The term "field theory" can refer to various mathematical frameworks used in physics. Here, I will explain the difference between quantum field theory and classical field theory, which is often referred to as "regular field theory" or "classical field theory."
Classical Field Theory (Regular Field Theory): Classical field theory describes the behavior of fields in classical physics. It is based on the principles of classical mechanics and classical field equations, such as Maxwell's equations for electromagnetism or the wave equation for scalar fields. In classical field theory, fields are treated as smooth continuous functions defined over spacetime. The dynamics of these fields are governed by deterministic equations, and their values at each point in spacetime are well-defined. Classical field theory provides a framework to describe the behavior of fields, including their propagation, interactions, and energy-momentum distribution. It does not incorporate quantum mechanics or the uncertainty inherent in the quantum world.
Quantum Field Theory: Quantum field theory (QFT) is a framework that combines the principles of quantum mechanics with special relativity. It aims to describe the behavior of quantum fields, which are quantized versions of the classical fields. In QFT, fields are treated as operators that act on a quantum state and create or annihilate particles. The dynamics of these fields are described by quantum field equations, such as the Klein-Gordon equation or the Dirac equation, which incorporate both the wave-like and particle-like properties of quantum systems. QFT provides a way to describe the interactions of particles and fields, as well as the creation and annihilation of particles, by quantizing the underlying field degrees of freedom. It also introduces concepts like vacuum fluctuations, particle-antiparticle pairs, and the uncertainty principle inherent in quantum mechanics.
In summary, the main difference between classical field theory and quantum field theory lies in the treatment of fields and the underlying principles used to describe their behavior. Classical field theory deals with smooth, continuous fields governed by deterministic equations, while quantum field theory incorporates quantization and treats fields as operators acting on quantum states, taking into account the probabilistic and uncertain nature of quantum mechanics.