Quantum field theory, the quantum mechanics of continuous systems, arose at the beginning of the quantum era, in the problem of black-body radiation. It became fully developed in quantum electrodynamics, the most successful theory in physics. Since that time, it has been united with statistical mechanics through Feynman's path integral, and its domain has been expanded to cover particle physics, condensed matter physics, astrophysics, and wherever path integrals are spoken.
This book is a textbook on the subject, aimed at readers conversant with what is usually called "advanced quantum mechanics," the equivalent of a first-year graduate course. Previous exposure to the Dirac equation and "second quantization" would be very helpful, but not absolutely necessary. The mathematical level is not higher than what is required in advanced quantum mechanics; but a degree of maturity is assumed.
In physics, a continuous system is one that appears to be so at long wavelengths or low frequencies. To model it as mathematically continuous, one runs into difficulties, in that the high-frequency modes often give rise to infinities. The usual procedure to start with a discrete version, by discarding the high-frequency modes beyond some cutoff, and then try to approach the continuum limit, through a process called renormalization.
Renormalization is a relatively new concept, but its workings were already evident in classical physics. At the beginning of the atomic era, Boltzmann noted that classical equipartition of energy presents conceptual difficulties, when one seriously considers the atomic structure of matter. Since atoms are expected to contain smaller subunits, which in turn should composed of even smaller subunits, and so ad infinitum, and each degree of freedom contributes equally to the thermal energy of a substance, the specific heat of matter would be infinite. The origin of this divergence lies in the extrapolation of known physical laws into the high-frequency domain, a characteristic shared by the infinities in quantum field theory.
Boltzmann's "paradox," however, matters not a whit when it comes to practical calculations, as evidenced by the great success of classical physics. The reason is that most equations of macroscopic physics, such as those in thermodynamics and hydrodynamics, make no explicit reference to atoms, but depend on coefficients like the specific heat, which can be obtained from experiments. From a modern perspective, we say that such theories are "renormalizable," in that the micro-structure can be absorbed into measurable quantities.
One goal of this book is to explain what renormalization is, how it works, and what makes some systems appear "renormalizable" and other not. We follow the historical route, discovering it in quantum electrodynamics through necessity, and then realizing its physical meaning through Wilson's path-integral formulation.
This book, then, starts with a thorough introduction to the usual operator formalism, including Feynman graphs, from Chapters 1-10. This is follows by Chapters 11-14 on quantum electrodynamics, which illustrates how to do practical calculations, and includes a complete discussion of perturbative renormalization. The last part, Chapters 15-19, introduces the Feynman path integral, and discusses "modern" subjects, including the physical approach to renormalization, spontaneous symmetry breaking, and topological excitations. I have entirely omitted non-Abelian gauge fields and the standard model of particle physics, because these subjects are discussed in another book:
K. Huang, Quarks, Leptons, and Gauge Fields, 2nd ed. (World Scientific, Singapore, 1992).
I have chosen to introduce path integrals only after the canonical approach is fully developed and applied. Others might want them discussed earlier. To accommodate different tastes, I have tried to make each chapter self-contained in as much as possible, so that a knowledgeable reader can pick and skip.
There is definitely a change in flavor when quantum field theory is conveyed through the path integral. Apart from the union with statistical mechanics, which mmeasurably enriches the subject, it liberates our imagination by making it possible to contemplate virtual but fantastic deformations, such as altering the structure of space-time. I am reminded of the classification of things as "grey" or "green" by Freeman Dyson, in his book Disturbing the Universe (Harper and Row, New York, 1979). He classified physics grey, (and I suppose that included quantum field theory,)as opposed to things green, such as poems and horse manure. In a private letter dated August 3, 1983, Dyson wrote, "Everyone has to make his own choice of what to call grey and green. I took my choice from Goethe:
tenerer freund, ist alle Theorie,
Und grun des Lebens Goldner Baum.
all theory is grey,
And green is the golden tree of life.)
I must admit that Hilbert space does seem a bit dreary at times; but, with Feynman's path integral, quantum field theory has surely turned green.