In the octonionic theory, why does a black hole radiate?

Are split octonions connected with black hole space-times?!

The context is that fermionic states can be defined using octonions, and using split octonions. A fermion lives in 8D space.

Octonions have signature (8,0) whereas split octonions have signature (4,4).

Black holes arise from the spontaneous localisation of fermions (i.e. octonions) when the gravitational radius of the system exceeds its effective Compton length.

We well know that in the interior of a black hole, space and time interchange roles. Could it then be that a black hole spacetime [interior + exterior] is the limiting case of a split octonionic space-time?

Assuming that it is, we ask the question:

In the octonionic theory, why does a black hole radiate?

In this theory, fermions are defined in 8D octonionic space using octonions and split octonions.

Elementary particles and black holes are the opposite limits of fermionic entities: particles are the quantum dominated limit, whereas black holes are the gravity dominated limit. There are two competing length scales associated with a collection of entangled fermions. A gravitational length scale Lg and an effective Compton length Lq, which obey Lg x Lq = L_p^2, where L_p is Planck length.

Non-black hole non-elementary particle entities such as stars and other classical macroscopic objects are entangled fermions caught mid-way between elementary particles and black holes - their proceeding to the BH state being slowed down by standard model forces. For now we can ignore these forces and focus only on elementary particles and black holes.

Quantum systems are those in which the effective Compton length exceeds the gravitational radius. And just the reverse for black holes.

In the trace dynamics of the octonion-valued fermions, a quantum system with no entanglement is at statistical thermodynamic equilibrium. It has maximum Boltzmann entropy.

Entanglement moves the system away from equilibrium, towards classicality, thereby reducing it's entropy. Entanglement is order; no entanglement is disorder. Entanglement reduces the effective Compton length while increasing the gravitational radius. Critical entanglement is when the gravitational radius becomes larger than the Compton length, the quantum-to-classical transition is competed, and the black hole forms. This is a non-equilibrium state, and the high entropy of the black hole notwithstanding, the entropy of the system is now lower than what it is when it is completely unentangled [thermodynamic equilibrium]. This is why a black hole radiates; it is attempting to disentangle and return to equilibrium.

We note that this quantum-to-classical transition is governed by the degree of entanglement; this physics is independent of energy scale. In the expanding very early universe, an energy scale shows up because only when the expanding universe has cooled sufficiently, critical entanglement becomes possible. But the key physical role is of degree of entanglement, not of energy.

Why does entanglement lead to classicality? In the octonionic theory, the Hamiltonian also has an anti-self-adjoint (ASA) part. This ASA is negligible when there is no entanglement, the self-adjoint part dominates, we have unitary quantum evolution and the equivalent of thermodynamic equilibrium. The system lives in an 8D octonionic space.

Entanglement enhances the ASA part relative to the self-adjoint part, bringing in some non-unitary component to the unitary evolution. Critical entanglement is when the ASA becomes dominant over the SA, spontaneous localisation breaks unitarity, and the black hole forms. This classical object is confined to a (coarse-grained) 4D subspace of the 8D octonion space (this 4D being the black hole interior). The black hole exterior is the other 4D half of the (coarse-grained) octonion subspace - it is our 4D spacetime. The split octonions are playing a role here.

What then of the information loss paradox? Conventional studies take a black hole as given a priori and then ask if the complete evaporation of a black hole into thermal Hawking radiation violates unitarity?

We would like to look at this process differently, and ask how did the black hole form in the first place? The initial state [thermodynamic equilibrium] has no information: maximum entropy, no entanglement. Entanglement is gain of information, reduction of entropy. If we remove observers from the scene, and consider the BH interior as well as exterior [the full 8D octonionic space] we know the information content. It is determined by the entanglement. By radiating, the BH is disentangling, spontaneously unlocalising, increasing entropy, and going back go equilibrium. The black hole is a far from equilibrium system, confined to the 4D subspace of 8D octonionic space (as if the molecules of gas in a box have all landed up in one half of the box). Gravitation is an emergent, far-from-equilibrium, thermodynamic phenomenon. By evaporating, the black hole is returning to the full 8D octonionic space, and returning to the unentangled equilibrium state.

(I) We began with zero information (no BH), (II) gained information (BH formed) and (III) went back to zero information (complete evaporation). The information loss paradox arises if we ignore step I and straightaway start at step II. But we must necessarily ask how the black hole got there in the first place? When we do that, we find there is no paradox.

How do fermions curve space-time?

Quantum gravity should not be viewed as the question: `how to quantise the gravitational field'?

But rather as the question: How do fermions curve space-time?

After all, Newtonian gravitation as well as classical general relativity answer this very question: what is the gravitational field / space-time curvature produced by material bodies and fields?

