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Dirac spinor
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==Energy eigenstate projection matrices== It is conventional to define a pair of [[projection (mathematics)|projection]] matrices <math>\Lambda_{+}</math> and <math>\Lambda_{-}</math>, that project out the positive and negative energy eigenstates. Given a fixed Lorentz coordinate frame (i.e. a fixed momentum), these are <math display="block">\begin{align} \Lambda_{+}(p) = \sum_{s=1,2}{u^{(s)}_p \otimes \bar{u}^{(s)}_p} &= \frac{{p\!\!\!/} + m}{2m} \\ \Lambda_{-}(p) = -\sum_{s=1,2}{v^{(s)}_p \otimes \bar{v}^{(s)}_p} &= \frac{-{p\!\!\!/} + m}{2m} \end{align}</math> These are a pair of 4Γ4 matrices. They sum to the identity matrix: <math display="block">\Lambda_{+}(p) + \Lambda_{-}(p) = I</math> are orthogonal <math display="block">\Lambda_{+}(p) \Lambda_{-}(p) = \Lambda_{-}(p) \Lambda_{+}(p)= 0</math> and are [[idempotent]] <math display="block">\Lambda_{\pm}(p) \Lambda_{\pm}(p) = \Lambda_{\pm}(p) </math> It is convenient to notice their trace: <math display="block">\operatorname{tr} \Lambda_{\pm}(p) = 2 </math> Note that the trace, and the orthonormality properties hold independent of the Lorentz frame; these are Lorentz covariants.
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