Fields in Physics

Historical notions of field

Already in ancient Greece, philosophers and mathematicians developed an intuition regarding, mostly, non-contact forces of something that existed in space and caused things to move or behave in certain ways. A precise mathematical formulation of this idea was not achieved until Newton formulated his theory of gravitation. Since then, the basic characteristics of fields came understood to be:

• Presence of a source responsible for the existence of the field. For instance, the usual source of the magnetic field is a magnet. In modern physics, advanced theories and measurements have led to the discovery of certain particles ‘carrying’ the interaction, such as photons in the case of the electromagnetic field.
• Spacelike character: the field might behave as a scalar quantity throughout space, showing only magnitude, or it might have a direction or other more complicated features. For example, gravity has a direction towards the earth but temperature matters only as a number measuring its intensity.
• Locality: this is the key characteristic of usual fields, which are functions of spacetime. That means that the most general field takes different values for different locations of space, and it evolves in time for each point. For instance, the temperature in a room changes over time and will be higher the closer we get to the heater.

Figure 1. Example of a direction field.

Although the spacelike character was quickly understood by scientists and philosophers, the nature of sources and of locality were widely questioned throughout history. Regarding the sources, it is true that, according to our understanding of fields, a proper source does not always exist, as, for instance, for temperature.

On the other hand, the issue of locality has posed a greater challenge when modelling and understanding fields. For example, it used to be believed that temperature was generated in ‘bubbles’ that floated around. Nowadays, although locality is a widely accepted principle, some non-local effects have led us to believe that it is approximately but not fully correct.

The importance of fields

Most physics problems in high school involve highly simplified forces acting in a very local way but only at a certain point in space. We find point-like particles to be the elemental objects that we work with. We have examples of these in the charges responsible for creating an electric field or in the majority of modern atomic models when considering the electron. Fields are a generalisation for the whole space of the local point-like forces we have mentioned. In the end, this is all thanks to the fact that interactions caused by fields occur in points smaller than any scale we consider.

What are the consequences of fields as physical objects?

We are going to review some of the implications of fields for physical theories. Specifically, we will briefly focus on a couple of mathematical aspects and then on some purely physical ones.

Mathematical implications of fields

The basic implication of fields and their nature is the ability to use differential calculus to study how fields work. This is generally beneficial since we have a lot of notions of statistical manipulations, transformations of expressions, etc., which allow us to progress in many directions and to work with computers when systems are too complex. Nevertheless, field theories have had mathematical issues, which are being (and have been) addressed in order to get sensible results.

Physical implications of fields

The physical implications, although not completely independent of the mathematical ones, are based on the fact that forces are now caused by an entity that permeates the whole space and evolves in time. The essence of a field is the abundance of information. Since, for each point, we have information about the strength of the field, its direction, its time evolution, etc., we can generalise the dynamics of any object subject to the influence of the field.

What are examples of physical fields?

We will, finally, list some examples of fields in physics and briefly explain their nature and role:

• Gravitational field: probably the most famous and, still, widely investigated. It is generated by the presence of mass and has a direction. The first mathematical formulation is due to Newton, and it is the most relevant field at interplanetary scales.
• Electromagnetic field: the runner-up in the contest of the fame of fields, it is generated by the presence of charge whose state of motion creates different kinds of electric fields, magnetic fields, or both. Modern theories of electromagnetism are based on particles, called photons, that are sourcing the field.
• Stream current/wind: these are very similar in that they propagate on material mediums, air and water, and usually have a direction restricted to a plane (they do not go up or down). They do not have a proper physical source as in the previous examples but are caused by environmental phenomena.