The Universe as a Material: How spacetime froze, and why the quantum world never did

Take a glass of water. Nothing in a single water molecule is wet. Wetness, flow, the slow swirl when you stir in sugar - none of these belongs to any molecule. They belong to the crowd. Ten trillion trillion molecules, each obeying simple rules, collectively produce behaviours the rules never mention. Cool the same molecules a little and the crowd reorganises: suddenly there is rigidity, and the ability to bear weight. Ice is not a new substance; it is the same substance in a new state of order. Physics learned something profound from water: the properties we experience are usually not properties of the constituents but of how the constituents organise - and when the organisation changes, at a freezing point or a critical point, the large-scale world changes character abruptly, though nothing about the molecules has changed at all.

This essay is about a proposal that the two deepest features of our world - the smooth spacetime of Einstein's gravity, and the probabilistic rules of quantum theory - are exactly this kind of thing. Not fundamental laws written into the bedrock of reality, but two phases of one underlying substance, as ice and liquid water are two phases of one substance. And that the transition between them happened at a definite temperature, at a definite moment in the early universe, much as a pond freezes on a winter night.

The molecules of reality

If spacetime and quantum behaviour are like ice and water, what plays the role of the molecules? In our proposal they are entities we call aikyons - atoms of space-time-matter. An aikyon is not a particle sitting in space, and not a piece of space containing matter, but something more primitive from which both notions are later carved out. Asking whether an aikyon is matter or geometry is like asking whether a water molecule is ice or steam: the question dissolves once you see that ice and steam are ways the molecules organise, not ingredients inside them.

Aikyons differ from water molecules in one crucial respect, and it is the source of everything quantum. The position of a molecule is an ordinary number; the corresponding attributes of an aikyon are matrices - objects for which the order of operations matters, so that doing A then B differs from doing B then A. There is no probability and no mystery at this level: aikyons evolve by deterministic laws as classical in spirit as Newton's. But their arithmetic is non-commuting, and that single structural fact, carried up through the statistics of the crowd, will surface as the uncertainty principle. One more ingredient completes the picture: a tiny fraction of each aikyon's dynamics does not conserve probability in the ordinary sense. In the liquid phase this whisper stays negligible. Like a trace impurity that becomes decisive at the freezing point, it is precisely this whisper that will drive the freezing of the world.

Quantum theory is the liquid

Here is the first half of the dictionary. Quantum theory, in this picture, is not a fundamental rulebook. It is the hydrodynamics of the aikyon fluid  - the statistical description of the substrate near equilibrium, just as the smooth flow of water emerges from molecular chaos when you average over the crowd. We usually treat quantum mechanics as the deepest layer; here it is itself one level up, the thermodynamics of something more fundamental. Quantum fluctuations are, quite literally, the thermal-like jitter of the underlying medium - the way Brownian motion, the jiggling of a pollen grain in water, is the visible fingerprint of invisible molecules. Einstein used Brownian motion to prove that atoms exist. In this programme, quantum fluctuations are the Brownian motion of the aikyons: the fingerprint of the layer below. This idea, pioneered by Stephen Adler under the name trace dynamics, comes with precise bookkeeping: when the aikyon fluid equilibrates, a conserved quantity spreads itself democratically among the degrees of freedom, as thermal energy equipartitions in a gas - and the mathematical shadow of that equipartition is exactly the commutation relations of quantum mechanics, the algebraic heart of the theory.

Gravity is the ice

When water freezes, the molecules surrender their freedom to wander and lock into a lattice. Each acquires what it never had in the liquid - a definite, persistent position - not because the molecule changed, but because the collective settled into an ordered state that assigns everyone a place. And the lattice acquires properties liquid water lacks: rigidity above all. Ice has stiffness; it transmits stress; it rings when struck.

