Difference between a classical and quantum field theory?

In summary, the conversation discusses the differences between quantizing a classical field theory and a point particle, and how the transition to creator/annihilator operators can be done from position/momentum operators. It also explores the equivalence between canonical quantization and path integral formalism, which is due to Nelson's theorem and the Wightman-Garding reconstruction theorem.
  • #1
gato_
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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:

[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?
 
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  • #2
gato_ said:
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.
 

1. What is the main difference between a classical and quantum field theory?

A classical field theory describes the behavior of a physical system using classical mechanics, which is based on Newton's laws of motion and assumes that matter behaves like particles. On the other hand, a quantum field theory takes into account the principles of quantum mechanics and describes the behavior of matter and energy at a subatomic level.

2. How do classical and quantum field theories differ in their treatment of fields?

In classical field theory, a field is treated as a continuous and deterministic entity, meaning that its value at any point in space and time can be precisely determined. In contrast, in quantum field theory, a field is considered to be made up of discrete and probabilistic particles, and its value at any point can only be described by a probability distribution.

3. Can classical and quantum field theories be used to explain the same physical phenomena?

Yes, both theories can be used to explain the same physical phenomena, but they may provide different levels of understanding and accuracy. Classical field theory is better suited for macroscopic systems, while quantum field theory is necessary for describing the behavior of particles at a microscopic level.

4. How do classical and quantum field theories differ in their predictions?

Classical field theory makes deterministic predictions, meaning that given the initial conditions of a system, its future behavior can be precisely calculated. In contrast, quantum field theory makes probabilistic predictions, meaning that it can only predict the likelihood of a particle's behavior, rather than its exact trajectory.

5. Why is quantum field theory considered a more complete theory than classical field theory?

Quantum field theory is considered more complete because it takes into account the principles of quantum mechanics, which have been extensively tested and proven to accurately describe the behavior of particles at a subatomic level. Classical field theory is limited in its ability to accurately describe certain phenomena, such as the behavior of particles at high energies.

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