Fermions, being described by spinors, cannot be expected to produce / live in Minkowski spacetime, except in an approximate sense.

Instead, they are expected to live in a complex spinor spacetime, which is the spinor analog of Minkowski spacetime, and their mass-energy curves this spinor spacetime.

And this is a statement independent of energy scale, and valid at low energies as well.

The standard model has 25 dimensionless constants whose origin we do not understand. We seem to believe that by doing experiments at higher energies we will find new physics and a theory of unification which will then explain, through RG flow, the values of the constants at low energies.

This may or may not be true. What if we have not correctly understood quantum field theory and spacetime structure at low energies, and that is what is responsible for these undetermined free constants?

Indeed, it turns out that when we describe fermions at low energies, the complex spinor spacetime dictates the symmetries of the standard model, and determines several of it's dimensionless constants. This is because this non-commutative analog of Minkowski space-time is much richer in structure than Minkowski. It constrains properties of fermions, and removes the arbitrariness allowed for them when we describe them using QFT on Minkowski spacetime.

In hind-sight, this is not at all radical. We have taken a square-root of Minkowski space-time at the same-time as taking the square-root of the Klein-Gordon equation to arrive at the Dirac equation. We have written the Dirac equation on a spinor spacetime. The Dirac equation on Minkowski shows that spin angular momentum exists and is quantised. Dirac equation on spinor spacetime shows that electric charge and mass are quantised as well, and take values as observed in experiments. This confirms that the description of fermions on spinor spacetime is correct.

The Koide formula revisited

Important update: The Left-Right symmetric extension of the standard model yields the Koide ratio to be exactly 2/3

The Koide formula is the following empirical observation about the experimentally measured masses of the three charged leptons (electron, muon, tau-lepton): if the sum of their masses is divided by the square of the sum of the square-roots of their masses, the result is the number

0.666661(7) ~ 2/3

Is there a theory in which this number is shown to be exactly 2/3? Yes.

The first important assumption, justifiably, is: the unification of gravity with the standard model is required at all energies, not just at the Planck scale. And the exact description of fermions is on a spinor spacetime, not on Minkowski spacetime. Such a description, unifying gravity with the standard model, has been set up on an octonionic space.

A quantum system such as an electron obeys such L-R unification at all energies, lives in a spinor spacetime, the neutrino is a Dirac neutrino, and the Koide ratio comes out to be exactly 2/3.

However, and this is crucial, when we make a measurement on the electron, L-R symmetry is broken, because we make this measurement not in a spinor spacetime but in the emergent Minkowski spacetime: classical gravity is present and RH sector is hidden. The neutrino appears as if Majorana, and we find the Koide ratio to be 0.669163, departing a little both from 2/3 and from the measured value quoted above.

So the big picture is this: before we make a measurement on the charged leptons to measure their masses, the Koide ratio is exactly 2/3. After the measurement is made, the theoretical prediction for the resulting value is 0.669163, whereas the measured value is smaller. [The uncertainty in the mass of the tau-lepton is such that by demanding the Koide ratio to be 2/3 one can predict the mass of the tau-lepton.]

The above is an important development as we now know when the Koide ratio is exactly 2/3 [it is when the electron is not being observed]. And we understand why the measured value of this ratio is not exactly 2/3. In principle, we could demand the measured value to be equal to the value in Minkowski spacetime, and thereby fix the mass of the tau-lepton.

We also realise that a quantum measurement breaks Left-Right symmetry !!

Reference: eqn. 63 in https://arxiv.org/abs/2108.05787 [hep-ph]

The significance of three right-handed sterile Majorana neutrinos

In the algebraic description of the standard model using complex split bi-octonions, the right-handed fermions are eigenstates of square-root mass in the Clifford algebra Cl(6). Square-root mass: This is given by a U(1) number operator which defines square-root mass, and has the eigenvalues (0, 1/3, 2/3, 1). These represent the right-handed sterile neutrino, the positron, the up quark, and the down quark, respectively. The right-handed fermions can be considered as excitations of the sterile neutrino.

Correspondingly, the left-handed fermions are excitations of the left-handed active neutrino, and the associated Clifford algebra Cl(6) has a U(1) number operator interpreted as electric charge, and having the eigenvalues (0, 1/3, 2/3, 1) which stand for the left-handed active neutrino, anti-down quark, up quark, and the positron. [Anti-particles are defined by complex conjugation of particle states].

In a very natural way, the Higgs boson arises in this algebraic description, as shown in the attached preprint, and the Higgs couples the right-handed fermions to the left-handed fermions, thereby giving mass to the charged fermions. In this setting it becomes physically clear as to why there is a universally prevailing Higgs field which gives masses to the particles.

