Feynman's path integral formulation

See this video for intuition and also this other one. It explain why it coincides with the principle of least action in the limit.

For a particle

See @baez1994gauge, page 137.
Suppose that we have a quantum-mechanical particle that starts out in the state $ψ (x)$ at $t = 0$ , that is,

| Ψ (0) ⟩ = \int ψ (x) | x ⟩ d x .

And we wish to compute its state $ϕ (x)$ at some other time $t = T$ , i.e.,

| Ψ (T) ⟩ = \int ϕ (x) | x ⟩ d x .

Let

P (a \to b) = {γ : [0, T] \to R^{3} : γ (0) = a, γ (T) = b}

denote the space of all paths that start at the point $a$ at time 0 and end at the point $b$ at time $T$ .
The probability of finding the particle at a given point $b$ at time $T$ is the sum of the amplitudes of all the paths that end at that point.

A_{total} = \int_{P (a \to b)} A [γ] D γ,

where $D γ$ is some sort of mysterious "Lebesgue measure" on the space $P (a \to b)$ . The probability amplitude of an individual path is given by the exponential of the action of the path divided by $ℏ$ .

A [γ] = \exp (\frac{i}{ℏ} S [γ])

where $S$ is the action of the path. The action is defined as

S [γ] = \int_{0}^{T} L (γ, \dot{γ}) d t

where $L$ is the Lagrangian of the system. If we discretize the process in "finite jumps", all this should be concluded (or, at least, visualized) from quantum approach to particle motion- an example. I have to develop this...

The quantity $A_{total}$ is usually denoted by $K (a, b, T)$ and called the propagator.

Finally, the evolved state $ϕ (x)$ is given by

\begin{aligned} ϕ (x) & = \int_{R^{3}} K (x^{'}, x, T) ψ (x^{'}) d x^{'} \\ = \int_{R^{3}} (\int_{P (x \to x^{'})} e^{\frac{i}{ℏ} S (γ)} D γ) ψ (x^{'}) d x^{'} . \end{aligned}

For a field

How does this connect to Quantum Field Theory?

The path integral formulation extends naturally to fields (quantum fields). Instead of summing over all possible paths of a particle, we sum over all possible field configurations. The action $S$ now involves integrals over space and time, and the Lagrangian $L$ is replaced by a Lagrangian density $L$ which is a function of the fields and their derivatives.
The amplitude for a field configuration is given by

A = \exp (\frac{i}{ℏ} S [ϕ])

where $ϕ$ represents the fields, and

S [ϕ] = \int d^{4} x L (ϕ, \partial_{μ} ϕ)

where $d^{4} x$ represents integration over spacetime.