Dirac spinor

A Dirac spinor is a mathematical term that is named after Paul Dirac . Dirac spinors are an element of the fundamental representation of the complexized Clifford algebra and are therefore a specific genus of spinors . They are a useful concept in quantum physics . ${\displaystyle \Delta }$ ${\displaystyle \mathrm {Cliff} (p,q)}$

Solutions of the Dirac equation are often referred to as Dirac spinors . These are Dirac spinor fields , which means that a four-dimensional Dirac spinor is assigned to each point in spacetime.

Mathematical construction

Be . The complexified Clifford algebra is isomorphic to the matrix algebra if is even, or isomorphic to if is odd. In any case, it has a canonical dimensional representation of that so for all signatures with exists and also a representation of the spin group is. This representation is called the spinor representation , the vectors of this representation space are called Dirac spinors . ${\displaystyle n=p+q}$ ${\displaystyle \mathrm {Cliff} (p,q)}$ ${\displaystyle Mat_{2^{k}}(\mathbb {C} )}$ ${\displaystyle n=2k}$ ${\displaystyle Mat_{2^{k-1}}(\mathbb {C} )\oplus Mat_{2^{k-1}}(\mathbb {C} )}$ ${\displaystyle n=2k-1}$ ${\displaystyle 2^{k}}$ ${\displaystyle (p,q)}$ ${\displaystyle n=p+q}$ ${\displaystyle \mathrm {Spin} (p,q)}$

In odd dimensions , this representation, viewed as a representation of , is reducible. Available in two so-called Weyl spinors of dimension are broken: . ${\displaystyle n}$ ${\displaystyle \mathrm {Spin} (p,q)}$ ${\displaystyle 2^{k-1}}$ ${\displaystyle \Delta =\Delta _{+}\oplus \Delta _{-}}$

Application in elementary particle physics

Dirac spinors in 3 + 1 space-time dimensions, i.e. to , are used in the context of quantum electrodynamics for the mathematical description of fermions with spin 1/2. In the Standard Model of particle physics, these Dirac fermions include all fundamental fermions. In this case the Dirac spinors are four-dimensional, belong to a representation of the Lorentz group and are solutions of the Dirac equation . ${\displaystyle \mathrm {Cliff} (3,1)}$

Majorana fermions , on the other hand, have not yet been found, but have been predicted by some unified field theories. They correspond to real representations of the Clifford algebras. In string theories and branch theories , Dirac spinors in higher dimensions are also considered.

literature

• Thomas Friedrich : Dirac operators in Riemannian geometry. Vieweg Verlag, ISBN 978-3-528-06926-1 .
• Michael E. Peskin , Daniel V. Schroeder: An Introduction to Quantum Field Theory . Addison-Wesley, 1995, ISBN 978-0-201-50397-5 (English).
• Pierre Ramond : Field Theory . A modern primer. Addison-Wesley, 1990, ISBN 978-0-201-54611-8 (English).