Why do elementary particles have such strange mass ratios? -- The role of quantum gravity at low energies

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infrared. Using the octonions as space-time coordinates, we construct a pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with $SU(2)_R$ chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time.


I. WHEN IS QUANTUM GRAVITY NECESSARY?
Consider a massive object in a quantum superposition of its two different classical position states A and B. The resulting gravitational field is also then in a superposition, of the field corresponding to position A and the field corresponding to position B. A clock kept at a field point C will not register a definite value of time, nor a measurement of the metric will yield a well-defined result [1]. Let us now imagine a thought experiment in which every object in today's universe is in a superposition of its two different position states. The space-time metric will then undergo quantum fluctuations. Now, the Einstein hole argument shows that in order for space-time points to be operationally distinguishable, the manifold must be overlaid by a (classical) metric [2]. Therefore, in our thought experiment, the point structure of space-time is lost, even though the energy scales of interest are much smaller than Planck scale, and the gravitational fields are weak.
When we describe microscopic systems by the laws of quantum theory, we take it for granted that the universe is dominated by classical bodies, so that a background space-time can be achieved and is available for defining time evolution of quantum systems. However, if everything were to be quantised at once, in the sense of the afore-mentioned thought experiment, no classical time will be available, and yet we ought to be able to describe the dynamics. This is an example of quantum gravity in the infra-red: the action of the gravitational field is much larger thanh (unlike for Planck scale quantum gravity), and yet the point structure of space-time is lost. The manifold has to be replaced by something non-classical: quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time.
Since the energy scale is not a relevant criterion for deciding whether gravity is classical or quantum, we propose that a gravitational field is quantum in nature when one or more of the following three (energy independent) criteria are satisfied: (i) the time scales of interest are of the order of Planck time t P ; (ii) the length scales of interest are of the order of Planck length L P ; and (iii) every sub-system has an action of the orderh (and is hence quantum and obeys quantum superposition). If (iii) holds but (i) and (ii) do not, we have quantum gravity in the infra-red. If (iii) holds along with (i) and (ii) then we have quantum gravity in the UV.
Put differently, there ought to exist a reformulation of quantum (field) theory, even at low energies, which does not depend on classical time. Such a reformulation is essential also for the standard model of particle physics. In fact we show that it helps us understand why the standard model has the symmetries it does, and why it's free parameters take the specific values they do, and also shows how to unify gravity with the other fundamental forces: electroweak and strong. We construct such a dynamics using Planck time t P , Planck length L P and Planck's constanth as the only three fundamental parameters in the theory. We note that in these units the low energy fine structure constant α f = e 2 /hc ≡ e 2 t P /hL P ∼ 1/137 is order unity and hence quantum gravitational in origin (QG in IR). On the other hand, particles masses m ∼ m P ≡ ht P /L 2 P are not, because 1. However, mass ratios (at low energies) can be, and in fact are, quantum gravitational in origin.
To achieve our goal, we build on Adler's pre-quantum theory, i.e. trace dynamics (TD) [3,4]. For a detailed explanation of the emergence of the classical universe the reader is referred to Section XIII of [9].
II. REPLACING THE POINT STRUCTURE OF SPACETIME BY THE NON-

COMMUTATIVE GEOMETRY OF THE OCTONIONS
Next, TD is generalised, so as to replace the 4D Minkowski space-time manifold by a higher dimensional non-commutative space-time, and incorporate matrix-valued pre-gravitation, thus taking TD to a pre-space-time, pre-quantum theory. Let us recall that in special relativity, given the four-vector V µ = dtt + dxx + dyŷ + dzẑ connecting two neighbouring space-time points having a separation (dt, dx, dy, dz), one can define the line element ds 2 = η µν V µ V ν and the four-velocity dq µ /ds of a particle having the configuration variable q µ = (q t , q x , q y , q z ). The action for the particle is mc ds and the transition to curved space-time and general relativity is made by introducing the metric g µν , i.e. ds 2 = g µν dx µ dx ν , and writing down the action Here, the first term is the Einstein-Hilbert action, and S Y M stands for the action of Yang-Mills fields, and also includes their current sources.
