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.

In addition to the big omissions I mentioned in the text, there is a host of subjects in field theory that I have neglected. I will mention them here, without detailed reference. In the days of Spires and Google detailed reference hardly seems necessary: all you need is a name. The large-N approximation, particularly the connection between large-N matrix field theories and string theory, is a big lacuna. I've barely mentioned the field of computational lattice gauge theory, which is slowly achieving a quantitative understanding of the hadron spectrum and other low-energy properties of QCD. The use of field theory techniques in condensed-matter physics produced the theory of superconductivity, of critical phenomena, and of the quantum Hall effect as well as a variety of other phenomena. The books by Parisi, Ma, Drouffe and Itzykson, and Zinn-Justin provide an entrée into this vast body of knowledge. Another big lacuna is the study of integrable two-dimensional theories and exact S-matrices. There has been a variety of attempts to study the high-energy fixed-momentum-transfer region of scattering amplitudes (the Regge region) by summing selected classes of diagrams in field theory. This has led to an effective field theory approach called Reggeon calculus as well as to a very interesting set of equations called the BFKL equations (this is not quite the Regge region). Then there is the use of two-dimensional conformal field theory to do perturbative string theory.