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Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett [16] and by Fritz Sauter. [17] [18] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
Another appealing feature of spinors in general relativity is the condensed way in which some tensor equations may be written using the spinor formalism. For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart.
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.
Here the coordinates of physical points are transformed according to ′ =, while , a matrix, is an element of the spinor representation (for spin 1/2) of the Lorentz group. In the Weyl basis, explicit transformation matrices for a boost Λ b o o s t {\displaystyle \Lambda _{\rm {boost}}} and for a rotation Λ r o t {\displaystyle \Lambda _{\rm ...
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + = ,where the curly brackets {,} represent the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.
The Majorana equation is similar to the Dirac equation, in the sense that it involves four-component spinors, gamma matrices, and mass terms, but includes the charge conjugate of a spinor . In contrast, the Weyl equation is for two-component spinor without mass.
This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons , quarks , gauge bosons and the Higgs boson .