Classes & Types of Fields
We will now start to highlight some of the fundamental differences in the class of fields, i.e. classical and quantum, which have developed in the post-war era. In this respect, we shall start by providing a general description of the basic differences between a classical field versus a quantum field.
Initially, classical fields, such as electric and magnetic fields, were primarily described in terms of fields of force, which direct the motion of particles. However, if these fields were capable of transporting energy and momentum, they might also be said to have some sort of independent physical existence. Such ideas eventually led to a classical theory of fields, as defined by Maxwell’s equations, in which the composite electromagnetic field was described in terms of two perpendicular vector fields in space. In a similar fashion, Einstein's theory of gravity, i.e. general relativity, can also be cited as another example of an essentially classical field theory in as much that the field is not quantized. So, in this context, classical field theories remain an active and valid description, when the issue of quantum properties do not arise. As such, classical field theory considers quantities that are generally a continuous function of time and space, which can be described in terms of wave-like physical phenomena, e.g. sound and light, or other continuous phenomena, e.g. fluid flow. As such, we might generalise the definition of a classical field by a set of real numbers for each point in spacetime and assume that, in principle, such fields can be observed without disturbance.
In contrast, a quantum field is more of a mathematical concept defined by an operator at each point in spacetime, rather than a real number, plus it is generally assumed that a quantum field cannot be observed without disturbance. While, in some circumstances, a classical field will be an adequate description, many will argue that a quantum field theory must ultimately underpin all physical phenomena, at least in principle. So, for example, when classical electrodynamics is subject to the idea of quantization, it should align to the basic tenets of quantum electrodynamics (QED), which experimental data suggests is capable of higher precision. However, while QED has been subject to experimental verification, it is still unclear as to whether it is capable of actually describing the underlying mechanisms at work. In this context, quantum electrodynamics sits under the overall umbrella of quantum field theory, which also sub-divides into quantum chromodynamics and the electroweak theory, which are collectively said to underpin the standard model of particle physics. It should also be noted that the classical field theory of gravity, i.e. general relativity, has yet to be successfully quantized. By way of example, a bosonic field is a quantum field whose quanta are bosons, which include scalar field description of spin-0 particles, such as the Higgs boson, and gauge fields description of spin-1 particles, such as the photon. In contrast, a fermionic field is a quantum field whose quanta are fermions, which describe fermion particles with spin-1/2, such as electrons, protons, quarks etc.
Note: Based on the introductory description above it might naturally follow that any quantum field theory might be thought of as an extension of classical field ideas, but for the complication of the fields being quantised. For example, in 1926, Born, Heisenberg and Jordan published one of the first papers on quantum mechanics addressing electromagnetic fields. In this paper, a formula for the electromagnetic field was presented as a Fourier transform, which used a commutation relationship to identify the coefficients of a Fourier transform, which were then described as operators that created and destroyed photons. In so doing, this quantization of the classical electromagnetic field would lead to a quantum theory of photons, which is now described in terms of QED. However, in the context of pre-war quantum mechanics, a solution to Schrodinger’s wave equation was generally interpreted to reflect the state of a particle, while later post-war QFT developments would lead to the idea of a solution to the wave equation, which did not reflect the state of a particle, but rather the operators that created and destroyed states.