The Characteristic Equation of The Exceptional Jordan Algebra: Its Eigenvalues, And Their Relation With Mass Ratios For Quarks And Leptons


 We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued La-grangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra J3 (O) [exceptional Jordan algebra] which in turn is related to the three fermion generations. The exceptional Jordan algebra [also known as the Albert algebra] is the finite dimensional algebra of 3x3 Hermitean matrices with octonionic entries. Its automorphism group is the exceptional Lie group F4. These matrices admit a cubic characteristic equation whose eigenvalues are real and depend on the invariant trace, determinant, and an inner product made from the Jordan matrix. Also, there is some evidence in the literature that the groups F4 and E6 could play a role in the unification of the standard model symmetries, including the Lorentz symmetry. The octonion algebra is known to correctly yield the electric charge values (0, 1/3, 2/3, 1) for standard model fermions, via the eigenvalues of a U (1) number operator, identified with U (1)em. In the present article, we use the same octonionic representation of the fermions to compute the eigenvalues of the characteristic equation of the Albert algebra, and compare the resulting eigenvalues with the known mass ratios for quarks and leptons. We find that the ratios of the eigenvalues correctly reproduce the [square root of the] known mass ratios for quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an F4 and E6 symmetry. In conjunction with the trace dynamics Lagrangian, the Jordan eigenvalues also provide a first principles theoretical derivation of the low energy value of the fine structure constant, yielding the value 1/137.04006. The Karolyhazy correction to this value gives an exact match with the measured value of the constant, after assuming a specific value for the electro-weak symmetry breaking energy scale.

In the pre-geometric, pre-quantum theory of generalised trace dynamics, the definition of spin requires 4D space-time to be generalised to an 8D non-commutative space. In this case, an octonionic space is a possible, natural, choice for further investigation. We found that the additional four directions can serve as 'internal' directions and open a path towards a possible unification of the Lorentz symmetry with the standard model, with gravitation arising only as an emergent phenomenon. Instead of the Lorentz transformations and internal gauge transformations, the symmetries of the octonionic space are now described by the automorphisms of the octonion algebra. Remarkably enough, the symmetry groups of this algebra, namely the exceptional Lie groups, naturally have in them the desired symmetries [and only those symmetries, or higher ones built from them] of the standard model, including Lorentz symmetry, without the need for any fine tuning or adjustments. Thus the group of automorphisms of the octonions is G 2 , the smallest of the five exceptional Lie groups G 2 , F 4 , E 6 , E 7 , E 8 . The group G 2 has two intersecting maximal sub-groups [33], SU (3) × U (1) and SU (2) × SU (2), which between them account for the fourteen generators of G 2 , and can possibly serve as the symmetry group for one generation of standard model fermions. The complexified Clifford algebra Cl(6, C) plays a very important role in establishing this connection. In particular, motivated by a map between the complexified octonion algebra and Cl(6, C), electric charge is defined as one-third the eigenvalue of a U (1) number operator, which is identified with U (1) em [6,8].
Describing the symmetries SU (3) × U (1) and SU (2) × SU (2) of the standard model [with Lorentz symmetry now included] requires two copies of the Clifford algebra Cl(6, C) whereas the octonion algebra yields only one such independent copy. It turns out that if boundary terms are not dropped from the Lagrangian of our theory, the Lagrangian describes three fermion generations [ [4] and Section III below in the present paper], with the symmetry group now raised to F 4 . This admits three intersecting copies of G 2 , with the SU (2)×SU (2) in the intersection, and a Clifford algebra construction based on the three copies of the octonion algebra is now possible [34]. Attention thus shifts to investigating the connection between F 4 and the three generations of the standard model. F 4 is also the group of automorphisms of the exceptional Jordan algebra [20,35,36]. The elements of the algebra are 3x3 Hermitean matrices with octonionic entries. This algebra admits an important cubic characteristic equation with real eigenvalues. Now we know that the three fermion generations differ from each other only in the mass of the corresponding fermion, whereas the electric charge remains unchanged across the generations. This motivates us to ask: if the eigenvalues of the U (1) number operator constructed from the octonion algebra represent electric charge, what is represented by the eigenvalues of the exceptional Jordan algebra? Could these eigenvalues bear a relation with mass ratios of quarks and leptons? This is the question investigated in the present paper and answered in the affirmative. Using the very same octonion algebra which was used to construct a state basis for standard model fermions, we calculate these eigenvalues. Remarkably, the eigenvalues are very simple to express, and bear a simple relation with electric charge. We describe how they relate to mass ratios. In particular we find that the ratios of the eigenvalues match with the It is known that since F 4 does not have complex representations, it cannot give a representation of the fermion states. It has hence been suggested that the correct representation could come from the next exceptional Lie group, E 6 , which is the automorphism group of the complexified exceptional Jordan algebra. This aspect is currently being investigated by several researchers, including the present author. However, the standard model free parameters certainly cannot come from the characteristic equation related to E 6 , because the roots of this equation are not real numbers in general. It is clear that the parameters must then come from the roots of the characteristic equation of F 4 , which in a sense is the self-adjoint counterpart of the equation for E 6 . It is in this spirit that the present investigation is carried out, and the results we find suggest that the present approach is indeed the correct one, as regards determining the model parameters. One must investigate E 6 for representations, but F 4 for the parameter values.
The plan of the paper is as follows. In the next section we recall the exceptional Jordan algebra, construct the octonionic representation of the three fermion generations, calculate the roots of the characteristic equation, and make some comments on mass-ratios and the roots. In Section III we construct the trace dynamics Lagrangian for three generations, along with the bosons, and we give a theoretical derivation of the asymptotic fine structure constant from first principles. In Section IV we calculate an additional set of eigenvalues for the fermions, generation wise; these provide evidence for violation of lepton universality.
