How do Gaussian solutions manifest in Dirac diffusion?

[jk22]

does anyone know how dirac diffusion looks like, i.e. The solution of dirac equation with no potential and initial condition for 1 spinor component psi(x,t0) being delta(x-x0) ?
Are the solution gaussian like the schroedinger case ?
Thanks.

 

[Bill_K]

What you're talking about is the Green's function or propagator. The exact form depends on which boundary conditions you want to use, but without going into too much detail, here's what it looks like. (It's not a Gaussian!)

Since a delta function source, δ⁴(x-x'), is relativistically invariant (looks the same in all reference frames), so is the Green's function. G(x, x') depends only on the invariant interval between x and x'. Define a variable z where z² = m²(x·x - c²t²). Then the Green's function for the Klein-Gordon Equation has the form of a Hankel function, G(x, x') ~ H₁²(z)/z. The one for the Dirac Equation is very similar, but of course is a spinor.

 

[jk22]

thanks a lot for your answer, though I am not familiar with hankel function.
Shall it be deduced that there is a singularity at x equals ct and -ct ?
And what happens further ?
Im interested in this since if the wave function does not vanish further, which would imply a discontinuity, then the probability the particle goes faster than light would be non zero which seems in conflict with relativity theory.
With schroedinger equation this is the case since its not relativistic so i wanted to know what about dirac ones.
Thanks if you can help me further.

 

[jk22]

I found some material on Wikipedia that says the spacelike part is not relevant for FTL since the commutator of the field vanishes, but I don't understand that.

 

[andrien]

are you really asking about green function.It seems that you are trying to determine the time evolution of wave function if it is specified for some time t?if it is green function then you will get hankel functions from it by first calculating the propagator in momentum space and then Fourier transform it to coordinate space.