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Following Feynman's lectures on gravitation, we consider the theory of the gravitational (massless spin-2) field in flat spacetime and present the third- and fourth-order Lagrangian densities for the gravitational field. In particular, we present detailed calculations for the third-order Lagrangian density. We point out that the expression for the third-order Lagrangian density which Feynman provided is not a solution of Feynman's condition that the third-order Lagrangian density must satisfy. However, Feynman's third-order Lagrangian density gives the correct perihelion shift.

The Feynman learning technique is an active learning strategy that helps learners simplify complex information through student-led teaching and discussion. In this paper, we present the development and usability testing of the Feynman Bot, which uses the Feynman technique to assist self-regulated learners who lack peer or instructor support. The Bot embodies the Feynman learning technique by encouraging learners to discuss their lecture material in a question-answer-driven discussion format. The Feynman Bot was developed using a large language model with Langchain in a Retrieval-Augmented-Generation framework to leverage the reasoning capability required to generate effective discussion-oriented questions. To test the Feynman bot, a controlled experiment was conducted over three days with fourteen participants. Formative and summative assessments were conducted, followed by a self-efficacy survey. We found that participants who used the Feynman Bot experienced higher learning gains than the Passive Learners' group. Moreover, Feynman Bot Learners' had a higher level of comfort with the subject after using the bot. We also found typing to be the preferred input modality method over s

We give a concise and pedagogical introduction to Feynman diagrams. After discussing a toy model which requires only undergraduate mathematics, we focus on relativistic quantum field theory. We review the derivation of Feynman rules from the Lagrangian of the theory and we discuss modern methods to compute tree and loop diagrams.

We introduce SOFIA, a Mathematica package that automatizes the computation of singularities of Feynman integrals, based on new theoretical understanding of their analytic structure. Given a Feynman diagram, SOFIA generates a list of potential singularities along with a candidate symbol alphabet. The package also provides a comprehensive set of tools for analyzing the analytic properties of Feynman integrals and related objects, such as cosmological and energy correlators. We showcase its capabilities by reproducing known results and predicting singularities and symbol alphabets of Feynman integrals at and beyond the high-precision frontier.

In this article, we numerically investigate the vortex nucleation in a Bose-Einstein condensate trapped in a double-well potential and subjected to a density-dependent gauge potential. A rotating Bose-Einstein condensate, when confined in a double-well potential, not only gives rise to visible vortices but also produces hidden vortices. We have empirically developed the Feynmans rule for the number of vortices versus angular momentum in Bose-Einstein condensates in presence of the density dependent-gauge potentials. The variation of the average angular momentum with the number of vortices is also sensitive to the nature of the nonlinear rotation due to the density-dependent gauge potentials. The empirical result agrees well with the numerical simulations and the connection is verified by means of curve fitting analysis. The modified Feynman rule is further confirmed for the BECs confined in harmonic and toroidal traps. In addition, we show the nucleation of vortices in double-well and toroidally confined Bose-Einstein condensates solely through nonlinear rotations (without any trap rotation) arising through the density dependent-gauge potential.

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.

Copositive matrices and copositive polynomials are objects from optimization. We connect these to the geometry of Feynman integrals in physics. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. Pólya's method makes this manifest. We study the copositive cone for the second Symanzik polynomial of any Feynman graph. Its algebraic boundary is described by Landau discriminants.

For correlators in $\mathcal{N}=4$ Super Yang-Mills preserving half the supersymmetry, we manifestly recast the gauge theory Feynman diagram expansion as a sum over dual closed strings. Each individual Feynman diagram maps on to a Riemann surface with specific moduli. The Feynman diagrams thus correspond to discrete lattice points on string moduli space, rather than discretized worldsheets. This picture is valid to all orders in the $1/N$ expansion. Concretely, the mapping is carried out at the level of a two-matrix integral with its dual string description. It provides a microscopic picture of open/closed string duality for this topological subsector of the full AdS/CFT correspondence. At the same time, the concrete mechanism for how strings emerge from the matrix model Feynman diagrams predicts that multiple open string descriptions can exist for the same dual closed string theory. By considering the insertion of determinant operators in $\mathcal{N}=4$ SYM, we indeed find six equivalent open-string descriptions. Each of them generates Feynman diagrams related to one another via (partial) graph duality, and hence encodes the same information. The embedding of these Kontsevich-lik

We present a subtraction scheme for ultraviolet (UV) divergent, infrared (IR) safe scalar Feynman integrals in dimensional regularization with any number of scales. This is done by the introduction of $u$-variables, which are a suitable generalization of dihedral coordinates on the open string moduli space to Feynman integrals. The subtraction scheme furnishes subtraction terms which are products of lower loop Feynman integrals deformed by order $ε$ powers of $u$-variables and deformations of the degree of divergence. The result is a canonical and algorithmic prescription to express the Feynman integral as a sum of convergent integrals dressed with inverse powers of $ε$.

