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arXiv:1308.0479 · Physical Review A

Localization of relativistic particles and uncertainty relations

Daniel R. Terno

Published Apr 21, 2014• 24 citations
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Abstract

Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.

DOI
10.1103/physreva.89.042111

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Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3 polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound \ensuremathΔp\ensuremathΔz=12\ensuremathℏ\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}\ensuremathΔp\ensuremathΔz=21​\ensuremathℏ.
We discuss relations between several relativistic spin observables and derive a Lorentz-invariant characteristic of a reduced spin density matrix. A relativistic position operator that satisfies all the properties of its nonrelativistic analog does not exist. Instead we propose two causality-preserving positive operator-valued measures (POVMs) that are based on projections onto one-particle and antiparticle spaces, and on the normalized energy density. They predict identical expectation values for position. The variances differ by less than a quarter of the squared de Broglie wavelength and coincide in the nonrelativistic limit. Since the resulting statistical moment operators are not canonical conjugates of momentum, the Heisenberg uncertainty relations need not hold. Indeed, the energy density POVM leads to a lower uncertainty. We reformulate the standard equations of the spin dynamics by explicitly considering the charge-independent acceleration, allowing a consistent treatment of backreaction and inclusion of a weak gravitational field.
We put forward an interpretation of scalar quantum field theory as relativistic quantum mechanics by curing well known problems related to locality. A probabilistic interpretation of quantum field theory similar to quantum mechanics is difficult if particle localization is defined using the Newton-Wigner position operator as it is non-local and non-covariant. An alternative bilinear covariant position operator is discussed which incorporates a time operator that can be exponentiated to a unitary operator. Moreover, it satisfies an algebra that unifies special relativity and quantum mechanics and has the same form for particles with spin. Higher power position operators are derived which yield Heisenberg's uncertainty relations. Our ideas are illustrated with a relativistic wave function whose probability density can be perfectly localized.
We discuss relations between several relativistic spin observables and derive a Lorentz-invariant characteristic of a reduced spin density matrix.A relativistic position operator that satisfies all the properties of its nonrelativistic analog does not exist. Instead we propose two causality-preserving positive operator-valued measures (POVMs) that are based on projections onto one-particle and antiparticle spaces, and on the normalized energy density. They predict identical expectation values for position. The variances differ by less than a quarter of the squared de Broglie wavelength and coincide in the nonrelativistic limit. Since the resulting statistical moment operators are not canonical conjugates of momentum, the Heisenberg uncertainty relations need not hold. Indeed, the energy density POVM leads to a lower uncertainty. We reformulate the standard equations of the spin dynamics by explicitly considering the charge-independent acceleration, allowing a consistent treatment of backreaction and inclusion of a weak gravitational field.
The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.
The Heisenberg inequality $\ensuremath{\Delta}X\ensuremath{\Delta}P>~\ensuremath{\Elzxh}/2$ can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wave functions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation. Results may be generalized to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to energy bounds, entanglement, Wigner functions, and optimal estimation of an observable from the measurement of a second observable are also given.