Path Integral Framework Research Articles

Faddeev-Jackiw quantization and the path integral

The method for quantization of constrained theories that was suggested originally by Faddeev and Jackiw along with later modifications is discussed. The particular emphasis of this paper is to show how it is simple to implement their method within the path integral framework using the natural geometric structure that their method utilizes. The procedure is exemplified with the analysis of two models: a quantum mechanical particle constrained to a surface (of which the hypersphere is a special case), and a quantized Schr\"odinger field interacting with a quantized vector field for both the massive and the massless cases. The results are shown to agree with what is found using the Dirac method for constrained path integrals. We comment on a previous path integral analysis of the Faddeev-Jackiw method. We also discuss why a previous criticism of the Faddeev-Jackiw method is unfounded and why suggested modifications of their method are unnecessary.

Path-integral derivation of the nonrelativistic scale anomaly

In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the $\ensuremath{\beta}$ function.

Quantifying selective pressures driving bacterial evolution using lineage analysis.

Organisms use a variety of strategies to adapt to their environments and maximize long-term growth potential, but quantitative characterization of the benefits conferred by the use of such strategies, as well as their impact on the whole population's rate of growth, remains challenging. Here, we use a path-integral framework that describes how selection acts on lineages -i.e. the life-histories of individuals and their ancestors- to demonstrate that lineage-based measurements can be used to quantify the selective pressures acting on a population. We apply this analysis to E. coli bacteria exposed to cyclical treatments of carbenicillin, an antibiotic that interferes with cell-wall synthesis and affects cells in an age-dependent manner. While the extensive characterization of the life-history of thousands of cells is necessary to accurately extract the age-dependent selective pressures caused by carbenicillin, the same measurement can be recapitulated using lineage-based statistics of a single surviving cell. Population-wide evolutionary pressures can be extracted from the properties of the surviving lineages within a population, providing an alternative and efficient procedure to quantify the evolutionary forces acting on a population. Importantly, this approach is not limited to age-dependent selection, and the framework can be generalized to detect signatures of other trait-specific selection using lineage-based measurements. Our results establish a powerful way to study the evolutionary dynamics of life under selection, and may be broadly useful in elucidating selective pressures driving the emergence of antibiotic resistance and the evolution of survival strategies in biological systems.

Electron transport properties in m-plane and c-plane AlN/GaN heterostructures with interface roughness and anisotropic in-plane strain scatterings

Anisotropic transport properties of a two-dimensional electron gas in nonpolar m-plane AlN/GaN heterostructures with the interface roughness coupled anisotropic in-plane strain scattering were investigated theoretically using a path-integral framework. The scattering potential was composed of the interface roughness and the effective field from the electron charge and the net piezoelectric polarization. We showed that the anisotropic biaxial strains generate only the net piezoelectric polarization along the [0 0 0 1]-direction and cause anisotropy in electron mobility with a magnitude lower than the -direction. We also showed that the anisotropy in electron mobility reduced with increasing electron density. Moreover, the anisotropic electron mobility disappeared when the anisotropic in-plane strain scattering was removed, and the relation for pure interface roughness scattering was reestablished. This formulation with existing roughness parameters gave a good description for the experimental results of polar c-plane AlN/GaN heterostructures.

Atomistic modeling of vibrational action spectra in polyatomic molecules: nuclear quantum effects.

The response of a polyatomic molecule to an infrared (IR) laser pulse of varying frequency has been simulated by classical molecular dynamics simulations and by quantum methods based on the path-integral framework (PIMD), as well as quantum thermal baths (QTBs). The outcome of the trajectories was subsequently processed to predict a dissociation spectrum, from the precalculated rate constant. Naphthalene described by a tight-binding potential energy surface was chosen as a testing ground for the present problem, possibly emitting an hydrogen atom after a 12 ps long pulse. At low field intensities, the heating efficiency of the pulse is found to vary similarly as the IR absorption spectrum for all methods considered, reflecting the validity of linear response in this regime. At fields that are sufficiently high to induce statistical dissociation over mass spectrometry timescales, marked differences appear with the spectral features exhibiting additional broadenings and redshift, especially for quantum mechanical descriptions of nuclear motion. Those excessive broadenings are mostly caused by anharmonicities but also convey the inherent approximations of the semiclassical QTB method and point at limitations of the PIMD simulations when used in such strong out-of-equilibrium conditions.

Path integral action and Chern–Simons quantum mechanics in noncommutative plane

In this paper, the connection between the path integral representation of propagators in the coherent state basis with additional degrees of freedom (Rohwer et al 2010 J. Phys. A: Math. Theor. 43 345302) and the one without any such degrees of freedom (Gangopadhyay and Scholtz 2009 Phys. Rev. Lett. 102 241602) is established. We further demonstrate that the path integral formalism developed in the noncommutative plane using the coherent state basis leads to a quantum mechanics involving a Chern–Simons term in momentum which is of noncommutative origin. The origin of this term from the Bopp-shift point of view is also investigated. A relativistic generalization of the action derived from the path integral framework is then proposed. Finally, we construct a map from the commutative quantum Hall system to a particle in a noncommutative plane moving in a magnetic field. The value of the noncommutative parameter from this map is then computed and is found to agree with previous results.

