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What is clear difference between say Psi_1,psi_2,....psi_4 and the U+- and V+- matrices in case of dirac fields or are u,v (or some book use U^(1),U^(2)) matrices some rep of the same

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If you edit this for better readability, your question is more likely to attract good answers. Use \$ signs around the math to have it rendered by Latex. – Neuneck Jun 18 '14 at 12:05
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The $u(p,s) $ and $v(p,s)$ describe the spin of the particle and antiparticle, while the $ \psi $ are the fields.A Dirac field can then be expanded in terms of these objects, \begin{equation} \psi = \int \frac{ \,d^3p }{ (2\pi)^3 } \frac{1}{ \sqrt{ 2E _p }} \sum _s \left( a _p ^s u ( p,s ) e ^{ - i p \cdot x } + b _p ^{s \,\dagger} v ( p ,s ) e ^{ i p \cdot x } \right) \end{equation} You may wonder why $ u $ and $ v $ are even needed since a complex scalar field can just be expanded as, \begin{equation} \phi = \int \frac{ \,d^3p }{ (2\pi)^3 } \frac{1}{ \sqrt{ 2E _p }} \sum _s \left( a _p ^s e ^{ - i p \cdot x } + b _p ^{s \,\dagger} e ^{ i p \cdot x } \right) \end{equation} The reason is a scalar field doesn't carry any spin and so can just be described by an ordinary function (think quantum mechanics where we need two component spinors to describe particle spin).

On the other hand, a Dirac fermion is actually made up of two particles - a left chiral fermion and a right chiral fermion. Each of these two fields can be either spin up or spin down and hence requires 2 degrees of freedom. In total it needs $ 2 \times 2 =4$ components to describe it.

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thanks just one more point so what you are saying is that say first component represent a left chiral spin up fermion ,the second comp rep left chiral spin down fermion and so on? – AMIT SINGH Jun 18 '14 at 11:59
It depends on your choice of basis for the gamma matrices, but that is correct in the Weyl basis. See here for more details,… – JeffDror Jun 18 '14 at 12:00

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