Abstract
This is probably the most technical chapter of this book. Discrete symmetries play a fundamental role in modern particle physics and cosmology. We have delayed their study until now to be able to develop all the tools required to explore some or their fascinating consequences. Specifically, we present an outline of the derivation of the CPT theorem from first principles and some of the consequences of the proof, in particular, the connection between spin and statistics.
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- 1.
Had we tried \(\psi(x)\rightarrow\Gamma\psi(x)\) we would find, using Schur’s lemma, that the Dirac equation is preserved only if \(\Gamma\) is proportional to the identity, so the transformation is trivial.
- 2.
Alternatively, the right-hand side of this expression can be written as \(i\gamma^{0}\gamma^{2}\overline{\psi}(x)^{T}.\)
- 3.
For the sake of simplicity we ignore other degrees of freedom. Their inclusion does not change the result.
- 4.
We should bear in mind that the Majorana neutrino break the global U(1) phase symmetry of the Lagrangian and therefore all three arbitrary phases of the charged leptons are independent. Notice also that in this case there is CP violation even with only two families, since having two charged leptons only allows for the elimination of two of the three phases of the mixing matrix.
- 5.
See Appendix B (Sect. B.4).
- 6.
This section is substantially more mathematical than the rest of the text and it can be safely skipped. Its purpose is to prove Eq. (11.82).
- 7.
The reader surely has noticed by now that in this book we have been assuming the spin-statistics theorem all the time since we have been consistently using the term “fermion” or “fermionic” to refer to spinor fields.
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Álvarez-Gaumé, L., Vázquez-Mozo, M.Á. (2012). Symmetries II: Discrete Symmetries. In: An Invitation to Quantum Field Theory. Lecture Notes in Physics, vol 839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23728-7_11
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