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.

Like

Comment

Share

Understanding quantum theory as an emergent phenomenon

When we say that there is an underlying theory from which quantum theory is emergent, we are interested in some expansion parameter, which can be used to relate the approximate theory to the more exact theory. The way speed of light relates special relativity to Newtonian mechanics, and \hbar relates quantum mechanics to classical mechanics.

However, the present case is not like these examples. We are not relating two mechanical theories. Quantum theory is an emergent phenomenon. It emerges as a statistical thermodynamics [equilibrium] approximation to the underlying dynamics which is trace dynamics. Just as the equilibrium thermodynamics of a fluid is emergent from the statistical thermodynamics of the underlying atomistic theory of molecular motion.

The coarse-graining parameter relating trace dynamics to the emergent quantum theory is a time scale. The underlying microscopic theory holds at Planck time scale resolution. When averaged over time scales much larger than Planck time, using the methods of statistical thermodynamics, the emergent theory is quantum theory.

The trace dynamics has a conserved charge of dimension of action

C = sum [q,p]_B - sum {q,p}_F

i.e. the sum over all bosonic degrees of freedom of the commutators [q_i, p_i] minus the sum over all fermionic degrees of freedom of the anti-commutators {q_i, p_i} At thermodynamic equilibrium this charge is equipartitioned, and each commutator /anti-commutator is set equal to \hbar. This is how QM emerges.

One way to glean information (about the microscopic dynamics) from the emergent theory is to study fluctuations about equilibrium.

Another way is to think of the QFT description of the standard model as a thermodynamic description. Which is what it actually is anyway, in this theory. The free parameters of the standard model are then getting determined and fixed by the underlying microscopic theory of the atoms of space-time-matter [octonionic theory / trace dynamics]. The fact that the underlying theory is able to do so is evidence that quantum theory is an emergent thermodynamic phenomenon.

But is there a control parameter analogous to speed of light, which lets us think of quantum theory as a leading order approximation to trace dynamics. It does not seem so. To my understanding, that is not how statistical thermodynamics works. But I could be wrong about this.

__________________

Attached is Einstein's prophetic view on quantum theory as an emergent phenomenon, as quoted in `Subtle is the Lord' [Pais]

Prof. Stephen Adler's book `Quantum theory as an emergent phenomenon' is a perfect reading of Einstein's mind. He befittingly held the chair Albert Einstein Professor of Physics at the Institute of Advanced Study, Princeton.

Removing the last bit of commutativity from our fundamental physical theories

In quantum theory, the configuration variable and it's canonical momentum become non-commutative [q,p] = i\hbar

An analogous non-commutativity is imposed in quantum field theory, on the field and its canonical momentum

When we quantise gravity, we invoke a commutator for the metric (or equivalent) and it's momentum

But in all this,

One thing which remains commutative, is the space-time manifold. Several quantum gravity theories work to remove this manifold, subsequent to the quantisation.

But suppose, with hindsight, we go back to basics, and make everything non-commutative in one go, retaining only Newton's absolute time (same as Connes' perhaps)

Replace the 4D Minkowski spacetime manifold by an 8D octonionic spacetime + absolute time

Nudge Newton and Einstein a bit. Raise all (suitably identified) q, p variables for fermions, gravity, gauge fields, to the status of matrices made of Grassmann numbers. Write down the following action principle.

If the claim holds, then this action describes a pre-spacetime, pre-quantum theory. From which QFT, standard model, and GR are emergent.

The Pebbles of Newton: gravitation and quantum theory as emergent phenomena

We are all the children of Newton. He gave us modern physics, while himself standing on the shoulders of Copernicus, Galileo and Kepler. Newton gave us calculus, inertia, forces, law of motion, law of gravitation, absolute space, absolute time.

It could be said that since his time, we have been incrementing Newton. The law of motion is derived from the principle of least action. Other forces have been discovered since, and the laws of motion have been generalised. However, I tend to disagree that relativity and quantum theory overthrew Newton; better to say there has been incremental generalisation.

Because if the attached action principle is to be believed, we are back to Newton, maybe a pebble little prettier, but very Newtonian still. There is absolute time, the Connes time. There is kinetic energy, and the same equation of motion as his; the second law. Gravity, electromagnetism, strong force and weak force have all been bundled together, as also relativity and quantum theory. His three dimensional space has become octonionic space, real numbers have been replaced by matrices made of Grassmann numbers. Maybe the picture will simplify further, and become a geometric algebra. But everything we have found since Newton is still very Newtonian, when captured in this attached action principle. Locality and causality are only emergent - in a refined Newtonian sense, we have action at a distance!!

PS: Adler's Trace Dynamics, from which the action principle below is inspired, already has a Newtonian feel to it, because it is a Lagrangian dynamics. However, it has a conserved charge, the Adler-Millard charge = Sum_i [q_i, p_i] which Newton's classical dynamics does not have, making this a pre-quantum theory. Newtonian in its Lagrangian dynamics, but more quantum than quantum theory! This action represents an elementary particle which carries all its fields around with itself, in a very particulate sense. When there is no commutative background physical space anymore, it may not be meaningful anymore to talk of fields. But a matrix-valued particle should be just fine.

Quantum Gravity in the Infra-red, and it's significance for the standard model of particle physics

When is a physical system quantum gravitational?

