Related papers: The Maslov dequantization, idempotent and tropical…
These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has…
We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum…
Endeavoring to formulate an exhaustive solution to the measurement problem in view of the theory of decoherence leads to a better understanding of the status of the collapse and of the emergence of classicality, thanks to a precise…
We review recent results on a mean-field model for relativistic electrons in atoms and molecules, which allows to describe at the same time the self-consistent behavior of the polarized Dirac sea. We quickly derive this model from Quantum…
We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli…
We propose that whatever quantity controls the Heisenberg uncertainty relations (for a given complementary pair of observables) it should be identified with an effective Planck parameter. With this definition it is not difficult to find…
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.…
The partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of quantum states. The transform contains partial transposition as a special case. The scaling transform corresponds to partial…
Many physical constants related to quantized gravity, e.g., the Planck length, mass, curvature, stress-energy, etc., are nonanalytic in G at G=0, and thus have expansions in powers of G whose terms are progressively more divergent with…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…
We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions…
In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative formulation of classical…
The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the…
Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical…
Consider a free electron gas in a confining potential and a magnetic field in arbitrary dimensions. If this gas is in thermal equilibrium with a reservoir at temperature $T >0$, one can study its orbital magnetic response (omitting the…
A precise physical description and understanding of the classical dual content of quantum theory is necessary in many disciplines today: from concepts and interpretation to quantum technologies and computation. In this paper we investigate…
Let $X = X_\Sigma$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformation theory of the LG model $(\check{X}, W)$, and…
Our knowledge of quantum mechanics can satisfactorily describe simple, microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen…
A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal…