Path Integral Research Articles

Entanglement entropy and the boundary action of edge modes

We consider an antisymmetric gauge field in the Minkowski space of d-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.

A Path Integral Approach for Allele Frequency Dynamics Under Polygenic Selection.

Many phenotypic traits have a polygenic genetic basis, making it challenging to learn their genetic architectures and predict individual phenotypes. One promising avenue to resolve the genetic basis of complex traits is through evolve-and-resequence experiments, in which laboratory populations are exposed to some selective pressure and trait-contributing loci are identified by extreme frequency changes over the course of the experiment. However, small laboratory populations will experience substantial random genetic drift, and it is difficult to determine whether selection played a roll in a given allele frequency change. Predicting how much allele frequencies change under drift and selection had remained an open problem well into the 21st century, even those contributing to simple, monogenic traits. Recently, there have been efforts to apply the path integral, a method borrowed from physics, to solve this problem. So far, this approach has been limited to genic selection, and is therefore inadequate to capture the complexity of quantitative, highly polygenic traits that are commonly studied. Here we extend one of these path integral methods, the perturbation approximation, to selection scenarios that are of interest to quantitative genetics. In particular, we derive analytic expressions for the transition probability (i.e., the probability that an allele will change in frequency from , to in time ) of an allele contributing to a trait subject to stabilizing selection, as well as that of an allele contributing to a trait rapidly adapting to a new phenotypic optimum. We use these expressions to characterize the use of allele frequency change to test for selection, as well as explore optimal design choices for evolve-and-resequence experiments to uncover the genetic architecture of polygenic traits under selection.

On the evaluation of the Ray-Singer torsion path integral

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves S1 bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.

Mental navigation in the primate entorhinal cortex

A cognitive map is a suitably structured representation that enables novel computations using previous experience; for example, planning a new route in a familiar space1. Work in mammals has found direct evidence for such representations in the presence of exogenous sensory inputs in both spatial2,3 and non-spatial domains4–10. Here we tested a foundational postulate of the original cognitive map theory1,11: that cognitive maps support endogenous computations without external input. We recorded from the entorhinal cortex of monkeys in a mental navigation task that required the monkeys to use a joystick to produce one-dimensional vectors between pairs of visual landmarks without seeing the intermediate landmarks. The ability of the monkeys to perform the task and generalize to new pairs indicated that they relied on a structured representation of the landmarks. Task-modulated neurons exhibited periodicity and ramping that matched the temporal structure of the landmarks and showed signatures of continuous attractor networks12,13. A continuous attractor network model of path integration14 augmented with a Hebbian-like learning mechanism provided an explanation of how the system could endogenously recall landmarks. The model also made an unexpected prediction that endogenous landmarks transiently slow path integration, reset the dynamics and thereby reduce variability. This prediction was borne out in a reanalysis of firing rate variability and behaviour. Our findings link the structured patterns of activity in the entorhinal cortex to the endogenous recruitment of a cognitive map during mental navigation.

Black hole wavefunctions and microcanonical states

We consider the problem of defining a microcanonical thermofield double state at fixed energy and angular momentum from the gravitational path integral. A semiclassical approximation to this state is obtained by imposing a mixed boundary condition on an initial time surface. We analyze the corresponding boundary value problem and gravitational action. The overlap of this state with the canonical thermofield double state, which is interpreted as the Hartle-Hawking wavefunction of an eternal black hole in a mini-superspace approximation, is calculated semiclassically. The relevant saddlepoint is a higher-dimensional, rotating generalization of the wedge geometry that has been studied in two-dimensional gravity. Along the way we discuss a new corner term in the gravitational action that arises at a rotating horizon.

Formalising the role of behaviour in neuroscience.

We develop a mathematical approach to formally proving that certain neural computations and representations exist based on patterns observed in an organism's behaviour. To illustrate, we provide a simple set of conditions under which an ant's ability to determine how far it is from its nest would logically imply neural structures isomorphic to the natural numbers . We generalise these results to arbitrary behaviours and representations and show what mathematical characterisation of neural computation and representation is simplest while being maximally predictive of behaviour. We develop this framework in detail using a path integration example, where an organism's ability to search for its nest in the correct location implies representational structures isomorphic to two-dimensional coordinates under addition. We also study a system for processing strings common in comparative work. Our approach provides an objective way to determine what theory of a physical system is best, addressing a fundamental challenge in neuroscientific inference. These results motivate considering which neurobiological structures have the requisite formal structure and are otherwise physically plausible given relevant physical considerations such as generalisability, information density, thermodynamic stability and energetic cost.

Improved reliability and availability of fundamental properties for all hydrogen isotopologues by Gaussian process regression using data from experiments and path-integral simulations

The thermodynamic and transport properties of hydrogen isotopologues in the solid, liquid, and gas phases are crucial for hydrogen energy exploitation, such as the design of the fuel cycle of nuclear fusion reactors. However, experimental data for tritium-containing species are often not available. Alternatively, path integral (PI) simulation can evaluate hydrogen properties taking into account nuclear quantum effects; however, concerns about its reliability are present. This study proposes a method of Gaussian process regression (GPR) using both experimental and PI simulation data to obtain practically plausible estimates for all isotopologues. Using liquid viscosity and diffusivity as test cases, we demonstrate that the present method has high and robust predictive performance. The method has broad applicability and can also be used to design experiments and calculations to efficiently improve the regression model, and thus is expected to contribute to enhancing the availability and reliability of the hydrogen isotopologue property database.