We now generalise this action to construct a pre-spacetime, pre-quantum action principle [5] from which the sought for quantum theory without classical time emerges, and whose symmetries imply the standard model of particle physics and fix its free parameters. The space-time coordinates (t, x, y, z) are replaced by a set {e i , i = 0, 1, 2, ..., m − 1} of m noncommuting coordinates, to be specified later in this section. The configuration variable q µ for a particle is replaced by a matrix q F whose entries are odd-grade Grassmann elements over the field of complex numbers (so as to represent fermions). q F has m components q i F , one for each of the coordinates e i , i.e. q F = (q 0 F e 0 + ....q (m−1) F e m−1 ). The point structure of spacetime is lost; instead we have a non-commutative geometry, and the matrix-valued velocity is defined as dq F /dτ ≡q F . Here, the newly introduced Connes time τ is a unique property of a non-commutative geometry; it is an absolute real-valued time parameter distinct from the non-commuting coordinates e i , and is used to describe evolution [6].
To introduce pre-gravitation into trace dynamics, we recall the spectral action principle of Chamseddine and Connes, according to which the Einstein-Hilbert action can be cast in terms of the eigenvalues of the square of the (regularised) Dirac operator D B on a Riemannian manifold, by making use of a truncated heat kernel expansion [7] T r [L 2 Here, the eigenvalues λ n of the Dirac operator play the role of dynamical variables of general relativity [8]. Following trace dynamics, each eigenvalue λ n is raised to the status of a canonical matrix momentum: λ n → p Bn ∝ q Bn /dτ ≡ D B , and the bosonic matrix q B (with even grade Grassmann elements as entries) is now the configuration variable, and it has m matrix components q m B over the non-commuting coordinates e i . Therefore we have N copies of the Dirac operator (n runs from 1 to N , with N → ∞). The trace Lagrangian [space-time part] of the matrix dynamics for the n-th degree of freedom is given by L 2 P T r (dq Bn /dτ ) 2 . The full action for the total matrix dynamics [space-time part] is S ∼ n dτ L 2 P T r (dq Bn /dτ ) 2 . Yang-Mills fields are expressed by the matrices q Bn , pregravitation by theq Bn , the fermionic degrees of freedom by fermionic matrices q F n and by their 'velocities'q F n . Each of the n degrees of freedom has a fundamental action, which is given by [9,22] where a 0 ≡ L 2 P /L 2 . The net action of this generalised trace dynamics is therefore the sum over n of N copies of the above action, one copy for each degree of freedom, and this new action replaces (1) in the pre-theory. This full action defines the pre-spacetime, prequantum theory, with each degree of freedom [defined by the above action] considered as an 'atom' of space-time-matter [an STM atom]. L is a length parameter [scaled with respect to L P ; q B and q F have dimensions of length] which characterise the STM atom, and α is the dimensionless Yang-Mills coupling constant. β 1 and β 2 are two unequal complex Grassmann numbers [9].
The subsequent analysis of this pre-space-time, pre-quantum theory is carried out analogously to the pre-quantum trace dynamics. Equations of motion are derived, and there is again a conserved Noether charge. Assuming that the theory is valid at the Planck time scale, the coarse-grained emergent low-energy approximation obeys quantum commutation rules and Heisenberg equations of motion, and this is also the sought for reformulation of quantum theory without classical time. The emergent dynamics is also the desired quantum theory of gravity in the infra-red. If a sufficient number of STM atoms get entangled, the anti-self-adjoint part of the Hamiltonian becomes important, and spontaneous localisation results; the fermionic part of the entangled STM atoms gets localised. There hence emerges a 4D classical space-time manifold (labelled by the positions of collapsed fermions), which is sourced by point masses and by gauge fields, and whose geometry obeys the laws of general relativity given above by (1). Those STM atoms which are not sufficiently entangled continue to remain quantum; their dynamics is described by the low energy pre-theory itself, or approximately by quantum field theory on the 4D space-time background generated by the entangled and collapsed fermions [these being the macroscopic bodies of the universe].
