Before going into this theory, I want to give a summary of my understanding of the four forces of nature found so far, and then at the end I'll discuss why I think that there are likely only four forces of nature (3 of which are time symmetric and 1 of which is time assymmetric), and why is this likely due to the fact that there are only 4 normed division algebras.
There are four known forces: gravity (G), electromagnetism(E&M), weak nuclear (WN) & strong nuclear (SN). There are many similarities between the forces, and some interesting differences between them, when they separate out at low enough temperatures.
Here is a summary before going into the details: (the mathematical terms below can be searched for in Wikipedia. I'll add links to them soon.)
Gravity: Probably no exchange particle, only positive mass (i.e. no negative mass bodies), associated division algebra is addition and multiplication by the real numbers (commutativity, associativity, and the vectors have length.) Superposition always holds. Associated Lie Algebra is SO(1)...the unit point. Parity is conserved and most likely time is reversible for all interactions involving gravity.
E&M: One exchange particle (the photon); positive or negative charge; associated division algebra is addition and multiplication of the complex number. (commutativity, associativity, and the vectors have length.) Superposition always holds. Algebra here is abelian. Associated Lie Algebra is U(1)...the unit circle. Parity is conserved and most likely time is reversible for all interactions involving E&M.
Weak Nuclear: Three exchange particles, W+, W-, & Z. associated division algebra is addition and multiplication of the quaternions (x+iy+jz+kw). (no commutativity, but associativity and the vectors have length.) Superposition does not always hold. The associated Lie Algebra is SU(2)...similar to the surface of a sphere. Non-abelian algebra. Parity is not conserved, and there is no time reflection symmetry. This means that interactions involving the weak nuclear force can be irreversible. x → -x and t → -t are not symmetry operators for this force. Other interesting facts about the weak nuclear force are the fact that the weak force can convert one type of quark into another type of quark (i.e. strangeness is not conserved in weak interactions) and that the weak nuclear force can distinguish between left and right handed particles. (i.e. parity is not conserved as mentioned above)
Strong Nuclear Force: Eight exchange particles, (the eight gluons). The associated division algebra appears to be the octonions (x+iy+jz+kw+...four more directions) (no multiplication commutativity, no multiplication associativity, but the vectors have length.) The associated Lie Algebra is SU(3). SU(3) has 8 operators, which would made sense with the eight gluons. So far, the evidence suggests that C,P,& T are valid symmetry operators of the strong nuclear force. The problem is that we don't really know why this is the case. The SU(3) Lie Algebra is much more complicated than SU(2) algebra, so it's not clear why the weak nuclear force is space-time asymmetric, whereas the strong nuclear force is space-time symmetric. THis is called the Strong CP Paradox.
So, here's the same info, but in more details.
E&M is a linear force in the sense that 'superposition' is always valid for charges that obey this force of nature. There is only one exchange particle for E&M, the photon. The symmetry describing E&M is U(1), which is the symmetry of the unit circle. There is only one variable to describe one's position on the unit circle (angle), and hence, there's only one exchange particle for the E&M force. (The photon has no mass, and I believe that this is partly due to the fact that E&M is a 'linear' force of nature, i.e. Abelian.) The mathematics associated with E&M is that of complex numbers (real number plus imaginary numbers). Commutativity and associativity hold for complex numbers, and therefore, they hold for E&M interactions. The E&M force is really like the wrinkles in a table cloth when you make a circular twist in the middle of the cloth, and hold the ends fixed. This is like changing the phase of an oscillating electron. The change in phase must be communicated to the rest of the world, and the photon is the carried of this information in the twisting of U(1) space-time at the location of the charged particle.