In our proposal, classical spacetime is the lattice, and gravity is the stiffness. When enough aikyons aggregate, the whisper in their dynamics is amplified, and the aggregate undergoes spontaneous localisation: it freezes out of the quantum liquid into a state with definite properties. Macroscopic objects are frozen. A table has a definite position for the same reason a molecule in ice does - not because definiteness is fundamental, but because the table is deep inside the ordered phase. The measurement problem of quantum mechanics - why do definite outcomes occur at all? - becomes the question of why ice forms, which is a question physics knows how to answer. The geometry of spacetime is the order of this frozen phase: distances and light cones are collective coordinates of the condensate, like the lattice directions of a crystal. Einstein's equations become what engineers call a constitutive law - the stress-strain relation of a material - and Newton's constant becomes an elastic modulus, a stiffness coefficient of frozen spacetime. Gravitational waves are the ringing of the ice; they cross the universe almost undamped because good crystals ring clean.

Savour the inversion of everyday intuition: we do not live in the warm liquid looking at frozen patches. We live inside the ice. The quantum systems in our laboratories - an electron holding itself in two places at once - are droplets of the primordial melt that never froze, persisting inside the frozen world like liquid inclusions in a glacier. Every interference experiment is a peephole into the liquid phase of reality.

The melting point of the world

Every phase transition has its critical temperature, and here the proposal becomes concrete and daring. The freezing point of spacetime, we suggest, is the electroweak scale - the energy at which the Higgs field switched on and elementary particles acquired their masses: about two million billion kelvin, a condition the universe experienced once, a tenth of a nanosecond after the Big Bang. Above that temperature there was no spacetime and no classical anything - only aikyon liquid. As the universe cooled through the electroweak epoch, several orderings happened together, the way freezing water simultaneously fixes molecular positions, picks crystal axes, and expels dissolved air: spacetime crystallised and gravity appeared as its stiffness; the Higgs condensed and particles got masses; and the pattern of those masses was selected like the orientation of the crystal axes. Gravitation, mass, and classicality, on this view, share one birthday. Whether they truly share it, rather than merely living near each other, is a sharp question - and the point of the new work is that it is a calculable question, not a slogan.

From analogy to instrument

Analogies are cheap in physics; instruments are not. The recent paper replaces every poetic phrase above with a number that can be computed and a test that can fail. Condensed matter physics spent a century building the toolkit for exactly this situation: order parameters that say what has frozen; susceptibilities that diverge as a transition approaches; the distinction between abrupt first-order freezing, with supercooling, and gentle continuous transitions; fluctuation-dissipation relations that hold in equilibrium and break, measurably, during a quench. The paper transplants this toolkit, item by item, to the aikyon substrate — and states bluntly the conditions under which the proposal would be dead.

And there is a demonstration one can hold in one's hands. In the paper, a small ensemble of random matrices — a stand-in for the aikyon fluid, simple enough to simulate on a laptop — is cooled through its own transition. On one side of the critical coupling: featureless matrix soup. On the other: the matrices spontaneously organise into a fuzzy sphere — an actual geometry, with a definite dimension, crystallising out of pure algebra. A quantum particle set loose on this ensemble literally hears the change: its energy spectrum, structureless in the soup, snaps into the unmistakable pattern of a two-dimensional surface once the geometry forms. The transition shows supercooling and hysteresis, exactly like real freezing, and the matter in the model is dragged along by the ordering - a miniature of masses appearing when spacetime does. None of this proves the universe works this way; the model is a stand-in, and the paper says so on every page. What it proves is that the question has become the kind physicists know how to answer: with ensembles, order parameters, and error bars, rather than metaphors alone.

That is the excitement. For a century, gravity and quantum theory have been treated as rival monarchs whose kingdoms refuse to merge. The materials view suggests they were never rivals: one is the liquid and one is the ice, and the task is not to unify two sets of laws but to find the single substance whose phases they are — then to seek, in experiments on collapsing quantum states and in the relics of the cooling universe, the fingerprints of the day the world froze. Landau built his tools for magnets and superfluids. It would be a fine irony, and a very physical one, if those same tools described the birth of space and time.

