What's the Energy Density of the Vacuum?

John Baez

June 10, 2011

Once upon a time, someone named Amw wrote:

I have heard widely varying numbers for so called "zero point energy", some as low as practically zero and some as high as astronomical. It gets to the point I am not sure what to think.

Yes, one hears lots of conflicting stuff about this. However, you've come to the right place to get to the bottom of it all.

Here's the deal. We have two fundamental theories of physics: quantum field theory and general relativity. Quantum field theory takes quantum mechanics and special relativity into account, and it's a great theory of all the forces and particles except gravity, but it ignores gravity. General relativity is a great theory of gravity, but it ignores quantum mechanics. Nobody knows how to reconcile these theories yet. That's what people working on "quantum gravity" are trying to do.

Now, the reason I'm telling you this is that quantum field theory and general relativity have really different attitudes towards the energy density of the vacuum. The reason is that quantum field theory only cares about energy differences. If you can only measure energy differences, you can't determine the energy density of the vacuum - it's just a matter of convention. As far as we know, you can only determine the energy density of the vacuum by experiments that involve general relativity - namely, by measuring the curvature of spacetime.

So, when you ask about the energy density of the vacuum, you get different answers depending on whether the person answering you is basing their answer on general relativity or quantum field theory. Let me run through the 5 most common answers, explaining how people reach these different answers:

We can measure the energy density of the vacuum through astronomical observations that determine the curvature of spacetime. All the measurements that have been done agree that the energy density is VERY CLOSE TO ZERO. In terms of mass density, its absolute value is less than 10-26 kilograms per cubic meter. In terms of energy density, this is about 10-9 joules per cubic meter. One can know something is very close to zero without knowing whether it is positive, negative or zero. For a long time that's how it was with the cosmological constant. But, recent measurements by the Wilkinson Microwave Anisotropy Probe and many other experiments seem to be converging on a positive cosmological constant, equal to roughly 7 × 10-27 kilograms per cubic meter. This corresponds to a positive energy density of about 6 × 10-10 joules per cubic meter. The reason they get a positive energy density is very interesting. Thanks to the redshifts of distant galaxies and quasars, we've known for a long time that the universe is expanding. The new data shows something surprising: this expansion is speeding up. Ordinary matter can only make the expansion slow down, since gravity attracts - at least for ordinary matter. What can possibly make the expansion speed up, then? Well, general relativity says that if the vacuum has energy density, it must also have pressure! In fact, it must have a pressure equal to exactly -1 times its energy density, in units where the speed of light and Newton's gravitational constant equal 1. Positive energy density makes the expansion of the universe tend to slow down... but negative pressure makes the expansion tend to speed up. More precisely, the rate at which the expansion of the universe accelerates is proportional to - ρ - 3P where ρ is the energy density and P is the pressure. (This isn't supposed to be obvious: there's a nontrivial calculation involved, and I'm just telling you the final result. The 3 is there because there are 3 dimensions of space, oddly enough.) But as I mentioned, for the vacuum the pressure is minus the energy density: P = -ρ. So, the rate at which the vacuum makes the expansion of the universe accelerate is proportional to 2 ρ From this, it follows that if the vacuum has positive energy density, the expansion of the universe will tend to speed up! This is what people see. And, vacuum energy is currently the most plausible explanation known for what's going on. Of course, to believe this argument at all, one must have some confidence in general relativity. To believe scientists' attempts to determine an actual value for the energy density of spacetime, one must have more confidence in general relativity, and also other assumptions about cosmology. However, the basic fact that the energy density of spacetime is very close to zero is almost unarguable: for it to be false, general relativity would have to be very wrong. We can try to calculate the energy density of the vacuum using quantum field theory. If we calculate the lowest possible energy of a harmonic oscillator, we get a bigger answer when we use quantum mechanics than when we use classical mechanics. The difference is called the "zero-point energy". The zero-point energy of a harmonic oscillator is 1/2 Planck's constant times its frequency. Naively we can try calculating the energy density of the vacuum by simply summing up the zero-point energies of all the vibrational modes of the quantum fields we are considering (e.g. the electromagnetic field and various other fields for other forces and particles). Vibrational modes with shorter wavelengths have higher frequencies and contribute more vacuum energy density. If we assume spacetime is a continuum, we have modes with arbitrarily short wavelengths, so we get INFINITY as the vacuum energy density. But there are problems with this calculation.... A slightly less naive way to calculate the vacuum energy in quantum field theory is to admit that we don't know spacetime is a continuum, and only sum the zero-point energies for vibrational modes having wavelengths bigger than, say, the Planck length (about 10-35 meters). This gives an ENORMOUS BUT FINITE vacuum energy density. Using E = mc2 to convert between energy and mass, it corresponds to a mass density of about 1096 kilograms per cubic meter! But there are problems with this calculation, too.... One problem is that treating the vibrational modes of our fields as harmonic oscillators is only valid for "free field theories" - those in which there are no interactions between modes. This is not physically realistic. However, while taking interactions into account changes the precise answer, we are still left with an enormous energy density. The ridiculous ratio between this density and what's actually observed is often called the cosmological constant problem. One way to put it is that in units of Planck mass per Planck length cubed, the cosmological constant is about 10-123. It's hard to make up a theory that explains such a tiny nonzero number. But there's an even bigger problem, too.... Quantum field theory as it is ordinarily done ignores gravity. But as long as one is ignoring gravity, one can add any constant to ones definition of energy density without changing the predictions for anything you can experimentally measure. The reason is that without measuring the curvature of spacetime, one can only measure energy differences. The big problem with calculations 2 and 3 is that they ignore this fact. If we take advantage of this fact we are free to redefine energy density by subtracting off the zero-point energy, leaving an energy density of ZERO. In fact this is what is ordinarily done in quantum field theory. An even less naive way to think about the vacuum energy density in quantum field theory is the following. In quantum field theory we are neglecting gravity. This means we are free to add any constant whatsoever to our definition of energy density. As long as we are free to do this, we can't really say what the vacuum energy density "really is". In other words, if we only consider quantum field theory and not general relativity, the vacuum energy density is NOT DETERMINED.

