Path Integral Research Articles

Generalized second law for non-minimally coupled matter theories

We establish the generalized second law (GSL) within the framework of higher curvature gravity theories, considering non-minimal couplings in the matter sector. Our proof pertains to the regime of linearized fluctuations around equilibrium black holes, aligning with previous works by Wall and Sarkar. Notably, while prior proofs addressed various gravity theories such as Lovelock theory and higher curvature gravity, they uniformly assumed minimally coupled matter sectors. In this work, we extend the proof of the linearized semi-classical GSL to encompass scenarios involving non-minimal couplings in the matter sector. Our approach involves a proposal for evaluation of the matter path integral in the expectation value of the stress tensor, adopting an effective field theory treatment for the higher derivative couplings. We leverage the recently established outcome regarding the linearized second law in such theories to substantiate our argument.

Dynamical Reweighting for Biased Rare Event Simulations.

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Control and recalibration of path integration in place cells using optic flow.

Hippocampal place cells are influenced by both self-motion (idiothetic) signals and external sensory landmarks as an animal navigates its environment. To continuously update a position signal on an internal 'cognitive map', the hippocampal system integrates self-motion signals over time, a process that relies on a finely calibrated path integration gain that relates movement in physical space to movement on the cognitive map. It is unclear whether idiothetic cues alone, such as optic flow, exert sufficient influence on the cognitive map to enable recalibration of path integration, or if polarizing position information provided by landmarks is essential for this recalibration. Here, we demonstrate both recalibration of path integration gain and systematic control of place fields by pure optic flow information in freely moving rats. These findings demonstrate that the brain continuously rebalances the influence of conflicting idiothetic cues to fine-tune the neural dynamics of path integration, and that this recalibration process does not require a top-down, unambiguous position signal from landmarks.

Subleading analysis for S3 partition functions of 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} = 2 holographic SCFTs

We investigate the 3-sphere partition functions of various 3d 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} = 2 holographic SCFTs arising from the N stack of M2-branes in the ’t Hooft limit both analytically and numerically. We first employ a saddle point approximation to evaluate the free energy F = – log Z at the planar level, tracking the first subleading corrections in the large ’t Hooft coupling λ expansion. Subsequently, we improve these results by determining the planar free energy to all orders in the large λ expansion via numerical analysis. Remarkably, the resulting planar free energies turn out to take a universal form, supporting a prediction that these S3 partition functions are all given in terms of an Airy function even beyond the special cases where the Airy formulae were derived analytically in the literature; in this context we also present new Airy conjectures in several examples. The subleading behaviors we captured encode a part of quantum corrections to the M-theory path integrals around dual asymptotically Euclidean AdS4 backgrounds with the corresponding internal manifolds through holographic duality.

Uncovering 2-D toroidal representations in grid cell ensemble activity during 1-D behavior

Minimal experiments, such as head-fixed wheel-running and sleep, offer experimental advantages but restrict the amount of observable behavior, making it difficult to classify functional cell types. Arguably, the grid cell, and its striking periodicity, would not have been discovered without the perspective provided by free behavior in an open environment. Here, we show that by shifting the focus from single neurons to populations, we change the minimal experimental complexity required. We identify grid cell modules and show that the activity covers a similar, stable toroidal state space during wheel running as in open field foraging. Trajectories on grid cell tori correspond to single trial runs in virtual reality and path integration in the dark, and the alignment of the representation rapidly shifts with changes in experimental conditions. Thus, we provide a methodology to discover and study complex internal representations in even the simplest of experiments.

NS5-brane backgrounds and coset CFT partition functions

Worldsheet string theory is solvable for a variety of backgrounds involving Neveu-Schwarz fivebranes, in terms of gauged nonlinear sigma models on group manifolds. We compute the worldsheet torus partition function of these models, and propose gauging of null isometries as a unifying principle and conceptual framework for this large family of string backgrounds. In the process, we explain how partition functions of asymmetrically gauged Wess-Zumino-Witten models can be computed from the path integral, and organize and systematize various results scattered throughout the literature.

Worldline approach for spinor fields in manifolds with boundaries

The worldline formalism is a useful scheme in Quantum Field Theory which has also become a powerful tool for numerical computations. It is based on the first quantisation of a point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field theory in a bounded manifold one needs to restrict the path integration domain of the point-particle to a specific subset of worldlines enclosed by those boundaries. In the present article it is shown how to implement this restriction for the case of a spinor field in a two-dimensional curved half-plane under MIT bag boundary conditions, and compute the first few heat-kernel coefficients as a verification of the proposed construction. This construction admits several generalisations.

Path integration solutions for stochastic systems with Markovian jumps

A path integration method for solving Markovian jump stochastic dynamical systems is presented. The Markovian jump process and the state vector of the system are combined into a new augmented state vector. The randomness of the Markovian jump is modeled by a stochastic process, which is merged with the stochastic perturbations to form the stochastic source affecting the evolution of the augmented system. Then a probability mapping is constructed, mapping the probability space of the stochastic source to the short-time transition probability density function of the augmented system. By solving this probability mapping, the short-time transition probability density function of the path integration method is obtained, thus a path integration method is customized for Markovian jump stochastic systems. Finally, a hydraulic relief valve system is used as an application example to demonstrate the availability of the path integration algorithm. The results suggest that the pre-compression parameters and coefficient of restitution of the plug can induce stochastic P-bifurcation phenomena. When the above two parameters are small, the system becomes unstable with the parameters taken in this example. Monte Carlo simulations prove that the proposed method effectively captures the transient and stationary responses of Markovian jump systems.

Path integrals, complex probabilities and the discrete Weyl representation

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of the Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. The method is illustrated by applying it to scattering theory and quantum field theory. The implications of these applications for quantum computing is discussed.

Coherent spin states and emergent de Sitter quasinormal modes

As a toy model for the microscopic description of matter in de Sitter space, we consider a Hamiltonian acting on the spin-j representation of SU(2). This is a model with a finite-dimensional Hilbert space, from which quasinormal modes emerge in the large-spin limit. The path integral over coherent spin states can be evaluated at the semiclassical level and from it we find the single-particle de Sitter density of states, including 1/j corrections. Along the way, we discuss the use of quasinormal modes in quantum mechanics, starting from the paradigmatic upside-down harmonic oscillator.

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.