The technical version of these ideas appears in: T. P. Singh, “Critical emergence of quantum theory, spacetime and gravity from generalized trace dynamics,” Zenodo preprint (2026), doi:10.5281/zenodo.21811639, submitted to the International Journal of Modern Physics D. The record includes the complete simulation code and data, together with a list of the ways the proposal could fail — which is, after all, what makes it physics.

Do extra dimensions of time resolve the puzzle of quantum non-locality?

Consider a quantum particle emitted from a source, such that the associated wave function spreads as a spherical wave front. Assume that at a large distance from the source there is a spherical shell of a very large number of detectors. When the wave front arrives at the shell of detectors, one and only one (randomly selected) detector clicks, and the wave function is said to have collapsed to the location of that detector. The said particle has been localized to the clicked detector and is obviously no longer anywhere else. Therefore,  at the instant of collapse, the wave function becomes zero at the location of every other detector. This vanishing of the wave function happens even before any signal (traveling at the speed of light) from the location of the clicked detector can arrive at the other detectors. How does such an influence take place, in apparent violation of the principles of special relativity? This is the puzzle of quantum non-locality. The puzzle can also be highlighted by considering a pair of correlated and entangled quantum particles emitted by a source, and traveling in opposite directions, to be received by two independent detectors (Alice, and Bob) far away from each other. The measurement on the state of one particle instantly causes the state of the other particle to collapse. This ability of one particle to influence the other particle is in violation of the principle of relativistic causality [`an event B can be influenced by event A only if B occurs after a light signal traveling from A has arrived at B’]. This is known as the Einstein-Podolsky-Rosen (EPR) paradox or, equivalently, the quantum non-locality puzzle.

Independently of the EPR paradox, we ask the following question: why does the universe have only one dimension of time, whereas it has three dimensions of space? Could it be that there are in fact three dimensions of time as well, and the two additional dimensions of time are compact and too tiny (in their time-radius) and hence have escaped detection so far? Our ongoing research on unification of forces suggests this to be the case, and hence that our `quantum’ universe has six space-time dimensions. It is only the classical universe which appears to have four spacetime dimensions. The four classical dimensions are curved by gravitation (according to the laws of the general theory of relativity), whereas the two additional timelike dimensions are curved by the weak force (according to the laws of the electro-weak theory). The electroweak symmetry breaking in the very early universe bifurcates the 6D spacetime into two overlapping copies of 4D spacetimes: one is ours, with three spatial and one timelike dimension. The other 4D spacetime has three timelike and one spatial dimension; and the two 4D spacetimes have one space and one time dimension in common. 

The presence of the other 4D spacetime provides a novel causal channel for influence to travel from Alice to Bob. This channel, which we may call a `quantum wormhole’, offers a resolution of the EPR paradox, without having to modify the rules of quantum mechanics or of special relativity. Spacetime distances between Alice and Bob, through this other 4D spacetime, are far, far smaller than through our spacetime. The two spacetimes have distinct metrics of their own. A causal signal traveling from Alice, through the quantum wormhole, arrives at Bob almost instantaneously, in comparison to the corresponding signal traveling through our 4D spacetime. Quantum locality is hence restored in the 6D spacetime, and the apparent quantum nonlocality in our 4D spacetime is only an illusion, caused by our lack of knowledge of the additional timelike dimensions. Bell’s inequalities continue to be violated: the reason for their violation is indeterminism. Local hidden variable theories continue to be ruled out, but local indeterministic theories are allowed.  This kind of a resolution of the EPR paradox does not work if the additional dimensions are spacelike. The quantum particle mediating the signal through the other 4D spacetime is the so-called massless  dark photon predicted by our unification theory; this particle couples to the square-root of mass and should be sought for in laboratory experiments.

References:

1. Time-like extra dimensions: non-locality, spin, and Tsirelson bound, Mohammad Furquan, Tejinder P. Singh and P Samuel Wesley, Universe 11, 137 (2025) https://doi.org/10.3390/universe11050137

2. Does our universe have more than one dimension of time? Tejinder P. Singh, https://www.preprints.org/manuscript/202505.1074/v1

3. Gravi-weak unification in a six-dimensional spacetime with signature (3,3), T.Asselmeyer-Maluga, F. Finster, N. Gresnigt, J. Isidro, A. Marciano, C. Paganini, T.P. Singh and P Samuel Wesley, in preparation (2025).

