ABSTRACT
The term "second quantization" is a bit confusing at first. How do you quantize something that is already quantized? The answer to this question, which is the topic of this chapter, is that in second quantization you are quantizing a different thing than in first quantization. We are all familiar with first quantization from ordinary nonrelativistic point particle quantum mechanics. If the position of a classical particle is x and its momentum p, we first quantize by making x and p operators on a Hilbert space. The elements of the Hilbert space describe the possible configurations or states of the one-particle system. The coordinate representation of a state is called the wave function for the system in that state. The wave function is just a function. In the process of second quantization, we take the states of the first quantized system and make them operators. The wave "functions" are no longer functions, but operators. x and p are no longer operators, but continuous indices for our new operators. Now we have a new set of operators but no states for them to operate on. So we find a new set of states, a direct sum of Hilbert spaces, and call it a Fock space. The result is a quantum field theory.
