Quantum gravity effects in the infrared: a theoretical derivation of the low-energy fine structure constant and mass ratios of elementary particles

Abstract

We have recently proposed a pre-quantum, pre-spacetime theory as a matrix-valued Lagrangian dynamics on an octonionic spacetime. This theory offers the prospect of unifying internal symmetries of the standard model with pre-gravitation. We explain why such a quantum gravitational dynamics is in principle essential even at energies much smaller than Planck scale. The dynamics can also predict the values of free parameters of the low-energy standard model: these parameters arising in the Lagrangian are related to the algebra of the octonions, which define the underlying non-commutative spacetime on which the dynamical degrees of freedom evolve. These free parameters are related to the exceptional Jordan algebra $J_3(8)$, which describes the three fermion generations. We use the octonionic representation of fermions to compute the eigenvalues of the characteristic equation of this algebra and compare the resulting eigenvalues with known mass ratios for quarks and leptons. We show that the ratios of the eigenvalues correctly reproduce the [square root of the] known mass ratios. In conjunction with the trace dynamics Lagrangian, these eigenvalues also yield a theoretical derivation of the low-energy fine structure constant.

Appendix: Physical motivation for the present theory: quantum (field) theory without classical time, as a route to quantum gravity and unification

In this appendix, we recall from earlier work [9] the motivation for developing a formulation of quantum theory without classical time, and how doing so leads to a pre-quantum, pre-spacetime theory which is a candidate for unification of general relativity with the standard model.

1.1 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 spacetime. Such a spacetime—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 spacetime! 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 constant $\hbar$. 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 $L_P/t_P = c = 3\times 10^8$ cm/s then the system is Planckian. If the energy of the system is of the order $\hbar / t_P = 10^{19}$ GeV, the system is Planckian. If the action of the system is of the order $\hbar$, the system is Planckian. If the charge-squared is of the order $\hbar c$, the system is Planckian. Thus in our concepts, the value 1/137 for the fine structure constant, being order unity in the units $\hbar c$, is Planckian. This explains why this pre-quantum, pre-spacetime theory knows the low energy fine structure constant.

A quantum system on a classical spacetime background is hugely non-Planckian. Because the classical spacetime is being produced by macroscopic bodies each of which has an action much larger than $\hbar$. The quantum system treated in isolation is Planckian, but that is strictly speaking a very approximate description. The spacetime background cannot be ignored - only 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 spacetime—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 spacetime 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 spacetime 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 spacetime/gravity from quantum [field] theory. Spacetime 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 know - namely 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 background spacetime is [justifiably] neglected.

1.2 Why a quantum theory of gravity is needed at all energy scales, and not just at the Planck energy scale? And how that leads us to partially redefine what is meant by Planck scale: Replace Energy by Action.

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 spacetime 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 spacetime has been removed from quantum theory. We also call it a pre-quantum, pre-spacetime theory.

What is meant by Planck scale, in this pre-theory?

Conventionally, a phenomenon is called Planck scale if: the time scale T of interest is of the order Planck time $t_P$; and/or length scale L of interest is of the order of Planck length $L_P$; and/or energy scale E of interest is of the order Planck energy $E_P$. According to this definition of Planck scale, a Planck scale phenomenon is quantum gravitational in nature. Since the pre-theory is quantum gravitational, but not necessarily at the Planck energy scale, we must partially revise the above criterion, when going to the pre-theory: replace the criterion on energy E by a criterion on something else. This something else being the action of the system!

In the pre-theory, a phenomenon is called Planck scale if: the time scale T of interest is of the order Planck time $T_P$; and/or length scale L of interest is of the order of Planck length $L_P$; and/or the action S of interest is of the order Planck constant $\hbar$. According to this definition of Planck scale, a Planck scale phenomenon is quantum gravitational in nature.