Beyond string theory: new ideas for unification

In a recent interview with Curt Jaimungal @TOEwithCurt Prof. Leonard Susskind admits that string theory does not describe the real world. And that string theorists have put the project of unification on the back-burner, and that new ideas beyond string theory are needed.

It has for long been said by several researchers that if we want to quantise gravity, and unify it with the standard model of particle physics, we need to first fix foundational problems of quantum theory. These problems include the following:

(i) The problem of time in quantum theory: Quantum theory assumes classical time to describe evolution. However, spacetime and its geometry can be assumed to be classical if and only if the universe is dominated by macroscopic classical bodies. Such classical objects are a limiting case of quantum systems. It therefore follows that the current formulation of quantum theory depends on its own limit. This can only be an approximate description, and an exact formulation will not depend on classical time. Such an exact formulation is sought at all energy scales, and not just at the Planck scale, for in principle even a low energy universe can be entirely devoid of classical objects. Such a formulation turns out to be a gateway to quantum gravity, and from thereon to unification. The key idea is to replace classical spacetime (labeled by real numbers) by a non-commuting spacetime labeled by quaternions / octonions. The use of octonions unifies space-time symmetries [GR] with internal (gauge) symmetries [standard model].

(ii) Why do macroscopic entangled systems not obey the quantum principle of linear superposition in spatial position? Even though such macroscopic systems are composed of microscopic ones which obey such superposition.

(iii) Why does the wave-function collapse during a quantum measurement? Why are the outcomes of the collapse random and why do they obey the Born probability rule?

(iv) How do correlated quantum systems manage to influence each other outside the light-cone? [i.e. the quantum non-locality puzzle, the EPR paradox].

It is true that the current formulation of quantum theory is extremely successful and is not contradicted by any experiment. However, from here it does not follow that the foundational problems can be ignored. They were ignorable in the standard model [SM] of particle physics described by quantum field theory - but even there only partly so. Because one does not know why the dimensionless coupling constants take the values they do. Foundations cannot be ignored if one is trying to quantise gravity and/or unify it with the SM.

String Theory has paid a heavy price for neglecting these foundational problems of quantum theory.

Many physicists are attempting to approach the unification problem by making the above foundational questions as the starting point of their investigation. One example of such an approach is described in the attached brief review. Here, the pre-quantum theory of Trace Dynamics [due to Stephen Adler and collaborators] is generalised to a pre-spacetime, pre-quantum theory. From here, gravitation and quantum theory both are emergent phenomena. The octonionic theory, as it has come to be known, addresses and answers the above foundational questions, and in the process arrives at a theory of quantum gravity and of unification. 

This is an ongoing research program.   

https://www.preprints.org/manuscript/202411.0351/v1

What every string theorist should know about physics

It is not possible to quantize gravity without first also unifying it with the forces described by the standard model of particle physics. This is because quantum gravity will be sourced by the energy-momentum of bosons and fermions, and these elementary particles take part also in standard model forces, all of which are stronger than gravity. Therefore, switching on quantum gravity necessitates switching on the other forces as well.

This argument has been used (correctly) in the past to criticize stand alone theories of quantum gravity such as loop quantum gravity. On the other hand, in recent times string theory is being presented as a successful UV-complete theory of quantum gravity (without making any reference to unification). But surely the aforesaid criticism applies equally well to stringy quantum gravity. To this some respond by saying that the different vibrations of the string do include fermions and gauge bosons of  Yang-Mills theories. However, since the standard model has not yet been derived from string theory, the UV-complete stringy quantum gravity cannot by itself be nature’s true/correct quantum gravity theory per se.

Without successful unification, stringy quantum gravity in itself suffers from the above criticism and can be conveniently forgotten. Nonetheless, a vibrating string whose different vibrations are various elementary particles is in itself a very attractive idea. Can the idea be saved? The answer is yes, provided it is arrived at in a foundational manner, as follows.

