This paper describes the states of quantum versions of finite dimensional Hamilton–Dirac systems in terms of (analogues of) the Wigner function (various definitions of Wigner functions for classical Hamiltonian systems can be found in [8]). We give a system of equations, which we call the Moyal–Dirac system, describing the evolution of this Wigner function. The passage from the Moyal equation describing the evolution of the quantum version of a classical Hamiltonian system to the Moyal–Dirac system describing the evolution of the quantum version of the corresponding Hamilton–Dirac system is similar to the passage from the Liouville equation describing the evolution of a classical Hamiltonian system to the Liouville–Dirac system of equations describing the evolution of the corresponding Hamilton–Dirac system. Thus, our Moyal–Dirac system is a generalization of both the Liouville–Dirac system and the Moyal equationThis paper describes the states of quantum versions of finite dimensional Hamilton–Dirac systems in terms of (analogues of) the Wigner function (various definitions of Wigner functions for classical Hamiltonian systems can be found in [8]). We give a system of equations, which we call the Moyal–Dirac system, describing the evolution of this Wigner function. The passage from the Moyal equation describing the evolution of the quantum version of a classical Hamiltonian system to the Moyal–Dirac system describing the evolution of the quantum version of the corresponding Hamilton–Dirac system is similar to the passage from the Liouville equation describing the evolution of a classical Hamiltonian system to the Liouville–Dirac system of equations describing the evolution of the corresponding Hamilton–Dirac system. Thus, our Moyal–Dirac system is a generalization of both the Liouville–Dirac system and the Moyal equation

Doklady MathematicsJan 1, 20151 citation

On the conection between the Liouville equation and the Schrodinger equation

Edelver Carnovali, H. M. França

We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant $α$ with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude $Ψ(x,t)$ can be expanded in a complete set of functions $Φ_n(x)$ defined in the configuration space; 2) the classical phase space distribution $W(x,p,t)$ obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant $α$ appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant $\hbar$ .We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant $α$ with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude $Ψ(x,t)$ can be expanded in a complete set of functions $Φ_n(x)$ defined in the configuration space; 2) the classical phase space distribution $W(x,p,t)$ obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant $α$ appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant $\hbar$.

arXiv:quant-ph/0512049Dec 6, 20053 citations

α

with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude

Ψ(x,t)

can be expanded in a complete set of functions

Φ_n(x)

defined in the configuration space; 2) the classical phase space distribution

W(x,p,t)

obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant

α

appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant

\hbar