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:(adsbygoogle = window.adsbygoogle || []).push({});

[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:

[itex]\partial_x=\partial_y=\partial_z=0[/itex]

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?

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# Effective mass of Dirac electron increased by electrostatic potential?

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