Path integral quantization of spinning particles interacting with crossed external electromagnetic fields

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path-integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The construction is complicated by the fact that the amplitudes (class-operators) calculated using a path integral typically do not satisfy the constraint equation everywhere, but we show how they may be systematically modified in such a way that they do satisfy the constraint. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method).

The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrodinger formulation. As will be seen in this project work, many elementary problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, starting from the well known quantum mechanics as formulated by Schrodinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time.

We present a fast, high precision and easily implementable path integral method for numerically solving Fokker–Planck equations. It is based on a generalized Trotter formula, which permits one to attain an adequate description of dynamical and equilibrium properties even though the time increment τ=t/N is rather large. A remarkable property of the symmetric Trotter splitting is used to systematically eliminate the lower-order errors resulting from time discretization. This means a significant reduction of the number of time steps that are required to retain a given accuracy for a given net increment t=Nτ, and, therefore, significantly increasing the feasibility of path integral calculations. Yet another attractive feature of the present technique is that it allows for equations with singular diffusion matrices that are known to present a special problem within the scope of the path integral formalism. The favorable scaling of the fast Fourier transform is used to numerically evaluate the path integral on a grid. High efficiency is achieved due to the Stirling interpolation which dynamically readjusts the distribution function every time step with a mild increase in cost and with no loss of precision. These developments substantially improve the path integral method and extend its applicability to various time-dependent problems which are difficult to treat by other means. One can even afford to extract information on eigenvalues and eigenfunctions from a time-dependent solution thanks to the numerical efficiency of the present technique. This is illustrated by calculating the propagator and the lowest eigenvalues of a one-dimensional Fokker–Planck equation. The method is also applied to a two-dimensional Fokker–Planck equation, whose diffusion matrix does not possess an inverse (a so-called Klein-Kramers equation). The numerical applications show our method to be a dramatic improvement over the standard matrix multiplication techniques available for evaluating path integrals in that it is much more efficient in terms of speed and storage requirements.

Recent laboratory and epidemiological results have stimulated interest in the hypothesis that human beings may exhibit biological responses to magnetic and/or electric field transients with frequencies in the range between 100 Hz and 100 kHz. Much can be learned about the response of a system to a transient stimulation by understanding its response to sinusoidal disturbances over the entire frequency range of interest. Thus, the main effort of this paper was to compare the strengths of the electric fields induced in homogeneous ellipsoidal models by uniform 100 Hz through 100 kHz electric and magnetic fields. Over this frequency range, external electric fields of about 25-2000 V/m (depending primarily on the orientation of the body relative to the field) are required to induce electric fields inside models of adults and children that are similar in strength to those induced by an external 1 microT magnetic field. Additional analysis indicates that electric fields induced by uniform external electric and magnetic fields and by the nonuniform electric and magnetic fields produced by idealized point sources will not differ by more than a factor of two until the sources are brought close to the body. Published data on electric and magnetic field transients in residential environments indicate that, for most field orientations, the magnetic component will induce stronger electric fields inside adults and children than the electric component. This conclusion is also true for the currents induced in humans by typical levels of 60 Hz electric and magnetic fields in U.S. residences.

By formulating a path integral in the SU(2) fiber bundle space, a nonrelativistic spinor particle moving in external electromagnetic fields is studied. This formulation is a direct extension of Schulman’s [Phys. Rev. 176, 1588 (1968)] continuous treatment for spin; it unifies all spin states to a spherical top model. This theory enjoys a definite classical theory with spin–orbital interactions and the corresponding semiclassical approximation for the path integral. This method is also applicable to a particle moving in a more general Yang–Mills field.

The Pauli algebra is used to study the relativistic motion of charged particles in external electromagnetic fields. Plane-wave solutions to Maxwell's equations, in particular standing waves with electric and magnetic fields which are everywhere parallel, are easily represented and analysed in a gauge-independent treatment. The authors derive the 4-momentum of electromagnetic fields and show how the integrals for fields of a charged particle can be evaluated with integration limits specified in the rest frame of the moving charge. The general solution of the Lorentz force equation is shown to be a Lorentz transformation with time-dependent rotation and boost parameters which, in the case of parallel fields, are trivially related to the magnetic and electric fields, respectively. With the Pauli algebra, an earlier derivation in the Dirac algebra of the 4-velocity in constant uniform fields is greatly simplified and extended to arbitrary initial motion, and solutions are found to motion in a Coulomb fields and in the presence of arbitrarily polarised (monochromatic, directed) plane waves.

This paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem via a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). Through Lagrangian relaxation, we convert the chance-constrained (risk-constrained) SOC problem to a risk-minimizing SOC problem, the cost function of which possesses the time-additive Bellman structure. We show that the risk-minimizing control synthesis is equivalent to solving an HJB PDE whose boundary condition can be tuned appropriately to achieve a desired level of safety. Furthermore, it is shown that the proposed risk-minimizing control problem can be viewed as a generalization of the problem of estimating the risk associated with a given control policy. Two numerical techniques are explored, namely the path integral and the finite difference method (FDM), to solve a class of risk-minimizing SOC problems whose associated HJB equation is linearizable via the Cole-Hopf transformation. Using a 2D robot navigation example, we validate the proposed control synthesis framework and compare the solutions obtained using path integral and FDM.

