Feynman Path Integral Method Research Articles

One-dimensional Dunkl quantum mechanics: a path integral approach

In the present manuscript, we employ the Feynman path integral method to derive the propagator in one-dimensional Wigner-Dunkl quantum mechanics. To verify our findings we calculate the propagator associated with the free particle and the harmonic oscillator in the presence of the Dunkl derivative. We also deduce the energy spectra and the corresponding bound-state wave functions from the spectral decomposition of the propagator.

Field theory description of the non-perturbative optical nonlinearity of epsilon-near-zero media

In this paper, we introduce a fully non-perturbative approach for the description of the optical nonlinearity of epsilon-near-zero (ENZ) media. In particular, based on the rigorous Feynman path integral method, we develop a dressed Lagrangian field theory for light–matter interactions and discuss its application to dispersive Kerr-like media with order-of-unity light-induced refractive index variations. Specifically, considering the relevant case of Indium Tin Oxide (ITO) nonlinearities, we address the novel regime of non-perturbative refractive index variations in ENZ media and establish that it follows naturally from a scalar field theory with a Born–Infeld Lagrangian. Moreover, we developed a predictive model that includes the intrinsic saturation effects originating from the light-induced modification of the Drude terms in the linear dispersion of ITO materials. Our results extend the Huttner–Barnett–Bechler electrodynamics model to the case of non-perturbative optical Kerr-like media providing an intrinsically nonlinear, field-theoretic framework for understanding the exceptional nonlinearity of ITO materials beyond traditional perturbation theory.

Study of energies spectra and thermodynamic properties of the relativistic Dirac equation using Feynman path integral method

In the current paper, the contribution of a Coulomb-like potential tensor interaction on the solution of the Dirac equation with a new generalized Morse-like potential is investigated using the Feynman path integral method. Relativistic and non-relativistic energy spectra were obtained. It has been established that the Coulomb-like potential eliminates the degeneracy of all pairs of spin doublets. Using the resulting non-relativistic energy eigenvalues equation, three diatomic molecules (H2, LiH, and HCl) were investigated and their thermodynamic properties, including mean energy, free energy, entropy, and specific heat capacity were shown. A comparison with the available literature shows that the thermodynamic plots obtained are consistent with previous work.

Feynman path-integral strong-field dynamics calculation method

The emergence and development of ultrafast intense lasers and attosecond measurement techniques have made it possible to observe and control the motions of electrons on a timescale of attoseconds and a spatial scale of atoms. With the improvement of experimental measurement accuracy, higher requirements are put forward for the accuracy of theoretical calculation methods. Extracting temporal and spatial information about ultrafast dynamics from experimental results through using theoretical models presents a significant challenge. Compared with the exact solutions of the time-dependent Schrödinger equation, the Feynman path-integral method for strong-field dynamics calculations offers a simpler model and higher computational efficiency. The electronic wave packet is regarded as a particle with different initial states, and by analyzing the motion of the particle, the causes of various nonlinear physical phenomena in strong fields can be clarified. This work introduces the saddle point approximation into strong field dynamics calculations based on the strong field approximation theory. Furthermore, the Coulomb-corrected strong field approximation method, trajectory-based Coulomb-corrected strong field approximation method, and Coulomb quantum trajectory strong field approximation method are presented in detail. This review aims to provide relevant methods and literature references for studying strong field dynamics theoretical calculations and also to present some ideas for developing new algorithms.

Strong simulation of tracking single photons with which-way-detectors in linear optics

Which-way-detectors (WWDs) are path-entangled detectors characterizing mutual exclusivity between path information and interference visibility in wave-particle duality experiments. We show surprisingly that WWDs allow to utilize single photons distinguishable in time domain to realize linear optical circuits where tracking their paths is exponentially hard for strong simulation analogous to rectangular lattice based Ising models. Distinguishable photons have scalability advantages of generation and detection compared with indistinguishable photons by promising both theoretical and experimental improvements in linear optical computing including boson sampling. We calculate strong simulation complexities by using variable elimination (VE) method for undirected graphs related to tensor network contraction for quantum circuits and recursive Feynman path-integral (RFPI) method to reduce space complexity. Two designs include either a single photon touring m times or m single photons propagating sequentially through an optical circuit composed of n beam splitters and phase shifters entangled with n WWDs. VE method for tracking results in undirected graphs matching with (2 m − 1) × (n + 1) and m × (n + 1) lattice Ising models with computational complexities of and in time and and in space for single and multi-photon based designs, respectively. We exploit RFPI method for m ≫ n to reduce space complexities to polynomial levels with respect to n and log m. Probability amplitude of specific cases of multi-photon design is represented in terms of Ising partition function with purely imaginary weights to characterize sampling complexity. Open issues about sampling complexity and experimental implementation of multi-WWD set-ups are discussed.

