Exploring the Dirac Equation: β,αk,pk, and Ψ(x,t)

[AleksanderPhy]

Hello there I have a problem about Dirac equation

So I want to know what is matrices β,αk,pk value. And is it right that with Dirac equation we can calculate every particle spin and how we take dervitative of Ψ(x,t) and what is Ψ(x,t) value.

 

[ddd123]

The matrices ##\beta##, ##\alpha_k## are 4x4 Hermitian matrices and are defined by the Clifford algebra, which means that in this notation they satisfy ( {,} is the anticommutator and 1 is the unity matrix):

##\{\alpha^i,\alpha^j\} = 2 \delta^{ij}##
##\{\alpha^i,\beta\} = 0##
##\beta^2 = 1##

Their explicit matrix elements depend on which of these you want to make diagonal (Dirac representation, Chiral representation, Majorana representation...). The ##p^k## is not a matrix but the momentum operator which becomes a quantum operator in the form of a spatial derivative: so the one you posted is the free Dirac equation in which ##\psi## is the eigenfuction unknown, a bispinor (4 component spinor). Solving for ##\psi## in the Dirac representation you find (after a lengthy calculation) the normalized eigenfunctions for positive and negative energy eigenvalues. And then you can change representations using the appropriate transformation matrices.

 

[AleksanderPhy]

Thanks(;