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Many physicists consider quantum gravity to be the Holy Grail of modern physics. It is one of the greatest puzzles of 20th century physics that, despite all the progress in quantum physics in explaining the behavior of the very small and short lived and all the progress in gravitational physics explaining the behavior of the very large and long lived, when the two are combined, the mathematics doesn't work. To put it more simply, we know how to fit the entire universe of matter and energy and forces into quantum physics, all EXCEPT gravity. That has motivated physicists for the last 30 years now to try to reconcile the two.
The progress has been less than stunning. While thousands of papers have been published on different theories, no consensus has been realized. No simple solution has been found. And no experiment has proven any of the many theories correct.
The impasse led me to look at the problem myself and led me to develop this theory. One of the most astonishing things about gravitational theory is how hard it is to do experiments. We can't do them for the most part, which makes gravity much harder to study than, say, electromagnetism or even quarks. We can make strong electromagnetic fields with electricity. We can make quarks in a particle accelerator, but we can't make planets, stars, galaxies, (large) black holes, or big bangs. We have to watch them in the sky with telescopes and satellites. Over the decades, especially in the last 20 years, we have found some incredibly good test subjects in the form of black holes, precise measurements of the cosmic microwave background, double pulsar systems, and strange galaxies such as the bullet cluster. But what all these test subjects share, so far, is that they are either (1) too imprecise to measure with good accuracy or (2) have gravity too weak to go beyond first order. By first order, I mean what is called the first post-Newtonian expansion, which is what you get when you simplify Einstein's equation down to Newton's laws PLUS a little extra term. Second post-Newtonian expansions, which is what you get when you take the first post-Newtonian and derive another little extra term, have never been observed in any precise way. This gave me the idea that maybe Einstein went too far and the way to a quantum theory was to show that some other theory had the same first post-Newtonian expansion as Einstein's theory in addition to some other things like satisfying the so-called Strong Equivalence Principle, and, most importantly, working as a quantum theory.
I started, therefore, with the framework that the other forces had been built on, which is called Yang-Mills theory or non-Abelian gauge theory which model particles as being part of a group (as in group theory). Names aside, it's a theory that explains three forces: (1) electromagnetism, (2) the weak force responsible for radioactive decay, and (3) the strong force responsible for holding the nuclei of atoms together. It seemed reasonable to me that Yang-Mills theory would be the obvious choice for anyone trying to build a gravitational theory from the ground up. Of course, Yang-Mills theory had been tried before. Unfortunately, this particular theory (conformal gravity) was shown not to agree with even Newton's laws, which was a big count against it. In my theory I wanted to avoid that.
Ultimately, what came out over a period of years was this theory which is based on the SO(4,1) or de Sitter group. De Sitter (pictured below) was a colleague of Einstein. The group is named after him because it was the basis of his work on what we now call de Sitter space, which is based on the group. Essentially, the de Sitter group is an extension of the Poincare group of 3+1-D translations and Lorentz transformations to a single "semi-simple" group of 4+1-D transformations similar to the extension of the Galilean group of 3-D translations and 3-D rotations to the Lorentz group of 3+1-D transformations.
The other problem I wanted to tackle was the expanding universe because it is such a big part of gravitational theory. One thing I found that astonished me is that the theory predicts that the universe is accelerating. The acceleration comes from the de Sitter group. To put it in slightly technical terms, what happens is that the gravitational field has spin which produces another gravitational field that has an accelerating effect on the universe, similar to how electron spin is affected by the Lorentz group in Thomas Precession. This also comes out of the theory naturally without any assumptions that aren't already present in a typical Robertson-Walker universe.
This description was a non-mathematical overview of the theory while the theory is described in mathematical terms in the paper.
Technical Abstract
The search for a quantum theory of gravity has become one of the most well-known problems in theoretical physics. Problems quantizing general relativity because it is not renormalizable have led to a search for a new theory of gravity that, while still agreeing with measured observations, is renormalizable. In this paper, a spin-1 Yang-Mills force theory with a SO(4,1) or de Sitter group symmetry is developed. By deriving the standard geodesic equation and the first post-Newtonian approximation equations, it is shown that this theory, coupled to Dirac fields, predicts all N-body and light observations of gravitational phenomena to within experimental accuracy. Furthermore, because of the separation of gauge covariance from coordinate diffeomorphism, the theory satisfies the strong equivalence principle while maintaining a Minkowski coordinate metric. Cosmology is also briefly addressed: Vacuum energy is the most common explanation for the accelerating expansion of the universe but suffers from the drawback that any reasonable prediction of it is 120 orders of magnitude too large. The de Sitter solution to the Einstein Field Equations is an alternative to vacuum energy as an explanation for the accelerating expansion of the universe but only if the universe is approximately a vacuum. The proposed gauge theory, however, avoids both these problems and, cosmologically, the accelerating expansion of the universe is shown as a consequence of the de Sitter group Lie algebra. In addition, with quantized mass, because it is a generic massless, semi-simple Yang-Mills theory, it is mathematically proved to be a perturbatively renormalizable quantum theory of gravity.