Note that the non-commutative coordinate system {e i ; i = 1, 2, ..., n} is not impacted by the coarse-graining. The averaging takes place only over the time-scale τ and hence over energy; therefore the non-commuting coordinates e i remain valid at low energies as well. What then, should we choose as our e i , in place of the four real numbers (t, x, y, z) with the Lorentz symmetry. Whereas, the octonions seem to be just right for that purpose! An octonion is defined as O = a 0 e 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 + a 4 e 4 + a 5 e 5 + a 6 e 6 + a 7 e 7 such that the a i are reals, e 2 0 = 1, each of the seven imaginary directions (e 1 , e 2 , ..., e 7 ) squares to −1, these directions anti-commute with each other, and their multiplication rule is given by the so-called Fano plane. Octonionic multiplication is non-associative. The imaginary directions form the automorphism group G 2 , which is the smallest of the five exceptional Lie groups G 2 , F 4 , E 6 , E 7 , E 8 all of which have to do with the symmetries of the octonion algebra. F 4 is the automorphism group of the exceptional Jordan algebra: the algebra of 3 × 3 Hermitean matrices with octonionic entries, and E 6 is the automorphism group of the complexified exceptional Jordan algebra [10]. The octonions are our sought for noncommuting coordinates e i on which the action principle (3) is constructed. They generate 10D space-time : SL(2, O) ∼ SO (1,9). The coordinate geometry of the octonions dictates the allowed symmetry groups, and definition and properties of fermions such as quantisation of electric charge [11], value of the low energy fine structure constant [12], and mass-ratios [15]. The parameter L and the coupling constant α in (3) are determined by the algebra of the octonions, not by the dynamics of q F and q B . This way, not only does the geometry tell matter how to move, it also tells matter what to be. The dynamical variables (q B , q F ) curve the flat geometry {e i }; however even before the dynamics is switched on, the low-energy standard model of particle physics is fixed by the e i , unlike when space-time is R 4 . The transition e i → q i F e i + q i B e i is akin to the transition η µν x µ x ν → g µν x µ x ν , with the important difference that the former transition takes place at the 'square-root of metric' level, as if for tetrads, and the matrices q B and q F incorporate standard model forces besides gravity, and also fermionic matter. In fact, with the redefinition˙ (3) can be brought to the elegant and revealing form, as if describing a two-dimensional (because β 1 = β 2 ) free particle: In this fundamental form of the action, the coupling constant α is not present. In fact α, along with mass ratios, emerges only after (left-right) symmetry breaking segregates the unifying dynamical variable˙ Q B into its gravitational partq B and Yang-Mills part q B .

III. SPINOR STATES FOR QUARKS AND LEPTONS, FROM THE ALGEBRA OF THE COMPLEX OCTONIONS
The automorphism group G 2 of the octonions has two maximal sub-groups SU (3) and SO(4) ∼ SU (2) × SU (2), the first of which is the element preserver group of the octonions, and the second is the stabiliser group of the quaternions inside the octonions [16]. The two groups have a SU (2) intersection. Keeping one of the seven imaginary directions, say e 7 , fixed, the remaining six directions can be used to form an MTIS (maximally totally isotropic subspace) and the following generators (along with their adjoints) for the Clifford algebra Cl (6): (This is a covariant choice as all the imaginary directions are equivalent and interchanging any of them does not change the analysis or results). From here, one can construct spinors as minimum left ideals of the algebra, by first constructing the idempotent ΩΩ † where Ω = α 1 α 2 α 3 . The eight resulting spinors are After defining the operator Q = (α † 1 α 1 + α † 2 α 2 + α † 3 α 3 )/3 as one-third of the U (1) number operator we find that the states V and V e+ are singlets under SU (3) and respectively have the eigenvalues Q = 0 and Q = 1. The states V ad1 , V ad2 , V ad3 are anti-triplets under SU (3) and have Q = 1/3 each, whereas the states V u1 , V u2 , V u3 are triplets under SU (3) and each have Q = 2/3. These results allow Q to be interpreted as electric charge, and the eight states represent a neutrino, three anti-down quarks, three up quarks and the positron having the standard model symmetries SU (3) color × U (1) em . Anti-particle states are obtained by complex conjugation. The eight SU (3) generators can also be expressed in terms of the octonions and represent the eight gluons, whereas the U (1) generator is for the photon.