The above is also a description of the standard model in a spinor space-time. The inclusion of the sterile neutrinos actually makes this the Left-Right symmetric extension of the standard model. And clearly, this L-R model is needed at all energy scales, or how else would the Higgs impart mass to the left-handed fermions?!

Quantum systems obey L-R symmetry at all energy scales but to see this we have to be able to experimentally measure and show that the Lorentz interaction (the right-handed counterpart of the weak force) violates parity. Such an experiment is extremely difficult to perform, because cosmological expansion prior to the L-R symmetry breaking [same as EW symmetry breaking] has set the particle masses to extremely low values, making gravity very weak, while no such corresponding reduction takes place in the value of the electric charge. As a result the fine structure constant remains order unity, the weak coupling is also measurable, and parity violating weak interaction is observed.

What then apparently breaks the L-R symmetry? It is the critical entanglement of sufficiently many fermions to form macroscopic objects and the concurrent emergence of 4D classical spacetime and gravity. Once gravity becomes classical, the parity violating spin-one Lorentz interaction is lost. What remains in view is the left-handed standard model. But this parity violation is apparent only when we study quantum systems on the classical Minkowski spacetime background.

How the electron sees the world is something very different from how we currently believe it sees the world. The truth is that even at today's low energies the electron sees a left-right symmetric universe, with the [difficult to measure] right-handed Lorentz interaction being would be gravity. This is how it is in the spinor spacetime, with the weak-Lorentz interaction spanning an equivalent 6D Minkowski spacetime. Color-electro spans 10D Minkowski spacetime. Together we have the unified color-elecro-weak-Lorentz force, a unification that is manifest even at low energies, provided we describe quantum systems on a spinor spacetime.

As the early universe cools, critical entanglement of fermions becomes possible, classical objects form, 4D spacetime and classical gravity emerge, and the L-R symmetry is broken. It is not broken for those fermions which are not critically entangled.

Reference: arXiv:2110.01858 [hep-ph]

Unification of gravitation and quantum theory and the standard model is needed at all energy scales, not just at the Planck energy scale

The 25 or so dimensionless constants of the standard model are data. Unexplained data. What makes us so sure that to explain this data we must perform experiments at energies higher than the one at which these constants have been measured?!

Could it be that we have not yet fully understood quantum field theory and space-time structure at the energy scales at which we know the standard model?

Just as the Dirac equation is a square-root of the Klein-Gordon equation, a spinor spacetime is a square-root of Minkowski spacetime. Describing the standard model on a spinor spacetime, instead of on Minkowski, could explain properties of the standard model.

Elementary fermions live on a non-commutative spinor spacetime. Describing them as living on a Minkowski space-time is an approximation. The particles by their presence curve the spinor spacetime. This curving feeds back on the particles and controls their properties. Just as the non-commutativity of position and momentum in quantum mechanics results in discrete energy levels for bound systems, the non-commutativity of space-time implies quantisation of electric charge and of mass.

The emergence of Minkowski spacetime from the underlying spinor spacetime is a consequence of critical entanglement amongst the fermions. Once sufficiently many fermions are entangled, classicalisation takes place, macroscopic objects form and 4D classical space-time emerges. However, elementary particles which are not critically entangled continue to live in their spinor spacetime, although we describe them, approximately, from the vantage point of our own Minkowski spacetime.

This same classicalisation process takes place in the very early universe. Once the expanding universe cools sufficiently, critical entanglement of fermions becomes possible, and classical spacetime emerges. Quantum systems continue to evolve in their spinor spacetime.

A little bit of historical background about the octonionic theory

Over the last two and a half decades or so, the following developments have taken place:

Stream I

Stephen Adler and collaborators proposed the theory of Trace Dynamics, i.e. quantum theory is an emergent phenomenon, arising in the thermodynamic limit of an underlying pre-quantum dynamics on a flat spacetime. Adler expressed the view that classical gravitation is also an emergent phenomenon, that gravity should not be quantised, and that the emergence of gravity might be shown from a generalisation of trace dynamics.

Stream II

Completely independent of Stream I, Chamseddine and Connes developed the spectral action principle: the Einstein-Hilbert action can be expressed as the spectrum of the Dirac operator.

Stream III

Independent of Stream I and II, several researchers showed that some properties of the standard model could be explained in an algebraic way, using Clifford algebras made from the algebra of the octonions. This is an ongoing research programme. Space-time here is flat four dimensional Minkowski spacetime, in the spirit of GUTs.

Stream IV

Independent of Streams I, II and III, I had been making naive attempts at developing a formalism from which quantum theory and gravitation both are emergent.

