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Effective mass of Dirac electron increased by electrostatic potential?

  1. Dec 24, 2011 #1
    The Dirac electron in the Higgs vacuum field [itex]v[/itex] and an electromagnetic field with vector potential [itex]A_\mu[/itex] is described by the following equation:

    [itex]i \gamma^\mu \partial_\mu \psi = g v \psi + e \gamma_\mu A^\mu \psi [/itex]

    where [itex]g[/itex] is the coupling constant to the Higgs field and [itex]e[/itex] is the coupling constant to the electromagnetic field.

    Let us assume that we are in the rest frame of the electron so that:


    Let us also assume that there is only an electrostatic potential [itex]A_0=\phi[/itex] so that:

    [itex]A_x = A_y = A_z = 0[/itex]

    So the simplified Dirac equation is now:

    [itex]i \gamma^0 \partial_t \psi = g v \psi + e \gamma_0 \phi \psi [/itex]

    Let us choose the Weyl or Chiral basis so that:

    [itex]\gamma^0 = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix} [/itex]

    where [itex]I[/itex] is the [itex]2\times2[/itex] unit matrix.

    In this representation:

    [itex]\psi=\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} [/itex]

    where [itex] \psi_L [/itex] and [itex] \psi_R [/itex] are left-handed and right-handed two-component Weyl spinors.

    Subtituting into the simplified Dirac equation above we get:

    [itex] i \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix} \begin{pmatrix} \partial \psi_L / \partial t \\ \partial \psi_R / \partial t \end{pmatrix} = g v \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} + e \phi \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix} \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} [/itex]

    This equation separates into two equations of two-component Weyl spinors:

    [itex] i \partial \psi_R / \partial t = g v \psi_L + e \phi \psi_R [/itex]

    [itex] i \partial \psi_L / \partial t = g v \psi_R + e \phi \psi_L [/itex]

    Now let us add these two equations together to obtain:

    [itex] i \frac{\partial}{\partial t} (\psi_L + \psi_R) = (g v + e \phi)(\psi_L + \psi_R) [/itex]

    My question is this:

    Does the state [itex]\psi_L + \psi_R[/itex] describe an electron with an effective mass given by [itex]gv + e \phi[/itex]?

    Does the presence of an electrostatic field increase the electron's mass over and above the mass induced by the Higgs vacuum field alone?
  2. jcsd
  3. Dec 24, 2011 #2


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    Science Advisor

    johne1618, Mass is more than just "energy at rest". To define what the mass is, you need to see how the energy changes when the object is slowly moving (as in p2/2m.) The nonrelativistic limit of the Dirac equation is a thoroughly studied topic. I suggest you google for material on the Foldy Wouthuysen representation.
  4. Dec 27, 2011 #3
    Prof Susskind has kindly answered my question in an attachment. Adding a constant potential is equivalent to an overall phase shift and has no effect on the electron's equation of motion.

    The file is attached to a reply to my similar thread in the Particle Physics Forum.

    Last edited: Dec 27, 2011
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