Dirac equation with inertia operator (Pauli-Lubanski-like term)

Question

This will be a difficult and disturbing question: so it is not for students.

Dirac equation

The Dirac equation in $m^{-1}$ units is (summation on repeated index):

$$ (i \gamma_{\mu} \partial_{\mu} - k_0) \boldsymbol{\Psi} = 0 $$

with $k_0$ electron Compton wavelength, $\partial_{\mu}$ 4-gradient and $\gamma_{\mu}$ gamma matrices.

Inertia term in the Schrödinger Hamiltonian in the frame of Bohm-De Broglie theory

In the book of Peter R. Holland the author talks about an additional inertia term to the Hamiltonian in the Pauli-Schrodinger equation (i.e the non-relativistic approx of Dirac equation) in chapter 9, linked to the electron spin $\boldsymbol{S}$ in the form of:

$$ \frac{\boldsymbol{S}^2}{2 I} $$

with $I$ a not well defined electron moment of inertia. This is in the context of the alternative interpretation of quantum mechanics by De Broglie and Bohm (pilot wave theory).

What should be the form of an hypothetic inertia term in the Dirac Hamiltonian ?

The relativistic 4-angular momentum is:

$$ M_{\mu \nu} = r_{\mu} \partial_{\nu} - r_{\nu} \partial_{\mu} $$

Because Dirac equation only contains linear momentum ($i \partial_{\mu}$ term), why not adding the inertia term (self rotation) in this very harsh manner:

$$ \left[ i \gamma_{\mu} ( \partial_{\mu} \pm \beta M_{\mu \nu} \partial_{\nu} ) - k_0 \right] \boldsymbol{\Psi} \stackrel{?}{=} 0 $$

with $\beta$ a non-dimensional complex constant to be determined. This is really rough to write it like this but does somebody ever see this kind of theory ? Because the new vector operator $M_{\mu \nu} \partial_{\nu}$ looks like a lot the Pauli-Lubanski pseudo-vector. So I have found it very intriguing and interesting.

Answer 1

Basically OP is trying to add a term with two derivatives to the Dirac equation. The only Lorentz invariant term with two derivatives is Klein–Gordon-like $\beta \partial^\mu\partial_\mu\psi$, while terms like $\beta\gamma^\mu\gamma^\nu\partial_\mu\partial_\nu\psi$ for $\mu \neq \nu$ drop out given the anti-commutation property of $\gamma^\mu\gamma^\nu$.

The modified/enhanced Dirac Lagrangian can be written as $$ L = i\bar{\psi}\not D\psi + \beta \bar{\psi}\partial^\mu\partial_\mu\psi- m\bar{\psi}\psi \tag{1} $$ where the Klein–Gordon-like term $\beta\bar{\psi}\partial^\mu\partial_\mu\psi$ is added to the original Dirac-like term $i\bar{\psi}\not D\psi$.

The Klein–Gordon-like term $\bar{\psi}\partial^\mu\partial_\mu\psi$ is a dimension-5 operator (3 from the the two dimension-$3/2$ spinors plus 2 from the two derivatives), and thus non-renormalizable. Contrary to the claims here, in principle such non-renormalizable term IS allowed in the Effective Field Theory framework. However, given that the Klein–Gordon-like term is a dimension-5 operator, it is suppressed by a factor of: $$ \frac{E}{M_{Planck}} $$ where $E$ is the energy scale of the physics process in concern, and $ M_{Planck}$ is the Planck scale.

In other words, the Klein–Gordon-like term can be safely ignored under normal circumstances, unless we are dealing with Planck energy scale physics.

Additionally, the Klein–Gordon-like term $\bar{\psi}\partial^\mu\partial_\mu\psi$ breaks the axial symmetry $$ \psi \rightarrow e^{\theta i\gamma_5}\psi $$ Hence this term is further suppressed due to t' Hooft's technical naturalness argument.

See more details here.