Effective mass of Dirac electron increased by electrostatic potential?

[johne1618]

The Dirac electron in the Higgs vacuum field $v$ and an electromagnetic field with vector potential $A_\mu$ is described by the following equation:

$i \gamma^\mu \partial_\mu \psi = g v \psi + e \gamma_\mu A^\mu \psi$

where $g$ is the coupling constant to the Higgs field and $e$ is the coupling constant to the electromagnetic field.

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

$\partial_x=\partial_y=\partial_z=0$

Let us also assume that there is only an electrostatic potential $A_0=\phi$ so that:

$A_x = A_y = A_z = 0$

So the simplified Dirac equation is now:

$i \gamma^0 \partial_t \psi = g v \psi + e \gamma_0 \phi \psi$

Let us choose the Weyl or Chiral basis so that:

$\gamma^0 = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix}$

where $I$ is the $2\times2$ unit matrix.

In this representation:

$\psi=\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$

where $\psi_L$ and $\psi_R$ are left-handed and right-handed two-component Weyl spinors.

Subtituting into the simplified Dirac equation above we get:

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

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

$i \partial \psi_R / \partial t = g v \psi_L + e \phi \psi_R$

$i \partial \psi_L / \partial t = g v \psi_R + e \phi \psi_L$

Now let us add these two equations together to obtain:

$i \frac{\partial}{\partial t} (\psi_L + \psi_R) = (g v + e \phi)(\psi_L + \psi_R)$

My question is this:

Does the state $\psi_L + \psi_R$ describe an electron with an effective mass given by $gv + e \phi$?

Does the presence of an electrostatic field increase the electron's mass over and above the mass induced by the Higgs vacuum field alone?

 

[Bill_K]

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.

 

[johne1618]

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.

https://www.physicsforums.com/showthread.php?t=562314