We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ and a skew-symmetric vorticity tensor ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$, where $\mathbf{\ensuremath{\xi}}$ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ in the presence of an effective magnetic field ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$. To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.
We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor g\ensuremathμ\ensuremathν(\ensuremathξ,t) and a skew-symmetric vorticity tensor F\ifmmode~\else˜\fi\ensuremathμ\ensuremathν(\ensuremathξ,t), where \ensuremathξ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric g\ensuremathμ\ensuremathν(\ensuremathξ,t) in the presence of an effective magnetic field F\ifmmode~\else˜\fi\ensuremathμ\ensuremathν(\ensuremathξ,t). To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.
We formulate equations of time-dependent density functional theory (TDDFT) in the comoving Lagrangian reference frame. The main advantage of the Lagrangian description of many-body dynamics is that in the comoving frame the current density vanishes, while the density of particles becomes independent of time. Therefore a comoving observer will see a picture which is very similar to that seen in the equilibrium system from the laboratory frame. It is shown that the most natural set of basic variables in TDDFT includes the Lagrangian coordinate, $\mathbf{\ensuremath{\xi}}$, a symmetric deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, and a skew-symmetric vorticity tensor, ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$. These three quantities, respectively, describe the translation, deformation, and the rotation of an infinitesimal fluid element. Reformulation of TDDFT in terms of new basic variables resolves the problem of nonlocality and thus allows us to regularly derive a local nonadiabatic approximation for exchange correlation (xc) potential. Stationarity of the density in the comoving frame makes the derivation to a large extent similar to the derivation of the standard static local density approximation. We present a few explicit examples of nonlinear nonadiabatic xc functionals in a form convenient for practical applications.
We present a constructive derivation of a time-dependent deformation functional theory---a collective variable approach to the nonequilibrium quantum many-body problem. It is shown that the motion of infinitesimal fluid elements (i.e., a set of Lagrangian trajectories) in an interacting quantum system is governed by a closed hydrodynamics equation with the stress force being a universal functional of Green's deformation tensor gij. Since the Lagrangian trajectories uniquely determine the current density, this approach can be also viewed as a representation of the time-dependent current-density functional theory. To derive the above theory, we separate a ``convective'' and a ``relative'' motions of particles by reformulating the many-body problem in a comoving Lagrangian frame. Then, we prove that a properly defined many-body wave function (and thus any observable) in the comoving frame is a universal functional of the deformation tensor. Both the hydrodynamic and the Kohn-Sham formulations of the theory are presented. In the Kohn-Sham formulation, we derive a few exact representations of the exchange-correlation potentials, and discuss their implication for construction of nonadiabatic approximations. We also discuss a relation of the present approach to a recent continuum mechanics of the incompressible quantum Hall liquids.
We extend a recent formulation of quantum continuum mechanics [J. Tao et al., Phys. Rev. Lett. 103, 086401 (2009)] to many-body systems subjected to a magnetic field. To accomplish this, we propose a modified Lagrangian approach, in which the motion of infinitesimal volume elements of the system is referred to the ``quantum convective motion'' that the magnetic field produces already in the ground state of the system. In the linear approximation, this approach results in a redefinition of the elastic displacement field u, such that the particle current j contains both an electric displacement and a magnetization contribution: $\mathbf{j}={\mathbf{j}}_{0}+{n}_{0}{\ensuremath{\partial}}_{t}\mathbf{u}+\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}({\mathbf{j}}_{0}\ifmmode\times\else\texttimes\fi{}\mathbf{u})$, where n0 and j0 are the particle density and the current density of the ground state and ${\ensuremath{\partial}}_{t}$ is the partial derivative with respect to time. In terms of this displacement, we formulate an ``elastic approximation'' analogous to the one proposed in the absence of magnetic field. The resulting equation of motion for u is expressed in terms of ground-state properties, the one-particle density matrix and the two-particle pair-correlation function, and in this form it neatly generalizes the equation obtained for vanishing magnetic field.
By means of a quantum mechanical phase space distribution function
introduced by von Roos, the Schroedinger equation for a non-relativistic
system of N identical particles with scalar interactions is transformed
into a quantum mechanical generalization of the Liouville equation,
thereby formulating the problem in terms of a generalized density in
phase space, a quantity of primary interest in most treatments of the
corresponding classical system (or plasma). This transformation permits
a parallel development of the theories of classical and quantum plasmas
and thus allows the quantum many-body problem to be discussed virtually
completely in classical terms. In particular, a kinetic theory of
quantum plasmas is obtained by deriving the quantum analogue of the BBGKY
hierarchy, and applying thereto approximation techniques similar to those
of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from
similar previous studies based on the Wigner distribution function is that
the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations,
in which case the quantum BBGKY system reduces to a simple quantum
generalization of the Vlasov equation. This equation is used to study
the response of spatially homogeneous systems to weak external forces,
and the associated problems of plasmon and spin-wave excitations. It is
also used to derive the quantum and exchange corrected equations of
inviscid hydrodynamical transport which are then applied to the problem
of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the
many-electron atom in the Hartree and Hartree-Fock approximations.
The relevant quantum Vlasov equations lead naturally to a statistical
theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi
models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and
exchange corrections to these models are simultaneously generated. The
quantum hydrodynamical theory developed earlier is used to determine
the influence of these corrections on the boundary conditions of the
model, and a theory of the compressed atom is consequently obtained.
Considered in somewhat less detail are the effects of non-zero temperature,
net orbital angular momentum, relativity and correlations, as well as
time dependent processes. The final part deals with the problem of the degenerate electron
gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with
parallel and anti-parallel spin are obtained by neglecting three particle
correlations. From these functions, a quantum-mechanical collision
integral is derived which differs from that obtained by Silin
and Guernsey and conjectured by Wyld and Pines in that dynamical exchange
effects are included. Also obtained from the pair function
is an expression for the correlation which reduces in the high
density limit to the result of Gell-Mann and Brueckner. At intermediate
densities an additional term appears in the energy due to the screening
of the exchange interaction by the dielectric properties of the medium.
It is evaluated in the high density limit and found to be -0.151 r_s ln r_s
Ryd/electron in marked disagreement with the corresponding value obtained
by DuBois.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.