So, I've given you 5 answers to the same question:

VERY CLOSE TO ZERO INFINITY ENORMOUS BUT FINITE ZERO NOT DETERMINED

The moral is: for a question like this, you need to know not just the answer but also the assumptions and reasoning that went into the answer. Otherwise you can't make sense of why different people give different answers.

References

For a calculation that explains why the vacuum having positive energy density means it has enough negative pressure to make the expansion of the universe accelerate, see the cosmological constant section of my website about the meaning of Einstein's equation. You may need to read a bunch of stuff in this website to understand the calculation - but it's fun stuff!

Framk B. Tatom helped me update this page. Here is how we got the numbers. Using the Λ-CDM model, the Wilkinson Microwave Anisotropy Probe estimates that Ω &Lambda = 0.726 ± 0.015. This means that the energy density of the vacuum is about 0.726 times the critical density. The critical density, in turn, is defined to be

ρ c = 3H2/8πG

where H is the Hubble constant and G is the gravitational constant. The WMAP data estimate the Hubble constant at 70.5 ± 1.3 kilometers per second per megaparsec, and the gravitational constant is known much more accurately, at 6.67384 ± 0.0008 × 10-11 meters3 per kilogram second2. This puts the critical density between 9.0 × 10-27 and 9.7 × 10-27 kilograms per cubic meter, and the energy density of the vacuum between 6.4 × 10-27 about 7.2 × 10-27 kilograms per cubic meter. Please check our math, and our data!

For more, see:

Table 7 of G. Hinshaw, et al. (WMAP Collaboration), Five-year Wilkinson Microwave Anisotropy Probe observations: data processing, sky maps, and basic results, The Astrophysical Journal Supplement 180 (Feb. 2009), 225-245. Also available as arXiv:0803.0732.

© 2010 John Baez

baez@math.removethis.ucr.andthis.edu