# Should the energy density of the vacuum be zero?

• I
• jcap
In summary: Dark Energy and the Accelerating Universe article, 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
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##.

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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?

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jcap said:
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

PeterDonis said:
Because there are no "negative energy modes".
What about the negative energy solutions to the Dirac equation?

Khashishi said:
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.

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

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.

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

## 1. What is the energy density of the vacuum?

The energy density of the vacuum is the amount of energy that exists in a given volume of empty space. It is a concept in quantum field theory that suggests that even in a vacuum, there is still some energy present.

## 2. Why is the energy density of the vacuum important?

The energy density of the vacuum is important because it has implications for our understanding of the universe and its evolution. It is also related to the concept of dark energy, which is believed to be responsible for the accelerating expansion of the universe.

## 3. Should the energy density of the vacuum be zero?

This is a matter of ongoing debate and research in the scientific community. Some theories suggest that the energy density of the vacuum should be zero, while others propose that it has a non-zero value. The answer to this question is not yet clear.

## 4. What evidence is there for a non-zero energy density of the vacuum?

One of the main pieces of evidence for a non-zero energy density of the vacuum is the observed accelerating expansion of the universe. This is also supported by measurements of the cosmic microwave background radiation and the distribution of galaxies in the universe.

## 5. Could the energy density of the vacuum change over time?

There are theories that suggest the energy density of the vacuum may have changed over time, and it may continue to change in the future. However, this is still a topic of ongoing research and there is currently no definitive answer to this question.

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