Hello. I would like to ask something that will help me understand a little better how we work with Dirac spinors' inputs.... I know that the dirac equation has 4 independent solutions, and for motionless particles, the (spinor) solutions are: [itex] u_{+}=(1,0,0,0)^{T} [/itex] electron +1/2 [itex] u_{-}=(0,1,0,0)^{T} [/itex] electron -1/2 [itex] v_{+}=(0,0,1,0)^{T} [/itex] positron +1/2 [itex] v_{-}=(0,0,0,1)^{T} [/itex] positron -1/2 Now suppose that I write a solution as: [itex] Ψ=(1,0,0,1)^{Τ}[/itex] Does that mean that I'm having 2 particles (one positron and one electron) involved?
In the single-particle first quantized theory, the four components of the spinor represent particle states with spin ±ħ/2 and energy ±E. So just as ψ = (1,1,0,0) represents a state with equal probability of the particle having spin up and down, so ψ = (1,0,0,1) represents a state with equal probability of the particle having spin up energy +E, and spin down energy -E. In the framework of this theory there is no antiparticle.