We hence see the standard model of particle physics emerging from the symmetries of the physical octonionic space, and the quantisation of electric charge is a consequence of the coordinate geometry of the octonions [11].
To see how the weak force (and electroweak) and chiral gravity emerge from the other maximal sub-group SO(4) ∼ SU (2)×SU (2) we must consider three fermion generations and the larger exceptional Lie group E 6 because these symmetries are shared pair-wise across fermion generations, as shown in Fig. 1. Furthermore, the neutrino will be assumed to be Majorana, because only then the correct values of mass ratios are obtained [15]. Also, notably, E 6 is the only one of the exceptional groups which has complex representations.

METRIC EXTENSION OF THE STANDARD MODEL OF PARTICLE PHYSICS
The 78 dimensional exceptional Lie group E 6 is the automorphism group of the complexified Jordan algebra, and admits the sub-group structure shown in Fig. 1, as motivated by the discussion in [17]. E 6 contains three intersecting copies of Spin(9, 1) ∼ SL(2, O) which have an SO(8) intersection, and the triality property of SO(8) motivates that there are exactly three fermion generations. In order to account for the symmetries of E 6 and to obtain chiral fermions, we now work with split bioctonions (instead of octonions, which are used in (5) i.e. before symmetry breaking) [18]. Embedded in the three Spin(9, 1) are  [20]. These obtain the respective square-root mass number (0, 1/3, 2/3, 1) explaining why the down quark is nine times heavier than the electron. The SU (2) R is RH chiral gravity (LQG?) [19] which reduces, in the classical limit induced by spontaneous localisation, to general relativity. The Lorentz group (whose Casimir invariant is the introduced mass number) is common with the LH particles, and its 6 dimensions, together with the 24 RH fermions and 12 new gauge bosons, when added to the LH sector, give the correct count of 78 for E 6 . The split complex number gives a scalar field which acts as the Higgs mediating between the LH charge eigenstates, and RH mass eigenstates. Is U (1) gravity the dark energy? This possibility is discussed further in Section V.
The group E 6 is also the symmetry group for the Dirac equation in 10D [17] for three fermion generations (either LH or RH). The eigenvalue and eigenmatrix problem for the Dirac equation is in fact the same as J 3 (8)X = λX where J 3 (8) is the exceptional Jordan algebra with symmetry group F 4 . Substituting the above-mentioned spinor states of LH fermions (these being eigenstates of electric charge) and solving this eigenvalue problem expresses the LH charge eigenstates as superpositions of the RH mass eigenstates (thus fixing α and L in (3)), and the ratios of the eigenvalues yield mass ratios of charged fermions as shown in Fig. 2; these exhibit very good agreement with the mysterious mass ratios, as shown in the table below [15].    Table I: Comparison of theoretically predicted square-root mass ratio with experimentally known range [15] Apart from the two mass ratios of charged leptons, other theoretical mass ratios lie within the experimental bounds [13]. On accounting for the so-called Karolyhazy correction [14] we might possibly get more accurate mass ratios for all particles including charged leptons.
This will be investigated in future work.
