Cross between helicoid, complex plane wave

[Spinnor]

Function kind of cross between a helicoid and a complex plane wave?

I would like to translate a mental picture into a mathematical expression if possible. The picture might be roughly thought of as a cross between a complex plane wave and a helicoid. A construction I think goes as follows, take some complex scalar plane wave in 4 dimensional spacetime Ψ = exp(-i[p⋅x]) where p is the energy-momentum 4-vector for a massless and spinless particle and x is the spacetime 4-vector. Consider an infinite line, L, parallel to the 3-momentum vector together with the time axis. Consider the infinite half-plane, S, defined by those two lines where the "edge" of the half-plane is the line L.

Edit, sorry for my mistake but the half plane I'm thinking of is defined by the line L above and a ray that starts at a point on L and is perpendicular to L in space and is not the time axis.

2nd Edit, seems I cut some of the original, sorry.

Let this half-plane S define a cutting of our function Ψ, called Ψ_cut. Now deform Ψ_cut as follows, shift one surface defined by this cutting forwards in time by 1/2 period and shift the other surface backwards in time by 1/2 period. Now glue the surfaces back together and allow the Ψ to "relax" (minimize curvature in some unique way?).

Was my description clear enough so that Ψ might now be given as a mathematical expression and be defined almost everywhere?

Suggestions on how to come up with the expression would be appreciated.

Now I would like to do the same thing again by shifting the two surfaces defined by this cutting forwards and backwards in time by 1/2 period but then wrap the surfaces around the line L till they meet and glue the surfaces back together and again (if done properly I think we create two "sheets"?). Now allow the Ψ to "relax".

Have I given a construction that could be expressed mathematically?

Thanks for any help.

 

[Spinnor]

I guess that the expression I am looking for should reduce to Ψ = exp(-i[p⋅x]) for points very far from line L?

 

[Spinnor]

So to simplify things let momentum be in the +z direction and let the line L above be the z axis. Let the positive x-axis be the ray perpendicular to line L. Our cut plane now is defined by the positive x-axis and the whole z axis. Now I guess we would want to use cylindrical polar coordinates?

 

[Spinnor]

Ψ(r,θ,z,t) = exp(-i[Et-pz+θ]) for the first case and Ψ(r,θ,z,t) = exp(-i[Et-pz+θ/2]) for the second case?

 

[zinq]

What does it mean to exponentiate a 4-vector? Are you using quaternions? (Likewise, what is the 'i" in that expression?)

 

[Spinnor]

What does it mean to exponentiate a 4-vector? Are you using quaternions? (Likewise, what is the 'i" in that expression?)

All very simple. The phase is just the Lorentz invariant product of two four-vectors which is just a number.

p⋅x = Et - p_1 x_1 - p_2 x_2 - p_3 x_3