Richard P. Dolan
dick.dolan@stanfordalumni.org
The inflaton spacetime model is described in detail in Dick's paper, "A Discrete Quantum Spacetime Model Underlying the Standard Models of Particle Physics and Cosmology." It is a discrete spacetime model, meaning that spacetime is composed of discrete points. Nothing else exists. Both time and space are discrete. An elementary particle is an excitation of a point, that is, the difference between the energy of an excited point and its ground state energy.
1. If spacetime is made of discrete points, 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 a particular model. In this model, 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 this 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.
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.
Now what happens if we turn these points loose? The bosons will tend to drift towards the same state, which in this case means the same 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, independent of the observer's state of motion. 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 usual continuous spacetime model, they have effective momentum, but this momentum has no effect on the behavior described here.
2. What is the inflaton (the field responsible for cosmic inflation? In the inflaton spacetime model, spacetime is self-generating. Spacetime itself is the inflaton. Points are defined recursively, with the result that the number of points increases from N to 2N at each discrete time step. This is an extremely rapid expansion. It never stops, so spacetime is always inflating. Initially, there is no matter and there are only a few points having random position quantum numbers. As time goes on, the proliferation of points, together with their gravitational attraction (see question 1), causes the distance between points to decrease. Fermionic points must avoid each other, so they have some mean free path, which is the average distance a fermionic point can travel in any direction without encountering another fermionic point. The inverse of this mean free path is a scalar field that plays the role of the inflaton in triggering a phase transition in which all matter is created. The potential of the inflaton field has a minimum where the force of gravity acting on the fermionic points equals the degeneracy pressure that keeps them apart. When the inflaton reaches this minimum, the result is coherent oscillations of the mean free path parameter throughout spacetime. These oscillations decay into matter particles and radiation by raising some points above their ground state energy. Points where there are matter particles are decoupled from spacetime as a whole. The very rapid inflation of the universe stops at the phase transition, but spacetime as a whole continues to inflate at a much slower, but still accelerating rate. For the subset of spacetime that contains matter, this acceleration has little effect at first, but becomes more significant as time goes on. See question 4b.
Most inflationary models predict that galactic distance scales were smaller
than the Planck length before inflation. This is a problem, since spacetime is believed to
be structureless at sub-Planckian scales. In the inflaton spacetime model,
all scales start out larger than the Planck length and stay that way. The
minimum distance, that is, the Planck length, is established only at the end
of inflation. Two other problems of conventional inflation models are: 1) they
have difficulty producing primordial density perturbations that are as small in
amplitude as those observed in the cosmic microwave background radiation (CMBR),
which are at the 10-5 level, and 2) they predict more large-angle
power in the CMBR than is observed. The inflaton spacetime model may be free of
these problems. The end of inflation in the inflaton spacetime model is more of
a crash than a smooth transition to the minimum of the inflaton potential. This
may produce density perturbations that are both smaller in amplitude and shorter
in wavelength.
3. Will the universe expand forever or eventually collapse? The recently observed acceleration of the expansion of the universe seems to say that the universe will expand forever. In the inflaton spacetime model, the structure of spacetime is determined by a lattice of fermionic points pushed together by gravity and held apart by the degeneracy pressure that comes from their fermionic nature, that is, they obey the Pauli exclusion principle. In astrophysics, it is well established that dying stars can have a similar balance of forces. If large enough, such stars will collapse to form black holes. If too small, they remain dead stars--neutron stars, for example. The spacetime of the inflaton spacetime model is expanding, so it is like a degenerate star that is getting bigger. Eventually, its gravity should overcome its degeneracy pressure, and the universe should collapse to a black hole. The final state is different from the initial state, so there would probably not be a bounce, leading to a new big bang.
4. The Cosmological Constant Problem (Dark Energy Problem) and the Cosmic Coincidence
Problem.
Measurements of the cosmic microwave background radiation confirm that the
universe is flat. There is not enough matter in it to explain this, so it is
thought that there must be some "dark energy" that acts like
Einstein's cosmological constant and has just the right value to make the
universe flat. Supernova measurements show that the universal expansion is
accelerating, and the dark energy is thought to be responsible for this effect
as well. Physicists have no idea what this dark energy might be, how it gets so
finely tuned as to make the universe perfectly flat, and why it happens to be of
the same order of magnitude as the mass density of the universe. The best
candidate, vacuum energy density, doesn't work, because the cosmological
constant is 120 orders of magnitude smaller than the vacuum energy density
calculated using the Standard Model.
