Field Theory

Field in Classical Physics

In nature many phenomenon imply a distant action such as gravitation, the electrostatic attraction and repulsion, magnetism, etc. Consider two magnets : when one moves closer the poles of opposite sign of the two magnets, they are attracted. The attraction force increases when the distance between both magnet decreases. Now suppose that we fix one magnet which is then referred to as the reference magnet and move around the other one (referred to as the sample magnet). For each location of the sample magnet one measures the intensity of the attractive force. It is possible to draw the "map" of the intensity of the force around the reference magnet. Now imagine we replace the sample magnet by a bigger one and we draw the attractive force intensity "map". The resulting map has the same shape as the first one. The only difference resides in the absolute value of the intensity at each point surrounding the reference magnet. This simple experiment points out a basic property of magnets : the shape of the attractive force intensity "map" of a given magnet is an intrinsic property of this magnet. So it is possible to assign each point in the space the value of the attractive force.

 

Now suppose that we have the "magic" capability to materialize a magnet wherever we want to. As soon as we materialize one at a given distance of the reference magnet, our new sample magnet is immediately attracted (or repulsed) by the reference one. Therefore we can say that the attraction force was potentially present at the location where the sample magnet materialized. In other words, the reference magnet seems to radiate an attractive potential force following the map we have drawn above. So the attractive force map is directly caused by the presence of the reference magnet, it is a property of the reference magnet. This map is called a field.

Mathematically speaking, a field is just the distribution throughout the entire space of mathematical entities such as numbers. When the entities are numbers, the field is called a scalar field. When the entities are vectors (numbers with a direction) the field is called a vector field and so on.

The attraction or repulsion of magnets are distant actions. Physicists call such distance action interaction (literally : "action between).

The classical description of fields raises the following remark : a field is continuous, i.e. the values of the field between two infinitely close points is continuous, without jumps.

 

Translation :

• Variation continue : continuos variation

• Variation discontinue : discontinuous variation

• Saut de valeur : value jump

In classical physics, fields fill up instantly the entire space. Imagine we have the capability to create a magnet from the vacuum. As soon as the magnet is created its magnetic field fills the entire space of the universe. This view is not compatible with the theory of relativity which states that the speed of light is a physical speed limit. Nothing can move faster than light. This principle, according to the theory of relativity is applicable to anything, even fields. Thus, according to the relativistic description of fields the field influence moves at the speed of light. If one creates a magnet, its field fills the space at the speed of light.

Field in Quantum Physics

To introduce the concept of field in quantum physics, let us consider an electron. According to the theory of electromagnetism the electron is the source of an electric field. The closer to the electron, the stronger the field is. Now suppose that we create a second electron very close to the first one. Both electrons are exposed to the intense repulsive electric field from the other one (repulsive because both electrons have a negative electric charge) and they move away with a high speed.

Moving an electron requires energy. The energy of the motion can only come from the field of each electron. In other words, the electrostatic repulsion is equivalent to a transfer of energy between both electrons. But remember, according to quantum physics, every energy transfer is made of quanta exchange. In other words, the repulsive electrostatic interaction between the electrons is caused by the exchange of quanta, i.e. an exchange of particles!

 

The particle exchanged during an interaction is called the interaction quantum. As electron cannot distinguished from each other therefore any electron-electron interaction involves the same kind of interaction quantum. Moreover, the interaction between electrons is caused by the electrons electric charge. Consequently any interaction between particles caused by the electric charge involves the same kind of interaction quantum. We will see that this interaction quantum is the photon!