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.

Relativistic weak quantum gravity, and it's significance for the standard model of particle physics

A question which has the attention of several experimentalists these days is: is gravity quantum? How to establish this through an experiment? A possibly doable experiment is to create a quantum superposition of two different position states of a massive object. And then to detect it's gravitational field and see if this field is non-classical, i.e. inconsistent with Newton's law of gravitation.

This is an example of non-relativistic weak quantum gravity. We might ask what could be an analogous relativistic scenario: relativistic matter fields in a quantum superposition of position states, and the consequent quantum gravitational fields they produce.

Nature already gives us such a setting - it is the standard model of particle physics! At low non-Planck energy scales. Say the electron satisfying the Dirac equation, which certainly obeys quantum linear superposition and we might be interested in the gravitational field it produces. How does it distort spacetime geometry? While carrying out an actual experiment could be next to impossible, we can try and make a theory for such relativistic weak quantum gravity, and see if we can predict something?

The electron is described by a spinorial wave function obeying the Dirac equation. It cannot be expected to produce a spacetime that is a superposition of many Minkowski spacetimes. Minkowski spacetime is vector based.

That is why we defined the electron state on a spinorial spacetime, described by quaternions, and more generally by the octonions. This spinorial spacetime is the square-root of Minkowski spacetime, same way that the Dirac equation is the square-root of the Klein-Gordon equation.

Remarkably, the non-commutative, non-associative nature of the square-root spacetime dictates the standard model of particle physics, predicting its observed symmetries, and predicting several of it's properties. For instance, quantisation of electric charge as observed [0, 1/3, 2/3, 1] and the value of the famous low-energy fine structure constant 1/137, and mass-ratios of the charged fermions. We interpret these findings as evidence for relativistic weak quantum gravity, and hence as evidence for the quantum nature of gravity. Further work is in progress.

From here, the extrapolation of the dynamics to higher energies can be carried out using conventional QFT on Minkowski spacetime. The significant new development is the realisation that the low energy free parameters of the standard model are being determined not by Planck energy scale physics, but at low energies itself. It is a quantum gravity effect in the infra-red. Relativistic weak quantum gravity comes into play whenever the matter field sources are relativistic, and quantum, i.e. having action of the order \hbar, and we want to know their spacetime geometry.

Reference: https://arxiv.org/abs/2110.02062

Is there a relation between Loop Quantum Gravity and the Octonionic Theory?

LQG builds on the SU(2) gauge invariant connection introduced by Ashtekar, which in turn is related to the triad [drei bein]. The setting is the ADM 3+1 formulation of general relativity, where the three-metric / triad is the configuration variable and the canonical momentum is made from the corresponding connection.

In the octonionic theory, would-be-gravity is the right-handed sector of the standard model, including three sterile neutrinos, and possessing the symmetry group SU(3)_grav X SU(2)_R X U(1)_grav The standard model is the left-handed sector SU(3)_c X SU(2)_L X U(1)_em Prior to the L-R symmetry breaking, which is also the electroweak symmetry breaking, there is unification of gravity and the standard model [also of electro and weak] in the framework of E_6 and E_8.

It is tempting and plausible that in spirit this SU(2)_R should be thought of as being the same as the SU(2) of LQG. Both describe gravity and Lorentz-invariant spacetime can be built by extending to SL(2,C). The SU(2) rotations are in isospin 3D space for the weak force, and in our familiar physical 3D space in the gravity case. There is a strong suggestion of gravity weak unification in the SU(2)_L X SU(2)_R ~ SO(4) sense, also suggested by our recent split biquaternion construction: SL(2,H) ~ SO(1,5) and writing of Cl(3) as a direct sum of two Cl(2)s - one for weak force, one for gravity. The split quaternion structure makes the weak interaction subtly different from gravity [only space, no time] and yet close enough to suggest gravi-weak unification in 6D spacetime.

If the above thoughts are correct, we see LQG nicely unifying with the standard model, and now also seen as a sub-structure in this revived description of string theory, with chiral fermions nicely added on to LQG. Other researchers have already pointed to the connection between Clifford algebras, fermions, something known as braids, and LQG. The present set of thoughts strengthens that connection. It also suggests that quantum gravity violates parity, and is right-handed, being the RH counterpart of the LH weak force. We also understand that parity violation is a consequence of symmetry breaking, and prior to that the parity symmetry is restored.

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