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.