An Introduction to Quantum Field Theory by Peskin and Schroeder

By Peskin and Schroeder

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77) explicitly. 78) is antisymmetric under the interchange of the labels 2 and 4, and also under the 52 Chapter 3 The Dirac Field interchange of 1 and 3. 78) with the identity linking and : : (3:80) ( ) = ( T) This relation is also straightforward to verify explicitly. 79), and the relation =4 (3:81) we can, for example, simplify horrible products of bilinears such as u2L)(u3L u4L ) = 2 u1L u3L ( u2L) ( u4L) (u1L = 2 u1L u3L u2L ( u4L ) = 2 (4)2 u1L u3L u2L u4L = 16(u1L u2L )(u3L u4L ): (3:82) There are also Fierz rearrangement identities for 4-component Dirac spinors and 4 4 Dirac matrices.

The helicity of a massive particle depends on the frame of reference, since one can always boost to a frame in which its momentum is in the opposite direction (but its spin is unchanged). For a massless particle, which travels at the speed of light, one cannot perform such a boost. The extremely simple form of u(p) for a massless particle in a helicity eigenstate makes the behavior of such a particle easy to understand. In Chapter 1, it enabled us to guess the form of the e+ e; ! + ; cross section in the massless limit.

2 (3:14) produces the representation of the rotation group given in Eq. 11). It is generally true that one can nd matrix representations of a continuous group by nding matrix representations of the generators of the group (which must satisfy the proper commutation relations), then exponentiating these in nitesimal transformations. For our present problem, we need to know the commutation relations of the generators of the group of Lorentz transformations. 12) follow straightforwardly. 15) is special to the case of three dimensions.

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