A string is a quantum entity right from the word go, and so is a collection of strings. Such a collection obeys the quantum superposition principle and can never give rise to a classical space-time. Not even at low sub-Planck energies, nor when gravity is negligible. This is because  a quantum particle can be in more than one place at the same time; hence its resultant gravitation is also in a superposition. It is then a consequence of the Einstein hole argument that the point structure of the underlying classical spacetime is destroyed. Therefore, string theory can never be successfully formulated as a perturbative quantum field theory around Minkowski spacetime. Not even at low energies. Trying to do so is the reason why the theory fails as a theory of unification.

Nor is it a sound starting principle to assume, without justification, that fundamental building blocks of nature are extended objects such as strings. It would be fruitful if extended objects can be motivated from some basic premise. Such a premise exists. And that is to demand that there should exist a reformulation of quantum theory which does not depend on classical spacetime (for reasons outlined above), even at low energies. Such a theory is Stephen Adler’s trace dynamics, which is a pre-quantum theory: it is a matrix-valued Lagrangian dynamics from which quantum field theory is an emergent approximation. Trace dynamics can be transformed into a pre-spacetime theory by replacing every point of spacetime by an octonionic space (more precisely split bioctonionic space). Consistency of equations of motion then demands that the fundamental building blocks of space-time-matter must be extended objects, whose different vibrations are bosons and fermions. There is no supersymmetry. The trace dynamics Lagrangian is assumed to have an E8 x E8 symmetry, and symmetry breaking reveals the standard model forces and gravity, and two new forces are predicted. We also predict three sterile neutrinos, a BSM charged Higgs, and that the mass ratios of the electron, up quark, and down quark is precisely 1:4:9 (in the asymptotic free limit). 

Elementary particle states are described not by complex numbers, but my complex octonions. There are necessarily three and only three fermion generations, and the Dirac equation describing them obeys the exceptional Jordan algebra. The eigenvalues of the characteristic equation of this algebra reveal values of (at least some of) the fundamental constants of the standard model. Space-time is obtained from the squaring of the split bioctonionic space, and via a dynamically induced quantum-to-classical transition. The extra dimensions are not compactified – their extent is of the order of the range of the strong force and the weak force. 

The predictions of our unification theory based on extended objects are not at the Planck scale. The Planck scale gets naturally reset to the TeV scale in our theory. While much remains to be done and tested before claiming a successful theory of unification, we are free of the troubles of string theory (compactification and non-uniqueness, non-predictability, inability to derive the standard model and its fundamental constants). This is possible because from the outset we forego Minkowski spacetime in favour of the non-commutative octonionic space – this is where fermions live. And we forego quantum field theory (which needs classical time) in favour of the pre-quantum theory of trace dynamics.

In this octonionic theory, we have a better theory of unification (based on extended objects) than string theory in its current shape is. It is no longer correct to say that string theory is the only known / leading candidate for quantum gravity and unification.

Does our universe possess a second 4D spacetime, with its own light-cone, which is accessible only to quantum systems, and in which distances are necessarily microscopic?

As we discuss briefly in our recent CKM matrix paper 2305.00668, the answer to all the above questions is yes! Just as gravitation is the geometry of our familiar 4D spacetime, the weak force is the geometry of the second 4D spacetime. Indeed, the weak force is a spacetime symmetry masquerading as an internal symmetry. Symmetry breaking of a 6D spacetime in the very early universe gives rise to these two 4D spacetimes, one curved by gravity and the other curved by the weak force. The size of the universe in the other spacetime is of the order of the range of the weak force. A beam of light going through the universe in this other spacetime would be back to the starting point in just 10^-24 seconds! This could help us understand quantum nonlocality and the EPR paradox. When Einstein said that if quantum nonlocality (assuming QM is complete) is true then special relativity must go, one way to interpret Einstein is to propose that our universe has two 4D spacetimes.