Why does this latter criterion make sense? If every degree of freedom has an associated action of order $\hbar$, together the many degrees of freedom cannot give rise to a classical spacetime. Hence, even if the time scale T of interest and length scale L of interest are NOT Planck scale, the system is quantum gravitational in nature. The associated energy scale $\hbar / T$ for each degree of freedom is much smaller than Planck scale energy $E_P$. Hence in the pre-theory the criterion for a system to be quantum gravitational is DIFFERENT from conventional approaches to quantum gravity. And this makes all the difference to the formulation and interpretation of the theory. For example, the low-energy fine structure constant 1/137 is a Planck scale phenomenon [according to the new definition] because the square of the electric charge is order unity in the units $\hbar c = \hbar L_P / t_P$

In our pre-theory, there are three, and only three, fundamental constants: Planck length $L_P$, Planck time $t_P$ and Planck action $\hbar$. Every other parameter, such as electric charge, Newton’s gravitational constant, standard model coupling constants, and masses of elementary particles, are defined and derived in terms of these three constants: $\hbar , L_P$ and $t_P$.

In the pre-theory the universe is an 8D octonionic universe, as shown in Fig. 3, the octonion, reproduced in Fig. 9. The origin $e_0=1$ stands in for the real part of the octonion [coordinate time] and the other seven vertices stand in for the seven imaginary directions. A degree of freedom [i.e. ‘particle’ or an atom of spacetime-matter (STM)] is described by a matrix q which resides on the octonionic space: q has eight coordinate components $q_i$ where each $q_i$ is a matrix. We have replaced a four-vector in Minkowski spacetime by an eight-matrix in octonionic space: and this describes the particle / STM atom. The STM atom evolves in Connes time, this time being over and above the eight octonionic coordinates. Its action is that of a free particle in this space: time integral of kinetic energy, the latter being the square of velocity $\dot{q}$, where dot is derivative with respect to Connes time. Eight octonionic coordinates are equivalent to ten Minkowski coordinates, because of $SL(2,O) \sim \mathrm{Spin}(9,1)$.

Fig. 9

The octonions [From Baez [29]]

The symmetries of this space are the symmetries of the (complexified) octonionic algebra: they contain within them the symmetries of the standard model, including the 4D-Lorentz symmetry.

The classical 4D Minkowski universe is one of the three planes (quaternions) intersecting at the origin $e_0 = 1$. Incidentally the three lines originating from $e_0$ represent complex numbers. The four imaginary directions not connected to the origin represent directions along which the standard model forces lie (internal symmetries). Classical systems live on the 4D quaternionic plane. Quantum systems (irrespective of whether they are at Planck energy scale) live on the entire 8D octonion. Their dynamics is the sought for quantum theory without classical time. This dynamics is oblivious to what is happening on the 4D classical plane. QFT as we know it is this pre-theory projected to the 4D Minkowski spacetime. The present universe has arisen as a result of a symmetry breaking in the 8D octonionic universe: the electroweak symmetry breaking. Which in this theory is actually the colour-electro – weak-Lorentz symmetry breaking. Classical systems condense on to the 4D Minkowski plane as a result of spontaneous localisation, which precipitates the electro-weak symmetry breaking in the first place. The fact that weak is part of weak-Lorentz should help understand why the weak interaction violates parity, whereas electro-colour does not. Hopefully the theory will shed some light also on the strong-CP problem.

1.3 What is Trace Dynamics? : Trace dynamics is quantisation, without imposing the Heisenberg algebra

In the conventional development of canonical quantisation, the two essential steps are:

• 1. Quantisation Step 1 is to raise classical degrees of freedom, the real numbers q and p, to the status of operators / matrices. This is a very reasonable thing to do.

• 2. Quantisation Step 2 is very restrictive! Impose the Heisenberg algebra $[q, p] = i \hbar$. Its only justification is that the theory it gives rise to is extremely successful and consistent with every experiment done to date. In classical dynamics, the initial values of q and p are independently prescribed. There is NO relation between the initial q and p. Once prescribed initially, their evolution is determined by the dynamics. Whereas, in quantum mechanics, a theory supposedly more general than classical mechanics, the initial values of the operators q and p must also obey the constraint $[q, p] = i \hbar$. 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 by Adler [6,7,8] - a pre-quantum theory, which we have generalised to a pre-quantum, pre-spacetime theory [15].

• 4. This matrix valued dynamics, i.e. trace dynamics, is more general than quantum field theory, and assumed to hold at the Planck scale, and also whenever background classical spacetime is absent, no matter what the energy scale. The Heisenberg algebra is shown to emerge at lower energies, or when spacetime emerges, after coarse-graining the trace dynamics over length scales much larger than Planck length scale. Thus, quantum theory is midway between trace dynamics and classical dynamics.

