Quantum Theory for Mathematicians by Brian C. Hall

Author:Brian C. Hall
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

By the same argument as in Chap.​ 11, must be a constant multiple of the function The function is then a multiple of The problem is that . After all, does not satisfy the periodic boundary condition that defines the domain of B. Thus, we cannot continue to apply α ∗ to obtain an orthogonal chain of vectors and the entire argument breaks down.

What we need, then, is some additional condition that will distinguish between the “good” cases of the canonical commutation relations and the “bad” cases. One possibility for this additional condition is the exponentiated form of the canonical commutation relations, which are discussed in the following section. Our rigorous proof (Sect. 14.3) of the Stone–von Neumann theorem will follow the same outline as the heuristic argument in this section, except that the unbounded operators α and α ∗ will be replaced by certain bounded operators, constructed by an analog of the Weyl quantization.

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