Axiomes de Wightman

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 early 1950s, but they were first published only in 1964 after Haag–Ruelle scattering theory affirmed their significance. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields. One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectrum of the four-momentum to the positive light cone (and its boundary). However, this isn't enough to implement locality. For that, the Wightman axioms have position-dependent operators called quantum fields, which form covariant representations of the Poincaré group. Since quantum field theory suffers from ultraviolet problems, the value of a field at a point is not well-defined. To get around this, the Wightman axioms introduce the idea of smearing over a test function to tame the UV divergences, which arise even in a free field theory. Because the axioms are dealing with unbounded operators, the domains of the operators have to be specified. The Wightman axioms restrict the causal structure of the theory by imposing either commutativity or anticommutativity between spacelike separated fields. They also postulate the existence of a Poincaré-invariant state called the vacuum and demand it to be unique. Moreover, the axioms assume that the vacuum is "cyclic", i.e., that the set of all vectors obtainable by evaluating at the vacuum-state elements of the polynomial algebra generated by the smeared field operators is a dense subset of the whole Hilbert space.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (2)

Concepts associés (10)

Cours associés (2)

Séances de cours associées (7)

Convergent momentum-space OPE and bootstrap equations in conformal field theory

Marc Gillioz

General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary function in a specific kinematic region. Using microcausality, we also formulate a bootstrap equation directly in terms of momentum space Wightman functions.

SPRINGER2020

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

Marc Gillioz

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell F-4 type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

SPRINGER2020

Axiomes de Wightman

Les axiomes de Wightman, proposés par le physicien et mathématicien Arthur Wightman, sont une tentative de formulation mathématique de la théorie quantique des champs. Formulés par Wightman dans les années 1950, ses axiomes ne sont publiés qu'en 1964 après que Rudolf Haag et David Ruelle ont proposé une formulation algébrique de la théorie quantique des champs axiomatique confirmant leur importance.

Théorie quantique des champs axiomatique

Dans les années 1950, avec les succès de la renormalisation perturbative en électrodynamique quantique, est apparu le besoin d'une formulation mathématiquement rigoureuse de la théorie quantique des champs à partir des quelques principes généraux, dont : les concepts de la mécanique quantique, l'invariance sous le groupe de Poincaré, la localité des champs. L'objectif était d'éclaircir le statut des équations de la théorie quantique des champs, et d'essayer de montrer qu'il existe des solutions à ces équations.

Théorie conforme des champs

Une théorie conforme des champs ou théorie conforme (en anglais, conformal field theory ou CFT) est une variété particulière de théorie quantique des champs admettant le comme groupe de symétrie. Ce type de théorie est particulièrement étudié lorsque l'espace-temps y est bi-dimensionnel car en ce cas le groupe conforme est de dimension infinie et bien souvent la théorie est alors exactement soluble.

PHYS-702: Advanced Quantum Field Theory

The course builds on the course QFT1 and QFT2 and develops in parallel to the course on Gauge Theories and the SM.

PHYS-739: Conformal Field theory and Gravity

This course is an introduction to the non-perturbative bootstrap approach to Conformal Field Theory and to the Gauge/Gravity duality, emphasizing the fruitful interplay between these two ideas.

Théorème de Wigner; Symétrie de traduction PHYS-314: Quantum physics II

Explore le théorème de Wigner, la symétrie de traduction, les états propres et les fonctions d'onde en physique quantique.

Élargissement des fonctions de corrélation PHYS-739: Conformal Field theory and Gravity

Explore l'échelle des fonctions de corrélation, les opérateurs, les dimensions, la fonction bêta, les possibilités de RG, et les QFT complets UV.

Mise à l'échelle et renormalisation en mécanique statistique PHYS-739: Conformal Field theory and Gravity

Explore l'échelle et la renormalisation en mécanique statistique, en mettant l'accent sur les points critiques et les propriétés invariantes.