FEYNMAN'S PATH INTEGRAL FORMULATION AND THE PRINCIPLE OF LEAST ACTION

Classical Mechanics and Least Action- 

Let's do a thought experiment, Shall we? I assume the silence to be a YES.
Okay, Let's suppose that you are somewhere in Tokyo and you want to go to someplace in the city. You open maps on your cell phone and it shows you a lot of paths. Which path are you going to choose? assuming that there is no bias involvement.

It is very obvious that the path which requires the least action to get to the destination will be the choice. This is somewhat the principle of least action and in some sense our universe is lazy.

According to the principle of least action, an object will always follow a path that minimizes the quantity called Action which is the integral of energy difference (kinetic and potential energy) of the Initial and final states of that object over the travel time.

Languranian Mechanics

The vector q here is a point in the configuration space of the system.

This is as much as fundamental the universe gets. Now let's see how quantum mechanics modify this principle. This is done by Dr. Richard Feynman and known as Feynman's Path Integral.

Feynman's Idea-

The path Integral tells us that a Quantum particle takes all the trajectory from one state to another state. The contribution from every trajectory is different depending on the complexity of that particular trajectory.

So, according to this, If you are a quantum particle and you want to go from state A to state B, there could be infinitely many ways that you can do it. But since you are a Quantum particle you will have all those paths and the resultant state will be the sum of all those trajectories in proportion to their complexity.

Feynman derived various doodles to demonstrate the path of a single event, which are famously known as Feynman's Diagrams. The most significant path would be the one that has minimum vertices (virtual particle interactions). Based on this, we can neglect the infinite many complex trajectories which have very low probability.

Below are some of Feynman's Diagrams to get a feel of it.

Here is a python code that I wrote for the path integral simulation. It illustrates how can a process have a variable path for reaching the final state from an initial one. But this path has random behavior if we lower the temperature, that is if we enter the Quantum Realm. 

Beta is a variable that if a function of Inverse Temperature. 

Thus, for low beta -> High temperature, the path looks like this

Whereas, for high beta -> low temperature, the path looks like this.

Now we are more closer to reality !!!