Axiomatic quantum field theory

The axiomatic quantum field theory is a research area in mathematical physics .

The term describes different approaches to describe the structure of quantum field theory with mathematical means. The attempt is usually made to set up the smallest possible set of axioms from which the properties of quantum field theories follow.

Early axiomatic quantum field theory

Field operators

The axiomatic descriptions of quantum field theory are based on the Heisenberg picture of quantum mechanics , in which the states are viewed as space-time independent, while the operators are space-time dependent. The quantum fields are described as space-time dependent field operators.

Two problems became apparent early on:

1. A field can have singularities, so describing it as an operator-valued function is not appropriate.
2. In addition, the effect of the field operators cannot be defined for all states.

The first problem can be solved by interpreting the field operators as operator-valued distributions . Distributions are more general objects than functions, which in particular enable simple handling of singularities. A distribution space is always defined for an associated function space , the test function space, and maps each test function to a number or, here, an operator. In quantum field theory, rapidly decreasing functions of space and time are chosen as test functions.

To solve the second problem, it is assumed - as with the observables of quantum mechanics - that the field operators are only defined on a dense subspace of the Hilbert space . The operators are then called densely defined .

The first axiomatic description of quantum field theories that included these aspects was developed by Lars Gårding and Arthur Strong Wightman in the form of the Gårding-Wightman axioms .

State space

As in quantum mechanics, the state space is assumed to be Hilbert space . In quantum field theory, however, special Hilbert spaces, so-called Fock spaces, are assumed as state spaces . These Hilbert spaces are similar to the state space of the quantum mechanical harmonic oscillator and ascending and descending operators can be defined analogously. In addition, there is a clear basic state in jib rooms.

The scalar field is described by the Klein-Gordon equation , the solutions of which correspond to those of the harmonic oscillator. A collection of harmonic oscillators is obtained with the frequencies , where m is the mass and k is the momentum of the field. Since the impulse amount can be any positive real number, one obtains an infinite number of oscillators from which the scalar field is composed. The ground state or the vacuum of the Fock space is the state in which all harmonic oscillators are in the ground state. All other states are obtained by applying products of the ascending operators to the vacuum. ${\ displaystyle \ omega _ {k} = {\ sqrt {k ^ {2} + m ^ {2}}}}$

N-point functions

Wightman developed the axiomatic theory further by stating that a quantum field theory can be uniquely described by its N-point functions. An N-point function is the expectation of the product of N field operators in a state of Fock space. These objects are therefore distributions in N arguments, so they map N test functions to a number. On the basis of Laurent Schwartz's nuclear theorem , a distribution of test functions in N variables can be clearly assigned to each distribution in N arguments, which considerably simplifies the mathematical treatment.

From the axioms for the field operators and the state space, Wightman deduced a set of properties of the N-point functions. If these properties are assumed for N-point functions, the state space and the field operators can be reconstructed from them. The properties that the N-point functions have to meet are called Wightman's axioms . N-point functions that satisfy these axioms are called Wightman functions, although they are actually distributions.

A set of Wightman functions clearly determines a quantum field theory via the reconstruction theorem. This makes it possible to define a quantum field theory without specifying field operators or a Fock space.

Causal perturbation theory

The approaches described so far cannot describe interacting quantum field theories. In particular, the results of the renormalized perturbation theory cannot be reproduced with it. In 1973, the physicists Henri Epstein and Vladimir Jurko Glaser developed the causal perturbation theory, a procedure that made it possible to develop a renormalized perturbation theory for interacting quantum field theories in a mathematically well-defined manner. In their original work they only examined spinless , scalar fields, but their approach has since been extended to other theories, in particular to gauge theories such as quantum electrodynamics .

Algebraic quantum field theory

As early as the late 1940s, Irving Segal had suggested that quantum mechanics and quantum field theory could be described using C * algebras . However, he was unable to formulate it precisely.

In 1961 Hans-Jürgen Borchers discovered that the Wightman functions are based on an algebraic structure. He constructed so-called Wightman functionals, which are composed of Wightman functions for all numbers of arguments N, and formulated the Wightman axioms for them. He discovered that the Wightman functionals form a topological * algebra. With this he laid the foundation for the development of a purely algebraic description of quantum field theories.

Rudolf Haag and Daniel Kastler investigated the algebraic structure of quantum field theories further and in 1964 formulated the Haag-Kastler axioms for networks of C * algebras. They also defined the concept of the algebraic state over a C * algebra, which denotes linear forms on algebra and generalizes the concept of the state in a Hilbert space. Using the GNS construction , representations of the C * algebras on Hilbert spaces can be constructed from algebraic states. These representations satisfy the Gårding-Wightman axioms for quantum field theories except for the existence of a vacuum and the requirement that the state space is a Fock space. A clear vacuum results for states that fulfill a certain property and are called pure states , while so-called quasi-free states induce a representation in a Fock space. Further significant work on algebraic quantum field theory was done by Huzihiro Araki .