Paper
SO(4,1) Yang-Mills Theory of Quantum Gravity, T. D. Andersen, 2013.
Frequently Asked Questions
Q: Doesn't your theory require a background metric?
A: The short answer is yes, like all Yang-Mills theories, mine has a Minkowski metric, but that doesn't get into why the question is important. A better question would be, since your theory requires a background metric, doesn't it admit a field with a diffeomorphic symmetry, and hence, doesn't it require general relativity or some other metric theory to explain that field? This can be broken into two issues, one more theoretical and one more philosophical.
The theoretical issue is whether we can have a background metric and not have it generate a separate theory of gravity based on the symmetry of the metric as one would expect from Noether's theorem. The answer to this question is that you can if the metric field's symmetry is broken. My Yang-Mills theory has a de Sitter group-based potential which, when the gauge is fixed, implies a particular coordinate system for the observer under acceleration or gravitational fields. To see why, imagine a theory with a Poincare gauge. A non-Abelian Poincare group gauge would imply a particular translation and rotation for every point in the universe; thus, it defines a frame for every point in the universe, including a given observer. The Poincare group can be converted to the de Sitter group the same way that the Galilean group is converted to the Lorentz group. Hence, the same argument applies for the de Sitter group. Now, you change the observer by changing the de Sitter gauge. Suppose the gauge is fixed in a frame K. Now, change the coordinate system, not the gauge, to reflect a frame K'. There is now a disagreement between the gauge, which is still in frame K, and the coordinate system, which is in K'. In essence, this means that the diffeomorphic symmetry of general relativity is broken by introducing the de Sitter Yang-Mills theory. A broken symmetry generates no physical source; hence, the background metric is just a convenient mathematical tool but doesn't represent a real, physical thing. This goes back to the Einstein-Kretschmann debate about the physicality of general covariance. My theory is generally covariant but, like any theory of gravity that is not based on tensor potentials but on pseudotensor potentials, it breaks any diffeomorphic symmetry.
The philosophical issue is that it seems inelegant to admit a background metric when general relativity needs no such construct. I have put a great deal of effort trying to resolve this issue. To do this I looked at the discrete version of the Yang-Mills theory in the Wilson formalism of plaquette loop action. I then determined that, rather than requiring a lattice on a background manifold as in the traditional discrete Yang-Mills theory, the de Sitter theory is able to generate its own metric on the lattice through its potential; hence, I was able to revert the discrete theory from a lattice on a manifold to a lattice graph with no background manifold and use the de Sitter potential itself to determine the coordinate system and metric. I was then free to impose an artificial Minkowski metric on the lattice graph as a yardstick against which to compare the potentials' implied metric. In other words, while the potential determined the real coordinates as given by Einstein's famous "rods and clocks" definition, I superimposed an artificial Minkowski metric in order to give a basis of comparison. This method is similar (but not identical) to measuring distance through a path in a network graph, say, highway links between cities, between a node A and a node B by physical distance (defined as the sum of distances on traversed edges) vs. number of links (such that each edge effectively has a distance 1). While the physical distance is measurable, the number of links is not if all you have are rods and clocks (and can't tell when you've reached a new edge).
A simple thought experiment demonstrates this: imagine you are blind and deaf and all you have is a yardstick and a clock (which you can read by touch). You are tasked to follow a road from a starting point where your friend Alice lives to an ending point where your friend Bob lives. Along the way, you are to measure the time and the distance with your clock and stick respectively. As you walk you are only aware of the road, which twists and turns, and yet, unbeknownst to you, you pass through several towns and villages on your way to the city where Bob lives. Because you stick to the road, you are not aware of when you have passed through these cities. You can only read your stick and your clock, which tell you only the distance and duration of your journey. When you arrive at Bob's city, you tell him proudly that the distance is 252 km and that it took 63 hours. (It was a long walk.) Bob then asks you how many towns you passed through on your way. You reply you don't know. You had no way of telling.
The thought experiment demonstrates what happens when we take the continuum limit, because the nodes and edges vanish into the continuum, making us effectively blind to them. All that's left is the physical distance. Likewise, when I took the continuum limit of my Yang-Mills theory, the background metric remained but, as I had shown, no experiment can measure this background; hence, it has no physical meaning. It is just a mathematical tool. Because this is outside the scope of the main paper, I have written it up in a separate paper which I will post soon.
Q: Yang-Mills theories have ghost modes. Don't these modes make your theory unworkable because of negative kinetic terms?
