The simplicity of the action principle in the octonionic theory

In the attached screenshot from my review, I show the fundamental action, by opening which the standard model Lagrangian (+gravity) emerges.

This action is nothing but a refined form of the action of a relativistic particle in curved spacetime, i.e.

S = mc \int ds

I try to explain, using one more screenshot from the review, attached below.

Eqn. 43 defines an octonion, whose eight direction vectors define the underlying physical space in which the `atom of space-time-matter' [the Q matrices = elementary particles] lives.

The form of the matrix is shown in Eqn. 44. The elementary particles are defined by different directions of octonions. The Q-matrices as shown in the action define the `kinetic energy' of the STM atom. The trace is a matrix trace. Noting that L is proportional to square root of mass m, the action in the screen shot can be written as

S ~ mc \int Tr [ .... ]

Our fundamental action is a relativistic matrix-particle in higher dimensions.

The universe is made of enormously many such STM atoms which interact through `collisions' and entanglement. From their interactions emerges the low energy universe we see.

There perhaps cannot be a simpler description of unification than this action principle. Note that Q_1 and Q_2 are two different matrices which together define one `particle' hence giving it the character of an extended object such as the string of string theory.

Once again, we see the great importance of Connes time \tau. The universe is a higher dimensional spacetime manifold filled with matter, all evolving in an absolute Connes time.

Reference for the review: https://arxiv.org/abs/2110.02062

String Theory 3.0

When it was proposed that elementary particles are not point objects, but extended like strings, one important conceptual issue was not clear. Why strings? What is the foundational principle / symmetry principle which compels us to consider extended objects?

In our work we started by demanding that there ought to exist a reformulation of quantum field theory which does not depend on classical time. This is the starting foundational question.

The symmetry principle then emerged as: physical laws must be invariant under general coordinate transformations of *non-commuting* coordinates.

The Lagrangian dynamics which implements these requirements requires (for consistency) that elementary particles be described by extended objects (strings). And it also requires that the theory be formulated in 10 spacetime dimensions, plus Connes time as an absolute time parameter.

This essentially is String Theory 3.0, with important accompanying changes:

(i) The vacuum is not 10D Minkowski vacuum. It is the octonionic 8D algebraic vacuum. This makes it easy to relate the theory to the standard model.

(ii) The Hamiltonian at the Planck scale is not self-adjoint. This permits compactification without compactification: 4D classical spacetime is recovered without curling up the extra six dimensions.

We thus provide a quantum foundational motivation for string theory. Also, the underlying dynamics is deterministic and non-unitary; thus bringing the deterministic and reduction aspect of quantum theory in one unified new dynamics. Someone once said that when string theory is properly understood, the quantum measurement problem will solve itself. In a manner of speaking, that is now seen to be true!

We have a pre-quantum, pre-spacetime dynamics, from which quantum field theory is emergent. The original aspects of string theory which still remain are: elementary particles are extended objects (strings) and they live in 10D Minkowski spacetime. Hence the octonionic theory could also be called the return of string theory, in a significantly improved and falsifiable avatar, which has predictive power, and which can be tested in the laboratory.

Reference:

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

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