Book Reviews

Abstract

We've taken advantage of Bill Faris's rare ability to explain modern physics several times previously. This time, his featured review of Gerry Folland's Quantum Field Theory and the second (of six) thousand-page tome on the subject by Eberhard Zeidler points out what bargains with the devil must be made. Ultimately, with such advanced tutoring, we should all gain an appreciation of renormalization theory and its role in everyday life! Other reviews cover neuron models, Kalman filtering, math finance, operator theory, functional analysis, random dynamical systems, image analysis, dynamical systems, and long time asymptotics. In this season, when even Alice Munro has written about Sophie Kovalevsky, it's good to have expert recommendations about the newest math monographs and their significance. BOOK REVIEWSquantum fields defined on space-time. There may also be mention of random fields defined on Euclidean space; there is a close mathematical connection between relativistic quantum fields and Euclidean random fields.Here are some basic ingredients of the world view according to relativistic quantum field theory:• The geometric background for the world is relativistic space-time.• The world itself is made of matter and force.• Matter consists of quarks and leptons.• The spin-statistics theorem says that particles with spin n/2 with n odd obey Fermi statistics, while particles with spin n/2 with n even obey Bose statistics. • Quarks and leptons have spin 1/2 and obey Fermi statistics. Force carriers have spin 1 and obey Bose statistics. • The description of the physical nature and interactions of matter and force in space-time is via quantum field theory. • A quantum field is a quantum-mechanical object regarded as a function (or generalized function) on space-time. • In scattering experiments the excitations of a quantum field can also describe particles. • The predictions of the theory are ultimately probabilistic. Some of these points deserve more comment. Matter is made of quarks and leptons, but how does this relate to the usual picture of atomic structure? The most common quarks are the up quark and the down quark. A proton is made of two up quarks and one down quark. A neutron is made of two down quarks and one up quark. And protons and neutrons are the familiar constituents of the atomic nucleus. The most common leptons, on the other hand, are the electron and the electron neutrino. The picture of electrons going around the nucleus is an image that has imposed itself on the general consciousness, and to the extent that this represents quantum theory, the theory has not gone totally unrecognized.The probabilistic predictions of quantum theory involve the ideas of observation and measurement, and there is a certain mystery about what quantities should be taken as fundamental. This problem also arises in quantum field theory, but there seems to be agreement that for scattering experiments the theory predicts where particles will hit detectors. That is enough for many physicists. The Folland Book: Quantum Field ...