We then explain how the first set of Jordan eigenvalues in fact act as a definition of mass, quantised in units of Planck mass. We then show that mass ratios of charged fermions are obtained from these eigenvalues. In the Appendix in Section V we recall the motivation in earlier work, for developing this pre-theory, and we also include a few new insights. In particular we report on a 4D quaternionic version of the pre-theory, which describes the Lorentz-weak interaction of the leptons, based on an extension of the Lorentz algebra by SU (2). In order to include quarks and the strong interaction, this 4D quaternionic pre-theory is extended to eight octonionic dimensions.  [21,29,30,35] It satisfies the characteristic equation [21,29,30] X 3 − T r(X)X 2 + S(X)X − Det(X) = 0; T r(X) = ξ 1 + ξ 2 + ξ 3 (2) which is also satisfied by the eigenvalues λ of this matrix Here the determinant is and S(X) is given by The diagonal entries are real numbers and the off-diagonal entries are (real-valued) octonions.
A star denotes an octonionic conjugate. The automorphism group of this algebra is the exceptional Lie group F 4 . Because the Jordan matrix is Hermitean, it has real eigenvalues which can be obtained by solving the above-given eigenvalue equation.
In the present article we suggest that these eigenvalues carry information about mass ratios of quarks and leptons of the standard model, provided we suitably employ the octo-nionic entries and the diagonal real elements to describe quarks and leptons of the standard model. Building on earlier work [6,7,9] we recently showed that the complexified Clifford algebra Cl(6, C) made from the octonions acting on themselves can be used to obtain an explicit octonionic representation for a single generation of eight quarks and leptons, and their anti-particles. In a specific basis, using the neutrino as the idempotent V , this representation is as follows [4,6]. The α are fermionic ladder operators of Cl(6, C) (please see Eqn. (34) of [4]).
The anti-particles are obtained from the above representation by complex conjugation [6].
Note: Eqn. (33) of [4] for the idempotent has an incorrectly written expression on the right hand side. Instead of ie 7 /2 as written there, the correct expression is (1 + ie 7 )/2 [37].
Hence the idempotent V in that paper should be (1 + ie 7 )/2, not ie 7 /2. It has now been found however, that identification of the neutrino with the idempotent V = (1 + ie 7 )/2 does not give the desired values for mass-ratios and coupling constants reported in the present paper [37]. We hence propose the Majorana particle interpretation for the neutrino, and identify the neutrino with (V − V cc )/2 where V cc is the complex conjugate of V . Hence the neutrino is [(1 + ie 7 ) − (1 − ie 7 )]/4 = ie 7 /2, so that the octonionic representation of the neutrino remains the same as shown in [4] and is the one used in the present paper.
Our results here seem to suggest that the neutrino is a Majorana particle, and not a Dirac particle.
Note: In Eqn. (34) of [4] the denominator in the expression for the positron should be 4, not 8. The correct expression for the positron is shown above in Eqn. (6).
In the context of the projective geometry of the octonionic projective plane OP 2 it has been shown by Baez [23] that upto automorphisms, projections in EJA take one of the following four forms, having the respective invariant trace 0, 1, 2, 3.
Since it has earlier been shown by Furey [6] that electric charge is defined in the division algebra framework as one-third of the eigenvalue of a U (1) number operator made from the generators of the SU (3) in G 2 , we propose to identify the trace of the Jordan matrix with the sum of the charges of the three identically charged fermions across the three generations. Thus the trace zero Jordan matrix will have diagonal entries zero, and will represent the (neutrino, muon neutrino, tau-neutrino). The trace one Jordan matrix will have di- and will represent (positron, anti-muon, anti-tau-lepton).
We have thus identified the diagonal real entries of the four Jordan matrices whose eigenvalues we seek. We must next specify the octonionic entries in each of the four Jordan matrices. Note however that the above representation of the fermions of one generation is using complex octonions, whereas the entries in the Jordan matrices are real octonions. So we devise the following scheme for a one-to-one map from the complex octonion to a real octonion. Since we are ignoring color, we pick one out of the three up quarks, say (e 4 + ie 5 ), and one of three anti-down quarks, say (e 5 + ie 4 ). Since the representation for the electron and the neutrino use e 7 and a complex number, it follows that the four octonions we have where the eight a-s are real numbers. By definition, we map this complex quaternion to the following real octonion: a 0 + a 1 e 1 + a 5 e 2 + a 3 e 3 + a 2 e 4 + a 4 e 5 + a 7 e 6 + a 6 e 7 (12) Note that the four real coefficients in the original complex quaternion have been kept in place, and their four imaginary counterparts have been moved to the octonion directions (e 1 , e 2 , e 3 , e 6 ) now as real numbers. Clearly, the map is reversible, given the real octonion we can construct the equivalent complex quaternion representing the fermion. We can now use this map and construct the following four real octonions for the neutrino, anti-down quark, up quark and the positron, respectively, after comparing with their complex octonion representation above. These four real octonions will go, one each, in the four different Jordan matrices whose eigenvalues we wish to calculate. Next, we need the real octonionic representations for the four fermions [color suppressed] in the second generation and the four in the third generation.
We propose to build these as follows, from the real octonion representations made just above for the first generation. Since F 4 has the inclusion SU (3) × SU (3), one SU 3) being for color and the other for generation, we propose to obtain the second generation by a 2π/3 rotation on the first generation, and the third generation by a 2π/3 rotation on the second generation.
By this we mean the following construction, for the four respective Jordan matrices, as below.
It is justified as follows: One of the two SU (3) We have used the conventional multiplication rules for the octonions, which are reproduced below in Fig. 2, for ready reference. Similarly, we can construct the top quark by a 2π/3 rotation on the charm: Next, we construct the anti-strange V as and anti-bottom V ab , by left-multiplication of the anti-down quark V ad by e 2πe 3 /3 .
