ABSTRACT
Using the results in the previous chapter for path integrals in quantum mechanics, we are now ready to generalize the definitions to the simplest field theories, the free fields. Recall that in first quantized systems that the number of particles is fixed, and that the position and momentum, xg and Pi, of each particle become operators. In the path integral representation we integrate over paths in phase space, (4(0, fi(t)). In a second quantized system, the wave function, 0(E, t), of the first quantized system is elevated to an operator. Therefore, in the path integral representation of a field theory we can expect that we will need to integrate over "paths" in a "phase space" of functions whose coordinates are (0(i, t), 7r(i, t)), 7r(i, t) being the appropriate field momentum.
