Path Integral Treatment Research Articles

Path Integral Treatment of a Linear Combination of Deformed Diatomic Molecule Potentials

Path Integral Treatment of a Linear Combination of Deformed Diatomic Molecule Potentials

Thermal properties of a one-dimensional dirac oscillator in a homogeneous electric field with generalized snyder model: path integral treatment

Thermal properties of a one-dimensional dirac oscillator in a homogeneous electric field with generalized snyder model: path integral treatment

Path integral treatment of a Klein Gordon particle with generalized inverse Quadratic Yukawa potential

We solved the Klein–Gordon equation for a generalized inverse quadratic Yukawa (GIQY) potential via path integrals approach. We applied approximations to deal with the terms 1/r 2 and 1/r. A path integral representation of Green’s function relating to a particle moving in a mixture of equal vector and scalar potentials was established. To integrate Green’s function, a space-time transformation was successfully used, and the present problem was reduced to a previously known modified Pöschl-Teller potential problem. The integrated Green’s function contains important information about the considered system, from which we obtained energy eigenvalues and the corresponding normalized eigenfunctions for various values of n and ℓquantum numbers. Numerical results, Schrödinger solutions, and special cases such as the modified screened Coulomb plus inversely quadratic Yukawa potential, Kratzer potential, Yukawa potential, and Coulomb potential were also considered. These results are largely consistent with previous studies.

Hierarchical equations of motion analog for systems with delay: Application to intercavity photon propagation

Over the last two decades, the hierarchical equations of motion (HEOM) of Tanimura and Kubo have become the equation of motion-based tool for numerically exact calculations of system-bath problems. The HEOM is today generalized to many cases of dissipation and transfer processes through an external bath. In spatially extended photonic systems, the propagation of photons through the bath leads to retardation/delays in the coupling of quantum emitters. Here, the idea behind the HEOM derivation is generalized to the case of photon retardation and applied to the simple example of two dielectric slabs. The derived equations provide a simple reliable framework for describing retardation and may provide an alternative to path-integral treatments.

Path Integral Molecular Dynamics of Liquid Water in a Mean-Field Particle Reservoir.

We present a simulation scheme for path integral simulation of molecular liquids where a small open region is embedded in a large reservoir of non interacting point‐particles. The scheme is based on the latest development of the adaptive resolution technique AdResS and allows for the space‐dependent change of molecular resolution from a path integral representation with 120 degrees of freedom to a point particle that does not interact with other molecules and vice versa. The method is applied to liquid water and implies a sizable gain regarding the request of computational resources compared to full path integral simulations. Given the role of water as universal solvent with a specific hydrogen bonding network, the path integral treatment of water molecules is important to describe the quantum effects of hydrogen atoms’ delocalization in space on the hydrogen bonding network. The method presented here implies feasible computational efforts compared to full path integral simulations of liquid water which, on large scales, are often prohibitive.

A perturbative expansion scheme for supermembrane and matrix theory

We reconsider the supermembrane in a Minkowski background and in the light-cone gauge as a one-dimensional gauge theory of area preserving diffeomorphisms (APDs). Keeping the membrane tension T as an independent parameter we show that T is proportional to the gauge coupling g of this gauge theory, such that the small (large) tension limit of the supermembrane corresponds to the weak (strong) coupling limit of the APD gauge theory and its SU(N) matrix model approximation. A perturbative linearization of the supersymmetric theory suitable for a quantum mechanical path-integral treatment can be achieved by formulating a Nicolai map for the matrix model, which we work out explicitly to mathcal{O}left({g}^4right) . The corresponding formulæ remain well-defined in the limit N → ∞; this result relies on a cancellation of infinities not present for the bosonic membrane, indicating that the N → ∞ limit does not exist for the purely bosonic matrix model. Furthermore we show that the map has improved convergence properties in comparison with the usual perturbative expansions because its Jacobian admits an expansion in g with a non-zero radius of convergence. Possible implications for unsolved issues with the matrix model of M theory are also mentioned.

