Abstract
The concept of symmetry is paramount in modern Physics. In this chapter we are going to deal with the implementation of symmetries in quantum field theory. After reviewing the relation between continuous symmetries and conservations laws, we study how symmetries are realized quantum mechanically and in which way different realizations reflect in the spectrum of the theory. Our aim is to describe the concept of spontaneous symmetry breaking, which is crucial to our current understanding of how particle masses emerge in the standard model.
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Notes
- 1.
Here we use the notation \(|{\fancyscript{U}}\alpha\rangle\equiv {\fancyscript{U}}|\alpha\rangle\) and \(\langle{\fancyscript{U}}\alpha|\equiv \langle\alpha|{\fancyscript{U}}^{\dagger}.\)
- 2.
A quick survey of group theory can be found in Appendix B.
- 3.
For simplicity we consider that the minima of V(x) occur at \(V=0.\)
- 4.
In condensed matter the idea had been previously considered by Anderson [8].
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Álvarez-Gaumé, L., Vázquez-Mozo, M.Á. (2012). Symmetries I: Continuous 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_7
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DOI: https://doi.org/10.1007/978-3-642-23728-7_7
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