Next, we construct the octonions for the anti-muon V aµ and anti-tau-lepton V aτ by left multiplying the positron V e + by e 2πe 1 /3 Lastly, we construct the octonions V νµ for the muon neutrino and V ντ for the tau neutrino, by left multiplying on the electron neutrino V ν with e 2πe 6 /3 We now have all the information needed to write down the four Jordan matrices whose eigenvalues we will calculate. Diagonal entries are electric charge, and off-diagonal entries are octonions representing the particles. Using the above results we write down these four matrices explicitly. The neutrinos of three generations The anti-down set of quarks of three generations [anti-down, anti-strange, anti-bottom]: The up set of quarks for three generations [up, charm, top] The positively charged leptons of three generations [positron, anti-muon, anti-tau-lepton] Next, the eigenvalue equation corresponding to each of these Jordan matrices can be written down, after using the expressions given above for calculating the determinant and the function S(X). Tedious but straightforward calculations with the octonion algebra give the following four cubic equations: Neutrinos: We get T r(X) = 0, S(X) = −3/4, Det(X) = 0, and hence the cubic equation and roots Anti-down-quark + its higher generations [anti-down, anti-strange, anti-bottom]: We get T r(X) = 1, S(X) = −1/24, Det(X) = −19/216, and the following cubic equation and roots Up quark + its higher generations [up, charm, top]: We get T r(X) = 2, S(X) = 23/24, Det(X) = 5/108 and the following cubic equation and roots: Positron + its higher generations [positron, anti-muon, anti-tau-lepton]: We get T r(X) = 3, S(X) = 3 − 3/8, Det(X) = 1 − 3/8 and the following cubic equation and roots: As expected from the known elementary properties of cubic equations, the sum of the roots is T r(X), their product is Det(X), and the sum of their pairwise products is S(X). Interestingly, this also shows that the sum of the roots is equal to the total electric charge of the three fermions under consideration in each of the respective cases. Whereas S(X) and Det(X) are respectively related to an invariant inner product and an invariant trilinear form constructed from the Jordan matrix, their physical interpretation in terms of fermion properties remains to be understood.
The roots exhibit a remarkable pattern. In each of the four cases, one of the three roots is equal to the corresponding electric charge, and the other two roots are placed symmetrically on both sides of the middle root, which is the one equal to the electric charge. All three roots are positive in the up quark set and in the positron set, whereas the neutrino set and anti-down quark set have one negative root each, and the neutrino also has a zero root. It is easily verified that the calculation of eigenvalues for the anti-particles yields the same set of eigenvalues, upto a sign. In other words, the Jordan eigenvalue for the anti-particle is opposite in sign to that for the particle. The roots are summarised in the table below, and we see that they are composed of the electric charge, and the octonionic magnitude associated with the respective particle. [The octonionic magnitude L 2 P /L 2 is the sum x i x i over the three identically charged fermions of three generations, which appears in Equation (5) above.] One expects these roots to relate to masses of quarks and leptons for various reasons, and principally because the automorphism group of the complexified octonions contains the 4D Lorentz group as well, and the latter we know relates to gravity.
Since mass is the source of gravity, we expect the Lorentz group to be involved in an essential way in any theory which predicts masses of elementary particles. And the group F 4 , besides being related to G 2 , and a possible candidate for the unification of the four interactions, is also the automorphism group of the EJA. We have motivated how the four projections of the EJA relate naturally to the four generation sets of the fermions. Thus there is a strong possibility that the eigenvalues of the characteristic equation of the EJA yield information about fermion mass ratios, especially it being a cubic equation with real roots. We make the following preliminary observations about the known mass ratios, and then provide a concrete analysis in Section IV.
The Jordan eigenvalues allow us to express the electric charge eigenstates of a fermion's three generations, as superpositions of mass eigenstates. That is why these eigenvalues determine mass ratios.
For the set (positron, anti-muon, anti-tau-lepton), the three respective masses are known to satisfy the following empirical relation, known as the Koide formula: For the three roots of the corresponding cubic equation (32) we get that For the up quark set though, we see a correlation in terms of square roots of masses.
In the case of the up quark set, the following approximate match is observed between the ratios of the eigenvalues, and the mass square root ratios of the masses of up, charm and top quark. For the sake of this estimate we take these three quark masses to be [2.3, 1275, 173210] in Mev [38].
Within the error bars on the masses of the up set of quarks, the two sets of ratios are seen to agree with each other upto second decimal place.
Considering that one of the roots is negative in the anti-down-quark set, we cannot directly relate the eigenvalues to mass ratios. The same is true for the neutrino set, where one root is negative and one root is zero. In section IV we propose that the correct quantity to examine is the square-root of mass (in dimensionless units), which can take both positive sign and negative sign: ± √ m. The Jordan eigenvalues relate to the square-root of either sign, with the eigenvalue for anti-particle being opposite in sign to that for the particle.
The case of the neutrino is especially instructive, and shows how non-zero mass could arise fundamentally, even when the electric charge is zero. In this case, the non-zero contribution comes from the inner product related quantity S(X), and therein from the absolute magnitude of the octonions in the Jordan matrix, which necessarily has to be non-zero. We thus see that masses are derivative concepts, obtained from the three more fundamental entities, namely the electric charge, and the geometric invariants S(X) and Det(X), with the last two necessarily being defined commonly for the three generations. And since mass is the source of gravity, this picture is consistent with gravity and space-time geometry being emergent from the underlying geometry of the octonionic space which algebraically determines the properties of the elementary particles. We note that there are no free parameters in the above analysis, no dimensional quantities, and no assumption has been put by hand.
Except that we identify the octonions with elementary fermions. The numbers which come out from the above analysis are number-theoretic properties of the octonion algebra.
These observations suggest a possible fundamental relation between eigenvalues of the EJA and particle masses. In the next section, we provide further evidence for such a connection, based on our proposal for unification based on division algebras and a matrix-valued Lagrangian dynamics.

A. A Lagrangian on an 8D octonionic space-time
The action and Lagrangian for the three generations of standard model fermions, fourteen gauge bosons, and four potential Higgs bosons, are given by [4] Here,˙ and˙ By defining we can express the Lagrangian as We now expand each of these four terms inside of the trace Lagrangian, using the definitions of q 1 and q 2 given above: In our recent work, we suggested this Lagrangian, having the symmetry group lie on this cube. At the same time the fermionic cubes make contact with the bosonic cube, enabling the bosons to act on the fermions.
