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Published online by Cambridge University Press: 05 June 2012
If you've successfully worked your way through this short but arduous journey to the world of quantum field theory, you should be exhilarated! The landscape is full of fabulous beasts (most of which seem to go by the last name-on) and elegant formulae. Most surprisingly, all of this elegance seems to give us a remarkably precise and general description of many facets of our world, from phase transitions in humdrum materials to the interiors of stars and the intergalactic medium. In this concluding section I want to emphasize again a few general lessons, and chart out for you the parts of the quantum-field-theory landscape we have NOT explored.
The first of the important lessons that you should take away from this book is the beautiful unification of the classical theories of fields and particles that is forced on us by combining relativity and quantum mechanics in a fixed space-time background. The second is the unification of the methods of quantum field theory and classical statistical mechanics, which is provided by the Euclidean path-integral formulation of field theory.
Next I would ask you to remember the difference between a symmetry and a gauge equivalence and the different meanings of the idea of spontaneous symmetry breakdown in the two cases. Spontaneous breakdown of a global symmetry is related to locality. A quantum field theory is defined by its behavior at short distances, but there may be different infrared realizations of the same short-distance operator algebra and Hamiltonian.
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