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In mathematics the spin group, denoted Spin(n), [1] [2] 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)
Spin representations can be analysed according to the following strategy: if S is a real spin representation of Spin(p, q), then its complexification is a complex spin representation of Spin(p, q); as a representation of so(p, q), it therefore extends to a complex representation of so(n, C).
That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group ...
In a year in which it seemed every great luminary got a moment under the documentary lens, it can be a bit difficult to parse out which were must-see. This list will have you covered.
Spin- 1 / 2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. One such effect that was important in the discovery of spin is the Zeeman effect , the splitting of a spectral line into several components in the ...
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group [26] (the half-spin representations, or Weyl spinors) via + =, =. When dim( V ) is odd, V = W ⊕ U ⊕ W ′ , where U is spanned by a unit vector u orthogonal to W .
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.