Here we present a study of the solutions and mathematical structure of the momentum-averaged Liouville (or Collisionless Boltzmann) equation in Fourier space. We show that the averaging procedure leads to a formalism identical to that of the density matrix of quantum mechanics. This mathematical mapping leaves the averages of all quantities unaltered and provides a unique way to construct the Hermitian version of a given operator. This seems to be the only method that resolves the ambiguity of Hermitization of operators that contain products of non-commuting variables. We also present a systematic perturbation scheme to evaluate correctly the classical solutions from the quantum ones and a formal proof of the approximate correspondence between the Poisson brackets and commutators.
Here we present a study of the solutions and mathematical structure of the momentum-averaged Liouville (or Collisionless Boltzmann) equation in Fourier space. We show that the averaging procedure leads to a formalism identical to that of the density matrix of quantum mechanics. This mathematical mapping leaves the averages of all quantities unaltered and provides a unique way to construct the Hermitian version of a given operator. This seems to be the only method that resolves the ambiguity of Hermitization of operators that contain products of non-commuting variables. We also present a systematic perturbation scheme to evaluate correctly the classical solutions from the quantum ones and a formal proof of the approximate correspondence between the Poisson brackets and commutators.
The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that this postulation is in fact unnecessary and may be replaced by a few deeper concepts, which ultimately lead to the derivation of Poisson bracket. One would only need to use Fourier transform pairs and Kramers-Kronig identities in the complex domain. We present a definition of Hermitian time-operator and discuss some of its basic properties.
Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann's equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p -conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely Δ x Δ p = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, when η is set equal to ℏ. This procedure seems to provide a statistical representation of quantum mechanics.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann’s equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p-conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely xp = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, whenη is set equal to h̄. This procedure seems to provide a statistical representation of quantum mechanics. PACS numbers: 03.65.Ta, 05.20.Dd, 45.05.+x, 03.65.Sq 1.
In the Meyer--Miller--Stock--Thoss mapping representation, the Liouville operator of the quantum--classical Liouville equation contains an excess coupling term in addition to the Poisson bracket. Neglecting this term yields the Poisson bracket mapping equation (PBME). We expand the mapping dependence of the density in Hermite functions about the Gaussian factor common to all physical mapping densities. In this basis, the excess coupling term acts purely as a raising operator and exactly cancels the raising contribution generated by the back--reaction force. Consequently, the exact dynamics does not generate higher Hermite orders and closes on a finite, sparse system of equations for the coefficient fields. Within the physical sector, these equations reduce to the subsystem quantum--classical Liouville equation. In contrast, the PBME retains the raising contribution of the force, modifying the back reaction within the physical sector while continuously generating higher Hermite orders. For Tully's dual avoided crossing model, direct propagation of the coefficient equations confirms this finite closure and agrees with independent solutions of the subsystem equation. Comparison with converged full quantum dynamics further shows that, across all regimes examined, the PBME is less accurate than the quantum--classical description for populations, coherences, and momentum distributions.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.