Physics Question #1. The model says spacetime is made of discrete points. OK, then what's between them? What keeps them apart? For that matter, what keeps them from flying away from each other? The first question has been used as a put-down ever since someone first suggested that spacetime might be discrete. The other questions are used in the same way. Actually, the answers to these questions are simple, although not obvious.
Spacetime points are elementary quantum objects and can exist in various states, which are identified by quantum numbers. To say that two spacetime points are discrete, separate, or nondegenerate is the same as saying that there is some quantum number that would be the same if the points were degenerate, and that this quantum number is different for these two points. That's all. There is no requirement that there be other points or other quantum objects with intermediate or intervening quantum numbers. There is no requirement that there be anything between these two points, no matter what we choose to call this quantum number. Actually, we call it position, but this name has meaning only in reference to the conventional spacetime model, in which points are seen as elements of the continuous space of all possible values for the position quantum number. This is an illusion. Points exist independently of this space, which at best has only a potential or virtual existence. It is never observed. Only real points are observed. There are enough real points that spacetime appears continuous and so we find the conventional model meaningful, but it is only a model, and position is only a name for an intrinsic quantum number of a quantum object that we call a point. This is the answer to the first question. Seen in a model-independent way, spacetime can be made of discrete points and it makes no sense to ask what is between them. Between them does not exist.
Time is another intrinsic quantum number of
points. Like position, it is discrete. Similar arguments apply to time and
position.
Do points have other quantum numbers? Yes,
there is at least one other very important one. Take two separate but otherwise
identical quantum objects. If we exchange them, the wave function for this
quantum system will either remain the same or be multiplied by -1, depending on
whether it is symmetric or antisymmetric, respectively. If the wave function is
antisymmetric, we call the objects fermions because they obey Fermi-Dirac
statistics: no two of them can exist in the same quantum state. If the wave
function is symmetric, we call the objects bosons because they obey
Bose-Einstein statistics: two of them are more likely to be found in the same
state than in different states. The quantum number that identifies whether an
object is a fermion or a boson we call spin. Now, spin is like position. It is
a model-dependent name that suggests rotation, but it is really just an
intrinsic quantum number of a quantum object. Are points fermions or bosons?
Nothing puts them in one category or the other, so quantum mechanics says that
there is some probability that on a given observation, any point will be a
fermion, and some probability that it will be a boson. In other words, points
are mixed states. In discrete spacetime, time is discrete and observations
occur only at time ticks. Thus, at any time tick, we see a spacetime composed
of a lot of fermionic points and a lot of bosonic points, and there is a good
deal of randomness about which points are which. We can also look at this
spacetime as consisting of a field of always fermionic points and a field of
always bosonic points. Since the points are identical, spacetime looks the same
whether points change their spin and not their position or their position and
not their spin. However, when we look at spacetime as two separate fields, we
see that the fields are coupled. The coupling results from the mixing: really,
every point is sometimes a boson and sometimes a fermion.
What happens if we turn these points loose? The
bosons will tend to drift towards the same state, which means the same quantum
numbers, including position. Because the two point fields are coupled, the
fermions will be dragged along. This looks like gravity, and it is. The
fermions will go along with this for a while, but they can go only so far,
because they all have to have different positions. Eventually, they are as
close together as they can comfortably be and everything becomes stable. We
have a lattice of fermionic points--a Fermi gas--embedded in a sea of bosonic
points, which end up evenly distributed throughout space because they are
coupled to the fermionic points. The positions of all points fluctuate because
these are quantum objects, but on the average, everything is stable. How close
together are the fermionic points? We can't tell, but we infer from
gravitational considerations that the mean distance between them is the Planck
length, about 10-33 cm. So this is what keeps the points apart and
what holds them together.
The spacetime structure we've just described is
entirely relative to the observer. Every observer sees the same structure with
itself at the center. There is no preferred or absolute structure, no aether.
Notice that momentum plays no role in this
picture. Points have no momentum. They can only be observed at time ticks and
at these instants they are stationary. Seen in terms of the conventional
continuous spacetime model, they do have momentum, but this momentum has no
effect on the behavior described here.