The concept of a propagator is useful and is a well-known object in diffusion NMR experiments. Here, we investigate the related concept; the propagator for the magnetization or the Green’s function of the Torrey-Bloch equations. The magnetization propagator is constructed by defining functions such as the Hamiltonian and Lagrangian and using these to define a path integral. It is shown that the equations of motion derived from the Lagrangian produce complex-valued trajectories (classical paths) and it is conjectured that the end-points of these trajectories are real-valued. The complex nature of the trajectories also suggests that the spin degrees of freedom are also encoded into the trajectories and this idea is explored by explicitly modeling the spin or precessing magnetization by anticommuting Grassmann variables. A pseudoclassical Lagrangian is constructed by combining the diffusive (bosonic) Lagrangian with the Grassmann (fermionic) Lagrangian, and performing the path integral over the Grassmann variables recovers the original Lagrangian that was used in the construction of the propagator for the magnetization. The trajectories of the pseudoclassical model also provide some insight into the nature of the end-points.

The path-integral method seems to be the most suitable for the quantization of gravity. One would expect the dominant contribution to the path integral to come from metrics which are near background metrics that are solutions of classical Einstein equations. The action of these background metrics gives rise to a new phenomenon in field theory, intrinsic quantum entropy. This is shown to be related to the scaling behavior of the gravitational action and to the topology of the gravitational field. The quadratic terms in the Taylor series of the action about the background metrics give the one-loop corrections. In a supersymmetric theory the quartic and quadratic but not the so-called logarithmic divergences cancel to give a one-loop term that is finite without regularization. From the one-loop term one can obtain the effective energy-momentum tensor on the background metric. In the case of an evaporating black hole, the energy-momentum tensor will be regular on the future horizon. The usual perturbation expansion breaks down for quantum gravity because the higher (interaction) terms in the Taylor series are not bounded by the quadratic (free) ones. To overcome this I suggest that one might replace the path integrals over the terms in the Taylor series by a discrete sum of the exponentials of the actions of all complex solutions of the Einstein equations, each solution being weighted by its one-loop term. This approach seems to give a picture of the gravitational vacuum as a sea of virtual Planck-mass black holes.

Path integral in constrained classical mechanics

The stability of different crystal lattices of two-dimensional superparamagnetic suspensions that are confined to a planar liquid-gas interface and exposed to a tilted external magnetic field is studied theoretically by lattice sum minimizations. The magnetic field induces magnetic dipoles onto the colloidal particles along its direction, whose strength can be controlled by the amplitude of the external field. The mutual interaction between the colloids is governed by dipole-dipole forces and a short-ranged repulsion having its physical origin at the presence of the colloidal cores. If the direction of the magnetic field is perpendicular to the liquid-gas interface, there is a purely repulsive interaction leading to stable triangular crystals. By tilting the external field, the interaction becomes anisotropic and a mutual attraction appears upon a threshold tilt angle. We have calculated the full phase diagram at zero temperature varying the tilt angle, the colloidal density, and the strength of the magnetic field. Apart from the triangular lattice we find a variety of stable crystal lattices including rectangular, oblique, chainlike oblique, and rhombic structures. We also present the accurate derivation of the Hamiltonian of two polarizable particles of finite arbitrary geometries in external magnetic and electric fields.

In this study, for the first time, the combined effects of external magnetic, electric, and Aharonov-Bohm (AB) flux fields on quantum levels of the hydrogen atom embedded in Debye and quantum plasmas modeled by the more general exponential cosine screened Coulomb (MGECSC) potential are investigated within cylindrical coordinate formalism using the asymptotic iteration method. The MGECSC potential includes four different potential forms when considering different sets of the parameters in the potential. The corresponding Schrödinger equation is solved numerically in order to examine both strong and weak regimes and confinement effects of external fields. The influence of screening parameters of the MGECSC potential on quantum levels of the hydrogen atom is also studied in detail in the presence of external magnetic, electric, and AB flux fields. As it is possible to model both Debye and quantum plasmas by using screening parameters in the MGECSC potential, the effects of each plasma environment on quantum levels of the hydrogen atom are also considered in the external fields. It is observed that there are important results of external fields on the total interaction potential profile, and the most dominant one in these fields is the magnetic field. Furthermore, the effects of confinement on the physical state of the plasma environment is a subject of this study. These details would be important in experimental and theoretical investigations in plasma and atomic physics fields.

Electromagnetic Environment Associated with Lightning Strikes to Tall Strike Objects

Contrary to other claims, we argue that bulk viscosity associated with the interactions of non- relativistic particles with relativistic particles around the time of the grand unified theory (GUT) phase transition cannot lead to inflation. Simply put, the key ingredient for inflation, negative pressure, cannot arise due to the bulk-viscosity effects of a weakly interacting mixture of relativistic and nonrelativistic particles.