David Maurice Brink. 20 July 1930—8 March 2021

David Brink was one of the leading theoretical nuclear physicists of his generation. He made major contributions to the study of all aspects of nuclear physics, embracing nuclear structure, nuclear scattering and nuclear instability. His wide-ranging interests and interactions with theorists and experimentalists alike helped him in providing theoretical analysis and interpretations and suggesting experiments. He had the gift of visualizing complex problems in simple terms and provided clear analysis of the underlying processes. He was an expert on the use of semi-classical methods that provided an intuitively clear picture of complex phenomena. His research work and books are characterized by scientific clarity, transparency and depth. David possessed outstanding skills in mathematical computation, and he was an expert on special functions, group theory and the Feynman path integral method. David had many research students and collaborated with a large number of scientists from across the world, for whom he was a source of scientific and human inspiration and admiration. His most fundamental belief was that research was a means of trying to discover and understand the beauties of nature and explain them in simple terms to others. His absolute belief in the value of truth and his unselfish and generous attitude in sharing knowledge makes him an outstanding figure in contemporary nuclear physics.

The propagators for time-dependent mass harmonic oscillators

In this paper, the propagator for a harmonic oscillator with mass obeying the function of m(t)=mtan2νt is derived by the Feynman path integral method. The wave function of this oscillator is calculated by expanding the obtained propagator. The propagator for a harmonic oscillator with strongly pulsating mass is evaluated by the Schwinger method. The propagator for a harmonic oscillator with mass rapidly growing with time is calculated by applying the integrals of the motion of quantum systems. The comparison between these methods are also discussed.

Cyclotron toroidal braids: A cure for portrayal composite fermions on a torus

We present an inspection of the statistics of particles including composite fermions on a torus starting from a braid group analysis. For this purpose we considered a system of electrons confined to the surface of a torus under the influence of a strong magnetic field and interacting through a general rotational invariant potential. An explanation of the appearance of the cyclotron braids as an effect of restriction imposed by magnetic field on braid trajectories which in analyzed case reduces the full braid group to one of its subgroups (i.e. cyclotron subgroups), is given. The modified Feynman path-integral method is also reproduced with some minor enhancements. We improve known results concerning on braid groups on a torus in two directions: we obtain new estimates in terms of cyclotron braid subgroups and cyclotron band generator, respectively; we demonstrate that only multi-loop generators are accessible in the fractional quantum regime well and we also formally explain the unique statistic of composite fermions by construct trial wave function for the system on a torus, based on this idea. The topological oddness of torus geometry can be driven by shifting of electrons between the two different group of generators allowed for an explanation in satisfactory accordance the both compact commensurability condition and some numerical calculations in toroidal geometry. Besides, our approach may shed some new light on few interesting aspects in better understanding the fractional quantum Hall effect.

Newtonian dynamics of imaginary time-dependent mean field theory

A Time Dependent Hartree-Fock (TDHF) based classical model is applied to sub-barrier fusion reactions using the Feynman Path Integral Method (FPIM). The fusion cross-sections and modified astrophysical S*-factors are calculated for the 12C+12C reactions and compared to direct and indirect experimental results. Different channels cross-sections are estimated from the statistical decay of the compound nucleus. A good agreement with the direct data is found. We suggest a complementary observable given by the (imaginary) action A easily derived from theory and experiments. When properly normalized by the action in the Gamow limit it has an upper value of 1 at zero beam energies. It becomes negative at the Coulomb barrier which is Vcb=5.05±0.05MeV from direct data and Vcb=5.5MeV from model calculations.

Calculation of the C12+C12 sub-barrier fusion cross section in an imaginary-time-dependent mean field theory

The 12C+12C sub-barrier fusion cross section is calculated within the framework of a Time Dependent Hartree-Fock (TDHF) based classical model using the Feynman Path Integral Method. The modified astrophysical S*-factor is compared to direct and indirect experimental results. A good agreement with the direct data is found. In the lower energy region, where recent analyses of experimental data obtained with the Trojan Horse Method (THM) lead to contrasting results, the model predicts an S* factor half way between those results. Low energy resonances revealed in the THM data are added to the calculation and the relative reaction rate in the Gamow region is calculated. The role of different resonances is discussed in detail and their influence on the reaction rate at temperatures relevant to stellar evolution is investigated.

