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# Generalized uncertainty relations: Theory, examples, and Lorentz invariance

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### 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 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 Poincar\'e group.

### Most cited references10

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### Statistical distance and the geometry of quantum states

(1994)
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### Phase and Angle Variables in Quantum Mechanics

(1968)
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• Record: found

### Geometry of quantum evolution

(1990)
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### Author and article information

###### Journal
1995-07-07
###### Article
10.1006/aphy.1996.0040
quant-ph/9507004
39 pages of text plus one figure; text formatted in LaTeX
quant-ph

Quantum physics & Field theory