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. They play an important role in the physical description of fermions such as the electron.
The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures.
The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
Let V be a finite-dimensional real or complex vector space with a nondegenerate quadratic form Q. The (real or complex) linear maps preserving Q form the orthogonal group O(V, Q). The identity component of the group is called the special orthogonal group SO(V, Q). (For V real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, SO(V, Q) has a unique connected double cover, the spin group Spin(V, Q).
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In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions. None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion).
En mathématiques, le groupe spinoriel de degré n, noté Spin(n), est un revêtement double particulier du groupe spécial orthogonal réel SO(n,R). C’est-à-dire qu’il existe une suite exacte de groupes de Lie On peut aussi définir les groupes spinoriels d'une forme quadratique non dégénérée sur un corps commutatif. Pour n > 2, Spin(n) est simplement connexe et coïncide avec le revêtement universel de SO(n,R). En tant que groupe de Lie, Spin(n) partage sa dimension n(n–1)/2 et son algèbre de Lie avec le groupe spécial orthogonal.
En mathématiques, les représentations des algèbres de Clifford sont aussi connues sous le nom de modules de Clifford. En général, une algèbre de Clifford C est une algèbre centrale simple sur une certaine extension de corps L d'un corps K sur lequel la forme quadratique Q définissant C est définie. La théorie algébrique des modules de Clifford a été fondée dans un article de M. F. Atiyah, R. Bott et A. Shapiro. Nous aurons besoin d'étudier les matrices qui anticommutent (AB = –BA) car les vecteurs orthogonaux anticommutent dans les algèbres de Clifford.
Presentation of Wightman's axiomatic framework to QFT as well as to the necessary mathematical objects to their understanding (Hilbert analysis, distributions, group representations,...).Proofs of
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions such as Quantum Electrodynamics.
Magnetic skyrmions are whirl-like spin configurations with particle-like properties protected by non-trivial topology. Due to their unique spin structures and dynamical properties, they have attracted tremendous interests over the past decade, from fundame ...
Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p ≥ 0 and let φ be a nontrivial p-restricted irreducible representation of G. Let T be a maximal torus of G and s ∈ T . We say that s is Ad-regular if α(s ...
2023
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We study the space of 2→2 scattering amplitudes of neutral Goldstone bosons in four space-time dimensions. We establish universal bounds on the first two non-universal Wilson coefficients of the low energy Effective Field Theory (EFT) for such particl ...