Combining SR and QM requires that the number of particles of some type be represented by a quantum field. It is therefore subject to uncertainty relations. We treat the vacuum as if it were filled with a sea of particle-antiparticle pairs blinking in and out of existence, giving a "zero-point energy".

This treatment is not just hypothetical. The vacuum energy in the E-M field depends on confinement by conductors. Therefore a force is exerted on the conductors (the Casimir force). It's measurable, rather precisely.

Remember, a force depends on differences in energy as a function of position. So measuring the Casimir force, etc, do not tell you what the energy density is of empty space, only how it changes when pieces of metal, etc, are moved around in it.

Absolute energies (as opposed to differences) enter into physics as the masses which give rise to gravity. So we have to ask whether we can calculate the energy density in space that serves as a source of gravitational effects simply by adding up the sorts of background energies that give the Casimir effect.

If we do so, we end up with an energy density some 10¹²⁵ times as big as the critical density of cosmology.

So obviously we're missing something in our understanding of how to treat this problem.

At high temperatures, the atoms are randomly. That's because there are many more quantum states with random-looking spins than there are with the spins in any simple pattern. The iron is not magnetized oriented and, on average, the material is isotropic.

The spins have lower energy when they line up. At low T, that energy can make more entropy off in the nearby reservoir than the spins lose by lining up. When the iron is cooled below about 770° (the Curie temperature), the spins do align, maximizing the TOTAL entropy. The iron becomes magnetized.

However, they cannot align without picking some particular direction to align along. Now there is a preferred direction, even though at heart the physical equations contained no special direction.

Rotational symmetry is broken in this example.

This spontaneous symmetry breaking has various effects:

Depending on the details of the symmetry broken, there can be a variety of new types of excitations around the broken-symmetry state. E. g.

There will be a new type of wave (a magnon) in which the spins twist a bit from the preferred orientation, and spring back due to torques from neighboring spins.

There can be boundaries between regions which break the symmetry in different ways.

• On the average, any position in a liquid is just like any other position. When the liquid cools, it can freeze into a solid, in which there is a distinction between the sites with atoms on them and the spaces in between. Spatial translation is the broken symmetry.
• Some sugars (and other big molecules) have left-handed and right-handed forms, which twist polarized light opposite directions. Crystals form which contain only one type or the other. Within the crystal, PARITY (mirror-image) symmetry is broken. (If the interconversion rate between the two forms is big enough, all the molecules can end up in one crystal, breaking parity for the whole batch.)

We saw that physical laws can have various other symmetries, besides these simple spatial symmetries. E.g. time-reversal, Lorentz transformations, …

Any underlying symmetry can be broken by the equilibrium physical state, just as the spatial symmetries were broken in the examples above.

The fact that the h particles interact with each other can give rise to interesting and unusual results. Consider a box containing these particles. How does the energy of the box vary as we increase the density of h particles in the box? Normally, we would expect it to increase:

The graph is not linear, because the particles interact.

Suppose the h particles have an attractive interaction such that above a certain density they begin to condense. Then the energy vs. density graph will look like this:

On the right, the state of lowest energy has nonzero density!

There is nothing terribly exotic about these pictures of energy vs density. Essentially identical pictures occur for ordinary gasses, when you try to calculate whether they form dense liquids or stay as rarified gas. The difference arises in that we treat the gas molecules as coming from some reservoir with a fixed number of particles. Here, in contrast, we're dealing with particles which can pop in or out of existence. Empty space itself provides the reservoir, and can never be emptied of the potential to produce more of the particles.

In equilibrium, the total entropy is maximized, which is equivalent to minimizing the total free-energy. The density (strength) of the Higgs field is zero at very early times, but as the temperature decreases, that is no longer the favored configuration. If the middle energy-density graph is correct, then the universe will become trapped in a false vacuum state, i.e. a state in which the free-energy is bigger than it has to be. Even if the last version is right, the universe will not instantly reach the true vacuum, but will spend a little time in false vacuum states. The false vacuum will cause the expansion of the universe to have unusual features.

(See Scientific American, Jan. 1999 for excellent articles describing the difference between getting stuck for a while in a false vacuum [middle picture] and gradually rolling out of a false vacuum [last picture], and for a cosmological update.)

As a gas expands, its density decreases, and it does work (PdV) so its energy also decreases. This is not the case with the excess energy due to the false vacuum. The energy density remains constant as the universe expands, so the energy in an expanding volume actually increases. That is, the vacuum will act like a gas with negative pressure.

Another argument about the gravitational effect of a fixed energy DENSITY, as opposed to a fixed mass.

Suppose you have a Newtonian gravitational problem, with mass M spread out over dimensions R. The gravitational potential energy will be something like (not worrying about factors of 2,…)

-GM²/R

The forces are the derivatives of energies wrt position, so these are of order

-GM²/R²

Producing inward accelerations of order:

GM/R².

Now what happens if the thing that's fixed is the density

The potential energy is of order

So the force (given by the derivative wrt R) is of order

The second derivative of R wrt time is proportional to R.

R will grow (or shrink) exponentially!

It turns out that this odd Newtonian fixed-density calculation captures the key behavior of the GR equations, which also show exponential growth of space if the density rather than the mass is fixed.

The frozen symmetrical field has, at least for a while, a fixed density.

An question is: what energy density is to be used as the source of the gravitational effects? The zero-point energy would in a simple picture be INFINITE, or (given small-scale cutoffs) some 10¹²⁵ bigger than what's found.

So the energy that counts is only the energy above some reference vacuum state. How does the universe know that there's a lower energy "real" vacuum, in inflation?

I.e. it's not the state of the vacuum that changes initially as an inflationary period starts but the POSSIBLE state that the vacuum hasn't yet found. Somehow that drives inflation!

Inflation solves several cosmological problems. Here are two:

There is no evidence for the proposed interactions, which require the existence of a new, previously unobserved, force. Inflation does make predictions for the amount of structure of various sizes in the universe, especially in the background radiation. These predictions will be tested in the next ten years or so. (so far so good- and getting better by the semester!)

The inflationary scenario requires some fine-tuning of the strength of competing terms in the interactions, in order for the inflationary period to end smoothly

So we have strong reason to think that inflation made the total energy density very close to the critical value. In fact, if the same inflation that made the universe homogeneous also made it flat, the radius of curvature should be bigger than (10⁵)^1/2 as large as the age of the universe times c. That would mean that the universe is extremely close to flat: right at the edge between open and closed.

Does that mean that we are missing 70% of the matter in the universe, when we only find 30% of the critical density in galactic clusters?

No: the expansion of the universe is STILL ACCELERATING according to pretty reliable measurements.

In other words, it looks like we're still in a period of very weak inflation!

Once you allow for background energy density, causing accelerating expansion, the simple connection between energy density and the fate of the universe is broken.

Depending on the ratio of the ordinary mass to the background density, you can have:

Open/ always expands

Closed/Collapses

Closed/always expands

Open/collapses

We seem to be almost exactly on the edge between open and closed, very near flat.

If the field apparently causing the current acceleration of expansion remains constant, the expansion will continue forever. If it were to fall to zero, the expansion would still continue forever. But we really don't know what it is or what it will do.