In the twistor picture of our spacetime, null lines are more fundamental than spacetime points. Consider two null lines (t-x) and (t+x). A point arises as the intersection of these two null lines. 4D spacetime can be arrived at by overlaying these two null lines on a complex plane (y + iz). Consider the 2x2 matrix

t-x y+iz

y-iz t+x

Its determinant gives the 4D line element.

To get the second 4D spacetime we overlay these two null lines on an independent complex plane (a+ib) and make a new 2x2 matrix

t-x a+ib

a-ib t+x

The determinant now gives the line-element of the second 4D spacetime.

Between them the two 4D spacetimes are labeled by six real numbers (t, x, y, z, a, b) which can be used to obtain a 6D spacetime.

We can try to visualise the second spacetime by thinking of a torus in which the horizontal circle is much much larger than the vertical circle. The horizontal circle is space of our spacetime, the vertical circle is space of the other spacetime. If we are at location A on the torus then a galaxy at location B on the larger circle is correspondingly at location B' on the smaller circle, and B is identified with B' : the smaller circle is a scaled down version of the bigger circle and in one to one correspondence with it. A photon starting from our location will reach the galaxy B much faster along the smaller circle.

Mathematically, the two spacetimes result because of the group theory relations SL(2, H) ~ SO(1,5) and SL(2,C) ~ SO(1,3) where H are the quaternions. And because the Clifford algebra Cl(3) is the direct sum of two copies of Cl(2). Each Cl(2) generates a 4D Lorentz algebra...one is for our spacetime. The other is for the second spacetime whose three rotations are the weak isospin rotations, and the three boosts give Lorentz transformations along the vertical circle of the torus. Cl(3) is the algebra of complex split biquaternions and one can make a 6D spacetime from it.

If there is indeed a second 4D spacetime in our universe, which obeys the laws of special relativity, has its own light cone, is microscopic and accessible only to quantum systems, it can offer a neat solution to the EPR puzzle. Along the other spacetime, the photon arrives causally at B much before than along A, and this looks nonlocal from our perspective. One could call this a much more believable version of ER=EPR.

The trillion rupee question is: can the second spacetime be used for communication? Can we talk to someone on Andromeda in real time? Perhaps by sending weak force waves, analogous to gravitational waves, along the second spacetime.

Spacetime, vector bundles, and unification: the coming together of ideas which have been around for the last three decades or so

I have attached here Ed Witten's nice summary of `our knowledge of physics' :

The attempts at unification in the style of a Kaluza-Klein theory try to bring the vector bundle and the spacetime into the fold of a higher dimensional spacetime with its own metric, and the theory is then to be quantised.

In spirit, the octonionic theory does the same thing, but the formalism is not directly about `metric and higher dimensional spacetime'. Instead, one can construct the Dirac operator on a curved spacetime and find its eigenvalues; these are alternatives to the metric as observables of general relativity, and these eigenvalues are the entities used in our Kaluza-Klein programme, instead of the metric. We consider the eigenvalues of the Dirac operator on a higher dimensional spacetime. And these not only unify gravitation with the gauge fields of the vector bundle, but also unify the fermions with gauge fields and gravity. One quantum of this unified entity is named an atom of spacetime-matter, or an aikyon. Classical fields are recovered as `condensates' (i.e. macroscopic entanglements) of many aikyons.

Furthermore, we do not work on a higher dimensional curved Minkowski spacetime. We work on the square-root of Minkowski, a spinor spacetime, i.e. a twistor space labelled by the octonions. These define an 8D non-commutative space. We construct the Dirac operator on this octonionic space. Its eigenvalues are the dynamical variables which describe the aikyons. The act of quantisation consists of raising the eigenvalues to the status of matrices/operators. One matrix for every eigenvalue: this is the dynamical variable for the aikyon. Which matrix/operator? The very Dirac operator of which it is the eigenvalue. This is important, because in the quantum-to-classical transition, each matrix collapses to one of the eigenvalues, and from the collection of all the collapsed eigenvalues one again recovers the Dirac operator and hence the classical theory (on a 4D spacetime, because the collapse is a spacetime-dimension-reducing process, with classical gauge fields living on the 4D spacetime).