A: This statement is equivalent to stating that Yang-Mills theories are unworkable. Indeed this was the case until De Witt, Fadeev, and Popov solved the problem with ghost fields. It is also possible to eliminate the negative kinetic modes by using the "lightcone" gauge which ensures that all polarizations are transverse-traceless.
Q: Your theory can't possibly mimic general relativity.
A: This isn't a question but is probably the most frequent statement made about the theory. The short answer is that the reason the paper is 40 pages instead of 4 is because it is devoted to answering the question. The answer is that it can, precisely because of the concordance I mentioned above between the first post-Newtonian equations of the two theories and because de Sitter gauge transformations can reproduce most of the coordinate transformation-based tricks of general relativity. Specifically, I show in the paper:
Equation of test particle motion equivalent to the geodesic equation in Sec. 2.4
The Strong Equivalence Principle in Sec. 2.5.
The spherically symmetric solution equivalent to linearized Schwarzchild metric (sufficient for all solar system measurements) in Sec. 3.1
Test particle trajectories within the solar system in Sec. 3.2
1st Post-Newtonian expansion which explains binary pulsars, the strongest N-body observations to date in sec 3.3.
Orbital speed-up and inspiral from gravitational radiation in Sec. 3.3.1
Other objections in this vain seem to have to do with the number of degrees of freedom in the Yang-Mills theory. Most of these degrees of freedom have to do with spin which causes some odd behavior, such as an asymmetric metric. These predictions don't have any experimental or observational data to test against, however, because nothing with sufficient spin has been measured with sufficient accuracy, and, without sufficient angular momentum and spin, the metric is virtually symmetric, and all the extra degrees of freedom are too small to affect the measurements. This same argument is used in string theory to explain why we can't observe all the extra dimensions, but, in the case of my theory, it has a very sound physical basis because these predictions would require a very strong gravitational field and a very high rate of rotation (or magnetic dipole). Thus, like many new theories, it is just an avenue for testing not a legitimate objection. In particular, black holes that have been observed rotating near the speed of light may provide a test for asymmetric metrics.
Q: Your Yang-Mills theory is spin-1. Isn't gravity a spin-2 theory?
A: This is an assumption going back at least to Pauli. There's no evidence that gravity is a spin-2 theory. It only seems to mimic the spin-2 form because of the metric tensor. In my theory, the potential is spin-1 but the "translation" part of the de Sitter group produces a gauge field that has four group indexes, each corresponding to a different coordinate (in addition to one for Lorentz transformations that has six more group indexes). Putting the spin-1 field together with its group indexes creates a 2 index pseudotensor. A big reason why gravity is thought to be spin-2 is because it is easy to show that in spin-1 theories like signed charges (i.e. sources) repel while in spin-2 theories they attract. Gravity is attractive, ergo spin-2. When you derive the source term from the Dirac action with my theory, however, it turns out that the sign on the source flips because of the two indexes on the pseudotensor potential. This is derived in the paper (Sec. 2.1) so you don't have to take my word for it. The upshot is that because the sign flips, like signed charges attract in my theory. In short, you don't need a spin-2 tensor to have an attractive force, just a two index pseudotensor.
Q: Are there any tests of your theory?
A: Yes, my new paper shows that De Sitter Yang-Mills theory predicts a divergence from General Relativity close to the Schwarzschild radius near black holes. In De Sitter theory, escape velocity does not reach the speed of light until zero radius, i.e. right at the singularity, meaning that escape is always possible. This is good news for the fight over event horizons and preservation of quantum information because it essentially says that anything can escape a black hole if it goes fast enough until it actually crashes into the singularity. Whether there is an actual singularity on something larger, e.g., the size of the Compton wavelength, is up for debate as I have only looked at the classical theory. (Note: The Wikipedia links are for more information. I make no claims as to their accuracy.)
The de Sitter group is a natural choice for a theory of gravity and had already been thought of before with the Macdowell-Mansouri theory of gravity. The advantage of the group is that you can derive conservation of momentum, energy, and angular momentum from it. It also provides a nice way to calculate the gravitational effect of particle spin, something that Einstein's original theory didn't have (but got later with the spin connection). Most of my work involved calculating all the predictions that had already been confirmed with Einstein's theory. This involved deriving the first post-Newtonian equations, among other things, to show that all the N-body behavior was correct. In other words, I had to show that the solar system, binary pulsars, etc. all behaved the same way in my theory that they did in Einstein's up to experimental error. I was able to do this in a surprising way because I found that I had to go all the way to the Dirac equation, which governs the behavior of virtually all astronomical matter we can see, to make the derivation work out. This established the 1st post-Newtonian equations on very solid ground. I didn't have to make any assumptions about how matter behaved. The Dirac equation showed it directly through the usual Yang-Mills derivative and the Euler-Lagrange equations.
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