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Because it is excelant representation of anticommutation relationships which he had established for his matrixes. Moreover 4-component wave functions actually represents spinors for pare electron-positron!
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Why dirac delta function is used?
Well Dirac delta functions have a loot of application in physics.... Suppose u want to depict the charge density or mass density at only a particular point and want to show that at any other point in space this density is nil, we use this dirac delta function to depict the position of this charge or mass... In general, Dirac delta function is used whenever the divergence for a field has different and contradicting values at the origin....esp used when the usual Divergence theorum is proved wrong due to contradicting values of the flux...
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