Relativistic Field Theory

(Probably this note has to be unified with Classical Field Theory)

Overview:
We generalize Lagrangian Mechanics formalism of classical particles in two ways:

To include relativistic particles. The action is changed, chosen to be Lorentz invariant (although the previous were not Galilean invariant!). When we take Taylor polynomial to approximate for slow particles we obtain the classical Lagrangian... This is relativistic Lagrangian mechanics.
To include fields, both classical and, mainly, relativistic ones. A field is a measurable quantity that depends on position in space and may also vary with time: $ϕ (t, x_{1}, x_{2}, x_{3})$ . The field can be a scalar field (temperature) or a vector field (wind).

A particle can be seen as a field applied in one point (0-dimensional space). The measurable quantity of the field (scalar or vectorial) is the position of the particle. So a general field is nothing but infinite particles evolving together. It is totally analogous to passing from one harmonic oscillator to a continuous oscillating string.
Consider one particle in 1 dimension in Classical Mechanics, whose position in time is described by $ϕ (t)$ . This would be a (0+1)-dimensional scalar field. Following Lagrangian Mechanics, this scalar field is determined by the principle of least action, that is, we have

Action = \int_{a}^{b} L d t

Remember that the simplest situation for a particle is a constant velocity motion (no potential energy is present), and the Lagrangian is

L = \frac{1}{2} (\dot{ϕ} (t))^{2}

and the Euler-Lagrange equation for this Lagrangian is

\frac{d^{2} ϕ}{d t^{2}} (t) = 0

The analogous for a field is the wave propagation! This is because the Lagrangian that naturally extends that of a 0-dimensional field (particle) to that of a spatial field in relativistic context include terms that lead to a wave equation.
(see @susskind2017special page 122 to understand this)

That is, in Euler Lagrange equations, compare

\frac{d^{2} ϕ}{d t^{2}} + \frac{\partial V (ϕ)}{\partial ϕ} = 0

with

\frac{\partial^{2} ϕ}{\partial t^{2}} - \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{\partial^{2} ϕ}{\partial y^{2}} - \frac{\partial^{2} ϕ}{\partial z^{2}} + \frac{\partial V}{\partial ϕ} = 0

(Klein--Gordon equation).

The Lagrangian would be something like

L = - \frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ - V (ϕ)

Another examples of fields whose equations are invariants through Lorentz group are:
(from this video)

Equation	Spin	Expression
Klein-Gordon Equation	Spin-0	$(\partial^{μ} \partial_{μ} + m^{2}) ϕ = 0$
Dirac Equation	Spin- $\frac{1}{2}$	$(i γ^{μ} \partial_{μ} - m) ψ = 0$
Proca Equation	Spin-1	$\partial_{μ} (\partial^{μ} A^{ν} + \partial^{ν} A^{μ}) + m^{2} A^{ν} = 0$
Maxwell Equations	Spin-1	$\partial_{μ} (\partial^{μ} A^{ν} + \partial^{ν} A^{μ}) = 0$