Path-integral treatment of quantum bouncers

The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π. We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.

Cosmological constant and the use of cutoffs

Of the contributions to the cosmological constant, zero-point energy contributions scale as ${\ensuremath{\delta}}^{4}(0)\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{4}$ where $\mathrm{\ensuremath{\Lambda}}$ is an ultraviolet cutoff used to regulate the calculations. I show that such contributions vanish when calculated in perturbation theory. This demonstration uses a little-known modification to perturbation theory found by Honerkamp and Meetz and by Gerstein, Jackiw, Lee, and Weinberg which comes into play when using cutoffs and interactions with multiple derivatives, as found in chiral theories and gravity. In a path integral treatment, the new interaction arises from the path integral measure and cancels the ${\ensuremath{\delta}}^{4}(0)$ contributions. This reduces the sensitivity of the cosmological constant to the high energy cutoff, although it does not resolve the cosmological constant problem. The feature removes one of the common motivations for supersymmetry. It also calls into question some of the results of the asymptotic safety program. Covariance and quadratic cutoff dependence are also briefly discussed.

Beyond the Fröhlich Hamiltonian: Path-integral treatment of large polarons in anharmonic solids

The properties of an electron in a typical solid are modified by the interaction with the crystal ions, leading to the formation of a quasiparticle: the polaron. Such polarons are often described using the Fr\"ohlich Hamiltonian, which assumes the underlying lattice phonons to be harmonic. However, this approximation is invalid in several interesting materials, including the recently discovered high-pressure hydrides which superconduct at temperatures above $200$K. In this paper, we show that Fr\"ohlich theory can be extended to eliminate this problem. We derive four additional terms in the Fr\"ohlich Hamiltonian to account for anharmonicity up to third order. We calculate the energy and effective mass of the new polaron, using both perturbation theory and Feynman's path integral formalism. It is shown that the anharmonic terms lead to significant additional trapping of the electron. The derived Hamiltonian is well-suited for analytical calculations, due to its simplicity and since the number of model parameters is low. Since it is a direct extension of the Fr\"ohlich Hamiltonian, it can readily be used to investigate the effect of anharmonicity on other polaron properties, such as the optical conductivity and the formation of bipolarons.

Loop quantum gravity, signature change, and the no-boundary proposal

Covariant models of loop quantum gravity generically imply dynamical signature change at high density. This article presents detailed derivations that show the fruitful interplay of this new kind of signature change with wave-function proposals of quantum cosmology, such as the no-boundary and tunneling proposals. In particular, instabilities of inhomogeneous perturbations found in a Lorentzian path-integral treatment are naturally cured. Importantly, dynamical signature change does not require Planckian densities when off-shell instantons are relevant.

Effective mass path integral simulations of quasiparticles in condensed phases

The quantum many-body problem in condensed phases is often simplified using a quasiparticle description, such as effective mass theory for electron motion in a periodic solid. These approaches are often the basis for understanding many fundamental condensed phase processes, including the molecular mechanisms underlying solar energy harvesting and photocatalysis. Despite the importance of these effective particles, there is still a need for computational methods that can explore their behavior on chemically relevant length and time scales. This is especially true when the interactions between the particles and their environment are important. We introduce an approach for studying quasiparticles in condensed phases by combining effective mass theory with the path integral treatment of quantum particles. This framework incorporates the generally anisotropic electronic band structure of materials into path integral simulation schemes to enable modeling of quasiparticles in quantum confinement, for example. We demonstrate the utility of effective mass path integral simulations by modeling an exciton in solid potassium chloride and electron trapping by a sulfur vacancy in monolayer molybdenum disulfide.

