Where does a quantum particle such as an electron live? Is E_6 the sought for symmetry group of unification?

Where does a quantum particle such as an electron live?

No, it does not live in spacetime, not 4D nor a higher dimension like 10D.

A quantum system even at low energies is an `atom of spacetime-matter' described by a pre-quantum pre-spacetime matrix valued Lagrangian dynamics. The Lagrangian has a certain symmetry group, say/perhaps E_6, and this is where the electron lives. It lives in E6. Now E6 is not a spacetime symmetry, neither is it an internal SU(n) symmetry. Beautifully, it is a unification of both, quite befitting an atom of space-time-matter, which is what an electron is.

How then does our classical 4D spacetime emerge? When a large number of quantum degrees of freedom living in E6 get entangled, the quantum to classical transition takes place. These emergent classical matter degrees of freedom get confined to a 4D spacetime [a subgroup of E6] and their localisation defines and gives rise to a 4D classical spacetime. This is compactification without compactification.

Quantum systems continue to live in E_6. We can describe their dynamics, from the vantage point of the emergent classical 4D spacetime, in the conventional language of quantum field theory.

Hence one must never try to compactify the extra dimensions - in a sense these extra dimensions are complex dimensions; they are needed to describe the internal symmetries.

Is E_6 the sought for symmetry group of unification?

We made three copies of Left-Right symmetric fermions using three copies of complex split bioctonions. These are also three copies of the Clifford algebra Cl(7), each made from two copies of Cl(6). This suggests the complexified exceptional Jordan algebra [whose automorphism group is E_6] of 3x3 matrices with octonionic entries.

The following sub-group structure is suggested:

(i) SU(3)_color X SU(2)_L x U(1) : color-electro-weak

8x3 = 24 LH fermions and 12 SM bosons

(ii) SU(3)_grav X SU(2)_R x U(1) : pre-gravitation

8x3 = 24 RH fermions and 12 bosons (8 gravi-gluons, 1 dark photon, two Lorentz bosons, one Higgs boson).

(iii) SO(1,3) ~ SL(2,C) is 6 dimensional, shared by particles in (i) and (ii). The Higgs mediates between LH and RH fermions.

Do the numbers add up?

24 + 24 = 48 fermions (including 3 RH sterile neutrinos)

12 + 12 = 24 gauge bosons

6 generators for SO(1,3) Lorentz group of 4D spacetime

These add up to 48 + 24 + 6 = 78

Now, E_6 is 78 dimensional. So is this a good indication?

By computing the eigenvalues of the exceptional Jordan algebra for three fermion generations, and in conjunction with the Lagrangian for a pre-quantum pre-spacetime dynamics we derived the low energy fine structure constant, and mass ratios for charged fermions.

Critically entangled fermions form macroscopic classical objects and descend to 4D spacetime with Lorentz symmetry.

Quantum systems [entanglement is sub-critical] live in E_6, i.e. partly in 4D spacetime and partly beyond in the complex internal dimensions.

Perhaps these are good signs ... ?

[There is no dark matter nor a non-vanishing cosmological constant in this theory. The emergent spacetime geometry shows evidence in favour of MOND and RMOND, and a long-range modification of GR playing the role of dark energy.]

The tension between quantum mechanics and relativity, and what it signifies for the standard model of particle physics?

The principle of quantum linear superposition is well-tested for elementary particles such as electrons. The theory of special relativity is also well tested. However, the two are not consistent with each other, except in an approximate sense.

We realise this when we ask the question: what kind of spacetime geometry and gravitation is produced by an electron? It is tempting to think of the produced gravitation as a very tiny (quantum) perturbation of the flat spacetime of special relativity. However this cannot be correct, precisely because of the validity of the principle of quantum linear superposition.

