Use Of Path Integrals Research Articles

Taking a shortcut: what mechanisms do fish use?

Path integration is a powerful navigational mechanism whereby individuals continuously update their distance and angular vector of movement to calculate their position in relation to their departure location, allowing them to return along the most direct route even across unfamiliar terrain. While path integration has been investigated in several terrestrial animals, it has never been demonstrated in aquatic vertebrates, where movement occurs through volumetric space and sensory cues available for navigation are likely to differ substantially from those in terrestrial environments. By performing displacement experiments with Lamprologus ocellatus, we show evidence consistent with fish using path integration to navigate alongside other mechanisms (allothetic place cues and route recapitulation). These results indicate that the use of path integration is likely to be deeply rooted within the vertebrate phylogeny irrespective of the environment, and suggests that fish may possess a spatial encoding system that parallels that of mammals.

Effect of repetition of vertical and horizontal routes on navigation performance in Australian bull ants

Solitarily foraging ant species differ in their reliance on their two primary navigational systems— path integration and visual learning. Despite many species of Australian bull ants spending most of their foraging time on their foraging tree, little is known about the use of these systems while climbing. “Rewinding” displacements are commonly used to understand navigational system usage, and work by introducing a mismatch between these navigational systems, by displacing foragers after they have run-down their path integration vector. We used rewinding to test the role of path integration on the arboreal and terrestrial navigation of M. midas. We rewound foragers along either the vertical portion, the ground surface portion, or across both portions of their homing trip. Since rewinding involves repeatedly capturing and releasing foragers, we included a nondisplacement, capture-and-release control, in which the path integration vector is unchanged. We found that rewound foragers do not seem to accumulate path integration vector, although a limited effect of vertical rewinding was found, suggesting a potential higher sensitivity while descending the foraging tree. However, the decrease in navigational efficiency due to capture was larger than the vertical rewinding effect, which along with the negative impact of the vertical surface, and an interaction between capture and rewinding, may suggest aversion rather than path integration caused the vertical rewinding response. Together these results add to the evidence that M. midas makes minimal use of path integration while foraging, and the growing evidence that they are capable of quickly learning from aversive stimulus.

Path integrals: From quantum mechanics to photonics

The path integral formulation of quantum mechanics, i.e., the idea that the evolution of a quantum system is determined as a sum over all the possible trajectories that would take the system from the initial to its final state of its dynamical evolution, is perhaps the most elegant and universal framework developed in theoretical physics, second only to the standard model of particle physics. In this Tutorial, we retrace the steps that led to the creation of such a remarkable framework, discuss its foundations, and present some of the classical examples of problems that can be solved using the path integral formalism, as a way to introduce the readers to the topic and help them get familiar with the formalism. Then, we focus our attention on the use of path integrals in optics and photonics and discuss in detail how they have been used in the past to approach several problems, ranging from the propagation of light in inhomogeneous media to parametric amplification and quantum nonlinear optics in arbitrary media. To complement this, we also briefly present the path integral Monte Carlo method, as a valuable computational resource for condensed matter physics, and discuss its potential applications and advantages if used in photonics.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

Pheromone cue triggers switch between vectors in the desert harvest ant, Veromessor pergandei.

The desert harvester ant (Veromessor pergandei) employs a mixture of social and individual navigational strategies at separate stages of their foraging trip. Individuals leave the nest along a pheromone-based column, travelling 3-40m before spreading out to forage individually in a fan. Foragers use path integration while in this fan, accumulating a direction and distance estimate (vector) to return to the end of the column (column head), yet foragers' potential use of path integration in the pheromone-based column is less understood. Here we show foragers rely on path integration both in the foraging fan and while in the column to return to the nest, using separate vectors depending on their current foraging stage in the fan or column. Returning foragers displaced while in the fan oriented and travelled to the column head location while those displaced after reaching the column travel in the nest direction, signifying the maintenance of a two-vector system with separate fan and column vectors directing a forager to two separate spatial locations. Interestingly, the trail pheromone and not the surrounding terrestrial cues mediate use of these distinct vectors, as fan foragers briefly exposed to the pheromone cues of the column in isolation altered their paths to a combination of the fan and column vectors. The pheromone acts as a contextual cue triggering both the retrieval of the column-vector memory and its integration with the forager's current fan-vector.

Sugar Intake Elicits Intelligent Searching Behavior in Flies and Honey Bees.