The matrix D for the aikyon has Grassmann numbers as its entries, and hence can be written as a sum of a bosonic matrix (even grade Grassmann) and a fermionic matrix (odd grade Grassmann). D plays the role of canonical momentum. Let us define a configuration variable Q whose time derivative (and hence velocity) is D. The Lagrangian of the aikyon is simply Trace[D^2] where Trace is matrix trace. The action is simply the time integral (Connes time) of this kinetic energy Tr[D^2]. This is nothing but Newton's free particle with a Ph. D. so to say Newton's 3D absolute Euclidean space has been replaced by curved octonionic space: the matrix-valued coordinate components D_i of D are analogous to metric tensor components, they determine the geometry of octonionic space, which encodes gauge fields of the standard model, gravity, and fermions. Newton's absolute time has been replaced by Connes time, the latter being a property of non-commuative geometries. This matrix-valued Lagrangian dynamics is Adler's trace dynamics: a pre-quantum pre-spacetime theory from which QFT on classical curved spacetime is emergent. The fundamental universe is made of a large number of aikyons.

To incorporate chiral fermions the octonionic space is generalised to split bioctonionic space, which is essentially the doubling of the octonionic space to 16D, with the second half being the parity reverse of the first, hence permitting chiral fermions to be introduced. The velocity Q-dot is used to define a new matrix variable q such that Q-dot = q_dot +q and the Lagrangian is rewritten in terms of q and further rewritten in terms of the bosonic and fermionic parts: q=q_B + q_F. Bosons and chiral fermions emerge. The dotted variables relate to gravity and to right-chiral fermions and are defined over the second half of the 16D space; the undotted ones relate to the standard model forces and to left chiral fermions and are defined over the first half of the 16D space. Left chiral fermions are eigenstates of electric charge; right chiral fermions are eigenstates of square-root mass: +sqrt{m} is matter; -sqrt{m} is antimatter. Our universe has only the former, having separated from antimatter in a breaking of scale-invariance (symmetry) in the very early universe. The Lagrangian prior to symmetry breaking is scale invariant. After the breaking, scale invariance is replaced by C, P, T: our universe violates CP and violates T. Together with the mirror antimatter universe which ours separated from, CPT is preserved. The mirror universe is a CP image and a T image of our universe.

The Lagrangian is assumed to have an E8 x E8 symmetry, with the first E8 symmetry being over the 8D half, and the second E8 over the parity reversed (split part) 8D half. The first 8D space has Euclidean signature and is equivalent to SO(10) space. The second 8D space has Lorentzian signature and is equivalent to SO(1,9) spacetime. We see the coming together of Witten's vector bundle and of spacetime into a unified entity, which is a physical reality. The vector bundle is not merely a mathematical construct; it is reality. The aikyon does not live in spacetime, but in E8 x E8 space: we could call it aikyon space. Only classical objects (these result from entanglement of many aikyons) live in spacetime (4D spacetime). Aikyons live in aikyon space, and the Higgs (implied naturally by the Lagrangian) couples left chiral and right chiral fermions.

Octonions are magical. Not only do they define spacetime and gauge field space, they also define elementary particles (SM fermions and bosons) and determine their properties such as quantisation of charge and mass. The octonionic coordinates and the Lagrangian work hand in hand in all this. Spinors made from Clifford algebras made from octonionic maps define quarks and leptons of the standard model. The first E8 branches as SU(3)_EuclideanSpace x SU(3)_ThreeGensLH x SU(3)_color X SU(2)_L x U(1)_Y. The second E8 branches as SU(3)_spacetime x SU(3)_ThreeGensRH x SU(3)_grav x SU(2)_R x U(1)_g Here SU(2)_R x U(1)_g lead to general relativity in the classical limit, whereas SU(3)_grav is new, and seems related to the conformal gravity modification of GR explicit in the Chamseddine-Connes spectral action principle (the heat kernel expansion of Tr[D^2]).