Path integral pricing of outside barrier Asian options

Using the path-integral framework to cast the pricing problem of the outside barrier Asian option into a Wiener functional integral form, we show that, after the introduction of a law-equivalent process and transformation of the new system, the deviation from the Monte Carlo price is seen to be widely reduced. Bypassing the path-partitioning step, we show that our results behave nicely with respect to increasing correlation. After putting forward empirical evidence of this improvement, we extend the scope to a double knock-out outside barrier, and derive there an original formula. In the latter setting, we propose a simple scheme to reduce the relative error due to a nearby knock-out barrier.

A simple but accurate potential for the naphthalene-argon complex: Applications to collisional energy transfer and matrix isolated IR spectroscopy

An explicit polarizable potential for the naphthalene-argon complex has been derived assuming only atomic contributions, aiming at large scale simulations of naphthalene under argon environment. The potential was parametrized from dedicated quantum chemical calculations at the CCSD(T) level, and satisfactorily reproduces available structural and energetic properties. Combining this potential with a tight-binding model for naphthalene, collisional energy transfer is studied by means of dedicated molecular dynamics simulations, nuclear quantum effects being accounted for in the path-integral framework. Except at low target temperature, nuclear quantum effects do not alter the average energies transferred by the collision or the collision duration. However, the distribution of energy transferred is much broader in the quantum case due to the significant zero-point energy and the higher density of states. Using an ab initio potential for the Ar-Ar interaction, the IR absorption spectrum of naphthalene solvated by argon clusters or an entire Ar matrix is computed via classical and centroid molecular dynamics. The classical spectra exhibit variations with growing argon environment that are absent from quantum spectra. This is interpreted by the greater fluxional character experienced by the argon atoms due to vibrational delocalization.

On Dirac-Coulomb problem in (2+1) dimensional space-time and path integral quantization

The problem of Dirac particle interacting with Coulomb potential in (2+1) dimensions is formulated in the framework of super-symmetric path integrals where the spin degrees of freedom are described by odd Grassmannian variables. The relative propagator is expressed through Cartesian coordinates in a Hamiltonian form by the use of an adequate transformation. The passage to the polar coordinates permitted us to calculate the fixed energy Green's function and to extract bound states and associating wave functions.

A NONLOCAL UNITARY VECTOR MODEL IN THREE DIMENSIONS

We present a unified analysis of single excitation vector models in three dimensions. We show that there is a family of first-order master actions related by duality transformations which interpolate between the different models. We use a Hamiltonian (2+1) analysis to show the equivalence of the self-dual and topologically massive models with a covariant nonlocal model which propagates also a single massive excitation. It is shown how the nonlocal terms appears naturally in the path integral framework.

Integrable modification of the critical Chalker-Coddington network model

We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an $\mathrm{SU}{(2)}_{r}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{r}/\mathrm{SU}{(2)}_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n=\ensuremath{-}2\mathrm{cos}[\ensuremath{\pi}/(r+2)]$, where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.

Two-dimensional electron transport in MgZnO/ZnO heterostructures: role of interface roughness

Mobility of two-dimensional electron gases in MgZnO/ZnO heterostructures with interface roughness effects was investigated theoretically using path-integral framework. We modelled the roughness-induced fluctuation by including two major effects, i.e. the electron and polarization-induced positive charge concentrations. We showed that both effects cause the scattering potential in the in-plane direction and hence affect the 2D mobility. In this work, we treated both electron and polarization-induced positive charge concentrations as equally important factors and then calculated the electron mobility and compared with the experimental result of Mg0.2Zn0.8O/ZnO heterostructure at high-electron concentrations. We found that the fitting parameters Δ = 0.26 nm, Λ = 2.5 nm gave good description to the mobility data. We also showed that neglecting the polarization-induced positive charge concentration led to overestimating the 2D mobility.

SELF-DUAL FORMULATIONS OF d = 3 GRAVITY THEORIES IN THE PATH-INTEGRAL FRAMEWORK

We study the connection, at the quantum level, between (d = 2+1)-dimensional self-dual models with actions of growing (from first to fourth) order, governing the dynamics of helicity ±2 massive excitations. We obtain identities between generating functionals of the different models using the path-integral framework, this allowing us to establish dual maps among relevant vacuum expectation values. We check consistency of these vacuum expectation values with the gauge invariance gained in each mapping.

