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In mathematical physics, the Dirac algebra is the Clifford algebra, ().This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- 1 / 2 particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry.
For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. The spin representation of the Lorentz group is encoded in the spin group Spin(1, 3) (for real, uncharged spinors) and in the complexified spin group Spin(1, 3) for charged (Dirac) spinors.
χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written =, where σ i are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial.
For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. [1] The Dirac spinor is specific to that of Minkowski spacetime and Lorentz transformations; the general case is quite similar. This article is devoted to the Dirac spinor in the Dirac representation.
These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above.
The spin connection arises in the Dirac equation when expressed in the language of curved spacetime, see Dirac equation in curved spacetime. Specifically there are problems coupling gravity to spinor fields: there are no finite-dimensional spinor representations of the general covariance group.
The spin of a charged particle is associated with a magnetic dipole moment with a g-factor that differs from 1. (In the classical context, this would imply the internal charge and mass distributions differing for a rotating object. [4]) The conventional definition of the spin quantum number is s = n / 2 , where n can be any non-negative ...