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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Effective mass of Dirac electron increased by electrostatic potential?

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**