a. Why is the vacuum energy density not much larger? The quantum spacetime of
the inflaton spacetime model consists of a quantum lattice of fermionic points embedded in a
sea of bosonic points. The virtual particle-antiparticle pairs that are known to
populate spacetime are simply unexcited point-antipoint pairs, and contrary to
conventional thinking, do not contribute any observable vacuum energy. The
observable vacuum energy comes from the zero point energy or quantum fluctuations of the vacuum,
which are quantum fluctuations in the positions of the
points from one discrete time tick to another. These positions are independent random variables. While the
energy density for a single stationary point is equal to the fourth power of the Planck
energy (1019 GeV), which is enormous, the energy density over any appreciable volume of
space, such as the universe, is near zero because it is the average of a very large number of independent
random variables, which tends towards zero. It is the discreteness of space that makes this happen in the
inflaton spacetime model. The vacuum energy density is so low, in fact, that it plays no role
at all in the cosmological constant, which is something else entirely.
b. What is the dark energy responsible for the cosmological constant? In the inflaton
spacetime model, the number of spacetime points expands at an enormous rate. An
initial inflationary period ends with a phase transition in which all matter is
created. Elementary fermions (leptons and quarks) are excited fermionic
spacetime points, that is, points that remain above their vacuum or ground state
energy after the phase transition. After the phase transition and the mutual
annihilation of particles and antiparticles, relatively few points are left
with particles. Spacetime as a whole continues to expand, although much more
slowly than during the inflationary period. At first the rate of expansion
decelerates because of the gravitational attraction of the matter, but
eventually it begins to accelerate as the matter density decreases and the
accelerating expansion of the point creation process dominates. In this state
of accelerating expansion the universe is continually driven to flatness. Thus,
spacetime is always essentially flat regardless of the amount of matter in it.
So, the cosmological constant is partly an exotic form of energy—the accelerating
expansion of the number of fermionic points—and partly geometry: spacetime as a
whole is inherently essentially flat.
c. Why is the energy density ratio of the cosmological constant,
omega_lambda, of the same order of magnitude as the mass density ratio omega_m?
In the inflaton spacetime model, omega_lambda and omega_m always add up to one, since spacetime
is inherently very nearly flat. The mass density starts out high and decreases as the
universe expands, so it is inevitable that at some time the mass density and the
cosmological constant will be comparable in magnitude. However, the inflaton
spacetime model does not explain why this is happening now. It may be a true
coincidence.
5. Why is gravity so weak? Gravity between spacetime points is actually quite strong, but points where elementary fermions are located are gravitationally decoupled from the overall spacetime by the ratio of the particle's mass to the Planck mass--22 orders of magnitude in the case of the electron. This makes gravity a very weak force for matter.
6. What is mass? For an elementary fermion (lepton or quark), mass is the inverse of the precision with which the location of a stationary particle can be known. This makes sense, because mass is defined as a measure of inertia or resistance to acceleration. Resistance to movement and having a known or fixed location are really the same thing. For a composite particle such as a baryon, mass is mostly binding energy (gluons), the masses of the elementary constituents (quarks) contributing very little to the baryon mass. For a massive gauge boson, mass is the inverse of the range of the force carried by the particle.
7. What is the Higgs field and how does it give mass to leptons, quarks, and massive gauge bosons? In the discrete spacetime of the inflaton spacetime model, fermionic spacetime points are allowed to move between time ticks. The point velocity has two eigenvalues. From one time tick to the next, the average position of a point (neglecting the ever-present random quantum fluctuations) can either remain stationary (velocity = 0) or move one space increment (velocity = c, the velocity of light). The Higgs field is the velocity field. It has a value equal to the Planck energy for a stationary point and a value of zero for a moving point. The vacuum expectation value of the Higgs field, 246 GeV, is a measure of the average ratio of moving points to stationary points, which turns out to be about 1017. This ratio limits the precision with which a stationary point can be located. If all points were stationary, the precision would be high; the electron would have the Planck mass, and nothing could move. If all points moved at the velocity of light, the only possible particles would be massless neutrinos and nothing could stand still. The vacuum expectation value of the Higgs field is closer to zero than to the Planck energy, so the electron has a very small mass and things can move relatively freely in spacetime..
8. The Hierarchy Problem. Why is the electroweak symmetry breaking energy scale so much lower than the Planck scale or the GUT scale? The electroweak symmetry breaking scale is determined by the vacuum expectation value of the Higgs field which, at 246 GeV, is much closer to zero than to the Planck energy. If this value were equal to the Planck energy, all points would be stationary, electrons would have the Planck mass, and nothing could move. This would be a very high-energy state. The Higgs field tends towards the lowest possible energy, a vacuum expectation value of zero, where all points move with the speed of light. Here, only neutrinos would be possible and nothing could stand still. However, this state does not appear to be permitted. Apparently there is an exclusion principle that prevents the fermionic points from all having the velocity of light. In effect, there is a delta function in the Higgs field's potential energy at a vacuum expectation value of zero. Thus, the field reaches the minimum of its potential at a vacuum expectation value close to but greater than zero.