Formation of intermediate coupling optical polarons and bipolarons in two-dimensional systems

The formation of the optical polaron and bipolaron in two-dimensional (2D) systems is studied in the intermediate electron–phonon coupling regime. The total energies of the 2D polaron and bipolaron are calculated by using the Buimistrov–Pekar method of canonical transformations. The obtained results are compared with other existing results obtained by using the Feynman path integral method and the modified Lee–Low–Pines unitary transformation method. It is shown that the electron–phonon correlation significantly reduces the total energy of the 2D polaron in comparison with the energy of the strong coupling (adiabatic) polaron. It is found that the polaron formation in 2D systems is possible when the electron–phonon coupling constant α is greater than the critical value αc≃2.94, which is much lower than a critical value of the electron–phonon coupling constant α in three-dimensional (3D) systems. The critical values of the Fröhlich coupling constant α and the ratio η=ε∞/ε0 (where ε∞ and ε0 are the high frequency and static dielectric constants, respectively), which determine the bipolaron stability region in 2D systems, are calculated numerically. It is interesting for application to the layered cuprate superconductors that the (bi)polarons are formed more easily in quasi-2D regions than in the bulk. It is argued that the high-Tc cuprate superconductivity can exist above the bulk superconducting transition temperature Tc as the persisting superfluidity of polaronic (bosonic) Cooper pairs and large bipolarons at quasi-2D grain boundaries or in the CuO2 layers above Tc.

Measurement of gravity acceleration by cold atoms in a harmonic trap using Kapitza-Dirac pulses

The interferometry of two Kapitza-Dirac (KD) pulses acting on cold atoms in a harmonic oscillator potential well is investigated by Feynman path integral method. The wave function and density distribution function of the system at a given time are calculated analytically by using the propagator under the action of an external field. The first KD pulse acts on cold atoms to produce a large number of modes in the harmonic oscillator potential well. The maximum value of wave packet of mode 0 is larger than those of other modes. These modes evolve along different paths. The external field changes the phase of each mode and makes the evolution path of the mode deviate from that without the external field. When the second KD pulse is added, it splits the mode of the first KD pulse, and thus generates more modes. These modes will evolve along different paths under the action of external field and harmonic potential well, and interfere with each other. At the moment of measurement, all the wave packets are separated without overlapping. The effect of the external field does not change the magnitude of the density distribution at the time of measurement, but makes the wave packet of each mode shift. The phase difference between adjacent modes decreases linearly with the increase of field intensity. When the external field is a gravity field, we calculate the Fisher information and the Cramer-Rao lowér bound. The Fisher information is proportional to the mass of atoms and inversely to the third power of harmonic potential well frequency. We can improve the measurement accuracy of interferometer by reducing the frequency of harmonic potential well and increasing atomic mass. When the initial state is the ground state of the harmonic potential well, the accuracy of the gravity acceleration measured by the interference device can be obtained to be 10<sup>–9</sup> by using the experimental parameters. The initial state is the ground state of the harmonic potential well and the external field, and the calculation result indicates that the measurement accuracy will decrease. At the same time, the enhancement of inter-atomic repulsion and attraction interaction will also lead the measurement accuracy to increase.

Pricing of stochastic volatility stock index option based on Feynman path integral

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Path integral for non-paraxial optics

In this paper, we have constructed the Feynman path integral method for non-paraxial optics. This is done by using the mathematical analogy between a non-paraxial optical system and the generalized Schrödinger equation deformed by the existence a minimal measurable length. Using this analogy, we investigated the consequences of a minimal length in this optical system. This path integral has been used to obtain instanton solution for such an optical system. Moreover, the Berry phase of this optical system has been investigated. These results may disclose a new way to use the path integral approach in optics. Furthermore, as such systems with an intrinsic minimal length have been studied in quantum gravity, the ultra-focused optical pulses can be used as an optical analog of quantum gravity.

A Quantum-Type Approach to Non-Life Insurance Risk Modelling

A quantum mechanics approach is proposed to model non-life insurance risks and to compute the future reserve amounts and the ruin probabilities. The claim data, historical or simulated, are treated as coming from quantum observables and analyzed with traditional machine learning tools. They can then be used to forecast the evolution of the reserves of an insurance company. The following methodology relies on the Dirac matrix formalism and the Feynman path-integral method.

Maximizing gravitational acceleration measurement accuracy by tuning the interaction time between atoms and gravity

We propose a technique using an atom interferometer to obtain the most accurate measurement of gravitational acceleration by changing the interaction time between the atoms and gravity in a harmonic trap. The analytical calculation procedure is based on the Feynman path integral method. The first Kapitza–Dirac (KD) scattering pulse splits the initial state into several modes with different momenta. Before adding the second KD pulse, the interaction time between the atoms and gravity is changed. These modes have a position shift as a result of the gravitational field and are coherently recombined by the harmonic potential. The second KD pulse splits the evolved modes a second time and separates them along different paths again. At the measurement time, we obtain the greatest accuracy from the quantum Fisher information. For the variation of gravitational acceleration on the surface of earth, an accuracy of $$10^{-9}$$ can be achieved at any value of gravity acceleration by tuning the interaction time between the atoms and gravity.

Integrals of the motion and Green function for dual damped oscillators and coupled harmonic oscillators

The application for the integrals of the motion of a quantum system in deriving Green function or propagator is presented. The Green function is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phase space. The exact expressions for the Green functions of the dual damped oscillators and the coupled harmonic oscillators are evaluated in co-ordinate representations. The relation between the integrals of the motion method and other methods such as Feynman path integral and Schwinger method are also presented.

Integrals of the motion and green functions for time-dependent mass harmonic oscillators

The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.

Effects of Donor Size and Heavy Doping on Optical, Electrical and Thermoelectric Properties of Various Degenerate Donor-Silicon Systems at Low Temperatures

In various degenerate donor-silicon systems, taking into account the effects of donor size and heavy doping and using an effective autocorrelation function for the potential fluctuations expressed in terms of the Heisenberg uncertainty relation and also an expression for the Gaussian average of <img width="32" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image001.png" />, a ≥ 1 <img width="18" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image002.png" /> being the kinetic energy of the electron, calculated by the Kane integration method (KIM), we investigated the density of states, the optical absorption coefficient and the electrical conductivity, noting that this average expression calculated by the KIM was found to be equivalent to that obtained by the Feynman path-integral method. Then, those results were expressed in terms of <img width="55" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image003.png" /> for total electron energy <img width="42" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image004.png" />, vanished at the conduction-band edge: <img width="42" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image005.png" />, and for <img width="42" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image006.png" /> exhibited their exponential tails, going to zero as <img width="120" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image007.png" />, and presenting the maxima, in good accordance with an asymptotic form for exponential conduction-band tail obtained by Halperin and Lax, using the minimum counting methods. Further, in degenerate d-Si systems at low temperatures, using an expression for the average of <img width="19" height="20" src="http://article.sciencepublishinggroup.com/journal/122/1221186/image008.png" />, p ≥ 3/2, calculated using the Fermi-Dirac distribution function, we determined the mobility, electrical conductivity, resistivity, Hall factor, Hall coefficient, Hall mobility, thermal conductivity, diffusion coefficient, absolute thermoelectric power, Thomson coefficient, Peltier coefficient, Seebeck thermoelectric potential, and finally dimensionless figure of merit, which were also compared with experimental and theoretical results, suggesting a satisfactory description given for our obtained results.

Spin states of electrons in quantum dots upon heating. Simulation by the Feynman path integral method. Magnetic properties

Temperature dependences of spin states and spin paramagnetic susceptibility in ellipsoidal quantum dots (QDs) containing two or three electrons are numerically simulated using ab initio calculations based on the Feynman path integral method. Limits of the thermal stability of spin states are estimated. Upon cooling, the pairing of spins of an electron pair is most intense in spherical QDs; notably, prolate QDs hinder the pairing more strongly than the oblate ones. When the spherical shape of a QD is distorted, a characteristic peak in the temperature dependence of the electron-pair magnetic susceptibility shifts to lower temperatures. A spin of the system of three electrons may either increase or decrease upon cooling, depending on the QD shape. In the case of three electrons, strong spatial anisotropy of the electron-confining field causes a relative decrease in the energy of states with large spin values.