• DOI:10.1006/aphy.1996.0040
• Corpus ID: 358923

Generalized uncertainty relations: Theory, examples, and Lorentz invariance

@article{Braunstein1996GeneralizedUR,
  title={Generalized uncertainty relations: Theory, examples, and Lorentz invariance},
  author={Samuel L. Braunstein and Carlton M. Caves and Gerard J. Milburn},
  journal={Annals of Physics},
  year={1996},
  volume={247},
  pages={135-173}
}

• S. Braunstein, C. Caves, G. Milburn
• Published 7 July 1995
• Physics
• Annals of Physics

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 on N identically 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.