• 5. The moral of the story is that we assume that quantum field theory does not hold at the Planck scale. Trace dynamics does. QFT is emergent.

• 6. The other assumption one makes at the Planck scale is to replace the 4-D classical spacetime manifold by an 8D octonionic spacetime manifold, so as to obtain a canonical definition of spin. This in turn allows for a Kaluza–Klein-type unification of gravity and the standard model. Also, an 8D octonionic spacetime is equivalent to a 10-D Minkowski spacetime. It is very rewarding to work with 8D octonionic, rather than 10D Minkowski - the symmetries manifest much more easily.

• 7. Trace dynamics plus octonionic spacetime together give rise to a highly promising avenue for constructing a theory of quantum gravity, and of unification. 4D classical spacetime obeying GR emerges as an approximation at lower energies, alongside the emergent quantum theory.

• 8. How is this different from string theory? In many ways it IS like string theory, but ${ without}$ the Heisenberg algebra! The gains coming from dropping $[q,p]=i\hbar$ at the Planck scale are enormous. One now has a non-perturbative description of pre-spacetime at the Planck scale. The symmetry principle behind the unification is very beautiful: physical laws are invariant under algebra automorphisms of the octonions. This unifies the internal gauge transformations of the standard model with the 4D spacetime diffeomorphisms of general relativity. The automorphism group of the octonions, the Lie group $G_2$, which is the smallest of the five exceptional Lie groups, contains within itself the symmetries $SU(3) \times SU(2) \times U(1)$ of the standard model, along with the Lorentz symmetry. The free parameters of the standard model are determined by the characteristic equation of the exceptional Jordan algebra $J_3(O)$, whose automorphism group $F_4$ is the exceptional Lie group after $G_2$.

1.4 Normed division algebras, trace dynamics, and relativity in higher dimensions. And how these relate to quantum field theory and the standard model

Let us consider the four normed division algebras ${\mathbb R, \mathbb C, \mathbb H, \mathbb O}$ [Reals, Complex Numbers, Quaternions, Octonions] in the context of the spacetimes associated with them, and how these algebras relate to trace dynamics. This can be understood graphically with the help of Fig. 7, which contains within itself a representation of all the four division algebras.

Let us start with the reals $\mathbb R$, represented in the above diagram by the origin $e_0 = 1$. This direction represents the time coordinate, in all the four different spacetimes associated with these four division algebras. The three lines emanating from the origin and connecting, respectively, to $e_3, e_5, e_6$ represent complex numbers $\mathbb C$. The three planes intersecting at the origin represent quaternions $\mathbb H$ and the full cube represents the octonions $\mathbb O$.

1.4.1 Galilean relativity and Newtonian mechanics:

This is related to the quaternions, and we assume each of the three planes intersecting at the origin represent absolute Newtonian space [say in the plane ($1, e_1, e_5, e_6$) we set $e_1 = \hat{x}, e_5 = \hat{y}, e_6 = \hat{z}$]. Galilean invariance is assumed, and the spatial symmetry group is SO(3), the group of rotations in three dimensional space; this is also the automorphism group $Aut(\mathbb H)$ 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 $\mathbf{q}$, which is a three-vector, is simply

$$\begin{aligned} S = \int dt \; \dot{\mathbf{q}}^2 \end{aligned}$$

(79)

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 $e_1^2 = e_2^2 = e_3^2 = 1$, 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 $\mathbb R \times \mathbb H$, in an approximate sense, which becomes precise only in special relativity, as discussed below.

[The algebra $\mathbb C$ represents a 2D physical space, and $\mathbb R \times \mathbb C$ represents a spacetime for Newtonian mechanics in absolute two-space represented by $\mathbb C$, and absolute time $\mathbb R$. The homomorphism $SL(2, \mathbb R)\sim SO(2,1)$ suggests that we can relate 2x2 real-valued matrices to a 2+1 relativistic spacetime. 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. Thus $\hat{\mathbf{q}}$ is a matrix-valued three-vector whose three spatial components ${\hat{\mathbf{q}}}_\mathbf{1}, {\hat{\mathbf{q}}}_\mathbf{2}, \hat{\mathbf{q}}_\mathbf{3}$ are matrices whose entries are real numbers. The Lagrangian for a free particle will now be the trace of the matrix polynomial ${\dot{\hat{\mathbf{q}}}}^2$, and hence the action is

$$\begin{aligned} S = \int dt \; Tr [{\dot{\hat{\mathbf{q}}}}^2] \end{aligned}$$

(80)

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).

