Should the energy density of the vacuum be zero?

[jcap]

According to [Dark Energy and the Accelerating Universe](https://ned.ipac.caltech.edu/level5/March08/Frieman/Frieman5.html) 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 anti-particles in the above sum, maybe we should include the negative square root terms? If we do then we find that the energy density $\rho_{vac}=0$.

 

[jcap]

Ok I was wrong. I accept that antiparticles have positive energy. However my initial question still stands. Why are the negative energy modes ignored in the vacuum energy density calculation?

 

[PeterDonis]

Why are the negative energy modes ignored in the vacuum energy density calculation?

Because there are no "negative energy modes". There is a minimum energy ground state of any quantum field; no field state can have energy lower than that. The ground state is represented by $k = 0$, or $E = m$. All other states have energy larger than that.

 

[jcap]

Thanks for your reply

 

[Khashishi]

Because there are no "negative energy modes".

What about the negative energy solutions to the Dirac equation?

 

[PeterDonis]

What about the negative energy solutions to the Dirac equation?

In quantum field theory they become antiparticles with positive energy. The energy is still bounded below.

 

[Demystifier]

One should distinguish energy from frequency. Frequency can be negative, but energy can't.

 

[Khashishi]

Ultimately, this question is related to the gravitational effects of antimatter. Due to the weakness of the gravitational force, we do not currently have direct experimental evidence of the gravitation between antimatter and matter. Antimatter could attract matter, or it could repel matter. So, antimatter could contribute the same as matter or opposite to matter in the (gravitational) vacuum energy.

There's plenty of reasons to believe that antimatter attracts matter. If antimatter repels matter, it suggests that antiphotons are not the same as photons. We should have detected antiphotons by gravitational lensing in astronomical measurements by now, if they exist separately from photons. But without a direct measurement, the jury is still out.

 

[bhobba]

You should also look into something called Normal Ordering:
https://en.wikipedia.org/wiki/Normal_order

The energy of the vacuum is by definition zero and how you do it is normal ordering.

This is entirely different from what you read except in what really matters - QFT textbooks - no wonder people get confused.

Even the ultra reliable John Baez isn't clear in QFT its usually taken as zero. But he is correct pointing out it is to some extent a matter of definition:
http://math.ucr.edu/home/baez/vacuum.html

QFT is hard enough that we don't want to make it harder than necessary - taking it as zero iss by far the easiest way.

Thanks
Bill