Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

Restricted Constant of Motion for the One-Dimensional Harmonic Oscillator with Quadratic Dissipation and Some Consequences in Statistic and Quantum Mechanics

A restricted constant of motion, Lagrangian and Hamiltonian, for the harmonic oscillator with quadratic dissipation is deduced. The restriction comes from the consideration of the constant of motion for the velocity of the particle either for v 0 or for v < 0. A study is done about the implications that these restricted variables have on the specific heat of a thermodynamical system of oscillators with this dissipation, and on the quantization of this dissipative system.     

Total Hamiltonian and Extended Hamiltonian for Constrained Hamilton Systems

For constrained Hamiltonian systems, the motion equations are deduced from total Hamiltonian and extended Hamiltonian with Lagrangian multipliers depending on time t and canonical variables q ⁱ and p _i . When the multipliers reduced to only depend on time t, the motion equations exactly agree with the old results. Under the same conditions (Lagrangian multipliers depend on time t and canonical variables q ⁱ and p _i ), the relation equations of coefficients in the generator of gauge transformation are deduced, but the equations have an additive term besides the well-known results. This additive term is from Lagrangian multipliers depending on canonical variables, and it might perform the gauge symmetries that needs to be discussed further. This project is supported by the fund of National Natural Science (10671086) and by National Laboratory for Superlattices and Microstructures (CHJG200605).     

QCD on the Light-Front

Light-front Hamiltonian theory, derived from the quantization of the QCD Lagrangian at fixed light-front time x ⁺ = x ⁰ + x ³, provides a rigorous frame-independent framework for solving nonperturbative QCD. The eigenvalues of the light-front QCD Hamiltonian H _LF predict the hadronic mass spectrum, and the corresponding eigensolutions provide the light-front wavefunctions which describe hadron structure, providing a direct connection to the QCD Lagrangian. In the semiclassical approximation the valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a single-variable relativistic equation of motion, analogous to the nonrelativistic radial Schrödinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. Remarkably, the potential U has a unique form of a harmonic oscillator potential if one requires that the chiral QCD action remains conformally invariant. A mass gap and the color confinement scale also arises when one extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory. In the case of mesons, the valence Fock-state wavefunctions of H _LF for zero quark mass satisfy a single-variable relativistic equation of motion in the invariant variable \({\zeta^2=b^2_\perp x(1-x)}\) , which is conjugate to the invariant mass squared \({{M^2_{q\bar q}}}\) . The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories \({M^2(n, L, S) = 4\kappa^2( n+L +S/2)}\) with the same slope in the radial quantum number n and orbital angular momentum L. Only one mass parameter \({\kappa}\) appears. The corresponding light-front Dirac equation provides a dynamical and spectroscopic model of nucleons. The same light-front equations arise from the holographic mapping of the soft-wall model modification of AdS₅ space with a unique dilaton profile to QCD (3 + 1) at fixed light-front time. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. We also discuss the implications of the underlying conformal template of QCD for renormalization scale-setting and the implications of light-front quantization for the value of the cosmological constant.     

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity () and a time-dependent external force (K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ?. The Heisenberg equations of motion utilizing the quantized Hamiltonian surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

It is shown that the degeneracy of the noncanonical Poisson bracket operating on the space of solenoidal vector fields that arises due to the freezing-in of the curl of the velocity [E. A. Kuznetsov and A. V. Mikhailov, Phys. Lett. A 77, 37 (1980)] is lifted when the vorticity Ω is represented in terms of vortex lines. This representation makes it possible to integrate the equation of motion of the vorticity for a system with the Hamiltonian H=∫∣Ω∣d r. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 12, 1015–1020 (25 June 1998)     

The trace identity and the planar Casimir effect

The familiar trace identity associated with the scale transformationx ^Μ → x′ ^Μ = e^-λ x ^Μ on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition Φ(t, x¹, x² = -a) = 0 is imposed. It is however respected in: (i) 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition Φ(t, x¹ = -a) = 0 holds and (ii) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where Φ(t, x¹, x² = -a) = 0, the modified trace identity is shown to be anomalous with a numerical coefficient for the anomalous term equal to the canonical scale dimension, viz. 1/2. The technique of Bordaget al [Ann. Phys. (N.Y.),165, 162 (1985)] is used to incorporate the said boundary condition into the generating functional for the connected Green’s functions.     

