# Difference between a classical and quantum field theory?

#### 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?

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#### DarMM

Gold Member
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?
Quantum Field theory involves changing $\phi$ the classical field and its conjugate momentum $\pi$ from being dynamical variables (generically sections of fiber bundles) to being operators on a Hilbert space. We then impose the following condition on these operators:
$$\left[\phi(\mathbf{x}),\pi(\mathbf{y})\right] = i\delta^{\left(3\right)}\left(\mathbf{x} - \mathbf{y}\right)$$

These conditions imply that the Fourier amplitudes of the free quantum field, now operators since the fields are, have the algebra of raising and lowering operators.

Why the canonical and path integral formalisms are equivalent is due to a result called the Nelson's theorem. This demonstrates that the moments of a random Euclidean classical field theory can be analytically extended to a set of complex functions. These functions then can be proven to inherit properties from the moments on Euclidean space that imply they have boundary values which are distributions on (tensor products of) Minkowski space.

These distributions can then be seen to obey the Wightman axioms and thus from the Wightman-Garding reconstruction theorem are the correlation functions of a quantum field theory obeying the Wightman-Garding axioms.

"Difference between a classical and quantum field theory?"

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