Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance
Abstract
The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter-e.g., elapsed time-may be determined via arbitrary data analysis of arbitrary measurements onNidentically prepared quantum systems. The limits are expressed as generalized Mandelstam-Tamm uncertainty relations, which involve the operator that generates displacements of the parameter-e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations involving the displacement and Lorentz-rotation parameters of the Poincare group.
- Publication:
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Annals of Physics
- Pub Date:
- April 1996
- DOI:
- 10.1006/aphy.1996.0040
- arXiv:
- arXiv:quant-ph/9507004
- Bibcode:
- 1996AnPhy.247..135B
- Keywords:
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- Quantum Physics
- E-Print:
- 39 pages of text plus one figure