Quantum field theory is the conceptual framework of modern particle physics, describing and predicting the most fundamental of physical processes with incredible success. Rising to popularity in the 1950s, the way in which QFT models particles and their interactions is both creative and notably distinct from previous theories. For every particle there is a corresponding field and excitations in the fields give rise to particles.
Motivations for QFT
QFT provides convenient solutions for the theoretical issues that plagued earlier theories. Special relativity demonstrated that information couldn’t be transferred across space faster than the speed of light. Classically, the interactions of particles via forces was formulated in terms of an “action at distance“, with the movement of one particle being communicated instantly to the second. Field theories remove this issue; the movement of a particle changes the shape of the field nearby but this change is only broadcast to the wider universe through the propagation of waves through the field at a finite speed.
A second argument for QFT is that all particles of a given type are the same. For example, every electron in the universe, regardless of how and when it was produced, is identical. The electrons travelling within the circuitry of your laptop are exactly the same as those produced in a supernova 5 billion years ago. Upon realising how remarkable this is, many were driven to postulate that there must be some sort of universal substance that all electrons arise from. An oscillating electron field does the job perfectly.
Photons and Electrons
Aside from characteristics such as charge and rest mass, electrons and photons are not all that different from a quantum mechanical viewpoint. Wave-particle duality is assigned to explain scenarios where electrons behave as waves and light behaves as particles. In reality, the distinction between these two objects is not significant.
The electron, however, is fundamental whereas the photon e.g. light was thought to emerge from the oscillatory motion of electric and magnetic fields. If quantum mechanics seems to suggest the two particles are so similar, why are their classical origins so distinct?
QFT aims to reconcile these ideas. There are two ways this could be done. Perhaps it’s the particles that are fundamental, with the fields emerging whenever a large enough number of particles congregate in a given space. Maybe the classical electric and magnetic fields are a consequence of a large ensemble of photons.
Alternatively, we could insist the fields are fundamental, and that quantising the fields gives rise to the existence of particles. Specific oscillations of the electric and magnetic fields produce photons, creating the illusion of independent particles. The latter description turns out to be most accurate and is the basis of QFT.
Quantising the Field
A classical field is an object that takes some value at every point in space. The value of the field varies with both space and time. The motion of the field can be understood as the sum of a set of fundamental modes of oscillation. A mode is a particular, independent oscillation of the field with definite frequency. Summing over all possible frequencies gives the emergent dynamics of the field. The frequency of a given mode is free to take any value within a classical field.
Many quantum mechanical systems impose constraints on the values of the physical quantities that describe the system. Dynamical quantities such as energy and momentum are often quantised, assuming one of a discrete set of values. The step from classical to quantum field theory consists of quantising the modes of oscillation.
A quantum field is an ensemble of quantised oscillators, existing at each and every one of the infinite points in space. Particles appear as a result of excitations in the respective field and interact due to the coupling of fields with each other.
Infinities and Zero-point energy
There is a striking issue with conceptualising a quantum field as an array of quantum oscillators. Counterintuitively, a quantum oscillator in its ground state possesses a non-zero energy, a phenomenon known as zero-point energy. Insisting on the existence of an oscillator at every point in space means the field itself consists of infinitely many oscillators. The ground state of the field, or the vacuum, therefore has an infinite energy associated with it.
The easy way around this problem is to suggest that we should only be concerned with the energy of a particle relative to the vacuum i.e. the energy difference. We can then set the energy of the ground state to zero by definition and do our best to ignore the fact that our theory predicts that an empty universe would have an infinite energy.