Spinors

The module Spinors defines the necessary functions for the structure Spinor{NS,G}, which is a NS-tuple with values in G.

The functions norm, norm2, dot, *, /, /, +, -, imm and mimm, if defined for G, are extended to Spinor{NS,G} for general NS.

For the 4d case where NS = 4 there are some specific functions to implement different operations with the gamma matrices. The convention for these matrices is

\[\gamma _4 = \left( \begin{array}{cccc} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ \end{array} \right) \quad \gamma_1 = \left( \begin{array}{cccc} 0 & 0 & 0 & -i\\ 0 & 0 & -i & 0\\ 0 & i & 0 & 0\\ i & 0 & 0 & 0\\ \end{array} \right)\]

\[ \gamma _2 = \left( \begin{array}{cccc} 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ \end{array} \right) \quad \gamma_3 = \left( \begin{array}{cccc} 0 & 0 & -i & 0\\ 0 & 0 & 0 & i\\ i & 0 & 0 & 0\\ 0 & -i & 0 & 0\\ \end{array} \right)\]

The function dmul implements the multiplication over the $\gamma$ matrices

dmul(Gamma{n}, a::Spinor)

The function pmul implements the $ (1 \pm \gamma_N) $ proyectors. The functions gpmul and gdagpmul do the same and then multiply each element by gand g^-1 repectively.

pmul(Pgamma{N,S}, a::Spinor)

gpmul(Pgamma{N,S}, g::G, a::Spinor) G <: Group

gdagpmul(Pgamma{N,S}, g::G, a::Spinor) G <: Group

Some examples

Here we just display some examples for these functions. We display it with ComplexF64 instead of SU3fund or SU2fund for simplicity.

julia> spin = Spinor{4,Complex{Float64}}((1.0,im*0.5,2.3,0.0))Spinor{4, ComplexF64}((1.0 + 0.0im, 0.0 + 0.5im, 2.3 + 0.0im, 0.0 + 0.0im))
julia> println(spin)Spinor{4, ComplexF64}((1.0 + 0.0im, 0.0 + 0.5im, 2.3 + 0.0im, 0.0 + 0.0im))
julia> println(dmul(Gamma{4},spin))Spinor{4, ComplexF64}((-2.3 - 0.0im, -0.0 - 0.0im, -1.0 - 0.0im, -0.0 - 0.5im))
julia> println(pmul(Pgamma{2,-1},spin))Spinor{4, ComplexF64}((1.0 + 0.0im, -2.3 + 0.5im, 2.3 - 0.5im, 1.0 + 0.0im))