From Special Relativity to Feynman Diagrams.pdf

Defining γ μν 1 2 γ μ γ ν we verify that γ

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proper Lorentz transformations, and with an additional minus sign under parity. Defining γ μν 1 2 [ γ μ , γ ν ] , we verify that γ μν transforms an antisymmetric tensor of rank two : S ( ) 1 γ μν S ( ) = 1 2 S 1 γ μ S , S 1 γ ν S = μ ρ ν σ γ ρσ (10.250) while γ 5 γ μν transforms like a pseudo- (or axial-) tensor, that is with an additional minus sign under parity as it follows from ( 10.248 ): S ( ) 1 γ 5 γ μν S ( ) = det ( ) μ ρ ν σ γ 5 γ μν . (10.251) These properties allow us to construct bilinear forms in the spinor fields ψ which have definite transformation under the full Lorentz group. Indeed if we consider a general bilinear form of the type: ¯ ψ( x μ 1 ...μ k ψ( x ) (10.252) as shown in Appendix G the independent bilinears are: ¯ ψ( x )ψ( x ) ; ¯ ψ( x μ ψ( x ) ; ¯ ψ( x μν ψ( x ) ; ¯ ψ( x 5 ψ( x ) ; ¯ ψ( x 5 γ μ ψ( x ). (10.253) To exhibit their transformation properties we perform the transformation ψ ( x ) = S ψ( x ) ψ ( x ) = S ψ( x ) = ψ ( x ) S γ 0 (10.254) and use the relation ( 10.92 ) of Sect.9.3.3 , namely γ 0 S γ 0 = S 1 (10.255) Using ( 10.247 ) and ( 10.248 ) it is easy to show that ¯ ψ( x )ψ( x ) is a scalar field while ¯ ψ( x 5 ψ( x ) is a pseudoscalar, i.e. under parity they transform as follows: ¯ ψ( x )ψ( x ) ¯ ψ ( x )ψ ( x ) ; ¯ ψ( x 5 ψ( x ) → − ¯ ψ ( x 5 ψ ( x ). (10.256) By the same token, and using ( 10.250 ) and ( 10.251 ) as well, we find analogous transformation properties for the remaining fermion bilinears. The result is summa- rized in the following table:
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10.8 Parity Transformation and Bilinear Forms 357 Bilinear P -transformed Kind ¯ ψ( x )ψ( x ) ¯ ψ( x P )ψ( x P ) Scalar field ¯ ψ( x 5 ψ( x ) ¯ ψ( x P 5 ψ( x P ) Pseudo-scalar field ¯ ψ( x μ ψ( x ) η μμ ¯ ψ( x P μ ψ( x P ) Vector field ¯ ψ( x 5 γ μ ψ( x ) η μμ ¯ ψ( x P 5 γ μ ψ( x P ) Axial-vector field ¯ ψ( x μν ψ( x ) η μμ η νν ¯ ψ( x P μν ψ( x P ) (Antisymmetric) tensor field where, in the second column, there is no summation over the μ and ν indices, and x P ( x μ P ) = ( ct , x ) . Reference For further readings see Refs. [3], [8] (Vol. 4), [9], [13]
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Chapter 11 Quantization of Boson and Fermion Fields 11.1 Introduction In the previous chapter we have examined the relativistic wave equations for spin 0 and spin 1/2 particles. The corresponding fields φ( x ) and ψ α ( x ) were classical in the same sense that the Schroedinger wave function ψ( x , t ) is a classical field. In contrast to the non-relativistic Schroedinger construction, we have seen that requir- ing relativistic invariance of the quantum theory, that is invariance under Poincaré transformations, unavoidably leads to serious difficulties when trying to interpret the field as representing the physical state of the system: It implies the appearance of a non-conserved probability density and, most of all, the appearance of negative energy states. Note that the latter difficulty is in some sense contradictory because if we just consider the field aspect of the wave equations, the field energy, expressed in terms of the canonical energy momentum tensor, is positive.
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