# Difference between a classical and quantum field theory?

1. Mar 6, 2010

### gato_

This may be a very basic question, but I've had now some background on the quantum theory, and I think I am missing something. Roughly speaking, I feel like the main difference is that quantizing involves going from field amplitudes to counting operators, implying that a quantum process involves exciting discrete lumps of energy. What I don't understand is how this is reflected on, say, the canonical quantization procedure. Take for example non relativistic theory:

$$[q_{i},p_{j}]=i\hbar \delta_{ij}$$

I don't see how that adds any restriction in a classical field theory. If you make the transition from a point particle to a field, then p is still the infinitesimal generator of translations, leading naturally to $$p_{i}=-i\hbar \partial_{i}$$, so for the field the relation is satisfied trivially. The transition to creator/anihilator operators can be done from the position/momentum ones, so they contain the same information. Why then counting operators lead to a different field theory?
Moreover, why are canonical quantization and path integral formalism equivalent?

Last edited: Mar 6, 2010