From their axioms, Haag and Kastler concluded that quantum field theories, the associated C * algebras of which are isomorphic, are physically equivalent , that is, give the same results for a sequence of measurements. It was thus possible to show for the first time that representations of quantum field theories that are not unitarily equivalent can also be physically equivalent. The central point of this consideration is the axiom of locality, which Borchers transferred from the Hilbert space formulation to algebraic quantum field theory.

Constructive quantum field theory

An approach to the explicit axiomatic construction of quantum field theories comes from Konrad Osterwalder and Robert Schrader . They developed the so-called Osterwalder-Schrader axioms , which a quantum field theory must fulfill in Euclidean space so that a quantum field theory in Minkowski space can be constructed from it. In doing so, they put the wick rotation on a mathematical basis.

There were also various endeavors, building on this work, to put the path integral on a solid mathematical basis. The work of Ludwig Streit and Sergio Albeverio on this topic is viewed as a mathematically consistent description of path integrals of non-interacting quantum field theories. You use the stochastic process of white noise .

Axiomatic S-matrix theory

One of the first successes of axiomatic approaches in quantum field theory was the LSZ reduction formula , which was derived from Harry Lehmann , Kurt Symanzik and Wolfhart Zimmermann . This formula allows the S-matrix to be traced back to time-ordered or causal n-point functions.

The axiomatic S-matrix theory took a different approach than the work of Wightman. Nikolai Nikolajewitsch Bogoljubow , Konstantin Mikhaĭlovich Polivanov and BV Medvedev took the view that the S matrix is ​​the only observable quantity in a quantum field theory and that quantum field theory must therefore be defined via the S matrix.

Topological quantum field theory

A more recent axiomatic approach is topological quantum field theory , which examines the topological invariants of quantum field theories on manifolds with nontrivial topology . Since we are interested in the topological invariants, we consider quantum field theories in which the n-point functions do not depend on the metric, but only on the topological structure of space. A well-known example of a topological quantum field theory is the Chern-Simons theory , which is used to explain the fractional quantum Hall effect .

An axiomatic characterization of these theories comes from Michael Francis Atiyah .

Successes of axiomatic quantum field theories

Above all, the axiomatic theories have achieved a mathematically well-defined formulation of the basic principles of quantum field theory. From these mathematical formulations, various theorems can be derived that satisfy all quantum field theories that satisfy the axioms.

Spin Statistics Theorem

The spin statistics theorem states that the behavior of a statistical ensemble depends on the spin of the constituent microscopic elements. In the context of a quantum field theory, this means that fields with an integer spin must fulfill commutator relations, while fields with half-integer spin fulfill anti-commutator relations.

The theorem was originally proven by Pauli based on the equations of motion of relativistic quantum mechanics, i.e. the Klein-Gordon equation and the Dirac equation , for non-interacting particles. His proof is based on the fact that the assumption of false statistics does not lead to a positive definite Hamilton operator.

In the axiomatic context it was possible to examine exactly which axioms are necessary for the proof. In the context of algebraic quantum field theory, which offers a much more abstract and general concept of quantum field theories than the Hilbert space approach, an analog theorem was proven that reproduces the spin statistics theorem in the corresponding special cases. In the algebraic context, sector theory has also shown that it is possible to reconstruct from the observables of a theory whether there is an underlying theory with spinor fields.

The spin statistics theorem was extended within the framework of algebraic quantum field theory to include quantum field theories on curved spacetime .

Individual evidence

1. ^ AS Wightman: Les Problèmes mathématiques de la théorie quantique des champs , Center National de la Recherche Scientifique, Paris (1959), pp. 11-19
2. ^ AS Wightman, L. Gårding: Fields as Operator-Valued Distributions in Quantum Field Theory
3. ^ Henri Epstein, Vladimir Jurko Glaser: The role of locality in perturbation theory , Annales Poincaré Phys. Theor. A19, p. 211, 1973
4. HJ Borchers: On the structure of the algebra of field operators , Nuovo Cimento, 24 (1962), pages 214-236
5. ^ R. Haag, D. Kastler: An Algebraic Approach to Quantum Field Theory , Journal of Mathematical Physics, Volume 5, Number 7 (1964), pages 848-861
6. K. Osterwalder, R. Schrader: Axioms for Euclidean Green's functions I & II , Comm. Math. Phys. 1973, 31, 83-112; 42: 281-305 (1975)
7. ^ W. Pauli, The Connection Between Spin and Statistics , Phys. Rev. 58, 716-722 (1940).
8. D. Guido, R. Longo, JE Roberts, R. Verch: “Charged Sectors, Spin and Statistics in Quantum Field Theory on Curved Spacetimes”, Rev. Math. Phys. 13, 125 (2001) doi : 10.1142 / S0129055X01000557

literature

• RF Streater, AS Wightman: PCT, Spin and Statistics, and all that , WA Benjamin, Inc. New York 1964
• N. Bogoliubov, A. Logunov, I. Todorov: Introduction to Axiomatic Quantum Field Theory , Benjamin Reading, Massachusetts, 1975
• H. Araki: Mathematical Theory of Quantum Fields , Oxford University Press, 1999