Elementary particles, and the space-time in which they live.

We are accustomed to the fact that space-time is four dimensional, and its coordinates are labeled by four real numbers (t, x, y, z). All material objects, such as the electron, and the fields they produce, are supposed to live in this 4D spacetime.

But what if this description of space-time is only an approximation? In Newton's world, material bodies as well as space and time, are described by real numbers. However, in quantum mechanics, material particles are described not by real numbers, but by non-commuting matrices, q-hat and p-hat, the position and momentum operators. Eigenvalues of these matrices correspond to the classical Newtonian values of position and momentum.

We assume in quantum mechanics that we can continue to describe space-time by real numbers, which commute, even when position and momenta have been made operators, and these latter do not commute. What if this is only an approximation, and the truth is that when q and p are made operators, spacetime should also be labeled by non-commuting coordinates? Could it be that when the space-time is having non-commuting coordinates, this very property of non-commutativity determines properties of elementary particles? Such as, why is the electric charge of the down quark one third that of the electron? Quantum theory and the standard model have no answer to this question. However, when we replace 4D spacetime by an 8D spacetime labeled by the octonions, we are able to prove this relation theoretically!

So then, what are our choices for non-commuting coordinates? Turns out, not many! If we choose to generalise the real number system and yet retain the property of division (i.e. every element should have an inverse) there are only three other possible number systems.

The first of the three are complex numbers (x + i y)

The next are the quaternions (a + b i-hat + c j-hat + d k-hat)

Here, a, b, c, d are real numbers. i, j, k are three imaginary units each of which square to minus one, but they do not commute with each other: ij = - ji, jk = -kj, ki = - ik, ij=k, jk=i, ki=j.

The last of the four division algebras are the octonions. An octonion is denoted as

a_0 + a_1 e_1 + a_2 e_2 + ... + a_7 e_7

The eight a-s are real numbers, the seven e_i are imaginary units each squaring to minus one, these anti-commute with each other, and have a multiplication table known as the Fano plane. Octonion multiplication is non-associative, besides being non-commutative.

An eight dimensional spacetime labeled by the octonions as their coordinates is known as an octonionic space-time. The usual 4D spacetime is a special case of the 8D spacetime.

When we put fermions on this spacetime, interesting things happen. Only eight types of fermions and their eight anti-particles are allowed [and exactly three generations]. Electric charge is quantised in units of 0, 1/3, 2/3 ad 1. The 1/3 and 2/3 are SU(3) triplets and identified as down and up quark. The 0 and 1 are SU(3) singlets and identified as the anti-neutrino and the positron. This way the standard model fermions arise, and only those ones are allowed.

Quantum theory on an octonionic spacetime is the exact quantum theory.