Joint position-momentum observables are defined to be covariant POV measures on the group ${\mathit{openR}}^{3}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{openR}}^{3}$ ^ of space translations and boosts. For elementary or finite quantum systems, it is shown that there is no covariant PV measure on ${\mathit{openR}}^{3}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{openR}}^{3}$ ^ into the von Neumann algebra generated by average position and momentum. So these cannot be measured sharply at the same time. For an infinite quantum system, we construct a covariant PV measure on ${\mathit{openR}}^{3}$ \ifmmode\times\else\texttimes\fi{} ${\mathit{openR}}^{3}$ ^ into the von Neumann algebra generated by average position and momentum. Therefore average position and momentum of an infinite quantum system can be measured sharply at the same time.Joint position-momentum observables are defined to be covariant POV measures on the group ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ of space translations and boosts. For elementary or finite quantum systems, it is shown that there is no covariant PV measure on ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ into the von Neumann algebra generated by average position and momentum. So these cannot be measured sharply at the same time. For an infinite quantum system, we construct a covariant PV measure on ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ into the von Neumann algebra generated by average position and momentum. Therefore average position and momentum of an infinite quantum system can be measured sharply at the same time.

Physical Review ADec 1, 19940 citations

The von Neumann Model of Measurement in Quantum Mechanics

Pier A. Mello

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Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

David Chester, Xerxes D. Arsiwalla, Louis Hirsch Kauffman +2 more

We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl’s theory. Compared with Dirac’s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl’s theory. Compared with Dirac’s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

SymmetryMar 6, 20246 citations

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Jeffrey Bub

Springer eBooksJan 1, 20016 citations

A reinterpretation of von Neumann's theory of measurement

P. A. Moldauer

Foundations of PhysicsJan 1, 19726 citations

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

David Chester, Xerxes D. Arsiwalla, Louis Hirsch Kauffman +2 more

We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian density inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras.We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian density inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

arXiv:2305.08864May 5, 20231 citation

Joint position-momentum observables are defined to be covariant POV measures on the group

{\mathit{openR}}^{3}

\ifmmode\times\else\texttimes\fi{}

{\mathit{openR}}^{3}

^ of space translations and boosts. For elementary or finite quantum systems, it is shown that there is no covariant PV measure on

{\mathit{openR}}^{3}

\ifmmode\times\else\texttimes\fi{}

{\mathit{openR}}^{3}

^ into the von Neumann algebra generated by average position and momentum. So these cannot be measured sharply at the same time. For an infinite quantum system, we construct a covariant PV measure on

{\mathit{openR}}^{3}

\ifmmode\times\else\texttimes\fi{}

{\mathit{openR}}^{3}

^ into the von Neumann algebra generated by average position and momentum. Therefore average position and momentum of an infinite quantum system can be measured sharply at the same time.