1 Two component formalism for Spin 12 Fermions That is the fields \u03c7 1 and \u03c7 2

1 two component formalism for spin 12 fermions that

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1 Two-component formalism for Spin-1/2 Fermions That is, the fields χ 1 and χ 2 obey the standard time reversal transforma- tion laws of a single two-component fermion field [eqs. (1.155)–(1.158)], but with opposite sign phase factors η T in each case. Had one chosen η T = η T in eqs. (1.175) and (1.176), then the time reversal transformation laws of χ 1 and χ 2 would have been more complicated [eqs. (1.179) and (1.180)]. But as before, one is free to make a further SO(2) rotation to transform χ 1 and χ 2 into new fields that do exhibit the simpler time reversal transformation laws [eq. (1.151)]. One can now work out the time reversal properties of bilinear covariants constructed out of the fields of a charged fermion pair. The results are listed in Table B.3. 1.10 Charge conjugation of two-component spinors Charge conjugation was introduced in section 1.7. The charge conjugation operator is a discrete operator that interchanges particles and their C - conjugates. Here, conjugation refers to some conserved charge operator Q , where C Q C 1 = Q . The conjugate of the conjugate field is the original field, so that C 2 = 1. For a single two-component fermion field C χ α ( x ) C 1 χ C α ( x ) = η C χ α ( x ) , (1.183) C χ ˙ α ( x ) C 1 χ C ˙ α ( x ) = η C χ ˙ α ( x ) , (1.184) where | η C | = 1. In this case, no conserved charge exists, so that charge conjugation is trivial. In particular, as noted in section 1.7, the invariance of the mass term χχ + χ χ implies that η C = η C , and hence η C = ± 1. 27 The behavior of the bilinear covariants under C is simple: χ C 1 O χ C 2 = η C 1 η C 2 χ 1 O χ 2 , (1.185) for any O = I 2 , σ µ , σ µ , σ µν and σ µν (where bars should appear over the appropriate χ i depending on the choice of O ). In theories with multiple two-component fermion fields with no global symmetries, the charge conjugation properties of each fermion is given by eqs. (1.183) and (1.184), with η C = ± 1. When interactions are included, if there is some choice of the η C such that the action is invariant under charge conjugation, then the theory is charge conjugation invariant. If there is an internal global symmetry ( χ i U ij χ j ) under which the Lagrangian is invariant, then there are relations among the charge conjugation transformations of the fermion fields. Here, we consider 27 In a theory with just one fermion field, no physical quantity can depend on the sign of η C since the fermion field must appear quadratically in the Lagrangian. So, without loss of generality, we can take η C = 1.
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1.11 CP and CPT conjugation of two-component spinors 41 the simplest case of two mass-degenerate fermion fields, χ 1 and χ 2 , and identify the conserved charge as Q = J 0 d 3 x , where the conserved current J µ is given in eq. (1.108). In terms of the linear combinations ξ and η [eqs. (1.103) and (1.104)] corresponding to the fields of definite charge, we define the charge conjugation transformations as follows.
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