Cross between helicoid, complex plane wave

1. Feb 15, 2016

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 origional, 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.

Last edited: Feb 15, 2016
2. Feb 15, 2016

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?

3. Feb 15, 2016

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?

4. Feb 15, 2016

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?

5. Feb 16, 2016

zinq

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

6. Feb 16, 2016

Spinnor

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