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Cross between helicoid, complex plane wave

  1. Feb 15, 2016 #1
    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. jcsd
  3. Feb 15, 2016 #2
    I guess that the expression I am looking for should reduce to Ψ = exp(-i[p⋅x]) for points very far from line L?
  4. Feb 15, 2016 #3
    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?
  5. Feb 15, 2016 #4
    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?
  6. Feb 16, 2016 #5
    What does it mean to exponentiate a 4-vector? Are you using quaternions? (Likewise, what is the 'i" in that expression?)
  7. Feb 16, 2016 #6
    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
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