String Theory 3.0 : Use the algebraic vacuum in eight octonionic dimensions, instead of the 10D Minkowski vacuum.

General Relativity is curvature of 4D Minkowski spacetime.

Octonionic Theory is the unified theory = `curvature' of 10D Minkowski spacetime = 8D octonionic spacetime. Except that the concept of curvature is more sophisticated now.

The new action is simply that for a free particle in octonionic 8D: equivalence principle holds, unified interaction as geometry of spacetime holds.

The octonionic theory could also be called String Theory 3.0 where the vacuum shifts from being 10D Minkowski vacuum to octonionic 8D algebraic vacuum. Also, we did not arrive at String Theory 3.0 by postulating ad hoc existence of extended objects such as strings. Rather, we arrive at strings by solving a foundational problem of quantum theory: find a reformulation of quantum field theory which does not depend on classical time. This leads us to strings in an inevitable unavoidable way, and this time as a successful theory of unification.

All this is made very clear and convincing by Clifford algebras and Jordan algebras, and by extension of the standard model to the Left-Right symmetric model. The Right sector is (would be) gravity.

Consider this connection between Jordan algebra, Lie algebra, and Minkowski spacetime symmetry in different dimensions:

J_2 (R) \sim SL(2,R) \sim SO(1,2) 2+1 Gravity

J_2 (C) \sim SL(2,C) \sim SO(1,3) 3 + 1 GR, Einstein gravity

J_2 (H) \sim SL(2,H) \sim SO(1,5) 5+1 Gravi-Weak theory !

J_2 (O) \sim SL(2,O) \sim SO(1,9) 9+1 Gravi-weak-electro-color theory [L-R symmetric model]

In the language of Clifford algebras: modulo Bott periodicity, one only has to go up to Cl(7) and every Cl(n) then connects to the standard model.

Cl(2) 4D Lorentz algebra made from complex quaternions using two of the three imaginary directions: home for GR, SL(2,C)

Cl(3) 6D Lorentz algebra made from complex quaternions using all three imaginary directions (complex split biquaternions) home for gravi-weak theory SL(2,H)

Cl(4) just the weak interaction

Cl(6) complex octonions : SU(3) X U(1) symmetry, use six out of seven imaginary octonionic directions : electro-color symmetry

Cl(7) Lorentz algebra SO(1,9) \sim SL(2,O) made using all seven imaginary octonionic directions (complex split bioctonions) : gravi-weak-electro-color unified symmetry.

What do we gain by going to algebras? We bring in gravity in an obvious way linking up to SO(1,9) and E_6. Above all, the characteristic equation of the exceptional Jordan algebra determines the free parameters of the standard model.

In summary

4D is GR

6D. unification of gravity with weak interaction

10D unification of gravi-weak with electro-color

all the time staying with forces as geometry, but beyond 4D inevitably going (pre)quantum.

This could be called the return of string theory if one so wishes, but with the limitations now removed because of important changes: use the algebraic vacuum of octonions, not the Minkowski vacuum.

For a detailed review of the Octonionic Theory please see

Quantum Theory without Classical Time: a route to quantum gravity and unification

https://www.tifr.res.in/~tpsingh/Penrose90singh.pdf

The impact of Bohr-Einstein dialogs on thinking in quantum gravity: BSM in IR

Bohr, to my understanding, suggested that quantum theory was exact, or at least good enough for studying problems of the day. Today this accepting of Bohr's belief is implicit in applications of quantum field theoretic methods to arrive at a quantum theory of gravity valid at the Planck scale. It could be called the UV first approach.

Einstein, on the other hand, thought that quantum theory is approximate. That quantum non-locality and special relativity contradict each other, and that an underlying theory will quantitatively explain the origin of quantum indeterminism. These are not just UV issues. They concern low energy quantum mechanics as well. Quantum relativists worry about the loss of classical spacetime when quantum systems are in a superposition of two different position states. This could be called the IR approach to quantum gravity.

Both approaches are important and necessary. However a major recent realisation for me has been that the free parameters of the standard model are determined, not in the UV region, but in the IR region, by replacing Minkowski spacetime by its non-commutative analog. With hindsight, this paradigm shift becomes inevitable. But it is impossible, almost, to convince physicists of the UV approach about this. There is complete disbelief on their part. That is a Bohr way of thinking. The IR way of thinking is the Einstein way: quantum theory and special relativity are incompatible, and this is not just about quantum non-locality. It is about understanding the standard model. By sticking only to the UV approach we will never know why the low energy FSC is 1/137 and why the mass ratios are what they are.

