Have you ever wondered?
Why exactly does light have to travel in a
straight line, when it can go like this, or
this?
Similarly, why does the electron have to follow
this one particular orbit around the nucleus
when it can go like this, or this?
From what you observed, you might think that
things like light and electrons prefer certain
paths over the others.
Now why is there such a preference?
Do they have a mind which tells them to prefer
one path over the other?
The answer to our questions is this equation
right here.
Now, don't worry I'll explain it in simple
terms.
This equation is Feynman's Path Integral Formulation
of Quantum Mechanics.
It's also called the Sum Over Histories method.
Path Integral literally means "sum of paths."
You have this point particle at A. To calculate
its probability of arriving in B, you have
to consider all of the possible paths in space
and time between A and B.
As Murray Gell-Mann said: "Everything not
forbidden is compulsory."
But, each compulsory path doesn't have an
equal probability of getting taken.
In this equation, we have each path multiplied
with this block, which indicates that path's
probability amplitude.
So, if a path has a greater probability amplitude,
its weight is greater.
Then, you sum all of these up!
As you sum them up, the paths with greater
amplitudes have a greater contribution to
the final sum of paths.
I just said "sum of paths."
How exactly do you sum paths up?
Actually, what you add in the paths are their
probability amplitudes.
And since amplitudes are properties of waves,
we should just be adding waves!
Here we have the perfect principle telling
us how to add waves: the Principle of Superposition!
When two waves interfere like these water
waves, they add up to produce a final wave.
Let's illustrate this with my device right
here.
We have two waves.
When crest meets crest, notice that the wave
sum they make is a wave with higher amplitude.
It's like their amplitudes reinforce each
other!
This is constructive interference.
Now when I push this farther such that crest
meets trough, they should produce destructive
interference.
Notice that their amplitudes cancelled out
into a flat line with zero amplitude!
Their amplitudes just mutually destroyed each
other.
Back to the question at the beginning of this
video.
Actually, what the light and the electrons
did is they tried all of the possible and
compulsory paths.
But, the preferred paths we saw, the straight
photon path and the circular electron path,
are just the result of the constructive, mutually
reinforcing, coherent, and synergistic interference
of the paths.
And because of their huge contribution to
the final amplitude, they had the most probability
of being observed by us.
Now, why do I consider the Path Integral as
perhaps the generalization in Physics?
When you think about it, the essential idea
behind the Path Integral can be applied to
all of the interacting systems in the Universe!
An entity, be it an electron, an atom, a person,
an organism, a corporation, or a country,
can do anything it likes.
However, only those constructive, coherent,
harmonious, and synergistic interactions survive
and thrive.
On the other hand, destructive and incoherent
interactions lead to mutual annihilation and
destruction.
So, as time goes by, we see more incoherent
entities disappear and the coherent ones remain.
This is evident in the beautiful and seemingly
orchestrated evolution of things, from atoms,
matter, organisms, and galaxies, to sociopolitical
structures and even human affairs.
Perhaps even to all interacting systems in
the entire Universe!
So, in summary, what the Path Integral teaches
us is that the most probable or the most preferred
path is the path towards harmony - constructive
interaction, coherence, and synergy.
