Is MOND (Modified Newtonian Dynamics) related to the Left-Right symmetric octonionic theory of the standard model and gravitation?

Is MOND (Modified Newtonian Dynamics) related to the octonionic theory? Possibly yes. At least this question is worth investigating further, for the reasons explained below.

MOND is an alternative to dark matter. Instead of proposing that the dynamical anomaly of galactic rotation curves is due to the presence of additional invisible matter, it is proposed that on sufficiently large distance scales, where gravitational acceleration falls below a critical value a_0, the law of gravitation departs from Newton's.

Roughly put, in MOND, the circular orbital acceleration v^2 / R outside a mass M is given as usual by Newton's GM/R^2, so long as the acceleration exceeds a_0. This gives the well-known Keplerian fall-off V^2 ~ 1/R, which is contradicted by the flat galaxy rotation curves.

As is known from observations, whenever the observed orbital acceleration falls below a_0, the velocity curve stays flat thereafter.

In 1983, Milgrom proposed that in the deep MOND regime, where the acceleration is much below a_0, the law of gravitation changes, so that the orbital acceleration is now given by

v^2 / R ~ (GM a_0)^1/2 / R

which explains the flatness of the rotation curve. This phenomenological modification of Newtonian gravitation at large distances is quite successful in explaining observations. Clearly a modification to general relativity is implied at large distances, including at the cosmological Hubble scale. Curiously, a_0 is numerically of the order of the cosmic acceleration c/H_0. A mysterious coincidence or a pointer to a relativistic theory underlying MOND? Until recently, the main and justified criticism of MOND was that there is no relativistic extension of the theory which can account for structure formation and in particular the CMB anisotropies. Something at which cold dark matter is highly successful. This may have changed last year, when two Czech physicists constructed RMOND, an action principle based phenomenological relativistic extension of MOND, which explains CMB data.

What could be the fundamental origin of MOND? This is where the octonionic theory comes in. What caught my attention is the acceleration being proportional to square-root of M, instead of M, in the MOND regime. In the O-theory as well, the would-be-gravity theory SU(3)_g x SU(2)_R x U(1)_g has as its charge square-root of m, rather than m, where m is the mass of the elementary particle. In the unbroken L-R symmetric regime, the interaction strength goes as sqrt{m}. When L-R symmetry is broken, squaring of would-be-gravity is enabled, GR and Newtonian gravitation emerge, and the interaction strength goes as m.

Where and how does a_0 enter the picture? We will identify a_0 with cosmic acceleration at the corresponding cosmic epoch (making it epoch dependent!). The L-R symmetry breaking [same as electro-weak symmetry breaking] is caused by spontaneous localisation of classical matter perturbations (primordial black holes??) as a result of which the emergent gravitational acceleration in the vicinity of compact objects exceeds the (pre symmetry breaking) cosmic acceleration a_0. This would be the origin of MOND. In the vicinity of compact objects, where acceleration exceeds a_0, the square of would-be-gravity, i.e. GR and Newton, hold. As for instance in the solar system and near stars and black holes. However, in low density regions, where acceleration is below the cosmic acceleration a_0, the unbroken would-be-gravity law holds, where acceleration is proportional to square root mass, not mass.

If this line of thinking were to be correct, the octonionic theory could explain the fundamental origin of MOND ! The critical acceleration a_0 then serves as the order parameter for a phase transition: the L-R symmetry breaking. Could it be that the U(1)_g of would-be-gravity is the sought after dark energy. i.e. dark photons? In a universe made only of matter, all particles have like charge root(m), and the U(1) vector interaction is repulsive.

MOND would then be more fundamental than Newtonian gravitation, with the latter becoming square of MOND! MOND is then same as would-be-gravity.

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.