Not only at the Planck energy scale. But whenever:

(i) Length scale is of order Planck length

and/or

(ii) Time scale is of order Planck time

and/or

(iii) Action is order Planck constant \hbar

------------

If (iii) is satisfied but not (i) nor (ii), then it is QG in IR

This is why QG is important for the standard model, even in IR.

______________

(iii) along with (i) or with (ii) is QG in UV, as expected

---------

QG in IR and the standard model

The elementary particles of the standard model obey the rules of quantum field theory.

As a result, they are not localised in space, and obey the quantum superposition principle.

Hence, the spacetime geometry they produce, howsoever weak, is not classical. It is quantum gravitational.

We have found that this quantum gravitational field acts back on the elementary particles, and restricts what properties they can have. It enforces charge quantisation as well as specific mass ratios for elementary particles, the value 1/137 of the low-energy fine structure constant, and perhaps more.

This is not high energy physics. It is low energy quantum gravity - if we treat this gravity as classical, then that is approximate, and gives us the standard model as we know it. If we take the quantum gravity effect into account, as we should, we learn significantly more about the standard model, without going to high energies.

The existence of three light sterile neutrinos is a clear-cut low energy prediction of this theory.

-----------

Klein-Gordon equation is to Minkowski spacetime what Dirac equation is to quaternionic / octonionic spacetime

When we take the square-root of the Klein-Gordon equation to arrive at the Dirac equation, if we also want to ask what kind of space-time manifold does the electron produce / live in, we must take the square-root of Minkowski spacetime as well. The desired square-root is not a tetrad based description of Minkowski spacetime, because this does not gel with the position-momentum uncertainty inherent in the Dirac equation. Rather, the desired square-root spacetime is quaternionic / octonionic, and inspired by the homomorphisms SL(2,H) ~ SO(1,5) and SL(2,O)~SO(1,9). We then define the spinorial state of the electron, not on Minkowski spacetime but on the [spinorial by nature] quaternionic / octonionic spacetime. We do the same for the other elementary fermions. Soon as we do this, we start to understand why the standard model has the observed symmetries, and charge quantisation and more. The group of transformations of the octonions is the symmetry group of the square-rooted spacetime, and is more fundamental than the spacetime diffeomorphisms of Minkowski spacetime.

Reference: arXiv:2110.02062

Rethinking the relation between space, time, and matter

The Newtonian world-view is that of material objects, and light, embedded in three dimensional absolute physical space, and evolving through it in absolute time.

In special relativity, the world-view is that of objects and light and all electro-magnetic radiation evolving along world-lines in space-time. Space-time is absolute and causal, but spatial intervals and time intervals are no longer separately absolute.

In general relativity, the four dimensional space-time manifold is absolute and causal, but its geometry is determined by the distribution of matter and radiation.

In quantum theory, the background space-time is as given by general relativity, including a background distribution of classical matter and radiation, which co-exists with the curved 4D spacetime manifold. On this substrate, quantum systems [i.e. those with action order \hbar] evolve according to the rules of quantum theory. The time parameter in quantum theory is prescribed by the background space-time geometry.

If we get rid of the classical substrate of matter and radiation, and all systems now have action \hbar, we lose the absolute 4D classical space-time manifold and with it we lose its spacetime geometry. We also lose the distinction between matter-radiation and space-time. The universe is now made of atoms of space-time-matter (also known as aikyons). The action principle for an aikyon is shown in the attached screenshot. The action is measured in units of Planck's constant. The aikyon has an associated length scale L measured in units of Planck length, and it evolves in the absolute Connes time tau measured in units of Planck time.

The aikyon is described by a pair of dimensionless matrices Q_1 and Q_2 written on a sedenionic space, and this action is equivalent to three subsystems on an octonionic space [the three fermion generations]. Thus, although there is a space (octonionic space) there is no distinction between the space and the matter-radiation. The degrees of freedom that define the space are the same as those which define matter-radiation. This is unlike in any of the previous stages listed above. The length parameter is not free - it takes only a few discrete rational number values determined by the octonion algebra. This is a conformally invariant theory. The aikyon is so much like the string of string theory!

The universe is made from many copies of these aikyons. All the earlier stages listed above are supposedly emergent from here, including the standard model and its free parameters. Three right-handed sterile neutrinos are predicted.

Reference

https://arxiv.org/abs/2110.02062

What changes, when made to string theory, change it into a predictive, falsifiable theory?

1. After writing down the action principle and obtaining the equations of motion, one must not quantise the dynamics. The theory is already pre-quantum and pre-spacetime, and gravitation, and quantum theory, are emergent in a statistical thermodynamics approximation.

2. One must not assume the Minkowski vacuum to be the ground state of quantum gravity, and construct a Fock space over it. Just as the ground state of a harmonic oscillator has a non-vanishing wave function, and a non-vanishing zero point energy, it can be expected that the average Riemann curvature will be non-zero in the quantum gravitational ground state.

3. It has been found that when elementary particles are defined as excitations of the algebraic vacuum in the octonion algebra [idempotent / projector] then one makes immediate contact with the standard model and with Lorentz symmetry. There is a procedure for calculating the free parameters of the standard model.

Thus, if we do not quantise the string theory Lagrangian, but take it as a pre-quantum Lagrangian dynamics, and develop string theory on an octonionic space, the theory becomes predictive and falsifiable. The derivation of the fine structure constant and mass ratios could as well be called results of a resurrected string theory

Reference:

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

https://arxiv.org/abs/2110.02062

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.