Cell-type-specific cholinergic control of granular retrosplenial cortex with implications for angular velocity coding across brain states.

Cholinergic receptor activation enables the persistent firing of cortical pyramidal neurons, providing a key cellular basis for theories of spatial navigation involving working memory, path integration, and head direction encoding. The granular retrosplenial cortex (RSG) is important for spatially-guided behaviors, but how acetylcholine impacts RSG neurons is unknown. Here, we show that a transcriptomically, morphologically, and biophysically distinct RSG cell-type - the low-rheobase (LR) neuron - has a very distinct expression profile of cholinergic muscarinic receptors compared to all other neighboring excitatory neuronal subtypes. LR neurons do not fire persistently in response to cholinergic agonists, in stark contrast to all other principal neuronal subtypes examined within the RSG and across midline cortex. This lack of persistence allows LR neuron models to rapidly compute angular head velocity (AHV), independent of cholinergic changes seen during navigation. Thus, LR neurons can consistently compute AHV across brain states, highlighting the specialized RSG neural codes supporting navigation.

Relativistic energies for the q-deformed Scarf potential with Feynman path integrals formulation

In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for an arbitrary spin-orbit quantum number κ, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this method is used to determine the spectrum of two diatomic molecules Li 2(61Π u ) and KRb(B −1Π). The obtained results are compared to the experimental and numerical ones.

Universality in logarithmic temperature corrections to near-extremal rotating black hole thermodynamics in various dimensions

The low-temperature thermodynamics of near-extremal rotating black holes has recently been revisited to incorporate one-loop contributions that are dominant in this regime. We discuss these quantum corrections to the gravitational path integral for asymptotically Anti de-Sitter black holes in four and five dimensions. In four dimensions we explicitly consider Kerr-AdS4, Kerr-Newman-AdS4 and the rotating black hole in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 gauged supergravity with two scalars and two electric charges turned on. In five dimensions we explicitly address the asymptotically flat Myers-Perry black hole and the Kerr-AdS5 black hole. In every case we find that tensor modes contribute 32logTHawking\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{2}\\log {T}_{\ extrm{Hawking}} $$\\end{document} to the low-temperature thermodynamics. We identify the root cause of this universality in two facts: (i) the universal presence of a SL(2, ℝ) subgroup of isometries in the near-horizon geometry and (ii) a set of cancellations in the Lichnerowicz operator. We show that these two conditions hold for near-extremal black holes in asymptotically flat and asymptotically AdS spacetimes of various dimensions.

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

Supersymmetry and trace formulas. Part I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

A Bionic Localization Memristive Circuit Based on Spatial Cognitive Mechanisms of Hippocampus and Entorhinal Cortex.

In this article, a bionic localization memristive circuit is proposed, which mainly consists of head direction cell module, grid cell module, place cell module and decoding module. This work modifies the two-dimensional Continuous Attractor Network (CAN) model of grid cells into two one-dimensional models in X and Y directions. The head direction cell module utilizes memristors to integrate angular velocity and represents the real orientation of an agent. The grid cell module uses memristors to sense linear velocity and orientation signals, which are both self-motion cues, and encodes the position in space by firing in a periodic mode. The place cell module receives the grid cell module's output and fires in a specific position. The decoding module decodes the angle or place information and transfers the neuron state to a 'one-hot' code. This proposed circuit completes the localizing task in space and realizes in-memory computing due to the use of memristors, which can shorten the execution time. The functions mentioned above are implemented in LTSPICE. The simulation results show that the proposed circuit can realize path integration and localization. Moreover, it is shown that the proposed circuit has good robustness and low area overhead. This work provides a possible application idea in a prospective robot platform to help the robot localize and build maps.

(Nonequilibrium) Dynamics of Diffusion Processes with Non-conservative Drifts

The nonequilibrium Fokker–Planck dynamics with a non-conservative drift field, in dimension N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, can be related with the non-Hermitian quantum mechanics in a real scalar potential Vand in a purely imaginary vector potential -iAof real amplitude A (Mazzolo and Monthus in Phys Rev E 107:014101, 2023). Since Fokker–Planck probability density functions may be obtained by means of Feynman’s path integrals, the previous observation points towards a general issue of “magnetically affine” propagators, possibly of quantum origin, in real and Euclidean time. In below we shall follow the N=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=3$$\\end{document} “magnetic thread”, within which one may keep under a computational control formally and conceptually different implementations of magnetism (or surrogate magnetism) in the dynamics of diffusion processes. We shall focus on interrelations (with due precaution to varied, not evidently compatible, notational conventions) of: (i) the pertinent non-conservatively drifted diffusions, (ii) the classic Brownian motion of charged particles in the (electro)magnetic field, (iii) diffusion processes arising within so-called Euclidean quantum mechanics (which from the outset employs non-Hermitian “magnetic” Hamiltonians), (iv) limitations of the usefulness of the Euclidean map exp(-itHquant)→exp(-tHEucl)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-itH_{quant}) \\rightarrow \\exp (-tH_{Eucl})$$\\end{document}, regarding the probabilistic significance of inferred (path) integral kernels in the description of diffusion processes.