We now try to understand the central bosonic cube. First we count the number of its elements: it gets a total of 3x10=30 elements from the three side cubes, which when added to its own 26 elements gives a total of 56. But there are a lot of common elements, so that the actual number of independent elements is much smaller, and we enumerate them now. Three points are shared two-way and three points shared three-way and the point e 0 is shared four-way; that reduces the count to 44. Nine lines are shared: three of them three way, and six of them two way, reducing the count to 32. The shared three planes reduce the count to 29. We now account for the assignment of bosons to these 29 locations.
The eight gluons are on the front right, marked by the pink points, and lines labelled g 1 to g 8 , and the photon is assigned to the plane (e 3 e 7 e 6 e 5 ) on the front right enclosed by the This diagram does suggest that one could investigate bosonic degrees of freedom as made from pairs of fermion degrees of freedom. With this tentative motivation, we return to our Lagrangian, and seek to write it explicitly as for a single generation of bosons, and three generations of fermions. Upon examination of the sub-equations in Eqn. (46) we find that the last column has terms bilinear in the fermions, and we would like to make it appear just as the second and third column do, so that we can explicitly have three fermion generations.
With this intent, we propose the following assumed definitions of the bosonic degrees of freedom, by recasting the four terms in the last column of Eqn. (46): where A and B are bosonic matrices which drop out on summing the various terms to get the full Lagrangian, With this redefinition, the sub-equations Eqn. (46) can be now written in the following form after rewriting the last column: The terms now look harmonious and we can see a structure emerging -the first column are bosonic terms and these are not triples. where We see that each of these four fermionic sets could possibly be related to a Jordan matrix, after including the adjoint part. We also see that different coupling constants appear in different sets with identical coupling in third and fourth set and no coupling in the first set.
The first set could possibly describe neutrinos, charged leptons and quarks (gravitational and weak interaction), the second set charged leptons and quarks, and the third and fourth set the quarks. To establish this explicitly, equations of motion remain to be worked out and then related to the eigenvalue problem. As noted earlier, L relates to mass, and this approach could reveal how the eigenvalues of the EJA characteristic equation relate to mass.
This investigation is currently in progress, and proceeds along the following lines. We take the self-adjoint part of the above Lagrangian, because that part is the one which leads to quantum field theory in the emergent approximation after coarse-graining the underlying [The operator terms of the form q B q F etc. in (52) have been correspondingly made dimensionless by dividing by L 2 P ]. We assume that ln α is linearly proportional to the electric charge, and that the proportionality constant is the Jordan eigenvalue corresponding to the anti-down quark. The electric charge 1/3 of the anti-down quark seems to be the right choice for determining α, it being the smallest non-zero value [and hence possibly the fundamental value] of the electric charge, and also because the constant α appears as the coupling in front of the supposed quark terms in the Lagrangian, as in Eqns. (53) and (54). We hence define α by where λ ad is the Jordan eigenvalue corresponding to the anti-down quark, as given by Eqn.
(30) and q ad is the electric charge of the anti-down quark (=1/3). In order to arrive at this relation for α, we asked in what way α could vary with q, if it was allowed to vary? We then made the assumption that dα/dq ∝ α. In the resulting linear dependence of ln α on q, we froze the value of α at that given by the smallest non-zero charge value 1/3, taking the proportionality constant to be the corresponding Jordan eigenvalue. This dependence also justifies that had we fixed α from the zero charge of the neutrino, α would have been one, as it in fact is, in our Lagrangian. We are investigating if this way of constructing α can be further justified from the Lagrangian dynamics.
As for the value of L P /L, we identify it with one-half of that part of the Jordan eigenvalue which modifies the contribution coming from the electric charge.
[For an explanation of the origin of the factor of one-half, see the next paragraph]. Thus from the eigenvalues found above, we deduce that for neutrinos, quarks and charged leptons, the quantity L 2 P /L 2 takes the respective values (3/16, 3/32, 3/32). These values are equal to one-fourth of the respective octonionic magnitudes. Thus the coupling constant C defined above can now be calculated, with α 2 as given above, and L 2 P /L 2 = 3/32. Furthermore, since the electric charge q, the way it is conventionally defined, has dimensions such that q 2 has dimensions (Energy × Length), we measure q 2 in Planck units E P l × L P =hc. We hence define the fine structure constant by C = α 2 L 4 P /L 4 ≡ e 2 /hc, where e is the electric charge of electron / muon / tau-lepton in conventional units. We hence get the value of the fine structure constant to be The where L P and t P are Planck length and Planck time respectively -obviously their ratio is the speed of light. In our theory, there are only three fundamental dimensionful quantities: Planck length, Planck time, and a constant with dimensions of action, which in the emergent quantum theory is identified with Planck's constanth. We now see that electric charge is not independent of these three fundamental dimensionful constants. It follows from them.
Planck mass is also constructed from these three, and electron mass will be expressed in terms of Planck mass, if only we could understand why the electron is some 10 22 times lighter than Planck mass. Such a small number cannot come from the octonion algebra.
In all likelihood, the cosmological expansion up until the electroweak symmetry breaking is playing a role here.  We have not addressed the question as to how these discrete order one eigenvalues might relate to actual low values of fermion masses, which are much lower than Planck mass.
We speculatively suggest the following scenario, which needs to be explored further. The universe is eight-dimensional, not four. The other four internal dimensions are not compactified; rather the universe is very 'thin' in those dimensions but they are expanding as well. There are reasons having to do with the so-called Karolyhazy uncertainty relation [39], because of which the universe expands in the internal dimensions at one-third the rate, on the logarithmic scale, compared to our 3D space. That is, if the 4D scale factor is a(τ ), the internal scale factor is a The universe began in a unified phase, via an inflationary 8D expansion possibly resulting as the aftermath of a huge spontaneous localisation event in a 'sea of atoms of space-timematter' [10]. The mass values are set, presumably in Planck scale, at order one values dictated by the eigenvalues reported in the present paper. Cosmic inflation scales down these mass values at the rate a 1/3 (τ ), where a(τ ) is the 4D expansion rate. Inflation ends after about sixty e-folds, because seeding of classical structures breaks the color-elctro-weak-Lorentz symmetry, and classical spacetime emerges as a broken Lorentz symmetry. The electro-weak symmetry breaking is actually a electro-weakLorentz symmetry breaking, which is responsible for the emergence of gravity, weak interaction being its short distance limit.