In quantum theory, even at low energies, assuming a point structure for spacetime is an approximation; it is because of this approximation that the standard model of particle physics has so many unexplained free parameters. When we replace this approximate description by a non-commutative spacetime, we find evidence that these parameter values get fixed. In particular, we derive the low energy fine structure constant [12,14] and mass ratios of charged fermions [15] from first principles. We do not need experiments at ever higher energies to understand the low energy standard model. Instead, we need a better understanding of the quantum nature of spacetime at low energies, such that the quantum spacetime is consistent with the principle of quantum linear superposition.
V. FURTHER DEVELOPMENTS, CLARIFYING REMARKS, AND CURRENT

STATUS OF THE PRESENT UNIFICATION PROGRAMME
The aforesaid essay is intended to give the reader a short overview of a new approach to quantum gravity and unification, details of which can be found in [9]. In the present section we report on a few new insights not described in our earlier work, and provide clarifying details on some of the statements in the previous sections.
One further way to motivate the present theory is to recall that when one takes the square root of the Klein-Gordon equation to arrive at the Dirac equation for spin-half fermions described by spinors, one does not take the corresponding square root of the four dimensional Minkowski spacetime labelled by four real numbers. But suppose one were to take the latter square root as well; then one arrives at a spinor description of spacetime, i.e. Penrose's twistor space, labelled by complex numbers, and one notes that SL ( [20]. A detailed investigation of the Lagrangian as regards its particle content is currently in progress.q B is the Dirac operator on octonionic space and q B , the Yang-Mills field, is the correction to the Dirac operator, as in conventional quantum field theory. The spectral action principle tells us the classical limit of the trace of the Dirac operator squared, when Yang-Mills fields are present. This classical limit has been discussed briefly in our earlier work [22] and is given by the following equation, from [23] where D Bnew ≡ D B + αA is the corrected Dirac operator resulting after including the Yang-Mills potential A: This is the expansion of the squared Dirac operator when gauge fields are included alongside gravity. We do not yet take into account the volume term, and conformal gravity, and Gauss-Bonnet gravity in our present work. It is however an issue of great significance that in the classical limit, general relativity is being modified by conformal gravity, and encourages us to relate our theory to MOND and RMOND, as an alternative to cold dark matter.
Interestingly, there so far seems to be no cold dark matter candidate particle in our theory, and MOND and sterile neutrinos seem to arise naturally in our approach to unification; please see section XIV of [9]; further work is currently in progress in this direction.
The matrix-valued equations of motion are easily written down after first defining where we have set a 0 ≡ L 2 P /L 2 . In terms of these two variables, the above trace Lagrangian can be written as where a 1 ≡h/cL P . The last term in the trace Lagrangian is a total time derivative, and hence does not contribute to the equations of motion, so that we can get dynamics from the Lagrangian: The canonical momenta are given by The Euler-Lagrange equations of motion arë In terms of these two complex variables, the 2-brane behaves like two independent complexvalued oscillators. However, the degrees of freedom of the 2-brane couple with each other when expressed in terms of the self-adjoint variables q B and q F . This is because q 1 and q 2 both depend on q B and q F , the difference being that q 1 depends on β 1 and q 2 depends on The trace Hamiltonian is and Hamilton's equations of motion arė It is understood that this generalised trace dynamics is defined over complex bioctonionic space, and elementary particles and gauge bosons including those for pre-gravitation are special cases of these dynamical variables, reminiscent of the different vibrations of the string in string theory. Thus, by bringing in gravitation, and trace dynamics, our work significantly expands the scope of earlier related research. This new scenario is summarised in Fig. 3. Below, we elaborate on some aspects of these new developments. Note that it could instead be the second fermion generation, or the third generation. Each generation has the same charge ratio (0, 1/3, 2/3, 1).
This same analysis can now be used to show that the square-root of the masses of electron, up and down are in ratio 1:2:3 All we have to do is to identify the eigenvalues of the number operator with the square-root of the mass of an elementary particle, instead of its electric charge. And we also get a classification of matter and anti-matter, after noting that complex conjugation now sends matter to anti-matter, as follows: However, the second and third fermion generations do not have the simple mass ratios (1,4,9) unlike the electric charge ratios which are same for all three generations. Why so?! Because mass eigenstates are not the same as charge eigenstates. We make our measurements using eigenstates of electric charge; these have strange mass ratios, e.g. the muon is 206 times heavier than the electron. If we were to make our measurements using eigenstates of square-root mass, we would find that all three generations have the mass ratios (1,4,9) whereas this time around the electric charge ratios will be strange. There is a perfect duality between electric charge and square-root mass. A free electron in flight -is it in a charge eigenstate or a mass eigenstate? Neither! It is in a superposition of both, and collapses to one or the other, depending on what we choose to measure. In fact the free electron in flight does not separately have a mass and a charge; it has a quantum number which could be called charge -sqrt mass, which is the quantum number for the unified force.
Unification is broken by measurement: if we measure EM effect then we attribue electric charge to the source. If we measure inertia or gravity, we attribute mass to the source. These statements are independent of energy scale. A classical measuring apparatus emerges from its quantum constituents as a consequence of sufficient entanglement: the emergence of such classical apparatus is the prelude to breaking of unification symmetry. In the early universe, sufficient entanglement is impossible above a certain energy [possibly the EW scale] and it appears as if symmetry breaking depends on energy. This is only an indirect dependence.
The true dependence of symmetry breaking is on the amount of entanglement. In our current low energy universe we have both low entanglement systems (quantum, unified) and high entanglement systems (classical, unification broken). From the first three rows, we see that the electron has the square-root mass ratio e=B1 = 1/3. The muon has the square-root mass ratio mu/e=(C1/A1)/(I1/G1) and this same ratio also holds for tau/mu.

Some further insights into
These ratios above are the same as those shown earlier in Fig. 2 We believe that the octonionic theory provides a reasonably good understanding of the observed mass ratios of charged fermions. The mixing of the down quark family and the electron family is possibly the result of a gauge-gravity duality and of the fact that the three generations are not entirely independent of each other but related by the triality property of SO(8) [20]. Another remarkable feature we observed is that the eigenmatrices corresponding to the Jordan eigenvalues for the charged fermions always have the diagonal entry as 1/3, irrespective of whether the associated quantum number is 1/3 or 2/3 or 1. This seems to suggest that all charged fermions are made of three base states that all have an associated quantum number 1/3. The possible consequences of these observations are currently being investigated.
Octonions and the Koide formula: The Koide formula is the following observation for the experimentally measured masses of charged leptons That is, this ratio is close to (and a little less than) 2/3, but not exactly 2/3. Remarkably, the octonionic theory explains when the ratio is exactly 2/3, and why it departs from that exact value.
Prior to the L-R symmetry breaking, we can consider that a left-handed-electron-righthanded-electron state has an associated electric-charge-square-rot-mass of 1, and the neutrino is a Dirac fermion. In this case, the Jordan eigenvalues, as we mentioned above, are (1− 3/2, 1, 1+ 3/2). These give the superposition amplitudes when RH mass eigenstates are expressed as superposition of LH mass eigenstates. The Koide ratio is then Thus the exact value 2/3 is realised prior to symmetry breaking and prior to when the RH electron and RH down quark switch places. This switch might help understand why the mass ratios for charged leptons know about the Jordan eigenvalues (1 + 3/8) and (1 − 3/8) which are otherwise associated with the down quark family.
Using our theoretical mass ratios for the charged leptons as reported in Fig. 2 we get the following theoretically predicted value for the Koide ratio which is greater than the experimentally measured value of the Koide ratio and also greater than 2/3. The departure from the exact value of 2/3 is a consequence of the L-R symmetry breaking and of the switch between the RH down quark and RH electron. (It remains to be seen if the Karolyhazy correction will predict an exact match between theory and experiment).