The weak nuclear is the first 'non-linear' force of nature. There are three exchange particles (Z ,W-, & W+), all of which have mass. The weak nuclear force is non-Abelian, which means that the order of operation effects of the outcome of multiplying operators (which could be represented by matrices, which are well known to be non-Abelian.) The WN force has the Lie Algebra symmetry of SU(2), which is similar to the symmetry of the surface of a sphere. A sphere's surface has two angles to describe one's location on the globe and one angle to describe the orientation of the globe about an axis through the center. (The operators associated with moving a given angle do not commute with each other, which can be easily demonstrated by rotating a book 90 degrees about one axis, then 90 degrees about another axis, then -90 degrees about the first axis, and finally -90 about the second axis. Note: you don't end up where you started.) Interestingly, other symmetries that hold for the E&M force, don't hold for the WN force, such as parity and superposition. And this is tied back, once again, to the fact that the WN force is non-Abelian. The mathematics associated with the WN force is quaternions, (real numbers plus the i,j,k axes) in which associativity holds, but where commutativity does not hold. The weak nuclear force explains how quarks change flavor, but not color. (i.e. the force can turn an up quark into a down quark, but it doesn't change the color of the quark.) Interestingly, the weak nuclear force does not obey spatial or temporary reflection symmetry. This means that the weak nuclear force is irreversible, and this may be one reason that time appears to go in one direction. In technical terms, the weak nuclear force does not have P or T symmetry. The weak nuclear force is a weird one because it only acts on left-handed particles or right-handed anti-particles. And another weird aspect of the weak nuclear force is that it can violate the conservation of strangeness or charmness because it can convert one type of quark into another quark.
The strong nuclear force is also 'non-linear' , but even more 'non-linear' in the fact that the mathematics of the SN force is similar to the octonions (real numbers plus i,j,k,l,m,n,o). Neither associativity nor commutativity hold for multiplying octonions together. There are eight exchange particles for the strong nuclear force (which binds together quarks). The force is so strong that it also binds together quark-sets (like protons and neutrons) and we have yet to see clear evidence of a lone quark. As with the WN, the principle of superposition does not hold for the SN force, which is part of the reason that it's difficult to solve problems here. You can't solve part of the problem and then add it to another part of the problem you solved earlier. (Like trying to solve non-linear differential equations.) One interesting question is why there have been no violations of P or T symmetry (Parity or Time reflection) for the strong nuclear force, given that it is even more non-linear than the nuclear force.
So, I haven't discussed gravity yet, even though I should have placed it first. In my understanding, there is likely no exchange particle for gravity, and it follows the symmetry SO(1)...a point... which really means that there is no variable (exchange particle) associated with the mathematics (force). The mathematics of gravity is similar to the mathematics of the real numbers. Associativity and commutativity both hold here, and the mathematics is even easier to learn and apply than the complex numbers. Superposition and parity always hold here as well. Gravity is the curving of space-time by Lorentz transformations and there is no exchange particle associated with this curving of space time. There is no plus or minus charge (i.e. mass), as there are positive and negative electrical charges in E&M. There's only mass, and this mass warps time&space. (Where mass is the sum of the "rest" mass of a particle and its mass due to the energy of the particle.) In some ways, Einstein's theory of general relativity can be seen as local gauge theory just like the local gauge theories discussed above for E&M, WN & SN. If the laws of physics are the same for all observers (even those observers that are accelerating), then there must be a gravitational field. The gravitational field is the curvature of space-time due to mass/energy.
But we still have some unanswered questions left in the realm of physics. There appears to be a missing force (or a missing equation) that would tell us how to predict the masses and charges of quarks (all six types), electrons (muons, taon) and neutrinos (all three types). There appears to be too much coincidence in the size of the families and in the charges of the 'elementary' particles (+/-1, +/-2/3, +/-1/3, and +/- 0) (Electrons, Up quarks, Down quarks, & neutrino, respectively). It appears that there is a particle with a charge of +/-1/3 or perhaps +/-1/6 of the charge of an electron that is even more elementary than the ones listed above. Pairs and/or groups of this elemental particle (and its antiparticle) would then determine the total charge, and whether the collective particles (listed above) feel the E&M, WN or SN forces. All of the particles feel the force of gravity because it is intrinsic to all particles with energy.
I believe that there might be a missing force that describes the bonding of particles that make up an electrons, neutrinos, up quarks, and down quarks (and their related families of particles.) My guess is that the electron is not a fundamental particle, but rather that it consists of smaller particles that 'bond' to form either electrons/muons/tauons. And how the particles are bonded determine whether it's in an electron, muon, or tauon state. The tauon would be an excited electron in a similar way that 1s5 is an excited state of argon. We don't say that 1s5 is a new atom, just an excited state that will decay back to the ground state of argon. It's not clear yet to me (since I don't know how to predict the masses of the tauon/muon/electron) whether there are higher energy state available to the electron.
I believe that the same holds true for the neutrinos and quarks. I think that the neutrinos and quarks might not be fundamental particles, but rather they are made up of particles bound together by another force, which might have the symmetry of SU(4). How the particles inside of the quarks are bound together determines their energy, and hence their rest mass.
