I am reading about HQET in Grozin's book http://www.amazon.es/Effective-Theory-Springer-Tracts-Physics/dp/3540206922. While constructing the Lagrangian he first consider the usual QCD Lagrangian with only one heavy quark and many lights. He then says that if we consider characteristic momenta of the heavy quark smaller than its mass we cann simplify the energy momentum dispersion relation to


and that as a result the bilinear part of the heavy quark Lagrangian can be taken to be


he then states that we can use two component spinors to describe spin instead of four component ones. Can somebody give me a justification for this last step? don't we lose degrees of freedom this way?


The point is that in the low energy limit, the particle and antiparticle pieces of the action are decoupled. This can be seen by the equations of motion: \begin{equation} \left( i \gamma ^0 \partial _0 - m \right) Q = 0 \end{equation} If we work in the Dirac basis for the gamma matrices, where particle nature of a Dirac field is manifest, then this takes the form \begin{equation} \left( \begin{array}{cc} i \partial _0 - m & 0 \\ 0 & - i \partial _0 - m \end{array} \right) \left( \begin{array}{c} \psi \\ \chi ^{\dagger } \end{array} \right) = 0 \end{equation} The point is that the equation for the particle ($ \psi $) and the antiparticle ($ \chi $) are completely decoupled from one another. Thus it makes sence to discuss particle and antiparticle interactions safely. Of course if you want to describe antiparticle interactions you need to consider the $ \chi $ Lagrangian, however, if you just care about particles you can forget about it completely.


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