Classical Field Theory

From a historical perspective, we might view the concept of a quantum field theory as part of the on-going transition away from the classical perspective, which takes onboard the idea of quantization. However, classical field theory may still help us visualise how one or more physical fields might interact with matter, where a physical field can be mathematically described in terms of a set of quantities at each point of spacetime.

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In a classical context, a idea of field theory is most commonly used to describe the underlying physics of electromagnetism and gravitation, which represent two of the four fundamental forces of nature. However, in many respects, any field description is primarily a mathematical model of reality that helps explain the effects observed rather necessarily being a physical description of an actual field. As such, a classical gravitational field is a model that maps Newton's law of universal gravitational force onto a field theory. In this context, a gravitational vector field surrounds any given mass [m], where each point in spacetime is a vector whose magnitude is defined by the inverse square law and points towards the centre of mass [m]. The magnitude of this vector can also be described in terms of a force per unit mass at any point in space and because this field of force is conservative, there is a scalar potential energy per unit mass at each point in space, which is called the gravitational potential. Another early example of a classical field can be attributed to Faraday’s description of an electric field in terms of lines of force.

Gravitational Fields

As indicated, the classical field description of gravity is based on Newton’s idea of gravitational force, which exists as a mutual attraction between two masses [M,m]. In the field model, a test mass [m] experiences a vector force [F], where the gravitational field strength [g] is defined by the force per unit mass [F/m]. Newton's famous law of gravitation being a reflection of the inverse square law:

[1]      1

However, we can also use Newton's 2nd law [F=ma] to lead us to a definition of the gravitational field strength due to a mass [m], which we might realise can also be interpreted as a gravitational acceleration [g]:

[2]      2

It is worth noting that [2] contains the definition of both inertial and gravitational mass in the form of Newton’s 2nd law of motion and his universal gravitational law respectively. Subsequent experimental measurements have proved that the inertial and gravitational mass are equal to all extent and purpose, which is an idea that underpins general relativity in the form of the equivalence principle.

 Electrostatic  Fields

We can also present a classical field description of an electric field based on Coulomb’s law of force, which exists as both a mutual attraction and repulsion between two charges [Q,q]. In the field model, a test charge [q] experiences a vector force [F]:

[3]      3

We can also define the concept of an electric field [E] strength, where [F=qE]. While this leads to the definition of an electric field due to a single charged particle, it might be argued that this field only exists in the presence of two or more charged particles.

[4]     4