9. Why are there no righthanded neutrinos? The neutrino has an average mass of zero and an average velocity equal to the speed of light. However, it oscillates between two phases: a lefthanded, slower-than-light phase and a righthanded, faster-than-light phase. In its lefthanded phase, it is massive and feels the weak interaction, so it can be detected and can undergo oscillations. In its tachyonic phase, it does not feel the weak interaction because it cannot be changed into a righthanded electron by an SU(2) rotation. Therefore it cannot be detected. Notice that the neutrino instantaneously violates Lorentz invariance, although on the average, it obeys this symmetry. By the way, the neutrino is a Dirac particle, not a Majorana particle.
10. What are quarks? What are leptons? Spacetime points do not have to be pure states. A point can be a mixture of n point states. However, to preserve symmetry, a point can only be a mixture of n points if the n points share n positions such that the mixing coefficients add up to an integer. The simplest mixing coefficients that work for all n are plus or minus 1, 2/3, and 1/3. The vacuum is a superposition or mixture of many vacua, each having a different value of n. All of these vacua look the same at any time tick. A resonance of a point in the n = 2 vacuum is a meson. A resonance of a point in the n = 3 vacuum is a baryon. The point-position partial resonances that make up these points are quarks. There are no free quarks because there cannot be partial points, only complete points. Quantum mechanically, points are indivisible. In the n = 1 vacuum, a resonance of a point is an electron or positron if the point is massive and an electron neutrino if the point moves at the speed of light. Higher values of n are possible but are unlikely to be observed because they are very heavy and unstable.
A symmetry that is a dual of the one that leads to quarks (which is called color symmetry) permits n points to form a mixed point at a single position if they form a group as above and their mixing coefficients at that point add up to an integer. A resonance of such a point for n = 2 is a muon. A resonance of such a point for n = 3 is a tau particle. The point-position partial resonances for muons and tau particles are something like quarks, but do not have a name because they are unknown in conventional physics. They are a prediction of the inflaton spacetime model.
11. Why are there three generations of leptons and quarks? Are they structureless? There are only three ways in which n points can share one position. and satisfy the dual symmetry mentioned in question 10. One leads to the electron and its neutrino, one leads to the muon and its neutrino, and one leads to the tau particle and its neutrino. Just as there can be mixed, n-point, n-position electron-type resonances for n = 2, n = 3, etc., there can also be similar muon-type and tau-type resonances. The point-position partial resonances in these point states are second- and third-generation quarks. First-generation quarks and leptons are structureless. Second- and third-generation leptons are composed of quark-like objects, and second- and third-generation quarks are also composite particles. The mass differences between one generation and the next are mainly binding energy, that is, gluon-like objects.
12. Can quarks and leptons change into each other? Can the proton decay? Quarks and leptons can change into each other within baryons, and the proton can decay, but with near-zero probability (for the proton to decay, the two up quarks must exchange a gauge boson with a mass of approximately the Planck mass, 1019 GeV)..
13. What is the correct model for neutrino oscillations? The correct model for vacuum oscillations is the most widely accepted one. The flavor eigenstates of the three known neutrinos are different from their mass eigenstates and the flavor and mass eigenstates are mixtures of each other. This results in oscillations from one flavor to another in flight.
14. The Dark Matter Problem. What is the dark matter in galactic halos? According to Newton's laws, the velocity of stars and gas clouds orbiting a galaxy outside the luminous core should fall off with distance. In fact, it is observed to remain constant out to very large distances. This is explained by a halo of nonluminous matter surrounding the galaxy, so that mass increases with distance. However, no one knows what the dark matter is. The inflaton spacetime model identifies a particle that could be the dark matter. It is a Standard Model particle, but one that is thought not to exist because it would have exotic characteristics that are impossible in continuous spacetime. However, in the discrete spacetime of the inflaton spacetime model, such a particle is possible. The particle is the zero-helicity state of the photon. The photon is a spin-1 boson and should have three helicity states: -1, 0, and +1. The zero-helicity state is thought not to exist, but in the inflaton spacetime model, it can exist as a photon-antiphoton bound state. It is massive, has zero charge, does not feel the weak interaction, and is its own antiparticle. It interacts only gravitationally, and is very cold and sluggish. Its characteristics are very close to those of the leading dark matter model--the cold, collisionless dark matter model.
15. Why do the parameters of the universe seem to be so finely tuned to support life? The answer to this is found in my metaphysics paper. Our universe is the result of an observation of a quantum ensemble of possible universes. Through a natural selection process similar to the one conceived by Lee Smolin in his book, The Life of the Cosmos (Oxford University Press, 1997), the population of possible universes comes to be overwhelmingly dominated by universes like ours, that is, universes that support life. Thus, it is highly likely that the observed universe will be one that supports life.