Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance

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 Lorentzrotation parameters of the Poincare group. 1996 Academic Press, Inc.