We present a comparison of the sugar-elicited search behavior in Drosophila melanogaster and Apis mellifera. In both species, intake of sugar-water elicits a complex of searching responses. The most obvious response was an increase in turning frequency. However, we also found that flies and honey bees returned to the location of the sugar drop. They even returned to the food location when we prevented them from using visual and chemosensory cues. Analyses of the recorded trajectories indicated that flies and bees use two mechanisms, a locomotor pattern involving an increased turning frequency and path integration to increase the probability to stay close or even return to the sugar drop location. However, evidence for the use of path integration in honey bees was less clear. In general, walking trajectories of honey bees showed a higher degree of curvature and were more spacious; two characters which likely masked evidence for the use of path integration in our experiments. Visual cues, i.e., a black dot, presented underneath the sugar drop made flies and honey bees stay closer to the starting point of the search. In honey bees, vertical black columns close to the sugar drop increased the probability to visit similar cues in the vicinity. An additional one trial learning experiment suggested that the intake of sugar-water likely has the potential to initiate an associative learning process. Together, our experiments indicate that the sugar-elicited local search is more complex than previously assumed. Most importantly, this local search behavior appeared to exhibit major behavioral capabilities of large-scale navigation. Thus, we propose that sugar-elicited search behavior has the potential to become a fruitful behavioral paradigm to identify neural and molecular mechanisms involved in general mechanisms of navigation.

E-Invariant Quantized Motion of Valence Quarks

In sub-proton space wave processes are impossible. The analog of the Klein–Gordon equation in sub-proton space is elliptical and describes a stationary system with a constant number of particles. For dynamical processes, separation of variables is used and in each quantum of motion of the quark two states are distinguished: a localization state and a translation state with infinite velocity. Alternation of these states describes the motion of a quark. The mathematical expectations of the lifetimes of the localization states and the spatial extents of the translation states for a free quark and for a quark in a centrally symmetric potential are found. The action after one quantum of motion is equal to the Planck constant. The one-sided Laplace transform is used to determine the Green’s function. Use of path integrals shows that the quantized trajectory of a quark is a broken line enveloping the classical trajectory of oscillation of the quark. Comparison of the calculated electric charge distribution in a proton with its experimental value gives satisfactory results. A hypothesis is formulated, according to which the three Grand Geometries of space correspond to the three main interactions of elementary particles.

Idiothetic Path Integration in the Fruit Fly Drosophila melanogaster

After discovering a small drop of food, hungry flies exhibit a peculiar behavior in which they repeatedly stray from, but then return to, the newly discovered resource. To study this behavior in more detail, we tracked hungry Drosophila as they explored a large arena, focusing on the question of how flies remain near the food. To determine whether flies use external stimuli, we individually eliminated visual, olfactory, and pheromonal cues. In all cases, flies still exhibited a centralized search behavior, suggesting that none of these cues are absolutely required for navigation back to the food. To simultaneously eliminate visual and olfactory cues associated with the position of the food, we constructed an apparatus in which the food could be rapidly translated from the center of the arena. Flies continued to search around the original location, even after the food was moved to a new position. A random search model based on measured locomotor statistics could not reproduce the centered nature of the animal's trajectory. We conclude that this behavior is best explained by a form of path integration in which the flies use idiothetic cues to search near the location of the food. We argue that the use of path integration to perform a centered local search is not a specialization of Drosophila but rather represents an ancient behavioral mode that is homologous to the more elaborate foraging strategies of central place foragers such as ants.

Homeward navigation in Pogonomyrmex occidentalis harvester ants

This study determined the relative importance of sources of navigational information in Pogonomyrmex occidentalis harvester ants. Displacement experiments, polarized light manipulations, and view manipulations tested the use of path integration, polarized light patterns, and visual guidance in the ability of Pogonomyrmex ants to orient homeward during foraging. In the displacement experiments, worker ants that were displaced from their original locations to new locations equidistant from the nest initially walked in the direction of the home vector that would be determined by a path integrator (PI). Sham treatments showed that handling and returning ants to their original location had no effect on homeward navigation. Neither a polarizing filter nor changing nearby landmarks affected orientation. However, placement of a wall behind the nest while an ant was navigating homeward caused it to become disoriented. Ants viewing the wall walked in random directions and took more time to find their nest. These results suggest that P. occidentalis workers rely on PI to navigate homeward after foraging, but do not follow the home vector indicated by the PI if the panorama has been significantly altered and, therefore, conflicts with the available PI information.

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.

