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:(adsbygoogle = window.adsbygoogle || []).push({});

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

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 [tex]p_{i}=-i\hbar \partial_{i}[/tex], 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Difference between a classical and quantum field theory?

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**