Dirac oscillator in a uniform electric field: Path integral treatment

In this paper, the propagator of a two-dimensional Dirac oscillator in the presence of a uniform electric field is derived by using the path integral technique. The fact that the globally named approach is used in this work redirects, beforehand, our search for the propagator of the Dirac equation to that of the propagator of its quadratic form. The internal motions relative to the spin are represented by two fermionic oscillators, which are described by Grassmannian variables, according to Schwinger’s fermionic model. Once the integration over the anticommuting variables (Grassmannian variables) is accomplished, the problem becomes the one of finding a non-relativistic propagator with only bosonic variables. The energy spectrum of the electron and the corresponding eigenspinors are also obtained in this work.

Parity anomaly from the Hamiltonian point of view

We review the parity anomaly of the massless Dirac fermion in $2+1$ dimensions from the Hamiltonian, as opposed to the path integral, point of view. We have two main goals for this note. First, we hope to make the parity anomaly more accessible to condensed matter physicists, who generally prefer to work within the Hamiltonian formalism. The parity anomaly plays an important role in modern condensed matter physics, as the massless Dirac fermion is the surface theory of the time-reversal invariant topological insulator (TI) in $3+1$ dimensions. Our second goal is to clarify the relation between the time-reversal symmetry of the massless Dirac fermion and the fractional charge of $\pm\frac{1}{2}$ (in units of $e$) which appears on the surface of the TI when a magnetic monopole is present in the bulk. To accomplish these goals we study the Dirac fermion in the Hamiltonian formalism using two different regularization schemes. One scheme is consistent with the time-reversal symmetry of the massless Dirac fermion, but leads to the aforementioned fractional charge. The second scheme does not lead to any fractionalization, but it does break time-reversal symmetry. For both regularization schemes we also compute the effective action $S_{\text{eff}}[A]$ which encodes the response of the Dirac fermion to a background electromagnetic field $A$. We find that the two effective actions differ by a Chern-Simons counterterm with fractional level equal to $\frac{1}{2}$, as is expected from path integral treatments of the parity anomaly. Finally, we propose the study of a bosonic analogue of the parity anomaly as a topic for future work.

Analytical solution of the Feynman Kernel for general exponential-type potentials

This paper presents an analytical path-integral treatment of the ℓ-states of an exponential-type potential. We propose a generalization of the Pekeris approximation of the centrifugal term adapted to deformed potentials. To obtain solutions of the radial Feynman Kernel for arbitrary angular number, we perform a nontrivial change of variable accompanied by a local time rescaling. Using Euler angles and the isomorphism between S3 and SU(2), we convert the radial path integral into a maniable one. Analytical expressions of the energy spectrum and the normalized ℓ-state eigenfunctions are derived from the Green function. Several potentials are obtained as special cases of the general exponential-type potential. Thus, their eigenvalues and eigenfunctions are deduced straightforwardly. Numerical results show that our technique improves the state-of-the-art.

A Path Integral Treatment of a System With Variable Mass

The propagator related to a system described by an Hamiltonien having the symetric form $\hat {H}=\frac {1}{4}\left (\frac {1}{m\left (\hat {x},t\right ) }\hat {p}^{2}+\hat {p}^{2}\frac {1}{m\left (\hat {x},t\right ) }\right ) +V\left (\hat {x},t\right )$ , is determined by using the path integral approach. As applications, two cases, where the mass and the potential are both dependent on time and position, are considered:

Local gravity theories in conformal superspace

The conformal method for the initial value formulation of GR is one of the most powerful and widely usedtools in both analytical studies and numerical simulations of solutions. As is well-known, it exploits a hidden spatial conformal symmetry that can be extracted from the equations. But why conformal symmetry? Motivated by these questions and by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they are indeed spatial conformal diffeomorphisms. The next question is: what sort of theories most naturally incorporate this symmetry? Demanding locality for an action for metric gravity compatible with these principles determines the allowed operators, both for the purely gravitational part as well as matter couplings. The symmetries guarantee that there are two gravitational propagating physical degrees of freedom, but no explicit refoliation-invariance. The simplest such system has a geometric interpretation as a geodesic model in infinite-dimensional conformal superspace. One example of solution to the equations of motion corresponds to a static Bianchi IX spatial ansatz. The unique coupling to electromagnetism forces the electromagnetic equations to be hyperbolic, enabling us to “build” a standard space–time causal structure. There are, however, deviations from the standard Maxwell equations when space–time anisotropies become too large. Regarding quantization, with the geometric interpretation and the lack of refoliation invariance, the path integral treatment of the symmetries becomes much less involved than the similar approaches to GR. The symmetries form an (infinite-dimensional) Lie algebra, and no BFV treatment is necessary. Moreover, one can use gauge-invariant variational principles for selecting the boundary conditions of the path integral. We find that the propagator around the homogeneous solution has up to 6th order spatial derivatives, giving it plausible regularization properties as in Horava–Lifschitz.

