In finding a lagrangian for a left-handed spinor field , a textbook claims that a kinetic term such as $ \partial_{\mu} \psi^a \partial^{\mu} \psi_a =\epsilon^{ab}\partial_{\mu}\psi_a \partial^{\mu}\psi_b$ (where $\psi$ is a left-handed spinor field and $\epsilon^{ab}$is an antisymmetric tensor) is unacceptable because the corresponding hamiltonian is unbounded below. I cannot understand why this is true.
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4$\begingroup$ Perhaps you can provide more information, including a reference to where you found the statement. $\endgroup$– flippiefanusCommented Oct 3, 2016 at 4:17
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1$\begingroup$ That term there is not how a spinor kinetic term looks anyway, no matter the handedness... $\endgroup$– ACuriousMind ♦Commented Oct 3, 2016 at 8:08
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2$\begingroup$ I found this statement in Srednicki's QFT text (Section36). $\endgroup$– ShoCommented Oct 4, 2016 at 3:58
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1$\begingroup$ @ACuriousMind - That is true. I think that is the point of the question. To show that you cannot use the scalar kinetic term as is and adapt it for spinors for reasons alluded to in the question. $\endgroup$– PraharCommented Oct 4, 2016 at 4:38
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2$\begingroup$ Does this answer your question? Why the terms like $\partial_\mu\psi\partial^\mu\psi + h.c.$ cannot be included in the Lagrangian for spinor fields? $\endgroup$– KvotheCommented Dec 1, 2023 at 15:56
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