There is no reheating after inflation; rather inflation resets the Planck scale in the vicinity of the electro-weak scale, and the observed low fermion mass values result. The electroweak symmetry breaking is mediated by the Lorentz symmetry, in a manner consistent with the conventional Higgs mechanism. It is not clear why inflation should end specifically at the electro-weak scale: this is likely dictated by when spontaneous localisation becomes significant enough for classical spacetime to emerge. It is a competition between the strength of the electro-colour interaction which attempts to bind the fermions, and the inflationary expansion which opposes this binding. Eventually, the expanding universe cools enough for spontaneous localisation to win, so that the Lorentz symmetry is broken. It remains to prove from first principles that this happens at around the electro-weak scale and also to investigate the possibly important role that Planck mass primordial black holes might play in the emergence of classical spacetime. I would like to thank Roberto Onofrio for correspondence which has influenced these ideas. See also [40].
We can also infer this corrected length as the four-dimensional space-time measure of the length, which differs from the eight dimensional octonionic value 3/32 by the amount δ f .
If we take l f to be 10 −16 cm, the correction δ f is of the order 2 × 10 −6 . The correction to the asymptotic value (57) of the fine structure constant is then GeV range, the derived constant agrees with the measured value at least to the sixth decimal place, which is reassuring. The purpose of the present exercise is to show that the Karolyhazy correction leads to a correction to the asymptotic value of the fine structure constant which is in the desired range -a striking fact by itself. In principle, our theory should predict the precise value of the electroweak symmetry breaking scale. Since that analysis has not yet been carried out, we predict that the ColorElectro-WeakLorentz symmetry breaking scale is 144.something GeV, because only then the theoretically calculated value of the asymptotic fine structure constant matches the experimentally measured value.
The above discussion of the asymptotic low energy value of the fine structure constant should not be confused with the running of the constant with energy. Once we recover classical spacetime and quantum field theory from our theory, after the ColorElectro-WeakLorentz symmetry breaking, conventional RG arguments apply, and the running of couplings with energy is to be worked out as is done conventionally. Such an analysis of running couplings will however be valid only up until the broken symmnetry is restored -it is not applicable in the prespacetime prequantum phase. In this sense, our theory is different from GUTs.
Once there is unification, Lorentz symmetry is unified with internal symmetries -the exact energy scale at which that happens remains to be worked out.
How then does the Planck scale prespacetime, prequantum theory know about the low energy asymptotic value of the fine structure constant? The answer to this question lies in the Lagrangian given in (49) and in particular the Lagrangian term (52) for the charged leptons. In determining the asymptotic fine structure constant from here, we have neglected the modification to the coupling that will come from the presence of q B and q F . This is analogous to examining the asymptotic, flat spacetime limit of a spacetime geometry due to a source -gravity is evident close to the source, but hardly so, far from it. Similarly, there is a Minkowski-flat analog of the octonionic space, wherein the effect of q B and q F (which in effect 'curve' the octonionic space) is ignorable, and the asymptotic fine structure can be computed. The significance of the non-commutative, non-associative octonion algebra and the Jordan eigenvalues lies in that they already determine the coupling constants, including their asymptotic values. This is a property of the algebra, even though the interpretation of a particular constant as the fine structure constant comes from the dynamics, i.e. the Lagrangian, as it should, on physical grounds.
On a related note about this approach to unification, we recall that the symmetry group in our theory is U (1) × SU (3) × SU (2) × SU (2). This bears resemblance to the study of a left-right symmetric extension of the standard model by Boyle [41] in the context of the complexified exceptional Jordan algebra. This L − R model has exceptional phenomenological promise, and it appears that the unbroken phase [prior to the ColorElectro-WeakLorentz symmetry breaking] of the L-R model is well-described by our Lagrangian (49) for three generations. This gives further justification for exploring the phenomenology of this Lagrangian.

B. More Jordan eigenvalues for quarks and charged leptons
Assuming that the mechanism for mass generation of neutrinos is different from that for the electrically charged fermions, we can set aside the neutrinos for the time being, and calculate additional new eigenvalues of the exceptional Jordan algebra in yet another way.
We club the three charged fermions of the first generation to make a 3 × 3 Jordan matrix, with the octonionic entries assigned as: x 1 is the anti-down quark, x 2 is the up quark, and The three Jordan matrices for which we are now calculating the eigenvalues are hence given as follows, one for each generation of two quarks and one charged lepton: GenII : GenIII : The notation and octonionic representation is the same as earlier in the paper. For each of the three generations the eigenvalues are given by the following set of three real roots, each of which is positive (hence a total of nine unequal roots): Here, the angle θ is defined by and the function Q is the same for each of the three generations: whereas the function R differs slightly amongst the three generations because the determinant is different for each of them: The angle θ in the case of the three generations can thus be calculated, and is given in radians by mass is justified by recalling that in our approach, gravitation is derived from 'squaring' an underlying spin one Lorentz interaction [4]. It is reasonable then to assume that the spin one Lorentz interaction is sourced by √ m, and to try to understand the origin of the square-root of the mass ratios, rather than origin of the mass ratios themselves.
At this stage, the above proposed quantised root-mass-ratios for the first generation are only an assumption; we do not have a proof for this assumption.
[We return to thus aspect in detail in a forthcoming publication [42], where we consider an SU (3)  Here also the cosine of the angle is taken, multiplied by 2 √ −Q and the result added to 2/3. In terms of these two angles the nine fermions are placed symmetrically on a 2-torus; yet the angles manage to give rise to the measured mass ratios which appear to be quite random otherwise. from above that the three generations are respectively characterised by these three angles These three angles can be taken to be the defining characteristic of the three generations.
