mathematical physics Mathematical physics is the development of mathematics, 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 de ...

, constructive 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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

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

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 co ...

, putting

calculus Calculus 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

on a mathematically rigorous foundation.

Weak Weak may refer to: Songs * Weak (AJR song), "Weak" (AJR song), 2016 * Weak (Melanie C song), "Weak" (Melanie C song), 2011 * Weak (SWV song), "Weak" (SWV song), 1993 * Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995 * "Weak", a son ...

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 ...

, and

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

forces of nature are believed to have their natural description in terms of quantum fields. 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, differential equations,

probability theory Probability theory or probability calculus 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 expre ...

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

. 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, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

. 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, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

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 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 experi ...

s had given these rules the name "

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

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

mathematical theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

. 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

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

Konrad Osterwalder Konrad Osterwalder (born June 3, 1942) 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 Techno ...

and Robert 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 Rudolf Haag and Daniel Kastler, called algebraic quantum field theory. There is a firm belief in the physics community that the gauge theory of C.N. Yang and Robert Mills (the

Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...

) 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