I can say a few things on this issue on the basis of very basic non-relativistic quantum mechanics. When we solve the time dependent Schrodinger equation, after separation of variable, the solution is of the form,

ψ(x,t)=χ(t) φ(x)= e i E t / ħ φ(x)

For a negative energy particle E → -E. However, as we have a product of energy and time in the exponential, we can redefine things in such a way that E is positive and the negative sign sits in front of time. That is,

ψ(x,t)=χ(t) φ(x)= e i E (-t) / ħ φ(x).

Then interpret the solution as that of a positive energy particle which is going backwards in time.

Thank you Sir, does that shows the logical possibility of possible existence of particles reducing the total energy of the universe? But that goes against the fact that Energy can't be destroyed. What you say Sir?

Ok, to answer I have to use relativistic quantum mechanics, specially the Dirac equation, which admits negative energy solutions, those are mentioned in your question. As negative energy states are available, a normal positive energy electron could radiate energy and occupy a negative energy state. Dirac thought that this situation is unphysical. He proposed that all negative energy states are occupied by a "sea of electrons" and due to Pauli's exclusion principle a positive energy free electron cannot radiate and fall into the negative energy sea.

Some times a few negative energy electrons can escape from "negative energy sea" by gaining suitable amount energy. In this case it leaves a hole in the sea, which can be interpreted as a positron (not proton).

That means negative energy is co-existing with positive energy, isolated from each other. Sir, do we have any measure of total energy of the universe?