Related papers: Tropicalized quantum field theory and global tropi…
We introduce tropical scalar field theory as a model for renormalizable quantum field theory, and examine in detail the case of quartic self-interaction and internal $O(N)$ symmetry. This model arises in a formally zero-dimensional limit of…
The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate…
The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the…
We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical…
We study anisotropic scaling limits of topological field theories using tropical geometry. The resulting topological field theories are characterized by foliated geometries and are invariant under foliation-preserving gauge transformations.…
Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great…
We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative…
Recent work has shown that quantum computers can compute scattering probabilities in massive quantum field theories, with a run time that is polynomial in the number of particles, their energy, and the desired precision. Here we study a…
A perturbative approach to quantum field theory involves the computation of loop integrals, as soon as one goes beyond the leading term in the perturbative expansion. First I review standard techniques for the computation of loop integrals.…
We consider a family of quantum spin systems which includes as special cases the ferromagnetic XY model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field. We prove that the partition function of any…
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…
In this work, we argue that the $\alpha'\to 0$ limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes…
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup…
Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to…
We continue our work on the study of spherically symmetric loop quantum gravity coupled to two spherically symmetric scalar fields, one that acts as a clock. As a consequence of the presence of the latter, we can define a true Hamiltonian…
We summarize our recently proposed approach to quantum field theory on noncommutative curved spacetimes. We make use of the Drinfel'd twist deformed differential geometry of Julius Wess and his group in order to define an action functional…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the $\phi^{4}$ scalar field theory…
Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral…