Path Integral Approach Research Articles

Gauss relations in Feynman integrals

Embedding Feynman integrals in Grassmannians, we can write Feynman integrals as some finite linear combinations of generalized hypergeometric functions. In this paper, we present a general method to obtain Gauss relations among those generalized hypergeometric functions. The hypergeometric expressions of Feynman integral are continued from a connected component to another by the inverse Gauss relations, then continued to the whole domain of definition by the Gauss-Kummer relations. The Laurent series of the Feynman integral around the space-time dimension D=4 is obtained by the Gauss adjacent relations, where the coefficient of the term with power of D−4 is given as the linear combinations of hypergeometric functions with integer parameters. As examples, we illustrate how to obtain the expressions for the Feynman integrals of the 1-loop self-energy and a 2-loop massless triangle diagram in the domains of definition. Published by the American Physical Society 2025

Unravelling AdS3/CFT2 near the boundary

We study correlation functions of spectrally-flowed vertex operators in bosonic string theory on AdS3 × \U0001d44b in the path integral formalism. By restricting the path integral to only include worldsheets which live near the asymptotic boundary, we compute correlation functions of spectrally-flowed vertex operators and find a precise agreement with the perturbative correlators in the recently-proposed dual CFT at all orders in conformal perturbation theory. We thus provide highly nontrivial evidence for the bulk/boundary duality.

Brownian motion in the Hilbert space of quantum states along with the Ricci flow and stochastically emergent Einstein–Hilbert action: Formulating a well-defined Feynman’s path-integral measure for quantum fields in the presence of gravity

In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton–Perelman Ricci flow. It has been shown that the Wiener measure automatically contains the Einstein–Hilbert action and the path-integral formulation of the scalar quantum field theory on curved space-time at the first order of local approximations. This provides a well-defined formulation of the path-integral measure for quantum field theory in the presence of gravity. However, we establish that the emergence of Einstein–Hilbert action is independent of the matter field interactions and is a merely entropic/geometric effect stemming from the nature of the Ricci flow of the universe geometry. We also extract an explicit formula for the cosmological constant in terms of the Ricci flow and Hamilton’s theorem for 3-manifolds. Then, we discuss the cosmological features of the FLRW solution in [Formula: see text]CDM Model via the derived equations of the Ricci flow. We also argue the correlation between our formulations and the entropic aspects of gravity. Finally, we provide some theoretical evidence that proves the second law of thermodynamics is the basic source of gravity and probably a more fundamental concept.

Path-integral approach to mutual information calculation for nonlinear channel with small dispersion at large signal-to-noise ratio

Path-integral approach to mutual information calculation for nonlinear channel with small dispersion at large signal-to-noise ratio

Path integral derivation of the thermofield double state in causal diamonds

Abstract In this article, we adopt the framework developed by Laflamme (1989 Physica A 158 58–63) to analyze the path integral of a massless—conformally invariant—scalar field defined on a causal diamond (CD) of size 2α in 1+1 dimensions. By examining the Euclidean geometry of the CD, we establish that its structure is conformally related to the cylinder S β 1 ⊗ R , where the Euclidean time coordinate τ has a periodicity of β. This property, along with the conformal symmetry of the fields, allows us to identify the connection between the thermofield double (TFD) state of CDs and the Euclidean path integral defined on the two disconnected manifolds of the cylinder. Furthermore, we demonstrate that the temperature of the TFD state, derived from the conditions in the Euclidean geometry and analytically calculated, coincides with the temperature of the CD known in the literature. This derivation highlights the universality of the connection between the Euclidean path integral formalism and the TFD state of the CD, as well as it further establishes CDs as a model that exhibits all desired properties of a system exhibiting the Unruh effect.

Two-loop planar master integrals for NNLO QCD corrections to W-pair production in quark-antiquark annihilation

The planar two-loop scalar Feynman integrals contributing to the massive NNLO QCD corrections for W-boson pair production via quark-antiquark annihilation can be classified into three family branches, each of which is reduced to a distinct set of master integrals (MIs), totaling 27, 45 and 15, respectively. These MIs are analytically calculated using the method of differential equations, with solutions expanded as Taylor series in the dimensional regulator ϵ. For the first two family branches, the differential systems can be successfully transformed into canonical form by adopting appropriate bases of MIs. This enables the MIs of these family branches to be expressed either as Goncharov polylogarithms (GPLs) or as one-fold integrals over GPLs, up to O(ϵ4). In contrast, the differential system for the third family branch can only be cast into a form linear in ϵ due to the presence of elliptic integrals. The solution to this linear-form differential system is expressed in an iterated form owing to the strictly lower-triangular structure of the coefficient matrices at ϵ = 0. Our analytic expressions for these MIs are verified with high accuracy against the numerical results from the AMFlow package.

Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach

We revisit one-loop corrections to the power spectrum of curvature perturbations ζ in an inflationary scenario containing a transient ultra-slow-roll (USR) period. In ref. [1], it was argued that one-loop corrections to the power spectrum of ζ can be larger than the tree-level one within the parameter region generating the seeds of primordial black holes during the USR epoch, which implies the breakdown of perturbation theory. We prove that this is not the case by using a master formula for one-loop corrections to the power spectrum obtained in ref. [2]. We derive the same formula within the path-integral formalism, which is simpler than the original derivation in [2]. To show the smallness of one-loop corrections, the consistency relations and the effective constancy of tree-level mode functions of ζ for super-Hubble modes play essential roles, with which the master formula gives a simple expression for one-loop corrections. For concreteness, we provide a reduced set of interactions including the leading-order one, while establishing the consistency relations in a self-consistent manner. We also show how the consistency relations of various operators hold explicitly, which plays a key role in proving the absence of large one-loop corrections.

Calculation of thermodynamic properties of helium using path integral Monte Carlo simulations in the NpT ensemble and abinitio potentials.

We apply the methodology of Lustig, with which rigorous expressions for all thermodynamic properties can be derived in any statistical ensemble, to derive expressions for the calculation of thermodynamic properties in the path integral formulation of the quantum-mechanical isobaric-isothermal (NpT) ensemble. With the derived expressions, thermodynamic properties such as the density, speed of sound, or Joule-Thomson coefficient can be calculated in path integral Monte Carlo simulations, fully incorporating quantum effects without uncontrolled approximations within the well-known isomorphism between the quantum-mechanical partition function and a classical system of ring polymers. The derived expressions are verified by simulations of supercritical helium above the vapor-liquid critical point at selected state points using recent highly accurate abinitio potentials for pairwise and nonadditive three-body interactions. We observe excellent agreement of our results with the most accurate experimental data for the density and speed of sound and a reference virial equation of state for helium in the region where the virial equation of state is converged. Moreover, our results agree closer with the experimental data and virial equation of state than the results of semiclassical simulations using the Feynman-Hibbs correction for quantum effects, which demonstrates the necessity to fully include quantum effects by path integral simulations. Our results also show that nonadditive three-body interactions must be accounted for when accurately predicting thermodynamic properties of helium by solely theoretical means.

Path integral approach for time-dependent Hamiltonians with applications to derivative pricing

Path integral approach for time-dependent Hamiltonians with applications to derivative pricing

An optimization-based equilibrium measure describing fixed points of non-equilibrium dynamics: application to the edge of chaos

Understanding neural dynamics is a central topic in machine learning, non-linear physics, and neuroscience. However, the dynamics are non-linear, stochastic and particularly non-gradient, i.e., the driving force cannot be written as the gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory. Still, the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed-form solutions in general. From the associated Fokker–Planck equation, the steady-state solution is generally unknown. Here, we treat searching for the fixed points as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, can the distribution of the original fixed points be achieved. The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos. Furthermore, the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics.

From Chaos to Integrability in Double Scaled Sachdev-Ye-Kitaev Model via a Chord Path Integral.

We study thermodynamic phase transitions between integrable and chaotic dynamics. We do so by analyzing models that interpolate between the chaotic double scaled Sachdev-Ye-Kitaev (SYK) and the integrable p-spin systems, in a limit where they are described by chord diagrams. We develop a path integral formalism by coarse graining over the diagrams, which we use to argue that the system has two distinct phases: one is continuously connected to the chaotic system, and the other to the integrable. They are separated by a line of first order transition that ends at some finite temperature.

Path integral for chord diagrams and chaotic-integrable transitions in double scaled SYK

We study transitions from chaotic to integrable Hamiltonians in the double scaled Sachdev-Ye-Kitaev (SYK) and p-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bilocal (GΣ) Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition. We also analyze the phase diagram for generic deformations, which in some cases includes a zero-temperature phase transition. Published by the American Physical Society 2024

Finite basis topologies for multiloop high-multiplicity Feynman integrals

In this work, we systematically analyze Feynman integrals in the ‘t Hooft-Veltman scheme. We write an explicit reduction resulting from partial fractioning the high-multiplicity integrands to a finite basis of topologies at any given loop order. We find all of these finite basis topologies at two loops in four external dimensions. Their maximal cut and the leading singularity are expressed in terms of the Gram determinant and Baikov polynomial. By performing an integration-by-parts reduction without any cut constraint on a numerical probe for one of these topologies, we show that the computational complexity drops significantly compared to the conventional dimensional regularization scheme. Formally, our work implies an upper bound on the rigidity of special functions appearing in the iterated integral solutions at each loop order in perturbative quantum field theory. Phenomenologically, the integrand-level reduction we present will substantially simplify the task of providing high-precision predictions for future high-multiplicity collider observables. Published by the American Physical Society 2024

Tensor reduction for Feynman integrals with Lorentz and spinor indices

We present an efficient graphical approach to construct projectors for the tensor reduction of multi-loop Feynman integrals with both Lorentz and spinor indices in D dimensions. An ansatz for the projectors is constructed making use of its symmetry properties via an orbit partition formula. The graphical approach allows to identify and enumerate the orbits in each case. For the case without spinor indices we find a 1 to 1 correspondence between orbits and integer partitions describing the cycle structure of certain bi-chord graphs. This leads to compact combinatorial formulae for the projector ansatz. With spinor indices the graph-structure becomes more involved, but the method is equally applicable. Our spinor reduction formulae are based on the antisymmetric basis of γ matrices, and make use of their orthogonality property. We also provide a new compact formula to pass into the antisymmetric basis. We compute projectors for vacuum tensor Feynman integrals with up to 32 Lorentz indices and up to 4 spinor indices. We discuss how to employ the projectors in problems with external momenta.