We recover the standard model and modified gravity in the emergent theory. The trace dynamics equations of motion, when reduced to an eigenvalue problem, give evidence for determining values of free parameters of the standard model.

Note that in unifying the vector bundle and the spacetime we never had to go to high energies. We have simply recast what we already know, onto a spinor spacetime, which when enlarged to higher dimensions, casts gauge fields and gravity into the aikyon space with E8xE8 symmetry. Quantum systems live in this aikyon space even at low energies. The quantum-to-classical transition that we observe around us all the time breaks E8xE8 because macroscopic objects are confined to 4D and their localisation is the very process which in the first place gives rise to the 4D classical spacetime and segregates the vector bundle (which is Euclidean space) from emergent spacetime.

A case for Adler's trace dynamics

Suppose we are asked to `quantise' classical dynamics, and are given the following two choices for how to do it. Which one should we choose, given that the second choice agrees with all experiments done so far, and the first one is untested because it is a Planck scale theory :

1. Trace Dynamics [Stephen Adler, 1996]

Starting from classical dynamics, raise all dynamical variables to the status of matrices / operators, and hence arrive at a Lagrangian which is a matrix polynomial. Take its matrix trace, and use this trace Lagrangian (a scalar) as the new Lagrangian in the action principle. You can now develop a matrix-valued Lagrangian dynamics, derive its matrix-valued equations of motion, and also the corresponding Hamiltonian dynamics. Everything proceeds as in conventional classical dynamics,

(i) except that

The dynamical variables, now being matrices, do not commute with each other. The commutator [q,p] evolves with time and is determined by the dynamics.

(ii) and except that

The new matrix dynamics has a conserved Noether charge, absent in classical dynamics, which is a result of the invariance of the trace Hamiltonian under global unitary transformations. This is coming about because we are now working with matrices and with a Hamiltonian which is a trace over matrices. The conserved charge is

Sum over all degrees of freedom i of the commutators

[q_i, p_i]

That is, whereas each [q_i. p_i] evolves with time, the sum of all such commutators is conserved. It is as if the d.o.f. exchange [q,p] with each other dynamically. This conserved quantity known as the Adler-Millard charge has the dimensions of action. Its existence is what makes trace dynamics into a pre-quantum theory. One never quantises trace dynamics; rather quantum theory emerges from it, as follows.

It is assumed that trace dynamics holds at some energy scale, not yet tested in the laboratory, say the Planck scale. We then ask what is the emergent dynamics at a lower energy scale, such as at the LHC, if one is not observing at the Planck scale. Techniques of statistical thermodynamics are employed to answer this question, and it is shown that in the emergent low energy theory, the Adler-Millard charge is equipartitioned over all d.o.f. As a result, for all coarse-grained d.o.f. the averaged commutator < [q,p] > takes the same value, and it is set equal to i\hbar. This is how one gets [q,p]=\ihbar, the Heisenberg algebra.

The averaged Hamilton's equations of motion of the underlying theory become Heisenberg equations of motion, and quantum field theory is recovered as a low energy emergent approximation to trace dynamics.

2. The second choice: Quantum Theory

Raise all classical dynamical variables to the status of matrices / operators, and impose by hand in an ad hoc way the Heisenberg algebra

[q, p] = i\hbar

The resulting quantum field theory agrees with all experiments done so far. But from a theoretical viewpoint imposing the Heisenberg algebra seems ad hoc. q and p do not commute once they are matrices. Shouldn't the dynamics determine the commutator [q,p] as in trace dynamics, with [q,p]=i\hbar emerging in an approximation?

With trace dynamics as a benchmark, one can now view quantum theory as a special case of trace dynamics. Using trace dynamics along with a spacetime described by the octonions opens up new possibilities for better understanding of the standard model of particle physics, and its unification with general relativity.

So which one do we choose: 1 or 2? Should the Heisenberg algebra be imposed a priori, or allowed to emerge from a more general theory which does not constrain the commutator [q,p] but lets it evolve dynamically?