Gerard 't Hooft states:
"The disappearance of the negative energy modes is very troublesome, however. In a single oscillator, one might still say that energy is conserved, and once it is chosen to be positive, it will stay positive. However, when two or more of these systems interact, they might exchange energy, and we will have to explain why the real world appears to have interactions only in such a way that only positive energy states are occupied. This is a very difficult problem, and, disguised one way or another, it keeps popping up throughout our investigations. It still has not been solved in a completely satisfactory manner, but we can try to handle this difficulty, and one then reaches a number of quite interesting conclusions. In short, our problem is this: in a deterministic theory, one can reproduce quantum-like mathematics in a multitude of ways, but in many cases one encounters hamiltonian functions that are either periodic (in case time is taken to be discrete), or not bounded from below (when time is continuous). Can the real world nevertheless be approximated by, or rather exactly reproduced in, some deterministic model? What then causes the hamiltonian of the real world to be bounded below, with a very special lowest energy state, the ‘vacuum’, as a result? Without this positivity of H , we would not have thermodynamics. The hamiltonian is conjugated to time. Is there something about time that we are not handling correctly?"
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I guess that I should have answerd you here, so I will repost a slightly edited version of the reply that I gave in the quantum cosmology thread:
The "extra" 1/2 in the eigenvalues of the harmonic oscillator Hamiltonian can be thought of as having a phase factor of -1, which represents vacuum energy as being equal to mass energy.
The magnitude of the negative pressure needed for energy conservation is easily found to be P=-u=-rho*c^2 where P is the pressure, u is the vacuum energy density, and rho is the equivalent mass density using E=m*c^2
But in General Relativity, pressure has weight, which means that the gravitational acceleration at the edge of a uniform density sphere is not given by g=GM/R^2=(4*pi/3)*G*rho*R, but is rather given by g=(4*pi/3*G*(rho+3P/c^2)*R .
This describs a quasi-static, expanding model, where g=0. We know that rho > 0, so by setting rho(vacuum) =0.5*rho(matter) we have a total density of 1.5*rho(matter) and a total pressure of -0.5*rho(matter)*c^2, since the pressure from ordinary matter is essentially zero (compared to rho*c^2). Thus rho+3P/c^2=0 and the gravitational acceleration is zero; g=(4*pi/3)*G*(rho(matter)-2*rho(vacuum))*R=0
Matter generation from vacuum energy drives expansion via vacuum rarefaction, and the quantum oscillator evolves 'backwards' from the normal expectation when tension between the vacuum and ordinary matter increase when the increasing matter density is off-set by increasing negative pressure.
This leads to a statement about causality and inflationary theory, because neither, a first cause, nor extra-ordinarily rapid inflation, are necessary when a universe with certain volume has periodic big bangs when tension becomes so great that the forces that bind the finite structure are compromised, aka, "thermalization".
The "initial state" is perpetually inherent, in other words, but the imbalanced system *evolves* closer to absolute symmetry each time that we have a big bang, so the next universe will be infinitesimally more-"flat" than this one is, via the above described, asymmetrical thermodynamic transitions that occur when you make massive particles from negative pressure energy.
Of course, there is more that can be said to justify this against cutting edge assumptions, but I hope that I am in the park with what you wanted me to say.
I am the student, here, and I am wide-open to any help that you can offer.
We will have to explain why the real world appears to have interactions only in such a way that only positive energy states are occupied.
Eduardo, it may be as simple as this;
The negative energy states *are* occupied in Einstein's model, so the Dirac Equation is expected to unify QM and GR in this case, and that might be the most important point that we might make.
Also, it is assumed that the vacuum expectation value of the Hamiltonian is infinite, but it is cut-off by new physics or no physics at the planck scale, and fields do not extend beyond the visible universe when the universe has finite boundaries.
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