Constructive quantum field theory
Encyclopedia
In mathematical physics
, constructive
quantum field theory is the field devoted to showing that quantum theory
is mathematically compatible with special relativity
. This demonstration requires new mathematics, in a sense analogous to Newton
developing calculus
in order to understand planetary motion and classical gravity. Weak, strong, and electromagnetic
forces of nature
are believed to have their natural description in terms of quantum fields.
Attempts to put quantum field theory
on a basis of completely defined concepts have involved most branches of mathematics
, including functional analysis
, differential equation
s, probability theory
, representation theory
, geometry
, and topology
, to name a few. 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
. This implies that existence theorem
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
involved is crucial. In spite of these impediments, tremendous progress occurred, spurred on by a long collaboration and extensive work of James Glimm
and Arthur Jaffe
who showed that with d < 4 many examples can be found. Along with work of their students, coworkers, and others, constructive field theory resulted in giving 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
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.
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, constructive
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...
quantum field theory is the field devoted to showing that quantum theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is mathematically compatible with special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. This demonstration requires new mathematics, in a sense analogous to Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
developing calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
in order to understand planetary motion and classical gravity. Weak, strong, and electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
forces of nature
Fundamental interaction
In particle physics, fundamental interactions are the ways that elementary particles interact with one another...
are believed to have their natural description in terms of quantum fields.
Attempts to put quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
on a basis of completely defined concepts have involved most branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, including functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, representation theory
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 studiesmodules over these abstract algebraic structures...
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, to name a few. 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
Wightman axioms
In physics the Wightman axioms 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.The axioms exist in...
, a type of object from mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
. This implies that existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
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
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
involved is crucial. In spite of these impediments, tremendous progress occurred, spurred on by a long collaboration and extensive work of James Glimm
James Glimm
James Gilbert Glimm is an American mathematical physicist, and Professor at the State University of New York at Stony Brook.James Glimm was born in Peoria, Illinois, USA on 24 March 1934.- Career :...
and Arthur Jaffe
Arthur Jaffe
Arthur Jaffe is an American mathematical physicist and a professor at Harvard University. Born on December 22, 1937 he attended Princeton University as an undergraduate obtaining a degree in chemistry, and later Clare College, Cambridge, as a Marshall Scholar, obtaining a degree in mathematics...
who showed that with d < 4 many examples can be found. Along with work of their students, coworkers, and others, constructive field theory resulted in giving a mathematical foundation
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
and exact interpretation to what previously was only a set of recipes
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
, also in the case d < 4.
Theoretical physicists had given these rules the name "renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
," 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
Wightman axioms
In physics the Wightman axioms 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.The axioms exist in...
. Osterwalder and Schrader
Schwinger function
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...
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
Rudolf Haag
Rudolf Haag is a German physicist. He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms...
and Kastler
Daniel Kastler
Daniel Kastler is a French theoretical physicist, working at University of Aix-Marseille on non-commutative geometry.He is best known for his 1964 article with Rudolf Haag on algebraic quantum field theory, which is one of the ....
, called algebraic quantum field theory
Local quantum field theory
The Haag-Kastler axiomatic framework for quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local quantum physics of C*-algebra theory. It is therefore also known as Algebraic Quantum Field Theory...
. There is a firm belief in the physics community that the gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
of Yang and Mills
Robert Mills (physicist)
Robert L. Mills was a physicist, specializing in quantum field theory, the theory of alloys, and many-body theory. While sharing an office at Brookhaven National Laboratory, in 1954, Chen Ning Yang and Mills proposed a tensor equation for what are now called Yang-Mills fields...
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
- Jaffe, Arthur; Constructive Quantum Field Theory (pdf) review article from Mathematical Physics 2000 p. 111.