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Lecture

Spinorial Fields: Lorentz Group Representation

In course

PHYS-431: Quantum field theory I

The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions. ...

Description

This lecture covers the most general representation of Lorentz group on spinorial fields, including left-handed and right-handed spinors, spinorial rotations, and the concept of spinorial fields in quantum field theory.

Instructors (2)

Riccardo Rattazzi

Riccardo Rattazzi was born in Novara (Italy) in 1964. He studied physics at the University of Pisa, where he received the Laurea cum laude in 1987, and at the Scuola Normale Superiore where he received the Diploma in Scienze and carried out graduate research in theoretical physics. After having been a post-doctoral research associate at the Lawrence Berkeley Laboratory, at Rutgers University and at CERN, in 1998 Riccardo obtained a permanent research position at the Istituto Nazionale di Fisica Nucleare in Pisa. From 2001 to 2006 he was a staff member at the Theory Division of CERN. In 2006 he was appointed professor of physics at EPFL.

Francesco Riva

Official source

Ontological neighbourhood

Physics

Quantum mechanics: Symmetry in quantum mechanics

Related concepts (32)

Dirac spinor

In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group. Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks.

Spin group

In mathematics the spin group Spin(n) is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2) The group multiplication law on the double cover is given by lifting the multiplication on . As a Lie group, Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).

Spin structure

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.

Spinor bundle

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors . A section of the spinor bundle is called a spinor field. Let be a spin structure on a Riemannian manifold that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering of the special orthogonal group by the spin group.

Spin representation

In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. Elements of a spin representation are called spinors.

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