According to Dark Energy and the Accelerating Universe quantum field theory says that the energy density of the vacuum, $\rho_{vac}$, should be given by $$\rho_{vac}=\frac{1}{2}\sum_{\rm fields}g_i\int_0^{\infty}\sqrt{k^2+m^2}\frac{d^3k}{(2\pi)^3}\approx\sum_{\rm fields}\frac{g_i k^4_{max}}{16\pi^2}$$ where $g_i$ is positive/negative for bosons/fermions and $k_{max}$ is some momentum cutoff.
My question is why do we only take the positive square root terms?
According to the Feynman-Stueckelberg interpretation a positive energy antiparticle going forward in time is equivalent to a negative energy particle going backwards in time. Maybe we cannot rule out negative energy virtual particles moving backwards in time?
Therefore, in order to include antiparticles in the above sum, maybe we should include the negative square root terms? If we do then we find that the energy density $\rho_{vac}=0$.
Addendum ok I accept that I was wrong - antiparticles have positive energy. But my original question still stands. Why do we ignore the negative energy modes in the above vacuum energy density calculation?
