In this blog, we will try to seek answer to three most basic yet important questions of fundamental Physics.

1. What are quantum fields and what is quantum field theory?
2. How quantum fields help us in predating properties of particles, and
3. Why we use quantum fields anyway?

Before knowing what and how, one first tries to know the answer to why, as it motivates reader. But answering 'why' is often impossible and contextual. For example: why does the nature behave the way it does? We must know, but we don't know, such questions can be very hard to explain or are still unknown to us.

But in our case, the answer to 'why Quantum fields?' can be explained to some extent so that reader finishes the article.
Let me phrase the question properly:

But what are these quantised fields anyway, and even before that let's ask "what are fields?". I will explain these in later sections, but straight answer to why we use quantum field theory is that, it works. It successfully explains the behaviour of fundamental particles in very small scales and very high momenta. This theory explains how particles come into existence, how particles are created and destroyed. It explains how particles scatter of each other by providing scattering cross-section area. It also explains the average life time of particles before they decay to another particles. It also explains the spectra of any multi particle bound states. It can predict new particles, and their properties (mass, charge, spin).

The quantum field theory is a mathematical model which is successful in some regime. Outside the regime (length scale and energy), we might need another theory.

But we would like to know from where the idea of quantising fields arose? Why to use the model with quantisation of object having infinite number of degree of freedom. It turns out that in order to describe the system with infinite number of particles we need infinite number of degrees of freedom. Fields have infinite number of degree of freedom.

One more reason is that particles are defined by unitary irreducible representations of Poincare group, and it turns out that no such finite dimensional representations exist, and these representations have to be infinite dimensional which are fields.

We needed a theory which is compatible with quantum mechanics (which describes physics of fixed number of particles), and spatial theory of relativity. Basically we were looking for quantum mechanics which is compatible with spatial theory of relativity.

The Schrodinger equation:

\[ \begin{eqnarray} i \hbar \frac{d\psi }{dt} = H \psi \\ i \hbar \frac{d\psi }{dt} = -\frac{\hbar^2}{2m} \nabla ^2\psi +V(x)\psi \\ \end{eqnarray} \]

We tried with relativistic quantum mechanics but even that theory describes fixed number of particles. In our nature particle number is not fixed. QM can not describe the dynamical system with varying numbers of particles.

It is time to get to know these fields.

Let us first understand fields. Fields are defined as continuous functions of space and time coordinates. These fields can be scalar fields, vector fields or tensor fields.

1. Scalar fields are scalar quantities at each space time coordinates. Example, temperature in room is a scalar field, density in room is scalar field. To make the quantum field theory compatible with quantum mechanics and spatial theory of relativity, we treat space and time on equal footing.