Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Conformal Transformations

Description

This lecture covers the topic of conformal transformations, focusing on scale invariance, critical points, and conformal symmetry. It explains how these transformations preserve angles and are coordinate transformations. The lecture also delves into the concept of conformal invariance and the implications of scale invariance on different dimensions.

Official source

Related concepts (32)

Conformal symmetry

In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation. Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations.

Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

Conformal group

In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.

Conformal map

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let and be open subsets of . A function is called conformal (or angle-preserving) at a point if it preserves angles between directed curves through , as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation.

Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale.

Related lectures (60)

Conformal Transformations: Part 1

Covers the topic of conformal transformations, including translations, dilations, rotations, and the conformal algebra.

Lecture 5: Conformal Transformations

Explores Conformal Transformations, emphasizing scale transformations and correlation functions.

Conformal Correlation Functions

Explores conformal correlation functions, cross-ratios, conformal transformations, algebra, and correlation functions of primary operators.

Quantization: Topological Operators

Covers the quantization process using topological operators and their role in generating symmetries.

Conformal Field Theory: Generators and Commutation Relations