mathematical physics 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 developm ...

constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...

quantum field theory is the field devoted to showing that

quantum field theory 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 ...

can be defined in terms of precise mathematical structures. This demonstration requires new

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in a sense analogous to classical

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

, putting

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

on a mathematically rigorous foundation.

Weak Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...

strong Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United S ...

, and

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...

forces of nature are believed to have their natural description in terms of

quantum fields 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 ...

. Attempts to put

on a basis of completely defined concepts have involved most branches of mathematics, including

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, and

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

. It is known that a ''quantum field'' is inherently hard to handle using conventional mathematical techniques like explicit estimates. This is because a quantum field has the general nature of an '' operator-valued distribution'', a type of object from

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

. The

existence theorem In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...

s for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. One discovery of the theory that can be related in non-technical terms, is that the dimension ''d'' of the

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

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 A recipe is a set of instructions that describes how to prepare or make something, especially a dish of prepared food. A sub-recipe or subrecipe is a recipe for an ingredient that will be called for in the instructions for the main recipe. Hist ...

, also in the case ''d'' < 4.

Theoretical physicist Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...

s had given these rules the name "

renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...

," but most physicists had been skeptical about whether they could be turned into a

mathematical theory A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reason ...

. Today one of the most important open problems, both in theoretical physics and in mathematics, is to establish similar results for

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...

in the realistic case ''d'' = 4. The traditional basis of constructive quantum field theory is the set of

Wightman axioms In mathematical physics, the Wightman axioms (also called Gårding–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 ear ...

. 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 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 . The axioms are stated in te ...

. There is a firm belief in the physics community that the gauge theory of Yang and

Mills Mills is the plural form of mill, but may also refer to: As a name * Mills (surname), a common family name of English or Gaelic origin * Mills (given name) *Mills, a fictional British secret agent in a trilogy by writer Manning O'Brine Places Uni ...

(the

Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using t ...

) 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.

External links

* * {{cite book , last=Baez , first=John , title=Introduction to algebraic and constructive quantum field theory , publisher=Princeton University Press , publication-place=Princeton, New Jersey , year=1992 , isbn=978-0-691-60512-8 , oclc=889252663 , url=http://math.ucr.edu/home/baez/bsz.html Axiomatic quantum field theory Functional analysis