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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4Perhaps you can provide more information, including a reference to where you found the statement. – flippiefanus Oct 03 '16 at 04:17
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1That term there is not how a spinor kinetic term looks anyway, no matter the handedness... – ACuriousMind Oct 03 '16 at 08:08
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2I found this statement in Srednicki's QFT text (Section36). – Sho Oct 04 '16 at 03:58
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1@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. – Prahar Oct 04 '16 at 04:38
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2Does 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? – Marten Dec 01 '23 at 15:56
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Apparently I can only flag one duplicate. This question is a duplicate of both https://physics.stackexchange.com/questions/584727/why-the-terms-like-partial-mu-psi-partial-mu-psi-h-c-cannot-be-included and https://physics.stackexchange.com/questions/146329/are-terms-with-spinors-analogous-to-partial-mu-phi-partial-mu-phi. (Only the first link has a correct (although kind off incomplete) answer. – Marten Dec 01 '23 at 16:02