Imagine for simplicity that the electron state is a superposition of two localised position states A and B. Had the electron been at A, it will produce a perturbation of flat space time, with the perturbation peaked at A. Similarly if it had been at B, the produced perturbation would have been peaked at B. If we now consider a point X in space, the field there will be a superposition of the fields due to the location A and the location B. Such a field is not a perturbation of flat spacetime ! If we want to write down the Schrodinger equation for a test particle, then what time parameter to use at location X? That determined by the clock rate fixed by location A or by location B? There is ambiguity. The concept of a classical spacetime is lost [even in a perturbative sense] the moment we consider the implications of quantum superposition for spacetime geometry. In order to describe quantum spacetime - the one produced by an electron - we must entirely give up on the flat spacetime of relativity, even at low energies.

But quantum field theory on a flat spacetime works extremely well. We are able to construct the highly successful standard model of particle physics, assuming the flat spacetime of relativity, and assuming Lorentz invariance. Does that not imply that flat spacetime is an excellent approximation in the limit of low gravity? No, it does not imply that. There are 26 free parameters in the standard model, which have to be put in by hand, after measuring them experimentally. Could it be that these parameters get determined uniquely, and are not arbitrary, if we describe the standard model, not on flat spacetime, but on that spacetime which elementary quantum particles produce. Gravity is tiny, but it is not a perturbation of flat spacetime. What then is it a perturbation of?

The gravity is a perturbation of a [non-commutative] spinor spacetime, from which the flat spacetime of relativity is derived as an approximation. When we describe the standard model on this spinor spacetime, we find evidence that the parameters of the standard model are not free, but take values as measured in experiments. Such a noncommutative spinor spacetime is compatible with the quantum superposition principle. We could say that when we take the square root of the Klein-Gordon equation to arrive at the Dirac equation, we should also take the square root of flat spacetime so as to arrive at the noncommutative spinor spacetime. Then we write down the Dirac equation on this spinor spacetime. Right away we find that electric charge is quantised [0, 1/3, 2/3, 1 : neutrino, down quark, up quark, electron]. And that the low energy fine structure constant is 1/137. It cannot be any other value.

Are scalar lepto-quarks [if they exist] evidence for (revised) string theory?

In the octonionic theory, prior to the Left-Right symmetry breaking [=EW symmetry breaking] mass and electric charge are not defined. Both these are emergent concepts. The fundamental entities from which quarks and leptons emerge are lepto-quarks, and although spin angular momentum can be defined from the matrix-valued Lagrangian dynamics, spin is not quantised in the pre-stage. Only a length scale is associated with a lepto-quark, and this is order unity in Planck length units.

It might then be plausible to identify these lepto-quark states as the fundamental extended objects of the octonionic theory, and one way to think of the octonionic theory is as a revised string theory. If current experiments are pointing to violation of Lepton Flavour Universality, and if this violation is due to the existence of lepto-quark states, then the BSM physics we are seeing might be evidence for octonionic theory/string theory. The Planck scale is reset to the electro-weak scale by the inflationary cosmological expansion just after the big bang.

A few popular references on the octonionic theory

If you are interested in the evolution of the octonionic theory, the following are some popular video lectures I made, some talk recordings, a blog, and a semi-technical 2017 article. Only in 2020, the octonions got added to this attempt at quantum gravity. Since then it has become an attempt at quantum gravity and unification.

Does nature play dice? (2014)

https://www.youtube.com/watch?v=wSiDsMKS_uU&ab_channel=TejinderSingh

Spontaneous quantum gravity (2019)

https://www.youtube.com/watch?v=lJk_mE8K8uw&ab_channel=TejinderSingh

Aikyons, octonions and unification (2020)

https://www.youtube.com/watch?v=RmrkmGVUzo4&ab_channel=TejinderSingh

Octonions, elementary particles and the unification of forces (2021)

https://www.youtube.com/watch?v=kL_JnD2PxJI&ab_channel=PhysicsandAstronomyClub%2CIITD

Elementary particles and the magic of the octonions (2021)

https://www.youtube.com/watch?v=bdyCW49uydg&ab_channel=PyCalcXOrganization

Quantum gravity and the atoms of space-time-matter (2021)

https://www.youtube.com/watch?v=BIEMTbgtzdk&ab_channel=SpaceOdyssey

Gravitation: from Newton to Padmanabhan and beyond (2021)

https://www.youtube.com/watch?v=W6OKgnOW1Uk&ab_channel=BREAKTHROUGHSCIENCESOCIETYINDIA

