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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


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It basically boils down to the term $e^{-\frac i\hbar E t}$, where the minus can either be included in the energy, making it negative, or into the time. But a negative charge moving backwards in time is exactly the same as a positive charge moving forwards in time, and that is much more sensible than negative energy.


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I find things are clearer using the dotted and undotted spinor notation. The L-spinors $\chi_{L}$ are dotted vectors $\chi^{\dot{A}}$ and the R-spinors $\xi_{R}$ are undotted vectors $\xi^{A}$ with index $A=1,2$. The parity operator has to be a tensor $P^{\dot{A}}_{B}$ and another tensor $P^{A}_{\dot{B}}$ in order to change the way each type of spinor ...


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You are looking for a unitary representation of partity on spinors. That it should be unitary can be seen from the fact, that partity commutes with the Hamiltonian. Compare this to time-reversal and charge conjugation, which anticommute with $P^0$ and hence need be antiunitary and antilinear. They involve complex conjugation. As demonstrated parity ...


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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 ...


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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. ...


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In general you cannot, but in your special case it works out. You should be aware of what the objects in your expression actually are, and how they relate to each other. $X$ is a bosonic field and as such does not feel the presence of gamma matrices at all. Your first line could be rewritten as $$ S = \int \mathrm d^2 \sigma \, \bar \epsilon \gamma^\alpha ...


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There are two other interpretation of the Pauli matrices that you might find helpful, although only after you understand JoshPhysics's excellent physical description. The following can be taken more as "funky trivia" (at least I find them interesting) about the Pauli matrices rather than a physical interpretation. 1. As a Basis for $\mathfrak{su}(2)$ The ...


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Let me first remind you of (or perhaps introduce you to) a couple of aspects of quantum mechanics in general as a model for physical systems. It seems to me that many of your questions can be answered with a better understanding of these general aspects followed by an appeal to how spin systems emerge as a special case. General remarks about quantum states ...


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Groups are abstract mathematical structures, defined by their topology (in case of continual (Lie) groups) and the multiplication operation. But it is almost impossible to talk about abstract groups. That is why usually elements of groups are mapped onto linear operators acting on some vector space $V$: $$ g \in G \rightarrow \rho(g) \in \text{End}(V), $$ ...


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The Dirac equation is not the relativistic limit of the Schrodinger equation, it is the high energy completion of it (for a spin 1/2 particle). Therefore, it explains higher energy electrons as well as low energy ones. Also in the above by taking, $i \partial_\mu \Psi = m \gamma_0 \Psi$, you assumed that $\vec{p}=0 $, so of course you found the electron is ...



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