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.]