Lecture

Conformal Field Theory & Operator Product Expansion

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In course

PHYS-739: Conformal Field theory and Gravity

This course is an introduction to holography, the modern approach to quantum gravity. ...

Description

This lecture delves into Conformal Field Theory (CFT) and Operator Product Expansion (OPE), exploring the relationship between symmetries in quantum field theory and the existence of topological operators. The instructor discusses the implications of symmetries on the structure of operators, emphasizing the role of conformal transformations. The lecture covers the quantization process in CFT, focusing on the choice of spacetime foliation and the importance of conserved charges. The concept of states in CFT is explained through integrals and the projection of states onto a basis of fields. The OPE is introduced as a method to expand product of operators, leading to a convergent sum with specific coefficients. The lecture concludes with a discussion on the convergence of OPE and the importance of understanding the specific nature of OPE in CFT.

Instructors (2)

Kyriakos Papadodimas

Grégoire Olivier Mathys

Official source

Related concepts (32)

Symmetry

Symmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.

Symmetry (physics)

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries.

Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles.

Symmetry group

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.

Symmetry (geometry)

In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable.

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