All the three angles lie in the second quadrant and hence have a negative cosine; therefore the largest root λ 1 in (68) for each of the three generations is identified with the quark we conclude from the roots given in (29), that the three angles are (π/6, 5π/6, 9π/6). The same angles also arise for the charged fermions, with the first angle for the GenIII particle, next one for GenI and largest angle for GenII. Also, in each case, R = 0, while −Q = 1/8. Figure 6 below shows these Jordan angles, along with the measured mass values, as well the square-root of the mass ratio taken with respect to mass of the anti-down quark. We now see that the nine fermions are placed symmetrically on the torus, as far as the angles are concerned. And yet these angles manage to give rise to strange-looking mass ratios.

The table in
Since the square-root-mass ratio of the anti-down quark has been set to unity, and predicted above to be 4.599 MeV (= 9 × 0.511 MeV), we will calculate the square-root-mass ratios of the other particles with respect to the anti-down-quark, and demonstrate a correlation of these ratios with the Jordan eigenvalues. Also, since a negative Jordan eigenvalue is to be associated with minus of square-root mass, for finding the mass-ratio, we take the absolute value of the anti-down-quark eigenvalue, which is negative.
• Anti-muon : Take the ratio of the first set of Jordan eigenvalues for the electron and the muon [see the table in Fig. 6]. Multiply by a factor representing the down quark (the first factor in the expression below). Then compare the resulting value with the square-root mass ratio of the muon mass with respect to the electron mass: • Anti-tau lepton : Using the first set of eigenvalues for the charged leptons, we get the ratio for tau-lepton to electron: These ratios made from the Jordan eigenvalues suggest a possible correlation with the squareroot mass ratios, and hence provide a plausible definition of a mass quantum number for standard model fermions. This definition is completely independent of trace dynamics and its Lagrangian, and is a property exclusively of the octonionic algebra. This is completely analogous to the fact that in the octonionic approach to the standard model, quantisation of electric charge is deduced from eigenvalues of the U (1) e m operator made from the Clifford algebra Cl (6). Hence, square-root of mass is treated on the same footing as electric charge: their quantisation is a property of the algebra, not of the dynamics. The difference between charge quantisation and mass quantisation is that for finding the mass eigenstates, all three generations must be considered together, not one at a time.
The square-root mass numbers for the charged fermions are shown in Fig. 8. These have the same fundamental status as quantised electric charge values 1/3, 2/3 and 1. The exceptional Jordan algebra is of significance also in superstring theory, where it has been suggested that there is a relation between the EJA and the vertex operators of superstrings, and that the vertex operators represent couplings of strings [43,44]. This intriguing connection between the EJA, string theory and aikyon theory deserves to be explored further.
Lastly we mention that the Lagrangian (45) that we have been studying closely resembles the Bateman oscillator [45] model, for which the Lagrangian is I thank Partha Nandi for bringing this fact to my attention. Considering that the Bateman oscillator represents a double oscillator with relative opposite signs of energy for the two oscillators undergoing damping, it is important to understand the implications for our theory.
In particular, could this imply a cancellation of zero point energies between bosonic and fermionic modes, thus annulling the cosmological constant? And also whether this damping is playing any possible role in generating matter-anti-matter asymmetry?
Acknowledgements: I would like to thank Carlos Perelman for discussions and helpful In this appendix, we recall from earlier work [4] the motivation for developing a formulation of quantum theory without classical time, and how doing so leads to a pre-quantum, prespacetime theory which is a candidate for unification of general relativity with the standard model.
A. Why there must exist a formulation of quantum theory which does not refer to classical time? And why such a formulation must exist at all energy scales, not just at the Planck energy scale.
Classical time, on which quantum systems depend for a description of their evolution, is part of a classical space-time. Such a space-time -the manifold as well as the metric that overlies it -is produced by macroscopic bodies. These macroscopic bodies are a limiting case of quantum systems. In principle one can imagine a universe in which there are no macroscopic bodies, but only microscopic quantum systems. And this need not be just at the Planck energy scale.
As a thought experiment, consider an electron in a double slit interference experiment, having crossed the slits, and not yet reached the screen. It is in a superposed state, as if it has passed through both the slits. We want to know, non-perturbatively, what is the spacetime geometry produced by the electron? Furthermore, we imagine that every macroscopic object in the universe is suddenly separated into its quantum, microscopic, elementary particle units. We have hence lost classical space-time! Perturbative quantum gravity is no longer possible. And yet we must be able to describe what gravitational effect the electron in the superposed state is producing. This is the sought for quantum theory without classical time! And the quantum system is at low non-Planckian energies, and is even non-relativistic. This is the sought for formulation we have developed, assuming only three fundamental constants a priori: Planck length L P , Planck time t P , and Planck's constanth. Every other dimensionful constant, e.g. electric charge, and particle masses, are expressed in terms of these three. This new theory is a pre-quantum, pre-spacetime theory, needed even at low energies.
A system will be said to be a Planck scale system if any dimensionful quantity describing the system and made from these three constants, is order unity. Thus if time scales of interest to the system are order t P = 10 −43 s, the system is Planckian. If length scales of interest are order L P = 10 −33 cm, the system is Planckian. If speeds of interest are of the order L P /t P = c = 3 × 10 8 cm/s then the system is Planckian. If the energy of the system is of the orderh/t P = 10 19 GeV, the system is Planckian. If the action of the system is of the orderh, the system is Planckian. If the charge-squared is of the orderhc, the system is Planckian. Thus in our concepts, the value 1/137 for the fine structure constant, being order unity in the unitshc, is Planckian. This explains why this pre-quantum, pre-spacetime theory knows the low energy fine structure constant.
A quantum system on a classical space-time background is hugely non-Planckian. Because the classical space-time is being produced by macroscopic bodies each of which has an action much larger thanh. The quantum system treated in isolation is Planckian, but that is strictly speaking a very approximate description. The spacetime background cannot be ignoredonly when the background is removed from the description, the system is exactly Planckian.
This is the pre-quantum, pre-spacetime theory.
It is generally assumed that the development of quantum mechanics, started by Planck in 1900, was completed in the 1920s, followed by generalisation to relativistic quantum field theory. This assumption, that the development of quantum mechanics is complete, is not necessarily correct -quantisation is not complete until the last of the classical elements -this being classical space-time -has been removed from its formulation.
