Are terms with spinors analogous to $ ( \partial_\mu \Phi )(\partial^\mu \Phi)$ forbidden in the Lagrangian?

Question

For scalar particles, the Lagrangian involves terms of the form $ ( \partial_\mu \Phi )(\partial^\mu \Phi)$, which is equivalent through integration by parts to $ ( \partial_\mu \partial^\mu \Phi )\Phi$. I was wondering if analogous terms for spinors are forbidden for some reasons and if not how they are interpreted? For example a term like:

$$ \partial^{\dot{a}b} \Psi_{c} \partial_{\dot{a}b} \Psi^{c}, $$

Some background:

It's possible to write four-vectors usign the spinor (Van-der-Waerden) notation:

$$ v^{a \dot b} = v^\mu \sigma_\mu^{a \dot b} ,$$ where $v^\mu$ can be seen to transform like a four-vector.

Therefore, the usual derivation operator, is in the spinor formalism $$ \partial^{a \dot b} = \partial^\mu \sigma_\mu^{a \dot b} $$

and Lorentz invariant terms in the Lagrangian involving first order derivatives are of the form:

$$ \Psi_{\dot{a}} \partial^\mu (\sigma_{ \mu})^{\dot{a}b} \Psi_b = (\Psi_L)^{\dagger} \sigma^\mu \partial_\mu \Psi_L $$ and $$ \Psi^{\dot{a}} \partial^\mu (\sigma_{ \mu})_{\dot{a}b} \Psi^{b} = (\Psi_R)^{\dagger} \partial^\mu \bar{\sigma}_\mu \Psi_R .$$

I was wondering if terms like

$$ \partial^\mu (\sigma_{ \mu})^{\dot{a}b} \Psi_{c} \partial^\nu (\sigma_{ \nu})_{\dot{a}b} \Psi^{c}, $$

which would be analogous to the term $ ( \partial_\mu \Phi )(\partial^\mu \Phi)$ in the scalar case, are forbidden for some reasons, and if not how they are interpreted?

Answer 1

I stumbled upon a pdf, in another question here, where it is stated that a term of the form $( \partial_\mu \Phi )(\partial^\mu \Phi)$ is forbidden for spinors, because it leads to a hamiltonian that is unbounded from below.

I will update this answer as soon as I have investigated this any further

Answer 2

Maybe the correct answer is that we don't need to introduce it. Formally this term refers to the free lagrangian, while free lagrangian must produce the equation of motion which corresponds to irreducible representation of Poincare group with mass $m$ and spin $s$. For spinors corresponds to 1/2-spin field, Dirac operator implements irrep of Poincare group.

Answer 3

Well, it'd be non-renormalizable. Observe, the mass dimension of the kinetic term should be...dimensionless. So, for a partial derivative $\partial$, its mass dimension should be $[\partial]=1$. The differential should be the opposite of this, so the 4-volume should have its mass-dimension be $[\mathrm{d}^{4}x]=-4$. Hence the action for a massless fermionic field ("the kinetic part of the action") $$ I_{\text{kinetic}}\sim\int\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi\,\mathrm{d}^{4}x $$ should be dimensionless, implying $[\psi^{2}]+1-4=0$ or equivalently $[\psi]=3/2$.

Observe now that the mass dimension for your expression is $$ [\int\psi\partial^{2}\psi\,\mathrm{d}^{4}x]=3+2-4=+1 $$ which causes renormalizability problems. As to why this causes nonrenormalizability issues, John Baez has a web page dedicated to it.