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Wightman axioms

Index Wightman axioms

In physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Lars Gårding and Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. [1]

73 relations: Anyon, Arthur Wightman, Borchers algebra, Braid statistics, BRST quantization, Cluster decomposition theorem, Cohomology, Commutative property, Constructive quantum field theory, CPT symmetry, Cutoff (physics), David Ruelle, Distribution (mathematics), Dot product, Effective field theory, Euclidean space, Eugene Wigner, Faster-than-light communication, Fock space, Four-momentum, Four-vector, Free field, Gauge theory, Haag's theorem, Hilbert space, Hilbert's sixth problem, John von Neumann, Klaus Hepp, Klein transformation, Lars Gårding, Light cone, Local quantum field theory, Lorentz group, Lorentz transformation, Mass gap, Millennium Prize Problems, Minkowski space, Observable, Parastatistics, Physics, Poincaré group, Principle of locality, Projective representation, Quantum field theory, Quantum gauge theory, Quantum mechanics, Quantum state, Ray Streater, Representation theory of the Poincaré group, Res Jost, ..., Rudolf Haag, S-matrix, Schwinger function, Self-adjoint operator, Separable space, Spacetime, Special relativity, Spin (physics), Spontaneous symmetry breaking, Standard Model, Statistical ensemble (mathematical physics), Superselection, Unbounded operator, Unitary operator, Unitary representation, Vacuum expectation value, Vacuum state, Weak topology, Wick rotation, Wightman axioms, Wigner's classification, Wigner's theorem, Yang–Mills existence and mass gap. Expand index (23 more) »


In physics, an anyon is a type of quasiparticle that occurs only in ''two''-dimensional systems, with properties much less restricted than fermions and bosons.

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Arthur Wightman

Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist.

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Borchers algebra

In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions.

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Braid statistics

In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group.

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BRST quantization

In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry.

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Cluster decomposition theorem

In physics, the cluster decomposition property is related to locality in quantum field theory.

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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Constructive quantum field theory

In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity.

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CPT symmetry

Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T).

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Cutoff (physics)

In theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation.

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David Ruelle

David Pierre Ruelle (born 20 August 1935) is a Belgian-French mathematical physicist.

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Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Effective field theory

In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Eugene Wigner

Eugene Paul "E.

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Faster-than-light communication

Superluminal communication is a hypothetical process in which information is sent at faster-than-light (FTL) speeds.

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Fock space

The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space.

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In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.

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In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation.

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Free field

In physics a free field is a field without interactions, which is described by the terms of motion and mass.

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Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

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Haag's theorem

Rudolf Haag postulated that the interaction picture does not exist in an interacting, relativistic quantum field theory (QFT), something now commonly known as Haag's theorem.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Hilbert's sixth problem

Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent.

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John von Neumann

John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.

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Klaus Hepp

Klaus Hepp (born 11 December 1936) is a German-born Swiss theoretical physicist working mainly in quantum field theory.

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Klein transformation

In quantum field theory, the Klein transformation is a redefinition of the fields to patch up the spin-statistics theorem.

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Lars Gårding

Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He has made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in Sweden 1952–1984. Together with Marcel Riesz, he was a thesis advisor for Lars Hörmander.

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Light cone

In special and general relativity, a light cone is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime.

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Local quantum field theory

The Haag–Kastler axiomatic framework for quantum field theory, introduced by, is an application to local quantum physics of C*-algebra theory.

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Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (nongravitational) physical phenomena.

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Lorentz transformation

In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other.

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Mass gap

In quantum field theory, the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state.

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Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.

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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

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In physics, an observable is a dynamic variable that can be measured.

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In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics).

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Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Minkowski (1908) as the group of Minkowski spacetime isometries.

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Principle of locality

In physics, the principle of locality states that an object is only directly influenced by its immediate surroundings.

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Projective representation

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity; scalar transformations). In more concrete terms, a projective representation is a collection of operators \rho(g),\, g\in G, where it is understood that each \rho(g) is only defined up to multiplication by a constant. These should satisfy the homomorphism property up to a constant: for some constants c(g,h). Since each \rho(g) is only defined up to a constant anyway, it does not strictly speaking make sense to ask whether the constants c(g,h) are equal to 1. Nevertheless, one can ask whether it is possible to choose a particular representative of each family \rho(g) of operators in such a way that the \rho(g)'s satisfy the homomorphism property on the nose, not just up to a constant. If such a choice is possible, we say that \rho can be "de-projectivized," or that \rho can be "lifted to an ordinary representation." This possibility is discussed further below.

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Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

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Quantum gauge theory

In quantum physics, in order to quantize a gauge theory, for example the Yang–Mills theory, Chern–Simons theory or the BF model, one method is to perform gauge fixing.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Quantum state

In quantum physics, quantum state refers to the state of an isolated quantum system.

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Ray Streater

Raymond Frederick "Ray" Streater (born 1936) is a British physicist, and professor emeritus of Applied Mathematics at King's College London.

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Representation theory of the Poincaré group

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.

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Res Jost

Res Jost (10 January 1918 – 3 October 1990) was a Swiss theoretical physicist, who worked mainly in constructive quantum field theory.

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Rudolf Haag

Rudolf Haag (17 August 1922 – 5 January 2016) was a German physicist.

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In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process.

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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.

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Self-adjoint operator

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.

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Separable space

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

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In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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Spin (physics)

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.

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Spontaneous symmetry breaking

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state.

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Standard Model

The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.

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Statistical ensemble (mathematical physics)

In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1902, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in.

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In quantum mechanics, superselection extends the concept of selection rules.

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Unbounded operator

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.

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Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

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Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

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Vacuum expectation value

In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum.

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Vacuum state

In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy.

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Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.

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Wick rotation

In physics, Wick rotation, named after Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.

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Wightman axioms

In physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Lars Gårding and Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory.

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Wigner's classification

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group which have sharp mass eigenvalues.

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Wigner's theorem

Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics.

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Yang–Mills existence and mass gap

In mathematical physics, the Yang–Mills existence and mass gap problem is an unsolved problem and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 to the one who solves it.

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Haag–Ruelle scattering theory, Operator valued distribution, Operator-valued distribution, Wightman distribution, Wightman function, Wightman functions.


[1] https://en.wikipedia.org/wiki/Wightman_axioms

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