The Stone - von Neumann Theorem.
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The purpose of this thesis is to establish the Stone - von Neumann Theorem for locally compact abelian groups. The Stone - von Neumann Theorem states the requirements for when unitary representations of a locally compact abelian group G and its character group are unitarily equivalent to translations and multiplication by a certain constant on L²(M). It will therefore be necessary to establish when Hilbert spaces are isomorphic to L²(M), for some measure space M. In order to prove the Stone - von Neumann Theorem we need to establish Stone's Theorem. For motivational reasons, this will first be done in the specific case when the group in question is the real line. The general theory also depends on harmonic analysis, though not all theorems from harmonic analysis will be proved in this thesis. After showing the general Stone's Theorem, the main theorem will be shown. To conclude, we will return to the real line and look at some theory from the viewpoint of quantum mechanics.