Why does vacuum have energy?

The following is a very naive idea from freshman physics: you need to define a reference when you talk about energy. Whatever energy you say your vacuum is present at can be defined as the reference! However, sometimes you need to do more than simply justify this. Also, from a freshman perspective, one is commonly under the belief that vacuum means "nothingness." In some advanced theories, however, the notion of vacuum is in fact very non-trivial.

The term "vacuum" is used in a wide variety of contexts. A vacuum typically refers to a ground state of the system. For example, in the context of a semiconductor, the valence band being completely filled and the conduction band being completely empty is considered a vacuum. When an electron from the valence band absorbs sufficient energy to overcome the band gap then it can jump into the conduction band. This electron-hole pair creation is analogous to particle-antiparticle creation from vacuum. One of the biggest triumphs of physics has been the most beautiful equation by Paul Dirac, which is named after him, and it is the relativistic version of the Schrodinger equation. There were many consequences of this. He presented a new interpretation of vacuum to solve the so-called problem of negative energy states in relativistic quantum mechanics (you can look that up) and also hypothesized the existence of antimatter (which was later experimentally confirmed by Carl Anderson). This new interpretation of vacuum was analogous to the semiconductor analogy I gave above, where the conduction and valence bands are the positive and negative roots of:

##E^2 = p^2 c^2 + m^2 c^4##

and the band gap is ##2 m c^2##. As far as your harmonic oscillator problem goes, I suggest you look into "Zero-point energy." Just google/wikipedia it. There are fine references where this is explained really well and there's no point me repeating it here. I will, however, make a few comments:

What you're probably calculating is this zero-point energy. Also, one is normally interested in energy density. So it makes sense that your total energy is scaling as volume. Just an aside: if you're interested and have the time, then you should also probably take a look at the Casimir effect

 

The vacuum, is basically made of virtual particles... everywhere! There is no such thing as an empty spacetime. This has been called many names, zero point energy being one, which has similarities to background temperatures.

Anyway, yes, if you integrate a small box space you will find a quantity of vacuum energy. Usually, the smallest vibrational energy in theory which can exist is on the scale of [tex]\frac{1}{2}\hbar \omega[/tex]. This is close to the zero point scale, the scale in which all particles would cease to have motion - but they never reach this state. It is called a limit, but usually limits can be obtained and this limit can never be reached. So this is why the vacuum has and always has energy.

 

In quantum mechanics and quantum field theory you can apply normal ordering to H; this normal ordered : has zero expectation value <0|:|0>; subtracting an infinite amount of vacuum energy may seem to be strange but is certainly reasonable; we do not observe any non-zero vacuum energy.

Taking gravity and cosmology into account things become rather obscure (cosmological constant, dark energy, ...)

 

Strange? I would say that an infinite vacuum energy is in principle, unphysical. In fact, no infinities even exist in nature, none we have observed anyhow.

 

The strange thing is that this vacuum energy could in principle contribute to the cosmological constant (for which we have at least indirect experimental hints), but that all attempts towards a reasonable calculation fail.

And I am not so sure if an integrated energy (in contrast to a purely local energy density) is the right thing we should look for. In general relativity it is known that in general spacetimes this integral cannot be defined, whereas the local energy density is well-defined, even if it has a constant, non-zero term. So perhaps we need not worry about the fact that the integral in QFT + SR is infinite; in GR we are not allowed to do this integration, so its value is irrelevant