Thinking about quantum gravity, video lecture series (2018)

https://www.youtube.com/watch?v=-TleM9cGBK4&list=PLdQ8xQzqTQoYfp0I-JdyB880gfpinZ1mX&ab_channel=TejinderSingh

This blog is also helpful:

https://qfqg.blogspot.com/

Helpful article

https://arxiv.org/abs/1707.01012

Elementary particles, black holes, and the octonions

In this approach, quantum theory, and gravitation, both are emergent thermodynamic phenomena, emerging from the same underlying, more general, theory.

If they are both emergent thermodynamic phenomena, in what way do they differ from each other?

Quantum theory (without classical time) holds at thermodynamic equilibrium, and evolution is unitary.

If there is critical entanglement amongst the fermions in the underlying theory, non-unitary evolution becomes significant, spontaneous localisation results in the formation of a black hole, and in emergence of classical spacetime obeying laws of gravitation. This is a far from equilibrium state, at the opposite end from quantum theory. Elementary particle and black hole are two related fermionic states. In particular the charged spinning black hole [Kerr-Newman] has the same gyromagnetic ratio (g) as the electron satisfying the Dirac equation; both being twice the classical value. [With the understanding that interactions will make the electron g value depart from 2].

Gravitation is hence a far from equilibrium thermodynamic phenomenon. A black hole radiates so as to return to thermodynamic equilibrium, described by unitary quantum evolution.

At equilibrium the elementary particles live in 8D octonion space, evolving in Connes time. They obey a Left-Right symmetric dynamics which is an extension of the standard model, with the RH symmetry being the precursor of gravitation. This is true at all energy scales.

This dynamics could be described on the background of the 4D classical spacetime provided by those fermions which have already undergone localisation to form black holes. Such a description justifiably ignores the pre-gravity of the unlocalised fermions, and is given by our conventional QFT for the standard model, which has LH symmetry for the weak interaction. The RH symmetry - the precursor of gravity - is very hard to detect because gravity is so weak, but this RH symmetry is present, even at low energies.

It could well be, keeping in mind that space and time interchange roles inside a black hole, that the inside of the black hole is one half of a split octonion space, the other half being the exterior spacetime. Let us call the inside of the black hole the inner half space. It seems to us that a BH interior is embedded in 4D spacetime; but it is not! It is an extension of the external 4D spacetime to four additional dimensions.

Elementary particles live in the entire 8D octonion space.

Non-black-hole classical objects have a penetration depth (into the inner half space) much less than Planck length. They occupy the ordinary 4D spacetime.

The BH interior maximally occupies the inner half space, distinct from the exterior 4D spacetime - the other half.

By emitting Hawking radiation a black hole returns towards thermodynamic equilibrium and towards unitary quantum evolution in the 8D octonion space.

The octonionic theory vs. stringy multiverses

In the octonionic theory, there are no free parameters and no fine tuning. The trace dynamics Lagrangian and the algebra of the octonions are supposed to determine all the free parameters of the standard model. In fact if we have to ever to introduce some freedom / fine tuning, we would prefer to discard the theory in totality, instead of trying to save it by some quick-fix. The octonionic theory transforms string theory into a predictive theory by retaining extended objects but modifying the dynamics.

The multiverse / anthropic / landscape way of making string theory `predictive' is the exact opposite of the octonionic theory. String theory can give rise to a very large number of different kinds of universes, with different values for the fundamental parameters. Our universe has it's particular parameter values because these values allow life to arise. Theory wise there is nothing special about these values, just as there is nothing special theoretically about the distance of the earth from the sun.

I think most people will agree that the first way of saving string theory, if it works, is the preferred one. I think most people will also agree that proponents of the second method should carefully examination proposals in the first category, when such proposals are brought to their attention.

I am at a loss to understand on what scientific basis string theorists have concluded that a solution of the first type is impossible.

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.