One discovery of the theory that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. Notable work in the field by James Glimm and Arthur Jaffe showed that with d < 4 many examples can be found. Along with work of their students, coworkers, and others, constructive field theory resulted in a mathematical foundation and exact interpretation to what previously was only a set of recipes, also in the case d < 4.
Theoretical physicists had given these rules the name "renormalization," but most physicists had been skeptical about whether they could be turned into a mathematical theory. Today one of the most important open problems, both in theoretical physics and in mathematics, is to establish similar results for gauge theory in the realistic case d = 4.
The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with d < 4 satisfy the Wightman axioms as well as the Osterwalder–Schrader axioms. They also fall in the related framework introduced by Haag and Kastler, called algebraic quantum field theory. There is a firm belief in the physics community that the gauge theory of Yang and Mills (the Yang–Mills theory) can lead to a tractable theory, but new ideas and new methods will be required to actually establish this, and this could take many years.
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.
Edward Witten is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Witten's work has also significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, for his 1981 proof of the positive energy theorem in general relativity. He is considered the practical founder of M-theory.
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics.
Arthur Strong Wightman was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatment of quantum field theories, he promoted research on various aspects of modern mathematical physics.
In physics, the Wightman axioms, named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scattering theory affirmed their significance.
Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's problems in mathematics that he presented in the year 1900. In its common English translation, the explicit statement reads:
Rudolf Haag was a German theoretical physicist, who mainly dealt with fundamental questions of quantum field theory. He was one of the founders of the modern formulation of quantum field theory and he identified the formal structure in terms of the principle of locality and local observables. He also made important advances in the foundations of quantum statistical mechanics.
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions and they are real-analytic, symmetric under the permutation of arguments, Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms. Schwinger functions are also referred to as Euclidean correlation functions.
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution.
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial perspective.
In theoretical physics, statistical field theory (SFT) is a theoretical framework that describes phase transitions. It does not denote a single theory but encompasses many models, including for magnetism, superconductivity, superfluidity, topological phase transition, wetting as well as non-equilibrium phase transitions. A SFT is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations. It is closely related to quantum field theory, which describes the quantum mechanics of fields, and shares with it many techniques, such as the path integral formulation and renormalization. If the system involves polymers, it is also known as polymer field theory.
Arthur Michael Jaffe is an American mathematical physicist at Harvard University, where in 1985 he succeeded George Mackey as the Landon T. Clay Professor of Mathematics and Theoretical Science.
Konrad Osterwalder is a Swiss mathematician and physicist, former Undersecretary-General of the United Nations, former Rector of the United Nations University (UNU), and Rector Emeritus of the Swiss Federal Institute of Technology Zurich. He is known for the Osterwalder–Schrader theorem.
In physics, a gauge theory is a type of field theory in which the Lagrangian does not change under local transformations according to certain smooth families of operations.
Res Jost was a Swiss theoretical physicist, who worked mainly in constructive quantum field theory.
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem.
Jürg Martin Fröhlich is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry breaking, and for pioneering the study of topological phases of matter using low-energy effective field theories.
Thomas C. Spencer is an American mathematical physicist, known in particular for important contributions to constructive quantum field theory, statistical mechanics, and spectral theory of random operators. He earned his doctorate in 1972 from New York University with a dissertation entitled Perturbation of the Po2 Quantum Field Hamiltonian written under the direction of James Glimm. Since 1986, he has been professor of mathematics at the Institute for Advanced Study. He is a member of the United States National Academy of Sciences, and the recipient of the Dannie Heineman Prize for Mathematical Physics.
Robert C. Hermann was an American mathematician and mathematical physicist. In the 1960s Hermann worked on elementary particle physics and quantum field theory, and published books which revealed the interconnections between vector bundles on Riemannian manifolds and gauge theory in physics, before these interconnections became "common knowledge" among physicists in the 1970s.
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of connections modulo gauge transformations. This situation contrasts with the four-dimensional case, where a rigorous construction of the theory as a measure is currently unknown.
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