Since unification already takes place at low energies (i.e. whenever the system is quantum and not yet measured upon) it follows that before we make a measurement on the charged leptons to measure their masses, the Koide ratio is exactly 2/3. After the measurement is made, the theoretical prediction for the resulting value is 0.669163, whereas the measured value is smaller than 2/3. The uncertainty in the mass of the tau-lepton 1776.86 (12) MeV is such that by demanding the Koide ratio to be 2/3 one can predict the mass of the tau-lepton to be 1776.969 MeV. At the upper limit 1776.98 MeV of the experimentally measured taulepton mass, the ratio is 0.66666728706, i.e. larger than 2/3, but smaller than our predicted theoretical value for Minkowski spacetime (the value realised after measurement).
The above is an important result as we now know when the Koide ratio is exactly 2/3 [it is when the electron is not being observed]. And we understand why the measured value of this ratio is not exactly 2/3. In principle, we could have demanded the measured value to be equal to the theoretical value, and thereby fix the mass of the tau-lepton. It turns out there are no such allowed values for the tau-lepton mass, which is further evidence that the measured value could lie between the spinor spacetime value (2/3) and the Minkowski spacetime value if the mass of the tau lepton is greater than 1776.969 MeV. This is also indirect evidence that sterile neutrinos exist.
Why is matter electrically neutral? When L-R symmetry breaking mechanism in the early universe separated matter from anti-matter, particles were segregated from their anti- In fact spacetime itself, along with gravitation, emerge after this symmetry breaking, as a result of the quantum to classical transition. Spacetime emerges iff classical matter emerges.
Prior to the symmetry breaking, dynamics is described by trace dynamics, there is no spacetime, and we have 'atoms' of space-time-matter. The concepts of electric charge and mass are not defined separately; there is only a charge-square-root mass [a hypercharge can also be defined, as for EW] and this is the source for a unified force in octonionic space.
Octonions, scale invariance, and a CPT symmetric universe: a possible explanation for the origin of matter-antimatter asymmetry: In the octonionic theory, prior to the so-called left-right symmetry breaking, the symmetry group is E 8 × E 8 and the Lagrangian of the theory is scale invariant. There is only one parameter, a length scale, which appears as an overall multiplier of the trace Lagrangian. Something dramatic happens after the symmetry breaking. Three new parameters emerge, to characterise the fermions: Electric charge, has two signs, sign change operation C is complex conjugation, and ratios are (0, 1/3, 2/3, 1). Chirality / spin, has two signs, sign change operation P is octonionic conjugation. Ratios (1/2, -1/2). Square-root of mass, has two signs, sign change operation T is time reversal t → −t . Ratios are (0, 1/3, 2/3, 1). Thus there are 2x2x2 = 8 types of fermions, based on sign of charge, square-root mass, and spin. This could possibly offer an attractive explanation for the origin of matter-antimatter asymmetry: a CPT symmetric universe. The four types of fermions which have positive square root mass become matter, our universe, moving forward in time. The other four types of fermions, which have negative square root mass, become anti-matter, a mirror universe moving backward in time! The forward moving universe and the backward moving universe together restore CPT symmetry. Our universe by itself violates T, and hence also CP. Matter and anti-matter repel each other gravitationally, thus explaining their separation. This also explains why gravitation in our universe is attractive, even though the underlying pre-gravitation theory is a vector interaction. Prior to the symmetry breaking, an octonionic inflation [scale invariant, time-dependent in Connes time] precedes the 'big bang' creation event, which is the symmetry breaking itself. Freeze out happens when radiation → matter-antimatter is no longer favorable. Segregation takes place; our matter universe has a one in a billion excess of matter over anti-matter. The backward in time mirror universe has a one in a billion excess of anti-matter over matter.
The maths of complex octonions naturally accounts for the C, P, T operations. Scale invariance is transformed into CPT invariance in the emergent universe. We hope to make this idea rigorous in forthcoming investigations. In an elegant proposal, Turok and Boyle [37] have also recently proposed a CPT symmetric universe [mirror universes]. They, however, did not use the octonions.

