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Unformatted text preview: Chapter 9 Quantum Field Theory 9.1 Introduction We have studied the properties of photons primarily as single particles. It was Einsteins great discovery to realize that particulate basis of light. Granted these are particles that are very different than those that we are used to. At the same time we realize that we have to develop a picture based on photons that adequately describes the many wavelike properties that are associated with light, the field properties. The theory that does that is quantum field theory. We want to make a quantum theory of the electromagnetic field. This is a rather complex field; it is a combination of two vector fields with a rather complex dynamics. For this reason, we will first discuss a simpler field, the stretched string. We will construct it by realizing that the phenomena that we identify with the field nature of light is characterized by energies that are large compared to ~ and therefore states with many photons. Also our study of the quantum oscillator indicated that to recover classical motion, we needed states composed of several stationary states. Our first problem will be to describe the many photon state. This is actually a subtle construction and will lead us in to a new definition of the identity of particles. We will cover also one of the great theorems of modern physics, the spin statistics theorem. 9.2 The Many Particle State These are the things that locally transfer discrete amounts of energy and momentum and other things. The example that we have been dealing with 253 254 CHAPTER 9. QUANTUM FIELD THEORY is the photon. It has a definite energy and momentum. The energy is related to the time evolution of the state and thus there is a frequency identified, = ~ . This frequency is related to the classical frequency. I remind you though that in the definite energy state of a quantum system nothing is moving back and forth. From the classical relationships we know that there is a relationship between the energy and momentum, = | ~ p | c . The polarization of the light was known from the classical case to be related to the angular momentum of the light. The photon is said to have an intrinsic angular momentum L = ~ . In fact we can do experiments that measure the angular momentum transferred by the absorption of photons....
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