Ensembles on Configuration Space pp 85-113 | Cite as

# Quantization of Classical Ensembles via an Exact Uncertainty Principle

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## Abstract

The usual Heisenberg uncertainty relation may be replaced by an exact equality, valid for all states. This can be shown by carrying out a decomposition of the momentum of a quantum state into classical and nonclassical components and choosing suitable measures of position and momentum uncertainty. The exact uncertainty relation obtained in this way is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation. The approach based on the exact uncertainty principle is conceptually very simple, being based on the core notion of statistical uncertainty, intrinsic to any interpretation of quantum theory. This quantization procedure is not restricted to particles but can also be used to derive bosonic field equations. It is remarkable that the basic underlying concept, the addition of nonclassical momentum fluctuations to a classical ensemble, carries through from quantum particles to quantum fields, without creating conceptual difficulties, although significant technical generalizations are needed. This logical consistency and range of applicability is a further strength of the exact uncertainty approach.

## Keywords

Momentum Density Classical Ensemble Jacobi Formulation Quantization Procedure Classical Component## References

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