The analogy for quantum field theory (QFT) is often referred to as the "field of waves" analogy. In this analogy, QFT treats particles as excitations or disturbances in their corresponding fields, similar to waves in a field.
Mathematically, QFT is described using a combination of quantum mechanics and special relativity. The basic framework involves associating a quantum field with each type of particle in nature. These fields are represented by mathematical functions that span all of space and time.
In QFT, the behavior of particles is described by operators acting on these fields. These operators create, annihilate, or modify the number of particles in a given state. The dynamics of the fields are governed by an equation called the field equation, which is derived from the principles of quantum mechanics and special relativity.
Physically, the fields in QFT are not mere mathematical constructs but are considered fundamental entities in nature. They permeate all of space and are responsible for the creation and interaction of particles. Particles are viewed as localized excitations or quantized vibrations in these fields. The interaction between particles is described by exchanging virtual particles, which are again manifestations of the underlying fields.
QFT provides a mathematical framework that allows physicists to describe and understand the behavior of elementary particles and their interactions in a consistent and unified way. It has been successful in predicting and explaining a wide range of phenomena, and it is a fundamental tool in modern theoretical physics, particularly in the study of particle physics and quantum field theories such as the Standard Model.