I try to explain this in the attached sketch .

One can go from Minkowski spacetime vacuum to QFT at high energies, and then Planck scale, and make spacetime non-commutative.

Or one can go from Minkowski spacetime to non-commutative Minkowski spacetime, and then go to high energies and Planck scale.

Only the second method is correct. Because the ground state of quantum gravity is not Minkowski spacetime. It is non-commutative Minkowski spacetime. When quantum gravity is switched off, only the gravity is switched off. The quantum still remains. And quantum systems give rise, not to Minkowski spacetime, but to non-commutative Minkowski spacetime, Only classical systems can give rise to Minkowski spacetime, in the weak gravity limit.

Quantum theory without classical time: a route to quantum gravity and unification

If you are interested in the current status of the of the octonionic theory, you might find this new review article useful:

Quantum theory without classical time: a route to quantum gravity and unification

https://www.tifr.res.in/~tpsingh/Penrose90singh.pdf

152 pages, 17 figures. Invited review article submitted to the special collection "Celebrating Sir Roger Penrose's Nobel Prize"

[AVS Quantum Science (AIP Publishing and AVS), Guest Editor: Hendrik Ulbricht]

Abstract

There must exist a reformulation of quantum field theory which does not employ classical time to describe evolution, even at low energies. To achieve this goal, we have proposed a pre-quantum, pre-spacetime theory, which is a matrix valued Lagrangian dynamics on an octonionic spacetime. This is a deterministic but non-unitary dynamics in which evolution is described by Connes time, a feature unique to non-commutative geometry. From here, quantum field theory and its indeterminism, as well as classical space-time geometry, are emergent under suitable approximations. In the underlying theory, the algebra of the octonions reveals evidence for the standard model of particle physics, and for its unification with a pre-cursor of gravitation, through extension to the Left-Right symmetric model and the symmetry group $E_6$. When elementary particles are described by spinors made from a Clifford algebra, the exceptional Jordan algebra yields a theoretical derivation of the low energy fine-structure constant, and of the observed mass ratios for charged fermions. We identify the Left-Right symmetry breaking with electroweak symmetry breaking, which also results in separation of emergent four-dimensional Minkowski spacetime from the internal symmetries which describe the standard model. This `compactification without compactification’ is achieved through the Ghirardi-Rimini-Weber mechanism of dynamical wave function collapse, which arises naturally in our theory, because the underlying fundamental Hamiltonian is necessarily non-self-adjoint. Only classical systems live in four dimensions; quantum systems always live in eight octonionic (equivalently ten Minkowski) dimensions. We explain how our theory overcomes the puzzle of quantum non-locality, while maintaining consistency with special relativity. We speculate on the possible connection of our work with twistor spaces and spinorial space-time, and with Modified Newtonian Dynamics (MOND). We point to the promising phenomenology of $E_6$, and mention possible experiments which could test the present proposal. In the end we outline further work that still remains to be done towards completion of this programme.

How does a pre-quantum, pre-spacetime theory know about the low-energy fine structure constant and mass ratios?

Because such a theory is needed both in UV and in IR, and the octonionic theory is such a theory. UV is obvious, but IR could do with some explaining.

Even at low energies, there can be a situation where for a given system, all sub-systems have action of the order \hbar. Then there is no background classical spacetime anymore, and the pre-quantum, pre-spacetime theory is required. e.g. when a massive object is in a quantum superposition of two position states and we want to know what spacetime geometry it produces.

The pre-theory is in principle required also for a more exact description and understanding of the standard model, even at low energies. And the octonionic theory achieves just that, thereby being able to derive the low-energy SM parameters. This is BSM in IR, and has implications for how we plan BSM experiments: these have to be not only towards UV, but also in the IR.

The O-theory has only three fundamental constants, and these happen to be such that both the UV and IR limits can be easily investigated. These constants are Planck length, Planck time and Planck's constant \hbar. Note that, as compared to conventional approaches to quantum gravity, Planck's constant \hbar has been traded for Planck mass/energy. And this is very important:

The pre-theory is in principle required whenever one or more of the following three conditions are satisfied: times scales T of interest are order Planck time, Length scales L of interest are order Planck length, actions S of interest are order \hbar.