The pre-quantum, pre-spacetime theory achieves that, giving also an anticipated theory of quantum gravity. What was not anticipated was that removing classical space-time from quantum theory will also lead to unification of gravity with the standard model. And yield an understanding of where the standard model parameters come from. It is clear that the sought for theory is not just a high energy Beyond Standard Model theory. It is needed even at currently accessible energies, so at to give a truly quantum formulation of quantum field theory. Namely, remove classical time from quantum theory, irrespective of the energy scale.
Surprisingly, in doing so, we gain answers to unsolved low energy aspects of the standard model and of gravitation.
The process of quantisation works very successfully for non-gravitational interactions, because they are not concerned with space-time geometry. However, it is not necessarily correct to apply this quantisation process to spacetime geometry. Because the rules of quantum theory have been written by assuming a priori that classical time exists. How then can we apply these quantisation rules to classical time itself? Doing so leads to the notorious problem of time in quantum gravity -time is lost, understandably. We do not quantise gravity. We remove classical space-time / gravity from quantum [field] theory.
Space-time and gravity emerge as approximations from the pre-theory, concurrent with the emergence of classical macroscopic bodies. In this emergent universe, those systems which have not become macroscopic, are described by the beloved quantum theory we knownamely quantum theory on a classical spacetime background. This is an approximation to the pre-theory: in this approximation, the contribution of the said quantum system to the We have argued above that there must exist a formulation of quantum theory which does not refer to classical time. Such a formulation must in principle exist at all energy scales, not just at the Planck energy scale. For instance, in today's universe, if all classical objects were to be separated out into elementary particles, there would be no classical space-time and we would need such a formulation. Even though the universe today is a low energy universe, not a Planck energy universe.
Such a formulation is inevitably also a quantum theory of gravity. Arrived at, not by quantising gravity, but by removing classical gravity from quantum theory. We can also call such a formulation pure quantum theory, in which there are no classical elements: classical space-time has been removed from quantum theory. We also call it a pre-quantum, prespacetime theory.
What is meant by Planck scale, in this pre-theory?
Conventionally In the pre-theory the universe is an 8D octonionic universe, as shown in the Fig. 3 Hopefully the theory will shed some light also on the strong-CP problem.
theory supposedly more general than classical mechanics, the initial values of the operators q and p must also obey the constraint [q, p] = ih. This is highly restrictive! 3. It would be more reasonable if there were to be a dynamics based only on Quantisation Step 1. And then Step 2 emerges from this underlying dynamics in some approximation.
This is precisely what Trace Dynamics is. Only step 1 is applied to classical mechanics.
q and p are matrices, and the Lagrangian is the trace of a matrix polynomial made from q and its velocity. The matrix valued equations of motion follow from variation of the trace Lagrangian. They describe dynamics. This is the theory of trace dynamics developed  this is also the automorphism group Aut(H) of the quaternions. The origin represents absolute Newtonian time, and we have Newtonian dynamics in which the action principle for the free particle represented by the configuration variable q, which is a three-vector, is The generalisation to many-particle systems interacting via potentials is obvious and well-known. Newtonian gravity can be consistently described in this framework. The dynamical variables, being real-number valued three-vectors, all commute with each other. The important approximation made in the physical space is that by hand we set e 2 1 = e 2 2 = e 2 3 = 1, instead of −1. This of course is what gives us the Newtonian absolute space (Euclidean geometry) and absolute time, and the manifold R 3 for physical space. The associated algebra is R × H, in an approximate sense, which becomes precise only in special relativity, as discussed below.
[The algebra C represents a 2D physical space, and R × C represents a space-time for Newtonian mechanics in absolute two-space represented by C, and absolute time R. The homomorphism SL(2, R) ∼ SO(2, 1) suggests that we can relate 2x2 real-valued matrices to a 2+1 relativistic space-time. This observation becomes very relevant when we relate normed division algebras to relativity.] To go from here to trace dynamics, we will raise all dynamical variables from three-vectors to three-matrices. Thusq is a matrix-valued three-vector whose three spatial componentŝ q 1 ,q 2 ,q 3 are matrices whose entries are real numbers. The Lagrangian for a free particle will now be the trace of the matrix polynomialq 2 , and hence the action is The underlying three-space continues to have the symmetry group SO(3) and the dynamics obeys Galilean invariance; this is implemented on the trace dynamics action via the unitary transformations generated by the generators of SO(3).
Special relativity, Complex quaternions, and the algebra R × C × H: Consider the quaternionic four vector x = x 0 e 0 + x 1 e 1 + x 2 e 2 + x 4 e 4 and the corresponding position four-vector for a particle in special relativity: q i = q 0 e 0 + q 1 e 1 + q 2 e 2 + q 4 e 4 .
One can define the four-metric on this Minkowski space-time whose symmetry group is the Lorentz group SO(3, 1) having the universal cover Spin(3,1) isomorphic to SL(2, C). The complex quaternions generate the boosts and rotations of the Lorentz group SO(3,1). They can be used to obtain a faithful representation of the Clifford algebra Cl(2) and fermionic ladder operators constructed from this algebra can be used to generate the Lorentz algebra SL(2, C). Also, Cl(2) can be used to construct left and right handed Weyl spinors as minimal left ideals of this Clifford algebra, and as is well known the Dirac spinor and the Majorana spinor can be defined from the Weyl spinors. Cl(2) also gives the vector and scalar representations of the Lorentz algebra. These results are lucidly described in Furey's Ph. D.
