The Nature of Fields

From the previous discussion of basic concepts, it appears that the idea of fields becomes an important part of post-war developments, especially in the context of any developing description of a quantum field theory (QFT). As such, this section is intended to provide a slightly wider introduction to the ideas of fields within the transition from classical to quantum physics. As such, it might be suggested that there are possibly two distinct classes of fields, i.e.

• a classical field
• a quantum field

We might recognise that a field will be defined in terms of some ‘quantity’ associated with each point of spacetime So based on the nature of this quantity, we might classified a field into a number of distinct types:

• a static field
• a scalar field
• a vector field
• a spinor field

By way of some initial starting point, rooted in classical physics, we might describe a plot of temperature in a given volume of air in terms of a scalar field, although we might question the physicality of this description a little further. For temperature, in this context, might be better described as a conceptual and statistical description of the kinetic energy of the air molecules in some given volume of air. As such, we might see an interchange between the implied semantics of fields and particles.

Note: As a slight aside, it is possibly worth remembering that when the temperature of a distance object, e.g. the sun, is measured, this temperature is inferred from the radiation spectra emitted by all objects above absolute zero. As an example, imagine an asteroid colliding with a planet, such that the kinetic energy of the asteroid is effectively converted into heat energy. This dissipation of energy would then be accounted for in terms of the kinetic energy of particle debris, i.e. n*(mv²/2), plus the spectra of radiation energy, i.e. n*(hf) emitted. However, eventually we might need to even reflect further on the concept of kinetic energy, which is defined in terms of mass [m] and velocity [v], if the idea of physical mass is ultimately to be questioned. Equally, from a wave model description, it might be argued that only potential energy is transported in spacetime and it is this energy that is then capable of causing kinetic effects at other remote points in space.

Alternatively, we might highlight the relative gravitational strength, and its direction, in a volume of space in terms of a vector field. In this context, we are essentially describing a potential energy vector field that must exist at all points in spacetime, which has both magnitude and direction.

For now, we shall simply say that the idea of a spinor field is to be linked to Dirac’s equation and the concept of spin. As such, this type of field is more abstract and is possibly more dependent on a mathematical description, which is addressed in later discussions. Within this general classification of field classes and types, we might simply allude to the possibility of a field being subject to either a classical or quantum description, where the former might be characterised by numeric values, while the latter may be linked to the definition of a quantum operator. However, before starting to look at this distinction, it might be worth asking a more basic question:

What is the nature of a field?

Conceptually, a classical field may be extended to encompass the entirety of spacetime, although in terms of any practical measurements the strength of any field would appear to diminish with its distance from a given source. For example, the gravitational field strength is described by the inverse square law and, as such, conceptually extends to infinity, although the strength of the Earth’s gravitational field quickly becomes non-existence on a cosmic scale.

But does a field have tangible existence?

Again, let us start by considering this question from the classical perspective of a gravitational or electric field in the vacuum of space. First of all, it might be said that all classical physics can be reduced to a description involving only kinetic or potential energy. However, unlike kinetic energy that can be assigned to a single particle, potential energy can only be described as existing between two, or more, particles. Therefore, in this respect, a classical field only exists between  two or more particles and described in terms of the mapping of the potential energy surrounding these particles; although the perception of this field is typically measured as a vector force. So while we might conceptually describe an electric field as existing around a single charge particle in isolation, in practice, the existence of this field always requires a secondary test particle to measure the strength of the field, i.e. a force, at any point in the surrounding space.

So is the reality of this field being questioned?

Let us consider this question in terms of a somewhat contrived example of two particles between which exists an attractive force, e.g. gravitational. These particles are isolated in a vacuum at locations [A] and [B], but separated by a huge distance, such that any propagation mechanism linked to some finite speed, e.g. [c], must incur a finite propagation delay. Within this conceptual configuration, we might assume both particles to  be tethered in position, so that any infinitely small force of attraction on each particle can be measured independently. Now let us assumed that the mass at [A] can be simply moved to [A’] and back again by some external mechanism:

How does the movement of [A] affect this system?

In order to move [A] to [A’] and back, there is the assumption that some external energy has to be input into this system. However, when the particle is returned to [A], it possesses no additional kinetic energy in its frame of reference, i.e. its velocity is zero, and its potential energy with respect to [B] is not obviously different, if [B] is still tethered in its original position. However, if we assume that the change to the field between [A] and [B] is subject to a finite propagation velocity of [c], then this change may not have yet reached [B].

So where is the external energy input into this system?

In the scope of this example, we might reasonably assume that the input energy must now exist as potential energy in transit within the field, i.e. it is in the process of propagating from [A] to [B] at a finite speed of [c]. If so, this description would appear to suggest that the field has some form of physical reality, which is capable of transporting potential energy in space and time. As such, classical physics often utilises the idea of a field in order to explain the interaction between particles. As such, it was generally accepted that electromagnetic and gravitational fields exist physically, as suggested by the following Richard Feynman quote:

Therefore, field theory is the description of the dynamics of fields, i.e. its value as a function of space and time, which typically encompasses the ideas associated with the Lagrangian [L] or the Hamiltonian [H] of a field, which is said to have an infinite number of degrees of freedom.