MN© in terms of these algebras.