Prospects of tests through particle physics experiments and phenomenology: In
Section XV of [9] we have briefly discussed some of the possible experimental predictions of this theory and prospects for their experimental tests. Below we discuss some particle Our theory predicts specific new particles, though much work remains to be done to predict their specific properties such as masses. We predict that the neutrino is Majorana and that there are three right-handed sterile neutrinos. At present we do not understand neutrino masses, though there is a very real possibility that the neutrino is massless and flavor oscillations are caused by spacetime being higher dimensional and octonionic, and the neutrinos being spacetime triplets [20]. This possibility could br verified if we can calculate the PMNS matrix from first principles in the octonionic theory; this will be attempted in future work. The Majorana nature of the neutrino also suggests neutrinoless double beta decay, experimental implications for which we will investigate further. The possibility that the sterile neutrino could be massive and light (hence hot dark matter) or heavy (hence cold dark matter) will also be investigated, although it is true that our theory favours MOND over cold dark matter.
Our theory also predicts a new charged Higgs boson, and possibly a doubly charged Higgs, although their masses and the fundamental origin of their scalar nature remains to be understood. Also of interest is whether the Higgs triplet in this theory can cause the W mass to depart from the standard model prediction. The hierarchy problem, and the exact mechanism of the electroweak symmetry breaking and the left-right symmetry breaking remain to be understood: these could be the same symmetry breaking and could perhaps be mediated by a quantum-to-classical phase transition. It is also of interest to try and predict the (g-2) anomaly, some related physics was discussed in [22] and this issue might also be related to our derivations of mass ratios and the low energy fine structure constant.
A general line of investigation of serious interest is to note that our predictions are made on octonionic twistor space, which is non-classical, whereas all measurements are in Minkowski spacetime, and based on quantum field theory calculations. In transiting from trace dynamics to emergent quantum field theory there could be important and smoking gun corrections, which could possibly be cast in the language of effective field theory.
As per our recent work on E 8 × E 8 symmetry, we have been able to account for 208 out of the 496 degrees of freedom [20]. While we have commented briefly in that work, on the unaccounted degree of freedom, more work needs to be done to see if they predict new particles which can confirm or rule out this theory.
Why there is no black hole information loss paradox in the octonionic theory?
There appears to be an information loss paradox because we are ignoring the physical process of the quantum-to-classical transition which keeps the black hole (made of enormously many quantum constituents) classical in the first place. Let us consider the following analogy: Consider a box of gas at thermodynamic equilibrium -this is the maximum entropy and minimum information state. Now, a sudden spontaneous fluctuation sends the entire set of gas molecules to one corner of the box. This is a transient low entropy high information and ordered state, far from thermodynamic equilibrium. At the very next instant, the gas molecules will return to equilibrium, spread all over the box, entropy will have been gained, and information lost. If we ignore the spontaneous fluctuation which sent the gas to a corner, then we have an information loss paradox! Obviously there is no paradox in reality: information gained during the spontaneous fluctuation is lost during the return to equilibrium. It is exactly the same physics, when we work with a fundamental theory (gen- How taking the square-root of Minkowski space-time paves the way for unification: In summary, we believe we have a promising theory of unification under development, as captured in Fig. 6 and explained briefly below.
It is like going from the surface of the ocean to the ocean bed. The ocean floor can exist without the surface, but the surface cannot exist without the floor. We live in a 4D Minkowski space-time curved by gravitation, in which standard model gauge fields and fermions reside. But there is a more precise description. We take the square-root of Minkowski space-time and arrive at Penrose's twistor space, described by complex numbers.
In this spinor space-time replace complex numbers by quaternions, then by octonions. More precisely complex split bioctonions. We arrive at a space with E 8 × E 8 symmetry whose geometry is a unified description of the standard model and pre-gravitation. The gauge group is SU (3) c × SU (2) L × U (1) Y × SU (3) grav × SU (2) R × U (1) g Coupling constants are determined by the geometry. In the classical limit, the 4D curved spacetime and the stan-