In this article we will see how to implement the tools of QFT to explain the behaviour of some system with indefinite number of particles.

Particles can be classified as bosons or fermions on the basis of statistics. Bosons are those particles (composite or fundamental) who follow Bose statistics, whereas fermions obey Fermi-Dirac statistics

Bosons are integer spin (s=0,1,2,3,4...), while fermions have half integer spin (\(s=\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}...\)).

Let us begin by understanding the behaviour of spinless or spin zero particles. Such particles don't have degenerate states and thus are simplest for our study.

We generally divide the subject of QFT in three parts.

1. Spin 0 (Scalars: 1 component)
2. Spin 1/2 (Spinnors, 4 component (L,R transform differently for massive spinnors))
3. Spin 1 (Vectors, 4 component)

1. Need of QFT:
   1. QM is not compatible with spatial theory of relativity as Schrodinger equation is first order in time and second order in space. An attempt was made to use relativistic Hamiltonian of particle (\(H=\sqrt{p^2+m^2}\) and using \(\hat{p}=-i\partial_{x},\hat{H}=i\partial_{t}\)) which resulted in Klein-Gordon equation (\((\partial^2+m^2)\psi=0\)) for wave function \(\psi\). This solution has the following problems.
      1. This attempt results in negative energy eigenvalues (using \(\psi=e^{ik.x}\) implies \(E=\pm\sqrt{p^2+m^2}\)), we can not ignore these negative energy states since once system is in contact with heat bath it is going to do transition in all possible states.
      2. For free particles, what does it mean to have negative energy?
      3. \(\psi\) have interpretation of probability density, it is not constant and is time dependent due to KG equation being second order in time. Dirac came up with first order equation in space and time (only for spinors) but still had the problem of negative energy solutions, which he tried explaining by stating that all negative energy states are pre occupied and there is no more state left for other spinors(spin 1/2) to go back to negative energy states. This solution offered explanation of positrons (if negative energy states jumps to positive energy states (electron), then the absence of such state is an positron). Dirac solution could only explain spin 1/2, but what about other kind of particles?
      4. Here x(space),t(time) are still not on equal footing, since 'x' is an operator (\(\hat{x}\)) whereas time 't' is just a label.
      5. It only describes spinless particles what about spin 1/2, spin 1?
   2. Schrodinger equation describes fixed number of particles, it does not explain particle creation and destruction. At very small scale (\(\Delta x \le \frac{1}{4m}\)) Heisenberg uncertainty principle implies that new particle anti-particle can be created. So we can not use QM at such small scales. You can not explain the process of creation of so many photons from the sun, tube light and all light sources without use of QFT.
   3. It also does not explain how identical particles are "identical" in all respect with no error in their formation.

There are two ways of field quantisation:

1. Canonical Quantisation
2. Path Integral Quantisation

Regardless of Approach, some steps are common and crucial.

1. Guess the Lorentz covariant equation of motion for the field. i.e. EOM shall transform in same way as the corresponding field. Examples: For scalars the field don't change and so does look for a differential equation with mass term which is Lorentz invariant. For spinors and vectors the field transform and thus look for differential equation with same transformation properties as the fundamental field itself.
2. Write the Lagrangian density for the above equation of motion for the field.
3. Impose commutation/anti commutation relations on fields and field momenta.
4. Write the fields in terms of creation and annihilation operators, write mode expansion of field.
5. Find two point correlation function (Feynman propagator (time ordered)) of free fields.
6. We know that the n-point correlation function of interacting fields (\(\phi(x)\)) can be written in terms of free fields (\(\phi_{0}(x)\)) and free vacuum(\(\ket{0}\)). Apply that to find scattering matrix elements, which further can be used to find scattering cross-section or decay constant. For real scalar fields, it becomes: \[ \boxed{\frac{\bra{\Omega}T[\phi(x_{1})\phi(x_{2})\phi(x_{3})...\phi(x_{n})]\ket{\Omega}}{\bra{\Omega}\ket{\Omega}} =\frac{\bra{0}T[\phi_{0}(x_{1})\phi_{0}(x_{2})\phi_{0}(x_{3})...\phi(x_{n})e^{-i\int_{-T}^{T}H_{I}(t)dt}]{\ket{0}}}{\bra{0}{T[e^{-i\int_{-T}^{T}H_{I}(t)dt}]\ket{0}}}\\} \]

where, Interaction field Hamiltonian is evolution of interacting part with free Hamiltonian

\[ \begin{align*} H_{I}(t) &= e^{iH_{0}t}H_{int}e^{-iH_{0}t}\\ & = e^{iH_{0}t}(\int d^3x \frac{\lambda}{4}\phi^4(\vec{x},0))e^{-iH_{0}t} \\ & = \int d^3x \frac{\lambda}{4}e^{iH_{0}t}\phi^4(\vec{x},0)e^{-iH_{0}t} \\ & = \int d^3x \frac{\lambda}{4}\phi_{0}^4(\vec{x},t) \\ & = \int d^3x \frac{\lambda}{4}\phi_{0}^4(x) \\ \end{align*} \]