Energy–time uncertainty relations and time operators
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It is proved that for any Hamiltonian in a separable Hilbert space, having a non-empty absolutely continuous spectrum, there exists a time operator densely defined in the subspace of absolutely continuous vectors. This result is obtained by using the Carbó-Dorca parameterized vector spaces and the spectral representation theorem for self-adjoint operators in Hilbert spaces. The restriction of the Hamiltonian to the absolutely continuous subspace and its time operator are incompatible. These results bring a completely new light on the energy–time uncertainty relations. The possible physical interpretations and related facts are also examined.
Keywordsuncertainty relations time operators energy–time uncertainty relations cyclic operators parameterized vector spaces
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