Resonant enhancement of the Fulde-Ferell-Larkin-Ovchinnikov state in three dimensions by a one-dimensional optical potential

We describe an imbalanced superfluid Fermi gas in three dimensions within the path-integral framework. To allow for the formation of the Fulde-Ferell-Larkin-Ovchinnikov state (FFLO state), a suitable form of the saddle point is chosen, in which the pairs have a finite center-of-mass momentum. To test the correctness of this path-integral description, the zero-temperature phase diagram for an imbalanced Fermi gas in three dimensions is calculated, and compared to recent theoretical results. Subsequently, we investigate two models that describe the effect of imposing a one-dimensional (1D) optical potential on the three-dimensional (3D) imbalanced Fermi gas. We show that this 1D optical potential can greatly enlarge the stability region of the FFLO state, relative to the case of the 3D Fermi gas without 1D periodic modulation. Furthermore, it is shown that there exists a direct connection between the center-of-mass momentum of the FFLO pairs and the wave vector of the optical potential. We propose that this concept can be used experimentally to resonantly enhance the stability region of the FFLO state.

An Iterative Path Integral Stochastic Optimal Control Approach for Learning Robotic Tasks

AbstractRecent work on path integral stochastic optimal control theory Theodorou et al. (2010a); Theodorou (2011) has shown promising results in planning and control of nonlinear systems in high dimensional state spaces. The path integral control framework relies on the transformation of the nonlinear Hamilton Jacobi Bellman (HJB) partial differential equation (PDE) into a linear PDE and the approximation of its solution via the use of the Feynman Kac lemma. In this work, we are reviewing the generalized version of path integral stochastic optimal control formalism Theodorou et al. (2010a), used for optimal control and planing of stochastic dynamical systems with state dependent control and diffusion matrices. Moreover we present the iterative path integral control approach, the so called Policy Improvement with Path Integrals or (PI2) which is capable of scaling in high dimensional robotic control problems. Furthermore we present a convergence analysis of the proposed algorithm and we apply the proposed framework to a variety of robotic tasks. Finally with the goal to perform locomotion the iterative path integral control is applied for learning nonlinear limit cycle attractors with adjustable land scape.

The path integral representation kernel of evolution operator in Merton-Garman model

In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

On the Puzzle of Odd-Frequency Superconductivity

Since the first theoretical proposal by Berezinskii, an odd-frequency superconductivity has encountered the fundamental problems on its thermodynamic stability and rigidity of a homogenous state accompanied by unphysical Meissner effect. Recently, Solenov {\it et al}. [Phys. Rev. B {\bf 79} (2009) 132502.] have asserted that the path-integral formulation gets rid of the difficulties leading to a stable homogenous phase with an ordinary Meissner effect. Here, we show that it is crucial to choose the appropriate saddle-point solution that minimizes the effective free energy, which was assumed {\it implicitly} in the work by Solenov and co-workers. We exhibit the path-integral framework for the odd-frequency superconductivity with general type of pairings, including an argument on the retarded functions via the analytic continuation to the real axis.

Free-energy landscapes from adaptively biased methods: Application to quantum systems

Several parallel adaptive biasing methods are applied to the calculation of free-energy pathways along reaction coordinates, choosing as a difficult example the double-funnel landscape of the 38-atom Lennard-Jones cluster. In the case of classical statistics, the Wang-Landau and adaptively biased molecular-dynamics (ABMD) methods are both found efficient if multiple walkers and replication and deletion schemes are used. An extension of the ABMD technique to quantum systems, implemented through the path-integral MD framework, is presented and tested on Ne38 against the quantum superposition method.

The Maslov correction in the semiclassical Feynman integral

Abstract The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) $$ in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.

Kinetic paths, time scale, and underlying landscapes: A path integral framework to study global natures of nonequilibrium systems and networks

We developed a general framework to quantify three key ingredients for dynamics of nonequilibrium systems through path integrals in length space. First, we identify dominant kinetic paths as the ones with optimal weights, leading to effective reduction of dimensionality or degrees of freedom from exponential to polynomial so large systems can be treated. Second, we uncover the underlying nonequilibrium potential landscapes from the explorations of the state space through kinetic paths. We apply our framework to a specific example of nonequilibrium network system: lambda phage genetic switch. Two distinct basins of attractions emerge. The dominant kinetic paths from one basin to another are irreversible and do not follow the usual steepest descent or gradient path along the landscape. It reflects the fact that the dynamics of nonequilibrium systems is not just determined by potential gradient but also the residual curl flux force, suggesting experiments to test theoretical predictions. Third, we have calculated dynamic transition time scales from one basin to another critical for stability of the system through instantons. Theoretical predictions are in good agreements with wild type and mutant experiments. We further uncover the correlations between the kinetic transition time scales and the underlying landscape topography: the barrier heights along the dominant paths. We found that both the dominant paths and the landscape are relatively robust against the influences of external environmental perturbations and the system tends to dissipate less with less fluctuations. Our general framework can be applied to other nonequilibrium systems.