Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s original proof relied on the specific form of then-common field theories, but subsequently generalized by a number of authors, notably Hall & Wightman, who concluded that no single, universal Hilbert space representation can describe both free and interacting fields. A generalization due to Reed & Simon shows that applies to free neutral scalar fields of different masses, which implies that the interaction picture is always inconsistent, even in the case of a free field.
Vladimir Gershonovich Drinfeld, surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago.
Pierre Émile Cartier is a French mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory.
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory, . Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.
In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse scattering method. The name Yangian was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang.
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.
Glen Earl Baxter was an American mathematician.
Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.
Shahn Majid is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a professor of mathematics at the School of Mathematical Sciences, Queen Mary, University of London.
In mathematics, a quantum affine algebra is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized.
Kevin Joseph Costello FRS is an Irish mathematician, since 2014 the Krembil Foundation's William Rowan Hamilton chair of theoretical physics at the Perimeter Institute in Waterloo, Ontario, Canada.
Robert Lee Grossman is an American computer scientist and bioinformatician at the University of Chicago. His primary research interests are data science and data-intensive computing.
Karen Amanda Yeats is a Canadian mathematician and mathematical physicist whose research connects combinatorics to quantum field theory. She holds the Canada Research Chair in Combinatorics in Quantum Field Theory at the University of Waterloo.
Sergio Doplicher is an Italian mathematical physicist, who mainly dealt with the mathematical foundations of quantum field theory and quantum gravity.
Roberto Longo is an Italian mathematician, specializing in operator algebras and quantum field theory.
Klaus Fredenhagen is a German theoretical physicist who works on the mathematical foundations of quantum field theory.