Variational path-integral approach to back-reactions of composite mesons in the Nambu–Jona-Lasinio model

For the investigation of back-reactions of composite mesons in the NJL model, a variational path-integral treatment is formulated which yields an effective action Aeff[Dσ,Dπ;S], depending on the propagators Dσ, Dπ of σ− and π−mesons and on the full quark propagator S. The stationarity conditions δAeff/δS=0, δAeff/δDσ=0, δAeff/δDπ=0, then lead to coupled Schwinger–Dyson (SD) equations for the quark self-energy and the meson polarization functions. These results reproduce and extend results of the so-called “Φ−derivable” approach and provide a functional formulation for diagrammatic resummations of 1/Nc−corrections in the NJL model. Finally, we perform a low-momentum estimate of the quark and meson loop contributions to the polarization function of the pion and on this basis discuss the Goldstone theorem.

A Wiener Path Integral Solution Treatment and Effective Material Properties of a Class of One-Dimensional Stochastic Mechanics Problems

AbstractA Wiener path integral (WPI)-based approximate technique is developed for determining the joint response probability density function (PDF) of a class of one-dimensional stochastic mechanic...

Path integral treatment of the deformed Schiöberg-type potential for some diatomic molecules

The bound state solution of the Feynman propagator with the deformed generalized Schiöberg potential is determined using an approximation of the centrifugal term. The energy eigenvalue expression is computed using Duru–Kleinert space–time transformation for both positive and negative deformation parameters of diatomic molecules. Besides, the rotation–vibration energy eigenvalues are numerically calculated for some diatomic molecules and compared with those given in the literature. The obtained results are in agreement with those given by state-of-the-art approximate and numerical methods.

Drell-Yan process in pA collisions: Path-integral treatment of coherence effects

In this work, we investigate production of Drell-Yan (DY) pairs in proton-nucleus collisions in kinematic regions where the corresponding coherence length does not exceed the nuclear radius, $R_A$, and the quantum coherence effects should be treated with a special care. The results for the nucleus-to-nucleon production ratio available in the literature so far are usually based on the assumption of a very long coherence length (LCL) $l_c\gg R_A$. Since the onset of coherence effects is controlled by the coherence length $l_c$, we estimated its magnitude in various kinematic regions of the DY process and found that the LCL approximation should not be used at small and medium c.m. collision energies ($\sqrt{s} \lesssim 200$ GeV) as well as at large dilepton invariant masses. In order to obtain realistic predictions, we computed for the first time the DY cross section using the generalised color dipole approach based on the rigorous Green function formalism, which naturally incorporates the color transparency and quantum coherence effects and hence allows to estimate the nuclear shadowing with no restrictions on the CL. In addition to the shadowing effect, we studied a complementary effect of initial state interactions (ISI) that causes an additional suppression at large values of the Feynman variable. Numerical results for the nuclear modification factor accounting for the ISI effect and the finite $l_c$ are compared to the data available from the fixed-target FNAL measurements and a good agreement has been found. Besides, we present new predictions for the nuclear suppression as a function of dilepton rapidity and invariant mass in the kinematic regions that can be probed by the RHIC collider as well as by the planned AFTER@LHC and LHCb fixed-target experiments.