Interference in Phase Space of Squeezed States for the Time-Dependent Hamiltonian System

We showed that the idea of Schleich and Wheeler (1987, Nature 326, 574) for the semiclassical approach of the interference in phase space of harmonic oscillator squeezed states can be extended to that of general time-dependent Hamiltonian system. The quantum phase properties of squeezed states for the general time-dependent Hamiltonian system are investigated by using the quantum distribution function. The weighted overlaps A _n and phases θ _n for the system are evaluated in the semiclassical limit.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.     

Finite Maxwell field and electric displacement Hamiltonians derived from a current dependent Lagrangian

ABSTRACT

In the common Ewald summation technique for the evaluation of electrostatic forces, the average electric field E is strictly zero. Finite uniform E can be accounted for by adding it as a new degree of freedom in an extended Lagrangian. Representing the uniform polarization P as the time integral of the internal current and E as the time derivative of a uniform vector field A, we define such an extended Lagrangian coupling A to the total current j _t (internal plus external) and hence derive a Hamiltonian resembling the minimal coupling Hamiltonian of electrodynamics. Next, applying a procedure borrowed from nonrelativistic molecular electrodynamics the j _t · A coupling is transformed to P · D form where D is the electric displacement acting as an electrostatic boundary condition. The resulting Hamiltonian is identical to the constant-D Hamiltonian obtained by Stengel, Spaldin and Vanderbilt (SSV) using thermodynamic arguments. The corresponding SSV constant-E Hamiltonian is derived from an alternative extended Lagrangian.     

A simplified form for the Hamiltonian and Lagrangian of the spin-independent gravitational two-body system

In general, the gravitational two-body Hamiltonian, to orderc ^–2, containsGP ²,G (P · r)², andG ² terms. We have previously shown [4–6] that a proper choice of coordinate system enables one to eliminate theG (P · r)² term. We now show that, making use of energy conservation, and coordinate transformations, we can eliminate either of the remaining two terms. In particular, we are able to write down a Hamiltonian and a Lagrangian that contain no mixed potential and kinetic terms.Laboratoire associé au Centre National de la Recherche Scientifique.     

Poisson–Lie T-Duality¶for Quasitriangular Lie Bialgebras

We introduce a new 2-parameter family of sigma models exhibiting Poisson–Lie T-duality on a quasitriangular Poisson–Lie group G. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form , where the generalised metric E is constant (not dependent on the field u as in previous models). We characterise these models in terms of a global conserved G-invariance. The models on G=SU ₂ and its dual G ^* are computed explicitly. The general theory of Poisson–Lie T-duality is also extended, notably the reduction of the Hamiltonian formulation to constant loops as integrable motion on the group manifold. The approach also points in principle to the extension of T-duality in the Hamiltonian formulation to group factorisations D=G⋈M, where the subgroups need not be dual or connected to the Drinfeld double. Received: 22 August 1999 / Accepted: 4 February 2000     

The energy—momentum tensor, the trace identity and the Casimir effect

The trace identity associated with the scale transformation x^Μ → x′^Μ = e^-ρx^Μ on the Lagrangian density for the noninteracting electromagnetic field in the co-variant gauge is shown to be violated on a single plate on which the Dirichlet boundary conditionA ^Μ(t, x¹, x², x³ = -a) = 0 is imposed. It is however respected in free space, i.e. in the absence of the plate. These results reinforce our assertions in an earlier paper where the same exercise was carried out using the Lagrangian density for the free, massive, real scalar field in 2 + 1 dimensions.     

Gravitational two-body problem with acceleration-dependent spin terms

We generalize our previous work, on the gravitational two-body post-Newtonian Lagrangian with spin and parametrized post-Newtonian parameters and , by addingaccelerationdependent spin terms corresponding to anarbitrary spin supplementary condition. For the purpose of constructing the corresponding Hamiltonian we make use of our recently developedmethod of the double zero. Using this method, we remove the acceleration-dependent spin terms from the Lagrangian and, in the process, create new non-accelerationdependent terms. Use of this new Lagrangian enables us to construct the Hamiltonian corresponding to the arbitrary spin supplementary condition. Energy constants of the motion are also discussed.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian H(x ¹,…,x ^N+2) which, in addition, has a conserved quantity V(x ¹,…,x ^N+2)=v, so that the Poisson bracket {H,V} vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V(x ¹,…,x ^N+2)=v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.