thesis [6][7][8] as well as also in her video lecture series on standard model and division algebras https://www.youtube.com/watch?v=GJCKCss43WI&ab˙channel=CohlFureyCohlFurey The above relation between the Clifford algebra Cl(2) and the Lorentz algebra SL(2, C) strongly suggests, keeping in view the earlier conclusions for Cl(6) and the standard model and the octonions [6][7][8], that the Cl(2) algebra describes the left handed neutrino and the right-handed anti-neutrino, and a pair of spin one Lorentz bosons. This is confirmed by writing the following trace dynamics Lagrangian and action on the quaternionic space-time of special relativity, thereby generalising the relativistic particle S = −mc ds: where a 0 ≡ L 2 P /L 2 . This Lagrangian is identical in form to the one studied earlier in the present paper, but with a crucial difference that it is now written on 4D quaternionic spacetime, not on 8D octonionic space-time. Thusq B and q B have four components between them, not eight: q B = q Be2 e 2 + q Be4 e 4 ;q B =q Be0 e 0 +q Be1 e 1 . Similarly, the fermionic matrices have four components between them, not eight. Thus q F = q F e2 e 2 + q F e4 e 4 ;q F = q F e0 e 0 +q F e1 e 1 This has far-reaching consequences. Consider first the case where we set α = 0. The Lagrangian then is By opening up the terms into their coordinate components, the various degrees of freedom can be identified with the Higgs, the Lorentz bosons, the neutral weak isospin boson, and two neutrinos. The associated space-time symmetry is the Lorentz group SO(3, 1) and the associated Clifford algebra is Cl(2), reminding us again of the homomorphism SL(2, C) ∼ SO(3, 1).
When α is retained, the Lagrangian describes Lorentz-weak symmetry of the leptons: electron, positron, two neutrinos of the first generation, the Higgs, two Lorentz bosons, and the three weak isospin bosons. To our understanding, the associated Clifford algebra is still Cl(2) but now all the quaternionic degrees of freedom have been used in the Lagrangian and in the construction of the particle states.. What we likely have here is the extension of the Lorentz algebra by an SU (2), as shown in Figure 8 below, borrowed from our earlier work [4]. It remains to be understood if now the homomorphism SL(2, H) ∼ SO(5, 1) comes into play. And also, whether a quaternionic triality [46] could explain the existence of three generations of leptons. These aspects are currently under investigation. It is now only natural that this trace dynamics be extended to the last of the division algebras, the octonions, so as to construct an octonionic special relativity. This amounts to extending the Lorentz algebra by U (3), as can be inferred from Fig. 8.
Octonionic special relativity, complex octonions, and the algebra R × C × H × O The background space-time is now an octonionic space-time with coordinate vector x = x 0 e 0 + x 1 e 1 + x 2 e 2 + x 4 e 4 + x 3 e 3 + x 5 e 5 + x 6 e 6 + x 7 e 7 , and the corresponding eight-vector for a particle in this octonionic special relativity is q i = q 0 e 0 + q 1 e 1 + q 2 e 2 + q 4 e 4 + q 3 e 3 + q 5 e 5 + q 6 e 6 + q 7 e 7 . In ordinary relativity, the q i are real numbers, but now in trace dynamics they are bosonic or fermionic matrices. The space-time symmetry group is the automorphism group G 2 of the octonions, shown in Fig. 8, along with its maximal sub-groups, which reveal the standard model along with its 4D Lorentz symmetry. The Lagrangian is the same as in (86) above, but now written on the 8D octonionic space-time. As a result, q B and q F have component indices (3,5,6,7) whereas their time derivatives have indices (0, 1, 2, 4). This is the Lagrangian analysed in the main part of the present paper and it now includes quarks as well as leptons, along with all twelve standard model gauge bosons plus two Lorentz bosons.
We note the peculiarity that the weak part of the Lorentz-weak symmetry of the leptons, obtained by extending the Lorentz symmetry, intersects with the electr-color sector provided by U (3) ∼ SU (3) × U (1). This strongly suggests that the lepton part of the weak sector can be deduced from the electro-color symmetry. This is confirmed by the earlier work of Stoica [9], Furey [8] and our own earlier work [4].
We see that this Lagrangian is a natural generalisation of Newtonian mechanics and 4D special relativity to the last of the division algebras, the octonions, which represent a 10D Minkowski space-time because of the homomorphism SL(2, O) = SO(9, 1).

Emergent quantum field theory
In the entire discussion above, relating generalised trace dynamics to the standard model, we have made no reference to quantum field theory. The pre-quantum, pre-space-time matrix-valued Lagrangian dynamics which we have constructed above, reveals the standard model and its symmetries (including the Lorentz symmetry) without any fine tuning.
Quantum field theory, and classical space-time, are emergent from this pre-theory, after coarse-graining the underlying theory over time-scales much larger than Planck time, in the spirit of Adler's trace dynamics.
String theory is pre-space-time, but not pre-quantum. Trace dynamics is pre-quantum, The diagram below lists the three main steps in which the octonionic theory is developed.
Current investigation is focused at the third step.
The emergence of standard quantum field theory on a classical space-time background is a result of coarse-graining and spontaneous localisation and has been described in our earlier papers [10,12]. Spontaneous localisation gives rise to macroscopic classical bodies and 4D classical space-time. From the vantage point of this space-time those STM atoms which have not undergone spontaneous localisation appear, upon coarse-graining of their dynamics, as they are conventionally described by quantum field theory on a 4D classical space-time.
Operationally, the transition from the action of the pre-spacetime pre-quantum theory is straightforward to describe. Suppose the relevant term in the action of the pre-theory is denoted as dτ T r[T 1 ] + T r[T 2 ] + T r[T 3 ] . Say for instance the three terms respectively describe the electromagnetic field, the action of a W boson on an electron, and the action of a gluon on an up quark. Then, the corresponding action for conventional QFT will be recovered as: The trace has been replaced by the space-time volume integral, and each of the three terms have correspondingly been replaced by the conventional field theory actions for the three cases: conventional action for the electromagnetic field, for the W boson acting on the electron, and for the gluon acting on the up quark. In this way, QFT is recovered from the . An STM atom is an elementary fermion along with all the fields that it produces. The action for an STM atom resembles a 2-brane in a 10+1 dimensional Minkowski spacetime. The fundamental universe is made of enormously many STM atoms. From here, quantum field theory is emergent upon coarse-graining the underlying fundamental theory. pre-theory.
However, by starting from the pre-theory, we can answer questions which the standard model cannot answer. We know now why the standard model has the symmetries it does, and why the dimensionless free parameters of the standard model take the values they do.
These are fixed by the algebra of the octonions which defines the 8D octonionic spacetime in the pre-theory. While this is work in progress, it provides a promising avenue for understanding the origin of the standard model and its unification with gravitation.