Feynman on the Fine Structure Constant (1985); and the corresponding result from the octonionic theory

Feynman on the fine structure constant (1985)

"There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.)

Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans. You might say the "hand of God" wrote that number, and "we don't know how He pushed His pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out – without putting it in secretly!"

— Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129

******************

This below is the result that the octonionic theory gives for the expression for the fine structure constant:

Quantum theory without classical time: octonions, and a theoretical derivation of the fine structure constant 1/137

https://www.worldscientific.com/doi/abs/10.1142/S0218271821420104

https://arxiv.org/abs/2110.07548

The paper also explains how the exact currently measured value is recovered from theory, by assuming a specific energy scale for the electro-weak symmetry breaking. The result below should be seen as a confirmation of quantum gravity effects in the infra-red.

Atoms of Space-time-matter & Space-time from [Quantum] Entanglement

Emergence of space-time from quantum entanglement is an important and much talked of idea these days.

Essentially the same thing happens in the pre-spacetime, pre-quantum octonionic theory, but with important differences. Since spacetime is emergent, that means there was no time before time emerged! So one could not have had quantum theory in the form we use it at present. That is why in our theory gravitation and quantum theory are both emergent.

They emerge from a pre-theory which is a matrix dynamics of atoms of space-time-matter [STM atom]. An STM atom is an elementary particle along with all the fields and along with the pre-spacetime geometry it produces. It is a universe in itself, and furthermore, it is impossible to distinguish the STM atom from the mathematics which defines it! Hence in the accompanying figure, we depict the pre-universe as a collection of the actions of enormously many STM atoms.

From here, as a result of entanglement, there emerge gravitation, and quantum theory as well. Entanglement is more general than quantum theory. Also, this emergence can happen at any energy scale, and in fact is happening all around us at this very moment, which is what is keeping the world classical! Spacetime emerges because entanglement is causing classical objects to emerge.

And those systems which are not critically entangled, such as the particles of the standard model, are in reality in their original form as STM atoms, in their quantum spacetime. There we understand why the standard model is what it is.

It also becomes clear that the idea `spacetime from (quantum) entanglement' can in principle be tested in the lab today, at low energies.

How do octonions manage to dictate the properties of elementary particles?

Answer: Just as the Dirac equation carries much more information than it's square i.e. the Klein-Gordon equation, the octonionic space carries much more information than it's square, the conventional Minkowski space-time. Octonionic space is to Minkowski spacetime what Dirac equation is to the Klein-Gordon equation.

In the space spanned by the octonion O, elementary particles are defined by different directions. This is achieved through spinors of a Clifford algebra constructed from octonionic chains. The symmetries of the octonion algebra coincide with those of the standard model, allowing the identification of quarks and leptons. There is no freedom.

The dynamical variable Q is a matrix valued quantity on this octonionic space.

The norm of O has the properties of a metric [Euclidean / Lorentzian]. We can make the a_i functions of position, and we will get the analog of the usual curved space metric. We can construct a QFT on this metric in the usual way, but if we did not know that this metric has come from an underlying octonionic space we would not realise that properties of the elementary particles have already been fixed by the O-space. This is like Klein-Gordon vs. Dirac. This is how octonionic dynamics knows more than QFT on Minkowski spacetime.

And this is happening at low energies, before the interactions are made strong by going to high energies. The latter is achieved by switching on the Q, which are the analog of [square-root] of the metric. The equivalent of the metric comes from taking the trace of a matrix polynomial - this is the Lagrangian in our theory.

The Q-matrices incorporate the known standard model interactions, and also the precursor of gravitation. The low energy limit is achieved by setting the Q matrices to identity.

In conventional QFT, even if one works with tetrads (i.e. the square-root of the metric) the manifold is still R^4, and hence commutative. Quantum space-time is achieved by replacing R^4 by the octonionic space. This is the completion of quantum theory, where quantumness is extended to spacetime as well. In so doing we realise we no longer have the freedom to choose symmetries and properties of elementary particles. The octonions dictate these.

This can be called lifting of general relativity to the quantum level. In GR spacetime tells matter how to move. Now (octonionic) spacetime is telling (quantum) matter what to be !! Matter still curves spacetime - this is what the Q-matrices do, but now the curving represents not just gravitation, but all the four known forces.

Quantum Theory, and Gravitation, as Emergent Phenomena

This way of presenting the octonionic theory is perhaps more attractive and insightful than saying `Quantum Theory without Classical Space-time' although both phrases mean the same thing as far as physics is concerned.

The most important takeaway from this title is that this emergence does not have anything to do with energy scale! It is not necessarily an emergence from a high energy Planck scale, to our present day low energy universe.

The emergence is from an underlying situation in which microscopic sub-systems in superposition give rise to superpositions of space-time geometries. A system is said to be microscopic if its action is of order \hbar. As we know, microscopic superpositions arise at low energies as well, and if they were to dominate, there would be no classical spacetime. We will then need a formulation of quantum mechanics which does not use classical time.

Consider sub-systems S_1, S_2, ...., S_n, in a system S. If most of them are microscopic, then we do not have a classical time, nor classical gravity.

Consider now the limiting case that: a sub-system S_i consists of a critically large number of entangled microscopic entities. Then, S_i behaves classically. If the system S is dominated by classical subsystems, then classical spacetime and classical gravitation emerge. On this susbstrate, sub-systems which are still microscopic are described by quantum mechanics as we know it.

It is in this sense that quantum theory, and gravitation, are emergent phenomena.

The octonionic theory describes the standard model of particle physics keeping in mind that all sub-systems are microscopic. Even at low energies. This yields more information than when we study the standard model (as we currently do) assuming a classical substrate. Doing the latter hides the information coded in the quantum spacetime being produced by the standard model particles. This quantum spacetime dictates and fixes the properties of the standard model - the allowed symmetry groups, charge quantization, mass ratios, number of fermion generations.

The extension of the standard model to the Left-Right symmetric model appends quantum spacetime to the standard model, via the right-handed (would be gravity) sector SU(3)_grav X SU(2)_R X U(1)_grav

I attach a useful figure from a previous post. Interestingly, you see the L-R symmetry in this diagram, though I never realised that while drawing it. In the upper level as well as the lower level, the Left is matter, the Right is gravity. General relativity couples left-handed matter (weak interaction violates parity) to (right-handed ?) gravity. This also implies that right-handed sterile neutrinos do exist.

Quantum Worlds, Classical Worlds

Our universe, as it is today, is dominated by classical bodies, which produce, and live in, a classical spacetime. This is the substrate shown in the figure below.

There is a sprinkling of systems for which we get to see the actual quantum behaviour. These are shown at the top left of the figure.

Then we realise that classical systems are a limiting case of quantum systems. This is shown by the curved arrowhead on the left of the figure.

Our current formulation of quantum systems embeds them in a classical spacetime, as depicted by the coloured arrow marked `Approximate'. Howsoever successful by the standard of current experiments, this formulation can only be approximate. Because the truth is, we do not really need the classical substrate, nor the classical spacetime, to describe quantum systems. Nor should we have to depend on the classical substrate - it is after all a limiting case of quantum systems and quantum spacetime, and a theory should not have to depend on its own limit, for its formulation.

Quantum systems, produce and live in a quantum spacetime, as shown in the top right. This is where the standard model truly lives, irrespective of what energy scale we study. However we currently describe the standard model only approximately, which is why our understanding of the standard model is only partial.

Note that this diagram makes no reference to the energy scale. It is true at all energies. UV as well as IR.

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