If T and L are respectively much larger than Planck time and Planck length, but S is of order \hbar, that requires the pre-theory in IR.

If T is order Planck time, then the energy scale \hbar / T is Planck energy scale. However, \hbar / T is in IR for T >> Planck time, and yet the pre-theory is required (for an exact in-principle description of SM) if all actions are order \hbar.

The BSM physics in IR is achieved by replacing 4D Minkowski spacetime by 8D octonionic non-commutative spacetime. This is the pre-theory analog of flat spacetime - and it has consequences - it predicts the low energy SM parameters, without switching on high-energy interactions in the UV.

Going to high energies is just like in GR. In GR we switch on the gravitational field around Minkowski spacetime and doing so takes us from IR to UV. Same way, in the O-theory we switch on SM interactions and would-be-gravity, *around* the `flat' octonionic spacetime [=10D Minkowski] and this takes us from IR to UV. But unlike in the GR case, we already learn a lot of BSM physics in the IR, because the spacetime is non-commutative. String Theory missed out on this important IR physics, because it continued to work with 10D Minkowski spacetime which is commutative, and from there went to UV. Should have looked at octonionic spacetime and Clifford algebras.

More on MOND and the octonionic theory

In what way might we think of Newtonian gravitation as the square of MOND?

To get some insight, let us write the acceleration a in a circular orbit as

a = [ GM / R^2 ] ( 1 + \beta (R) ]

The function \beta (R) depends on a and goes to zero for a >> a_0, thus recovering Newtonian gravitation. For a in the vicinity of a_0 we approximate the last bracket to a_0 / a, thus yielding MOND. At large cosmic distances, relativistic effects become important. GM/R^2 is replaced by its GR counterpart, and (1+\beta) becomes the MOND induced modification of GR. It is important to ask if dark energy is a manifestation of this MOND induced modification. Were this to be so, we will have a common original cause for flat galaxy rotation curves and cosmic acceleration, without dark matter or dark energy, and because GR and Newtonian gravity are limiting cases of a more general law of gravity. On the clusters scale MOND will need warm dark matter such as sterile neutrinos, or dark baryons.

What then is this more general law of gravity? Which we demand must come from first principles. The Left-Right symmetric octonionic theory proposes SU(3)_g X SU(2)_R X U(1)_g as would-be-gravity, or square-root-gravity, this is the right-hand counterpart of the broken L-R symmetric theory, whose left-handed counterpart is the standard model. The gauge theory of would-be-gravity on an 8D octonionic space-time is proposed as the more general law of gravity, which explains the origin of the critical acceleration a_0 and the emergence of GR, MOND, and Newton as special cases.

Cosmology and the scale a_0: In the L-R symmetric theory, the very early universe undergoes an inflationary expansion having a time-dependent cosmic acceleration a_0(t). This inflationary expansion is halted (and converted to a power law expansion) when significant seeding of density perturbations causes a quantum-to-classical transition, L-R symmetry breaking, and emergence of 4D classical spacetime. The gravitational acceleration in the vicinity of the seeded relativistic density perturbations exceeds the then a_0, and GR emerges from would-be-gravity as its square. MOND is the transition zone between would-be-gravity and Newton/GR. Thus would-be-gravity can seed the scale-invariant matter perturbations whose effect is seen in the CMB (hence relativistic MOND).

In today's universe, away from compact objects, would-be-gravity (relativistic MOND) dominates because accelerations are smaller than a_0 and tend to the current cosmic a_0. In this deep MOND regime there is space-time scale invariance, and the universe tends to de Sitter.

Would-be-gravity when squared yields GR at high accelerations and the condensation of SU(2)_L and SU(2)_R into 4D spacetime geometry is indicated. SU(2)_L mediated on small scales by heavy weak bosons is the weak interaction. The electro-weak symmetry breaking is in reality an L-R symmetry breaking same as

QCD Color + U(1)_em - Grav Color + U(1)_g

breaks from - breaks from

Weak SU(2)_L - SU(2)_R

It appears that if we do cosmology with the L-R symmetric octonionic theory and its emergent approximations, all will be well without cold dark matter and without cosmological constant as dark energy. In this theory the cosmological constant (zero point energy of vacuum) is strictly zero. Cosmic acceleration is caused by U(1)_g

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.