1.4.2 Special relativity, complex quaternions, and the algebra $\mathbb R \times \mathbb C \times \mathbb H$

Consider the quaternionic four vector $\mathbf{x} = x_0 e_0 + x_1 e_1 + x_2 e_2 + x_4 e_4$ and the corresponding position four-vector for a particle in special relativity: $\mathbf{q_i} = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_4 e_4$. One can define the four-metric on this Minkowski spacetime 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 $SL(2,\mathbb C)$. 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 [11,12,13] 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 [11,12,13], 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 spacetime of special relativity, thereby generalising the relativistic particle $S = - mc \int ds$:

$$\begin{aligned} \frac{S}{C_0} = \frac{a_0}{2} \int \frac{d\tau }{\tau _{Pl}} \; Tr \bigg [\dot{q}_B^{\dagger } + i\frac{\alpha }{L} q_B^\dagger + a_0 \beta _1\left( \dot{q}_F^\dagger + i\frac{\alpha }{L} q_F^\dagger \right) \bigg ] \times \bigg [ \dot{q}_B + i\frac{\alpha }{L} q_B+ a_0 \beta _2\left( \dot{q}_F + i\frac{\alpha }{L} q_F\right) \bigg ] \end{aligned}$$

(81)

where $a_0 \equiv L_P^2 / L^2$. 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 spacetime. Thus $\dot{q}_B$ and $q_B$ have four components between them, not eight: $q_B = q_{Be2}\; e_2 + q_{Be4} \; e_4; \; \dot{q}_B = \dot{q}_{Be0} \; e_0 + \dot{q}_{Be1}\ e_1$. Similarly, the fermionic matrices have four components between them, not eight. Thus $q_F = q_{Fe2}\; e_2 + q_{Fe4} \; e_4; \; \dot{q}_F = \dot{q}_{Fe0} \; e_0 + \dot{q}_{Fe1}\ e_1$

This has far-reaching consequences. Consider first the case where we set $\alpha =0$. The Lagrangian then is

$$\begin{aligned} \frac{S}{C_0} = \frac{a_0}{2} \int \frac{d\tau }{\tau _{Pl}} \; Tr \bigg [\dot{q}_B^{\dagger } + a_0 \beta _1 \dot{q}_F^\dagger \bigg ] \times \bigg [ \dot{q}_B + a_0 \beta _2 \dot{q}_F \bigg ] \end{aligned}$$

(82)

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 spacetime symmetry is the Lorentz group SO(3, 1) and the associated Clifford algebra is Cl(2), reminding us again of the homomorphism $SL(2,\mathbb C)\sim SO(3,1)$.

When $\alpha$ 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 10, borrowed from our earlier work [9]. It remains to be understood if now the homomorphism $SL(2, \mathbb H)\sim SO(5,1)$ comes into play. And also, whether a quaternionic triality [54] could explain the existence of three generations of leptons. These aspects are currently under investigation.

Fig. 10

The maximal sub-groups of $G_2$ and their intersection [From Singh [9]]. Strictly speaking, the maximal subgroups of $G_2$ are SU(3) and SO(4). The U(1) arises from the number operator made from generators of the Clifford algebra Cl(6)

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. 10.

