Spin group

In mathematics the spin group Spin(n) 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) In mathematics the spin group Spin(n) 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) 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). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I . Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(n). A distinct article discusses the spin representations. The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; but of course, space is not zero-dimensional, and so the spin group is used to define spin structures on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection is useful as it can simplify and bring elegance to many intricate calculations in general relativity. The spin connection in turn enables the Dirac equation to be written in curved spacetime (effectively in the tetrad coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of Hawking radiation (where one of a pair of entangled, virtual fermions fall past the event horizon, and the other does not). In short, the spin group is a vital cornerstone, centrally important for understanding advanced concepts in modern theoretical physics. In mathematics, the spin group is interesting in its own right: not just for all these reasons, but for many more. Construction of the Spin group often starts with the construction of a certain Clifford algebra over a real vector space V. The Clifford algebra is the quotient of the tensor algebra TV of V by a two-sided ideal. The tensor algebra (over the reals) may be written as The Clifford algebra Cl(V) is then the quotient algebra where ‖ v ‖ {displaystyle Vert vVert } is the norm of a vector v ∈ V {displaystyle vin V} . The resulting space is naturally graded, and can be written as