It appears useful to employ the spectral action principle to generalise trace dynamics to include gravitation. Each of the eigenvalues in the spectrum of the Dirac operator is raised to the status of a matrix [in the spirit of trace dynamics], permitting one to construct an action principle for a pre-spacetime, pre-quantum theory which generalises trace dynamics so as to include gravitation. Because of the phenomenon of spontaneous localisation, it seems possible that both quantum theory and classical gravitation are emergent in this theory, in a thermodynamic sense.

For various reasons, it appears useful to assume that the pre-quantum pre-spacetime degrees of freedom live on an 8D space labelled by the octonions, and evolve in this space in an absolute time known as the Connes time.

This suggests a connection with stream III namely that the elementary particles of the standard model, described by the algebra of the octonions, live on this physical octonionic space, in the spirit of a Kaluza-Klein theory. This suggests an approach to unification, in which the non-commutative space determines the dimensionless parameters of the standard model, without appealing to physics at very high energy scales which are not currently accessible by present day accelerators. This is relativistic weak quantum gravity impacting on the standard model: unification of gravity with the other forces is a process independent of the energy scale: a collection of physical systems each having an action of order \hbar, between themselves give rise to (weak) quantum gravitation [even at low energies] and this gravitation must be described in a unified way with the physical systems which source it. The non-commutative nature of the space transforms elementary particles from points to extended objects with extent of the order of Planck length, making it a quantum gravitational situation [also because action is order \hbar so that quantum geometries are superposed]. However, the time scales of interest (in Connes time) are much larger than Planck time, so that Planck energy scales are not probed, nor are Planck length scales actually probed. This makes it a low-energy quantum gravitational situation.

It might be possible to show from this framework, following the principles of trace dynamics and spontaneous localisation, that 4D classical spacetime and classical gravitation, quantum field theory, and the standard model as we know it, are emergent from this framework. This is of course extremely challenging and ambitious. A modest effort in this direction is currently in progress.

A spinor spacetime, and the standard model

Just as the Dirac equation is the square-root of the Klein-Gordon equation, a spinor spacetime is the square-root of Minkowski spacetime.

If we are not interested in the gravitational field of an electron, it is perfectly fine to work with the Dirac equation written on a Minkowski spacetime.

However, if we want to know the gravitational field produced by the electron, we must first define the electron states on a spinor spacetime.

Octonions define a spinor spacetime, which has eight octonionic dimensions. It's square is a ten-dimensional Minkowski spacetime.

Using Clifford algebras, the spinorial states for fermions can be defined on 8D octonionic spacetime. The symmetries of the octonionic space restrict what properties the fermions can have. Charge and (square-root) mass are both defined as eigenvalues of certain symmetry operators of the 8D space, and take discrete values consistent with what is observed experimentally in the standard model. The allowed properties show that the only fermions possible are left-handed and right-handed quarks and leptons of the three generations. This includes three right handed sterile neutrinos, one per generation.

The gravitational effect of an electron is equivalent to curving of this 8D octonionic space-time, and is described by the equations of trace dynamics.

The description of the standard model using the laws of QFT on Minkowski spacetime, while extremely successful, is an approximate description. It does not tell us why the standard model is what it is, and why the dimensionless constants take those particular values which we see in experiments.

On the other hand, when we describe the dynamics of elementary particles using trace dynamics on a spinor spacetime, the symmetries of the standard model and its dimensionless constants are determined by the algebraic properties of the 8D octonionic spacetime. There is no freedom.

This description in terms of a spinor spacetime is available at all energy scales, low as well as high. That is the reason why the low energy fine structure constant gets determined in this theory. By a pre-spacetime pre-quantum theory we do not just mean a pre-theory at Planck scale energies. We also employ this pre-theory at low energies to understand the standard model at low energies. We can call this relativistic weak quantum gravity coupled to the standard model.

QFT on Minkowski spacetime can be recovered from trace dynamics on a spinor spacetime.

Associahedra vs. the Octonions

These two slides appear at the beginning of Prof. Arkani-Hamed's engaging talk at TIFR yesterday. He described a very rich mathematical framework for the emergence of scattering amplitudes in spacetimes. However, it is not at all obvious how one can do something down to earth like obtaining the standard model, and/or make predictions for experiments, in this framework.

In contrast, the octonionic theory is relatively much more simple minded, and yet is a pre-spacetime pre-quantum theory from which GR and quantum theory are explicitly emergent. The theory incrementally departs from the standard model and from general relativity, and makes falsifiable predictions, while at the same time explaining several aspects of the standard model. It is not a multiverse theory - quite the opposite, it has no free parameters at all. All parameter values are determined by the algebra of the octonions.