A funny thing about this, though, was that there are four solutions of the equation. Two of them correspond nicely to the two spin states of the electron. The other two solutions, however, extend to a strange prediction that there is an infinite set of quantum states where the electron has negative energy.
The two equations,
E=mc^2 and E^2=m^2*c^4
...are only different if there is a physical meaning to the negative mass and negative energy values, where the second equation allows for both positive and negative mass-energy solutions.
The expression arises from the fact that the magnitude^2 of the momentum four-vector is given by....
In the case of a body at rest, p=0, which leads to...
The concept of negative mass arises by analogy with electric charges, where the formula for the energy of a relativistic particle...
...derives that a particle with a certain positive energy but no momentum could theoretically have a positive or negative mass!
This brings us to his "hole theory", where Dirac rationalized the negative energy solutions by reinterpreting the vacuum state so that all of the negative energy states are filled, and all of the positive energy states are empty.
Dirac's theory was flawed in-spite its success at predicting the existence of the "Positron", because it can't fully account for particles of negative energy, since it is restricted to positive energy particles, but the Quantum Field Theory representaton for this is not an accurate representation of Dirac's negative energy states.
Dirac's hole theory and quantum field theory are generally thought to be equivalent. In fact field theory can be derived from hole theory through the process of second quantization. However, it can be shown that problems worked in both theories yield different results.
And therin lies the problem that has not been resolved by *any* "reinterpretation" of the vacuum state, and so the quantum expectation for the energy density of the vacuum is about 120 plus orders of magnitude^ greater than it should be without the assumption of an "ad hoc" suppression mechanism to cover this "little" discrepency.
The problem goes back to the negative mass absurdity that falls from the Dirac Equation, and QFT's "ad hoc" assumption is what I meant when I previously said that science has assumed the flaw, rather than to fix it, which only carries and compounds the problems when extended to Quantum Gravity theories of all varieties.
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