Note the \ frac { 1-\ gamma ^ 5 } { 2 } factors in the weak couplings : these factors project out the left handed components of the spinor fields.
22.
By means of the 2 & times; 2 Pauli matrices, and is not just a scalar wavefunction as in the non-relativistic Schr�dinger equation, but a two-component spinor field:
23.
The Dirac equation is true for all ] ] } } particles, and the solutions to the equation are spinor fields with two components corresponding to the particle and the other two for the antiparticle.
24.
Here is a four-component spinor field, which is conventionally split into two two-component spinors in the form : Again this notation is not necessarily standard, the more advanced literature usually writes
25.
Each index takes the values 1, 2, 3, or 4, so there are components of the entire spinor field, although a completely symmetric wavefunction reduces the number of independent components to.
26.
For a " massive " particle of spin, there are components for the particle, and another for the corresponding antiparticle ( there are possible values in each case ), altogether forming a-component spinor field:
27.
In relativistic quantum mechanics, wavefunctions are no longer single-component scalar fields, but now 2 ( 2 " s " + 1 ) component spinor fields, where " s " is the spin of the particle.
28.
Here, "'J "'and " ? " are the current and charge density of the " matter Dirac electron given by the four-component Dirac spinor field " ? ", the current and charge densities have form:
29.
The massless Rarita Schwinger equation has a fermionic gauge symmetry : is invariant under the gauge transformation \ psi _ \ mu \ rightarrow \ psi _ \ mu + \ partial _ \ mu \ epsilon, where \ epsilon \ equiv \ epsilon _ \ alpha is an arbitrary spinor field.
30.
In a given spin manifold, that is in a Riemannian manifold ( M, g ) admitting a spin structure, the Lie derivative of a spinor field \ psi can be defined by first defining it with respect to infinitesimal isometries ( Killing vector fields ) via the Andr?Lichnerowicz's local expression given in 1963:
