Properties
Typically, the standard quantum-mechanical representation of the Heisenberg group is through its (Lie Algebra) generators: a pair of self-adjoint (Hermitian) operators on some Hilbert space, such that their commutator, a central element of the group, amounts to the identity on that Hilbert space,
the quantum Canonical commutation relation. The Hilbert space may be taken to be the set of square integrable functions on the real number line (the plane waves). One may go beyond Hilbert spaces and work in a more general Schwartz space. Depending on the space involved, various results follow:
- If f is a real-valued function, then its Weyl-map image Φ is self-adjoint.
- If f is an element of Schwartz space, then Φ is trace-class.
- More generally, Φ is a densely defined unbounded operator.
- For the standard representation of the Heisenberg group by square integrable functions, the map Φ is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).
Read more about this topic: Weyl Quantization
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