3.5.1 Classical and Quantum Fields
A field in physics is a region or space in which a given effect exists, such as magnetism, static electricity, gravity, or even temperature. In classical physics, a field is a quantity, represented by a number or tensor, that has a value for each point in space and time. In the modern framework of the quantum theory of fields, a field occupies space, contains energy, and its presence precludes a classical “true vacuum”.
This led physicists to consider electromagnetic fields to be a physical entity, making the field concept a supporting paradigm of the edifice of modern physics. As Feynman said:
The fact that the electromagnetic field can possess momentum and energy makes it very real ... a particle makes a field, and a field acts on another particle, and the field has such familiar properties as energy content and momentum, just as particles can have. Feynman et al. (1963)
In general, a field can be classified as a scalar field, a vector field, a spinor field or a tensor field, according to the represented physical quantity. A field has a unique tensorial character in every point where it is defined, so a field cannot be a scalar field somewhere and a vector field in other regions. However, for all these categories, a field can be classical or a quantum field, depending on whether it is characterized by numbers or by quantum operators.
A classical field is a function defined over some region of space and time, such as gravitation and classical electromagnetism. Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom, called classical oscillators. However, classical field theory does not account for the quantum-mechanical aspects of such physical phenomena. Quantum Field Theory, therefore, define the field as a function over space and time, but in a way that accommodates the observations of Quantum Mechanics.
To achieve this task, the infinity of classical oscillators representing the modes of the classical fields are promoted to quantum harmonic oscillators. This means that they are functions whose values are abstract operators acting on an abstract state space. In such space, which is called Hilbert space, the observables represent physically observable quantities, and the state space represents the possible states of the system under study.
There are two common ways of handling a quantum field: canonical quantization and the path integral formalism.
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