So, what can we say about such a force? My guess is that this force has the symmetry of SU(4) or U(4), which means that there would be either 15 or 16 exchange particles with this force. The mathematics associated with this next force is probably the hexagonions (also called sedenions) and has 16 axes. This mathematics is even 'weirder' than octonions because, not only do associativity and commutativity not hold, but the there is no 'normed' vector space, which means that we can't use Euclidean geometry to determine the length of a vector in this 16-D space. In hexagonion algebra, you can multiply two non-zero numbers together and obtain the zero element, which is impossible for real numbers, complex numbers, quaternions or octonions. Length has no real meaning in this sixteen dimensional vector space. And this is a major reason to think that there is no force associated with such a twisting of SU(4) space. (This is a major reason why)
The hexagonion algebra is called a non-ring division algebra, and understanding it is even more difficult than octonions. But just because it's difficult to understand, doesn't mean that we can completely ignore it. We need to understand how to predict the masses of the electron/muon/tauon, the neutrinos, and the quarks. The mathematics associated with the force holding together the particles that make up an electron or a down quark might be described by the hexagonions, and it will therefore be quite difficult to make sense of what's going on. (especially because there's no normed vector space associated with this force.)
And so this line of reasoning begs the question, are there more forces past this SU(4) force? I'm not really sure, and beyond this already unseen force, we'll have to wait to see if we can ever find the points/strings/loops that make up an electron, a neutrino or a quark. Since it's difficult/impossible to find a quark by itself, I'm guessing that we'll find the particles inside an electron first. So, are these particles composed of even smaller particles that obey a force similar to SU(5) or U(5)? We have no hope right now of determining the answer to that question, but we can speculate.
Speculations goes as follows: The number of exchange particles seems to follow a rule of n squared, (or n squared minus one) (gravity, n=0, and no exchange forces; E&M, n=1, and one exchange particle; WN, n=2, and 3 exchange particles; SN, n=3, and 8 exchange particles; S(4) force, n=4, and 15/16 exchange particles, ??, n=5, 24/25 exchange particles.) You can see a close (but not perfect) relation between the number of exchange particles and the size of Cayley-Dickinson algebras 1=real, 2=imaginary, 4=quaternions, 8=octonions; and the continuing set... 16=hexagonions, 32=trigintaduonions, .when n<5. n squared and 2 to the power n start to diverge quickly starting with n=5. As well, the Cayley-Dickinson algebras of m=32, 64, 128 start to lose even more structure associated with the algebras that we are 'familiar with.' For example, once you reach the Cayley-Dickinson algebras with at least 32 operators, you start losing the rules of associativity in addition, and therefore, I believe the it's unlikely that the higher Cayley-Dickinson algebras will have corresponding forces of nature; but just because we're not familiar with it, doesn't mean it doesn't exist.
So, while I strongly believe that we will be able to eventually predict the color, masses, and charge of quarks, electrons and neutrinos, I'm still not sure whether there are more forces beyond the four known forces. There are real reason to believe that we are missing a really strong force is that we can not predict the masses or charges of tauons/muons/electrons, neutrinos and all of the quarks, but we need to see whether this new force can predict the masses, and if not, then we can start looking into forces beyond the SU(4) force discussed above.
While the charge of the neutron, quark and electron seem way too coincidental, there are reasons to believe that electrons are point-particles (rather than composite particles). For example, QED (and its ability to predict electron-photon interacts down to the ~8th decimal) assumes that electrons are point particles. But still, why do quarks have -1/3, and 2/3 of the charge of an electron. We are left with the feeling that there is still something very fundamental that we don't understand about the forces of nature, the cause of the rest mass of particles, and the cause of the charge of particles.
I'd like to conclude with the following open questions:
(1) Why are there only 4 dimensions to space-time? Is this related to the fact that there are only 4 forces of nature, which is most likely due to the fact that there are only 4 normed division algebras (over the real numbers...gravity, over the imaginary numbers...E&M, over the quaternions...weak nuclear, and over the octonions...strong nuclear.)
(2) Are more than 4-dimensions of space-time not possible because there are only 4 normed division algebras?
(3) Is the reason that there are 3 surface dimensions and 1 radial dimension (to our 4D sphere) due to the face that 3 of the forces of nature of space-time reversible (gravity, E&M, and strong nuclear) and one of the forces is space-time irreversible (weak nuclear)?