Spatiotemporal structure of foraging and path integration errors by fiddler crabs, Leptuca pugilator

Path integration is the navigational process by which animals construct a memory of a previous location by continuously measuring and summing their movements to form a single home vector pointing to the starting location. It is intrinsically error prone, subject to random errors and, potentially, to systematic errors in either measurement or the summing algorithm. Both types of errors lead to an incorrect vector memory and thus to an error in homing. Because the errors are incurred when animals move, they are theoretically predictable from the movements. We analyzed the behavior of fiddler crabs (Leptuca pugilator) as they performed foraging excursions followed by homing with varying degrees of error. From video recordings we measured body orientations and locations and computed these spatiotemporal path characteristics: duration, distance, turns, bearing and arc sector. These were analyzed for their effect on, separately, the magnitude, and the direction, of crabs’ homing error. The magnitude of the homing error was predicted by arc sector, Δbearing and path length, and several interactions. The direction of the homing error was predicted by interactions including arc sector x Δbearing, arc sector x turns, and Δbearing x turns. Covariance among these factors results in a path that traces a large arc while maintaining body orientation toward the burrow direction and leads to an error with the same clockwise/counterclockwise sign as the arc and the body turns. These results place L. pugilator’s path integration mechanism among others with known systematic errors.

Quantum key distribution (QKD) for wireless networks with software‐defined networking

AbstractNetwork management has been significantly transformed by software‐defined networking (SDN), which consolidates control and improves adaptability. As a result of this paradigm shift, security concerns have emerged concurrently. Within SDN environments, maintaining service path integrity is of utmost importance because malicious entities can exploit network flows to gain access to data, resulting in security breaches. Thus, QKD networks require a management plane that can control and manage the QKD resources. Quantum key distribution networks can be controlled and managed separately by SDN, allowing quantum key forwarding to be separated from their control and management. The paper provides an overview of QKD networks enabled by SDN, along with an overview of its architecture, interfaces, and protocols. This paper elaborates on the overall architecture and related interfaces and protocols of QKD networks assisted by SDN. Then, three important paradigms for simulation are presented using three use cases.

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

Nest excavators’ learning walks in the Australian desert ant Melophorus bagoti

The Australian red honey ant, Melophorus bagoti, stands out as the most thermophilic ant in Australia, engaging in all outdoor activities during the hottest periods of the day during summer months. This species of desert ants often navigates by means of path integration and learning landmark cues around the nest. In our study, we observed the outdoor activities of M. bagoti workers engaged in nest excavation, the maintenance of the nest structure, primarily by taking excess sand out of the nest. Before undertaking nest excavation, the ants conducted a single exploratory walk. Following their initial learning expedition, these ants then engaged in nest excavation activities. Consistent with previous findings on pre-foraging learning walks, after just one learning walk, the desert ants in our study demonstrated the ability to return home from locations 2 m away from the nest, although not from locations 4 m away. These findings indicate that even for activities like dumping excavated sand within a range of 5–10 cm outside the nest, these ants learn and utilize the visual landmark panorama around the nest.

Not seeing the forest for the trees: combination of path integration and landmark cues in human virtual navigation.

In order to successfully move from place to place, our brain often combines sensory inputs from various sources by dynamically weighting spatial cues according to their reliability and relevance for a given task. Two of the most important cues in navigation are the spatial arrangement of landmarks in the environment, and the continuous path integration of travelled distances and changes in direction. Several studies have shown that Bayesian integration of cues provides a good explanation for navigation in environments dominated by small numbers of easily identifiable landmarks. However, it remains largely unclear how cues are combined in more complex environments. To investigate how humans process and combine landmarks and path integration in complex environments, we conducted a series of triangle completion experiments in virtual reality, in which we varied the number of landmarks from an open steppe to a dense forest, thus going beyond the spatially simple environments that have been studied in the past. We analysed spatial behaviour at both the population and individual level with linear regression models and developed a computational model, based on maximum likelihood estimation (MLE), to infer the underlying combination of cues. Overall homing performance was optimal in an environment containing three landmarks arranged around the goal location. With more than three landmarks, individual differences between participants in the use of cues are striking. For some, the addition of landmarks does not worsen their performance, whereas for others it seems to impair their use of landmark information. It appears that navigation success in complex environments depends on the ability to identify the correct clearing around the goal location, suggesting that some participants may not be able to see the forest for the trees.

An essay on semiclassical analysis for microlocal singularities, turbulence intensity and integration of singularities by Schrödinger equation in probabilistic behavior

The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.