On factorization hierarchy of equations for banana Feynman integrals

We present a review of the relations between various equations for maximal cut banana Feynman diagrams, i.e. integrals with propagators substituted with δ-functions. We consider both equal and generic masses. There are three types of equation to consider: those in coordinate space, their Fourier transform and Picard-Fuchs equations originating from the parametric representation. First we review the properties of the corresponding differential operators themselves, mainly their factorization properties at the equal mass locus and their form at special values of the dimension. Then we study the relation between the Fourier transform of the coordinate space equations and the Picard-Fuchs equations and show that they are related by factorization as well. The equations in question are the counterparts of the Virasoro constraints in the much-better studied theory of eigenvalue matrix models and are the first step towards building a full-fledged theory of Feynman integrals, which will reveal their hidden integrable structure.

Conformal four-point integrals: recursive structure, Toda equations and double copy

We consider conformal four-point Feynman integrals to investigate how much of their mathematical structure in two spacetime dimensions carries over to higher dimensions. In particular, we discuss recursions in the loop order and spacetime dimension. This results e.g. in new expressions for conformal ladder integrals with generic propagator powers in all even dimensions and allows us to lift results on 2d Feynman integrals with underlying Calabi-Yau geometry to higher dimensions. Moreover, we demonstrate that the Basso-Dixon generalizations of these integrals obey different variants of the Toda equations of motion, thus establishing a connection to classical integrability and the family of so-called tau-functions. We then show that all of these integrals can be written in a double copy form that combines holomorphic and anti-holomorphic building blocks. Here integrals in higher dimensions are constructed from an intersection pairing of two-dimensional “periods” together with their derivatives. Finally, we comment on extensions to higher-point integrals which provide a richer kinematical setup.

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 role in a given allele frequency change. Predicting allele frequency changes under drift and selection, even for alleles contributing to simple, monogenic traits, has remained a challenging problem. 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. We derive analytic expressions for the transition probability (i.e., the probability that an allele will change in frequency from x to y in time t) 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.

One-dimensional Dirac oscillator with generalized Snyder model in a thermal bath

Following the path integral approach and within in the context of nonrelativistic Snyder–de Sitter algebra, we formulate the Green function for Dirac oscillator particle in ([Formula: see text]) space-time dimension. Using the momentum space representation [Formula: see text], we determine the energy eigenvalues and the eigenfunctions, where the wave functions can be given in terms of Gegenbauer polynomials. In addition, the high-temperature thermodynamic properties of the relativistic harmonic oscillators are analyzed, which we found that the heat capacity is not equal two times greater than the heat capacity of the one-dimensional harmonic oscillator for higher temperatures in SdS model. The special cases are found and discussed.

Predicting Feynman periods in ϕ4-theory

We present efficient data-driven approaches to predict the value of subdivergence-free Feynman integrals (Feynman periods) in ϕ4-theory from properties of the underlying Feynman graphs, based on a statistical examination of almost 2 million graphs. We find that the numbers of cuts and cycles determines the period to better than 2% relative accuracy. Hepp bound and Martin invariant allow for even more accurate predictions. In most cases, the period is a multi-linear function of the properties in question. Furthermore, we investigate the usefulness of machine-learning algorithms to predict the period. When sufficiently many properties of the graph are used, the period can be predicted with better than 0.05% relative accuracy.We use one of the constructed prediction models for weighted Monte-Carlo sampling of Feynman graphs, and compute the primitive contribution to the beta function of ϕ4-theory at L ∈ {13, … , 17} loops. Our results confirm the previously known numerical estimates of the primitive beta function and improve their accuracy. Compared to uniform random sampling of graphs, our new algorithm is 1000-times faster to reach a desired accuracy, or reaches 32-fold higher accuracy in fixed runtime.The dataset of all periods computed for this work, combined with a previous dataset, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.

Dynamical theory for adaptive systems

Abstract The study of adaptive dynamics, involving many degrees of freedom on two separated timescales, one for fast changes of state variables and another for the slow adaptation of parameters controlling the former’s dynamics is crucial for understanding feedback mechanisms underlying evolution and learning. We present a path-integral approach à la Martin–Siggia–Rose-De Dominicis–Janssen to analyse non-equilibrium phase transitions in such dynamical systems. As an illustration, we apply our framework to the adaptation of gene-regulatory networks under a dynamic genotype-phenotype map: phenotypic variations are shaped by the fast stochastic gene-expression dynamics and are coupled to the slowly evolving distribution of genotypes, each encoded by a network structure. We establish that under this map, genotypes corresponding to reciprocal networks of coherent feedback loops are selected within an intermediate range of environmental noise, leading to phenotypic robustness.