Today we are going to see why there is a ground-state energy and why there is more matter than anti-matter. Oddly enough, it has to do with Heisenberg's Uncertainty Principle and the Principle of Least Action. At least that's what the mathematics suggest.
We begin our action with the Lagrangian (L): Kinetic energy minus potential energy (m=mass; v=velocity; k=Hooke's coefficient; x=position). We set L to zero:
If we take the integral of L with respect to time (dt, t), we get the action (S).
We set the constant C to zero. Doing so will help us yield the least action physically possible. How do we know this is true? We recognize the action is energy times time (Et). We want the least action, so what is the least (Et) we can think of? How about the right side of Heisenberg's equation?
Thus the action (S) shall be equal to or greater than h-bar (Planck's constant) over two.
From here we can determine the energy of the ground state (f=frequency):
We get an interesting result. Equation (10 shows that L doesn't equal zero after all. There's something rather than nothing. This odd fact of nature could explain why matter and anti-matter didn't totally annihilate each other in the distant past.
If we express matter and anti-matter in terms of energy (+E, -E), we can show that we don't net zero as expected when we subtract anti-matter from an equal amount of matter. Let's commence with Einstein's energy equation (p=momentum; c=light speed):
We have positively charged energy (matter) and negatively charged energy (anti-matter). We get the net energy (J for Jackson) when we add them together. Does J really equal zero?
To find the answer we previously used the action (S). This time we'll use Sj. Sj stands for ... are you ready for this? ... Action Jackson!
The above math is similar to what we did before, and, like before, we end up with something rather than nothing. The amount of matter left over (in terms of energy) is greater than or equal to the ground state energy. Observations tell us it's "greater than" ... enough to fill our universe.