Hierarchical use of chemical marking and path integration in the homing trip of a subsocial shield bug

The female shield bug Parastrachia japonensis provisions its young by dragging fruit to its burrow. Field observations showed that the bug took a winding path when searching for a suitable fruit, but took the shortest route when homing to the burrow. Displaced homing bugs always walked straight towards the fictive burrow, suggesting that they use path integration to orient. After a homing bug neared the entrance of its burrow, it stopped and started beating the surface of fallen leaves with its antennae. To determine whether these bugs use navigational cues other than those used for path integration, when in the vicinity of the burrow, we blocked their sensory organs and presented them with their own burrows in a laboratory experiment. Although nearly all bugs whose eyes had been blocked found their burrows, the antennae-blocked bugs did not. Homing bugs encountering various experimentally manipulated burrows, such as those containing their own nymphs with an alien burrow's substrate or their own burrow's substrate with alien nymphs, entered burrows only if they were made of the original substrate. When we presented bugs with their own burrows along the homing route, they entered their burrows at every homing point, even at the earliest stage of the homing process. These results suggest that the chemical cues marked around a female's burrow can suppress the use of path integration, and that P. japonensis uses cues hierarchically to accomplish precise homing.

On the use of path integrals in micro‐electronic design

AbstractThe design of micro‐electronic processes is done by running process splits that either consist of laboratory or clean‐room experiment or of computer experiments. In order to obtain optimal designs, in terms of high yield and having the specifications on target, the design space is explored using design‐of‐experiment and response‐surface methods. In this paper we present a new technique for performing continuous interpolation between the experimental points in order to determine non‐explored input settings. The method eliminates all bias which is traditionally built into the surface construction by assuming polynomial behavior by using the maximum entropy principle and by constructing numerical interpolation hyper surfaces. The technique is generic and can be applied also outside the field of microelectronic design.

The use of path integration to guide route learning in ants

Cataglyphid ants travelling between their nest and feeding site follow familiar routes along which they are guided by views of the surrounding landscape1,2,3,4,5,. On bare terrain, with no landmarks available, ants can still navigate using path integration6. They continually monitor their net distance and direction from the nest, so that they can return home from any point using their computed ‘home vector’7. Here we ask whether path integration also provides signals to reinforce the learning of visual landmarks. A fall in the value of the home vector indicates when a homing ant moves in roughly the correct direction, and that it is appropriate to store those views that can guide subsequent trips to the nest8. We tested this hypothesis by training the ant Cataglyphis cursor to negotiate a variety of mazes that led from a feeding site back to the nest. Efficient passage of each maze required an ant to discriminate between different pairs of shapes9. We show that if the value of the home vector drops while the ant approaches and passes a shape, the shape's appearance is learnt, but if the vector grows, or is absent, no visual learning occurs. Path integration may both help ants navigate through an unfamiliar landscape, and assist them to become familiar with it.

Path integral calculation of the writhe for circular semiflexible polymers

Earlier obtained numerical and analytical (non-path-integral) results for the writhe of a circular polymer are reproduced exactly employing the available path integral models for the semiflexible chains. It is confirmed analytically that the average writhe scales with the root of the chain length. We also briefly discuss related topological problems (e.g. knotting of circular polymers), which could be formulated and solved with the use of path integrals.

ON A NEW METHOD FOR COMPUTING TRACE ANOMALIES

I describe a new method for computing trace anomalies in quantum field theories which makes use of path-integrals for particles moving in curved spaces. After presenting the main ideas of the method, I discuss how it is connected to the first quantized approach of particle theory and to heat kernel techniques.

Path integrals in N=2 supersymmetric quantum mechanics

By use of path integrals in N=2 supersymmetric quantum mechanics, a formula for the kernel of the supersymmetric time evolution operator is found. Structurally it is similar to the Feynman–Kac formula used in the path integrals of quantum mechanics. Crucial to the development of this formula is a generalization of the Wiener measure based upon the free supercharges rather than the free Hamiltonian.

Symmetry and combinatorics in the delta expansion for QED.

Combinatorics and symmetry are used to solve calculational problems in applying the {delta}-expansion method to QED. Adequate Feynman rules are used to simplify previous calculations and Ward identities are obtained. In the case with no external fermions an equivalence to the conventional loop expansion is established, both through the use of path integrals and through explicit calculations of the effective vertices. The Ward identities are reduced in this case to the conventional ones implied by gauge invariance.

Evaluation of matrix elements for nonorthogonal states with use of path integrals.

Based on path-integral formalism we derive expressions for nonorthogonal matrix elements. The result reduces to few integrals in a multivalued complex plane. The present approach represents an improvement over earlier work in that the matrix elements are much more accurate.

On the most probable path for diffusion processes

The Lagrangian which determines the most probable path for diffusion processes is obtained by a simple and intuitive technique based entirely on the use of path integrals.

The use of path integrals in guided wave theory

In a stratified wave guiding system the propagation of electromagnetic waves is usually described by the use of modes. This method however involves summations which are a formidable task if many modes are excited and attenuated equally. A new theoretical method using path integrals is applied to this problem and avoids the use of modes. As a consequence large summations are avoided and in some cases compact analytical expressions for the field or the impulse response are obtained.