1.4.3 Octonionic special relativity, complex octonions, and the algebra $\mathbb R \times \mathbb C \times \mathbb H \times \mathbb O$

The background spacetime is now an octonionic spacetime with coordinate vector $\mathbf{x} = x_0 e_0 + x_1 e_1 + x_2 e_2 + x_4 e_4 + x_3 e_3 + x_5 e_5 + x_6 e_6 + x_7 e_7$, and the corresponding eight-vector for a particle in this octonionic special relativity is $\mathbf{q_i} = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_4 e_4 + q_3 e_3 + q_5 e_5 + q_6 e_6 + q_7 e_7$. In ordinary relativity, the $q_i$ are real numbers, but now in trace dynamics they are bosonic or fermionic matrices. The spacetime symmetry group is the automorphism group $G_2$ of the octonions, shown in Fig. 10, along with its maximal sub-groups, which reveal the standard model along with its 4D Lorentz symmetry. The Lagrangian is the same as in (82) above, but now written on the 8D octonionic spacetime. 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 electro-colour sector provided by $U(3) \sim SU(3) \times U(1)$. This strongly suggests that the lepton part of the weak sector can be deduced from the electro-colour symmetry. This is confirmed by the earlier work of Stoica [14], Furey [13] and our own earlier work [9].

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 spacetime because of the homomorphism $SL(2, \mathbb O) = SO(9,1)$.

Left-Right symmetry breaking separates LH electric charge eigenstates from RH square-root mass particle eigenstates. The square-root of mass carries two signs: plus for matter, and minus for antimatter. CPT operations are mathematically defined as follows: complex conjugation for C, octonionic conjugation for P, and time-reversal operator T corresponds to change of sign of square-root mass. Pre-gravitation, i.e. $SU(2)_R$ symmetry, is a vector interaction which is attractive for matter-matter, attractive for anti-matter anti-matter, but repulsive for matter-antimatter. It is possible that the L-R symmetry breaking separates matter from antimatter in the very early universe, and this could be a possible explanation for the origin of matter-antimatter asymmetry. At the epoch of creation, our universe, made of matter, moves forward in time, whereas the antimatter universe moves backward in time. The two universes together obey CPT symmetry [for an earlier elegant proposal of CPT symmetric cosmology see [55]). Prior to this symmetry breaking the universe is scale invariant.

1.4.4 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-spacetime 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 spacetime, 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-spacetime, but not pre-quantum. Trace dynamics is pre-quantum, but not pre-spacetime. The octonionic theory [O-theory] is pre-spacetime and pre-quantum. It generalises trace dynamics to a pre-quantum, pre-spacetime theory. The O-theory is not intended as an alternative to quantum field theory. Rather, it is applicable in those circumstances when a background classical time is not available for writing down the rules of QFT. Then, the O-theory also reveals itself to be pre-quantum. When a background classical time becomes available, O-theory coincides with QFT and is no longer pre-quantum. O-theory reveals the symmetries of the standard model without any fine-tuning, and also shows a route for determining the free parameters of the standard model. This comes about because the background non-commutative spacetime fixes the properties of the allowed elementary particles. In this way, O-theory has a promising potential to tell us, in a mathematically precise way, where the standard model, and classical spacetime, come from. The O-theory is not a Grand Unified Theory [GUTs]. GUTs determine internal symmetries by making specific choices for the internal symmetry group, while classical spacetime and QFT are kept intact. In contrast to this, the O-theory retains neither QFT nor a classical spacetime. The symmetries of O-theory are a unification of internal and spacetime symmetries, in the spirit of a Kaluza-Klein theory.

The diagram in Fig. 11 lists the three main steps in which the octonionic theory is developed. Current investigation is focused at the third step.

Fig. 11

The pre-spacetime, pre-quantum octonionic theory in three key steps. The degrees of freedom are ‘atoms of spacetime-matter’ [STM]. 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

The emergence of standard quantum field theory on a classical spacetime background is a result of coarse-graining and spontaneous localisation and has been described in our earlier papers [15, 17]. Spontaneous localisation gives rise to macroscopic classical bodies and 4D classical spacetime. From the vantage point of this spacetime 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 spacetime. 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 $\int d\tau \; \bigg [ Tr [T_1] + Tr [T_2] + Tr [T_3] \bigg ]$. 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:

$$\begin{aligned} \int d\tau \; \bigg [ Tr [T_1] + Tr [T_2] + Tr [T_3] \bigg ] \quad \longrightarrow \quad \int d\tau \; \int d^4x \bigg [ [T_{1QFT}] + [T_{2QFT}] + [T_{3QFT}] \bigg ] \end{aligned}$$

(83)

The trace has been replaced by the spacetime 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 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.