In this paper, we study the singularities of Feynman integrals using homological techniques. We analyse the Feynman integrals by compactifying the integration domain as well as the ambient space by embedding them in higher-dimensional space. In this compactified space the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second-type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure.

In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of $A_\infty$ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit.

This talk reviews recent developments in the field of analytical Feynman integral calculations. The central theme is the geometry associated to a given Feynman integral. In the simplest case this is a complex curve of genus zero (aka the Riemann sphere). In this talk we discuss Feynman integrals related to more complicated geometries like curves of higher genus or manifolds of higher dimensions. In the latter case we encounter Calabi-Yau manifolds. We also discuss how to compute these Feynman integrals.

In this thesis we will study Feynman integrals from the perspective of A-hypergeometric functions, a generalization of hypergeometric functions which goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This point of view was recently initiated by the works [74] and [150]. Inter alia, we want to provide here a concise summary of the mathematical foundations of A-hypergeometric theory in order to substantiate this viewpoint. This overview will concern aspects of polytopal geometry, multivariate discriminants as well as holonomic D-modules. As we will subsequently show, every scalar Feynman integral is an A-hypergeometric function. Furthermore, all coefficients of the Laurent expansion as appearing in dimensional and analytical regularization can be expressed by A-hypergeometric functions as well. Moreover, we can derive an explicit formula for series representations of each Feynman integrals, which is in particular suitable for an algorithmic approach. In addition, the A-hypergeometric theory enables us to give a mathematically rigorous description of the analytic structure of Feynman integrals (also known as Landau variety) by means of principal A-determinants

We show that momentum space Feynman diagrams involving internal massless fields can be cast as conformal integrals. This leads to a classification of all Feynman diagrams into conformal families, labelled by conformal integrals. Computing the conformal integral in each family suffices to compute all the Feynman diagrams in the family exactly. Using known exact results for some conformal integrals, we present the solutions to the other Feynman diagrams in the corresponding families. These are answers to either finite or dimensionally regularized Feynman diagrams to all orders in the regularization parameter.

We present a new computer program, $\texttt{feyntrop}$, which uses the tropical geometric approach to evaluate Feynman integrals numerically. In order to apply this approach in the physical regime, we introduce a new parametric representation of Feynman integrals that implements the causal $i\varepsilon$ prescription concretely while retaining projective invariance. $\texttt{feyntrop}$ can efficiently evaluate dimensionally regulated, quasi-finite Feynman integrals, with not too exceptional kinematics in the physical regime, with a relatively large number of propagators and with arbitrarily many kinematic scales. We give a systematic classification of all relevant kinematic regimes, review the necessary mathematical details of the tropical Monte Carlo approach, give fast algorithms to evaluate (deformed) Feynman integrands, describe the usage of $\texttt{feyntrop}$ and discuss many explicit examples of evaluated Feynman integrals.

We give a brief overview of the Yangian symmetry of Feynman integrals. After a short introduction to the Yangian and integrability, we motivate the emergence of integrable structures for Feynman integrals via the fishnet limit of AdS/CFT. We discuss the resulting Yangian differential equations for massless fishnets in four dimensions as well as generalizations to massive propagators and generic dimensions. We also comment on the relation to momentum space conformal symmetry and on examples in dimensional regularization. Finally we sketch the recent application to fishnet integrals in two spacetime dimensions and the curious identification of Yangian invariants with period integrals of Calabi-Yau geometries.

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain GKZ-systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the GKZ-system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the 2-loop self-energy diagrams which are convenient to embed in the subvarieties of Grassmannians.

We introduce a novel compositional description of Feynman diagrams, with well-defined categorical semantics as morphisms in a dagger-compact category. Our chosen setting is suitable for infinite-dimensional diagrammatic reasoning, generalising the ZX calculus and other algebraic gadgets familiar to the categorical quantum theory community. The Feynman diagrams we define look very similar to their traditional counterparts, but are more general: instead of depicting scattering amplitude, they embody the linear maps from which the amplitudes themselves are computed, for any given initial and final particle states. This shift in perspective reflects into a formal transition from the syntactic, graph-theoretic compositionality of traditional Feynman diagrams to a semantic, categorical-diagrammatic compositionality. Because we work in a concrete categorical setting -- powered by non-standard analysis -- we are able to take direct advantage of complex additive structure in our description. This makes it possible to derive a particularly compelling characterisation for the sequential composition of categorical Feynman diagrams, which automatically results in the superposition of all possib

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, and segment by segment. This approach leads to a fully algorithmic method for computing Feynman integrals from differential equations, which does not require the manual determination of boundary conditions. Furthermore, the most complicated topology that appears in the method often has less master integrals than the original one. We illustrate the strength of our method with a five-point two-loop integral family.