E_8, octonions, standard model, gravitation, and quantum theory

There is some renewed activity in the field. Dray, Manogue and Wilson have a paper out on the arXiv few days back: Octions, an E_8 description of the standard model. They will give a talk on this coming Monday, you can find out more about it here:

https://www.furey.space

The typical goal here is to show that the standard model gauge symmetries, and the quarks and leptons, are embedded in the larger of the exceptional groups. Most of the time, the attention is focused on a GUTS type scheme, with Lorentz symmetry included, but not gravitation.

I have landed in this octonion space from very different considerations. Those of Quantum Foundations.

There must exist a reformulation of quantum theory which does not make any reference to classical space-time labelled by four real numbers. Such a space-time exists if and only if the universe is dominated by classical macroscopic objects, just as today's low energy universe is. Such classical objects are however a limiting case of quantum systems and therefore, in invoking spacetime, quantum theory depends on its own classical limit. This is approximate, no matter how well it agrees with current experiments. We can in principle imagine a low energy universe devoid of classical objects, and hence devoid of classical spacetime. How to describe the dynamics then?

We can also ask what is the ground state of low energy quantum gravity? It is not 4D classical spacetime. This latter is the ground state of classical general relativity sourced by classical matter fields. When the sources are quantum fermions, they have a non-trivial ground state and a zero point energy - this cannot give rise to a spacetime with a point structure labelled by real numbers. We must resort to a coordinate geometry with non-commuting coordinates. And write the rules of a (pre-) quantum theory on this non-commuting coordinate geometry. On general grounds, the coordinates themselves can be expected to be complex, not real. From here the familiar quantum theory on classical spacetime must emerge as an approximation.

One way to construct the pre-dynamics is to raise the classical dynamical variables to the status of operators/matrices, but not impose the Heisenberg commutation variables. Instead we have a matrix-valued Lagrangian dynamics, and an accompanying Hamiltonian dynamics, from which the Heisenberg algebra and Heisenberg equations of motion are emergent in an approximation.

In this Lagrangian dynamics, it should be possible to define spin as a dynamical variable corresponding to an angular canonical variable. Such an angle clearly cannot be a part of the familiar 4D spacetime nor of its non-commuting version. The doubling of the non-commuting spacetime coordinates from four dimensions to eight is strongly indicated. This is (one of the many reasons) as to why we land up with the octonions. Octonions label the (non-commuting) coordinate geometry of the pre-quantum theory, and all the work being done on octonions and the standard model is seen from this light. The standard model forces as well as gravitation are symmetries of this physical space, which is equivalent to ten dimensional Minkowski spacetime SO(9,1). The octonionic space is like a square-root of this 10D Minkowski spacetime, analogous to tetrads, only now non-commuting. The ground state of quantum gravity is this octonionic space. Quantum gravity and quantum standard model forces are switched on as the curving of this octonion geometry. In the same way that the curving of 4D classical Minkowski spacetime is switched on by classical gravity through the metric and the Riemann tensor.

However, the geometry of octonions has a very fundamental difference from that of Minkowski spacetime. Unlike in the latter, fermions cannot have arbitrary properties in octonion space. Even before the curving of octonion geometry is turned on by switching on interactions, the non-commutative geometry already dictates properties of fermions. There are correct number of quarks and leptons. Electric charge is quantised, just as observed. These properties have been proved earlier by other researchers, but viewed by them in an algebraic context [SM and GUTSs in terms of the algebra of the octonions]. They do not interpret their results as consequences of this new spacetime geometry of the octonions. But such a spacetime interpretation is most essential - it adds dynamics and quantum theory to the algebraic approach. Not only does spacetime tell matter how to move; it also tells matter what to be. This is a natural extension of general relativity, and also of quantum theory - in the latter theory, quantisation of energy levels e.g. is dictated by the dynamics. So it is not a surprise if a non-trivial coordinate geometry dictates particle properties.

The standard model is a mystery when looked at from a 4D Minkowski spacetime labelled by four real numbers. But not when viewed from the `square' root of 10D Minkowski spacetime - the octonionic space.

The extra dimensions are complex and never compactified. Quantum systems probe these extra dimensions, always. Only classical systems live in 4D. The curving of the higher dimensions represents the standard model forces - equivalence principle is not obeyed because the curving is proportional to electric charge, not to mass.

String theory and the octonionic theory, compared.

String theory does not predict the values of the free parameters of the standard model, not yet. Assuming that these parameters take unique values which should be derivable from an underlying theory, string theory is not true, not yet. In fact the theory gives very many different possible values for these parameters, and it is not clear what criteria might favour the observed values. In contrast, the octonionic theory makes unique predictions for these parameters, which are derivable from an underlying theory. The action of the theory, and its symmetries and dynamics, are well defined. 

 It is also quite clear why the octonionic theory succeeds where string theory fails .. actually the two theories are quite similar but have crucial differences, making it quite clear which of the assumptions of string theory are wrong. Flat 10D Minkowski space-time is not the correct ground state on which to define the Fock space. Rather, the correct choice is the equivalent octonionic space SL(2, O) where O is an octonion. The Hamiltonian of the theory is not self-adjoint on the Planck scale. The extra spacetime dimensions are complex... they define symmetries of the standard model. These extra dimensions are never compactified. Only classical systems live in 4D space-time. Quantum systems live in octonionic spacetime, even at low energies. These few basic but key changes set string theory on the right track...that being the octonionic theory.

ER=EPR : an idea in search of a theory, and Einstein's small-print

After experiments confirmed that quantum non-locality is for real, it has often been said that Einstein has been proven wrong.

However, what Einstein said was more precise than is made out to be. Namely, if reality is non-local, then either quantum mechanics is incomplete, or special relativity (and its description of spacetime structure) is approximate. This second part is entirely overlooked when talking of the `spooky action at a distance'.

Most mainstream physicists took the experimental confirmation of non-locality as matter of fact, with nothing more to be done about it. Clearly Einstein knew better: there is a quantum influence outside the light-cone which special relativity does not allow, and this means quantum systems travel through spacetime in ways which are not consistent with assuming that spacetime is 4D and has a causal light-cone structure. This needs to be explained with new physics. One cannot get away simply by saying that since such non-local influences do not permit information to be transferred faster than light, all is well between quantum mechanics and special relativity, no tension, and nothing more is to be done. This is a misreading of Einstein - physicist Roger Penrose and philosopher Tim Maudlin , amongst others, have repeatedly emphasised the need for an overhaul of how we see spacetime, given that nature is non-local.

Some years ago mainstream physicists Susskind and Maldacena vouched for Einstein, and suggested that quantum systems have an extra channel available to them for travel through spacetime (the Einstein-Rosen bridge of general relativity, the wormhole) which classical systems do not have access to. Quantum processes are only apparently non-local : the wormhole allows essentially instantaneous but causal travel.

This is a good idea in search of a theory: for the ER=EPR idea to be convincing, there has to be a theory of quantum gravity which is such that: classical spacetime is emergent but only classical systems live in 4D spacetime with its causal light-cone structure. Quantum systems live in a space (perhaps higher dimensional) which is in a sense a combination of 4D spacetime and a wormhole like feature. Through these extra dimensions, travel time is of the order of L/c, and with L of the order of 10^{-13} cm this is about 10^{-23} s, even if Alice and Bob are billions of light years apart.

The octonionic theory was not developed to explain quantum non-locality, but manages to explain it, as a byproduct of its inevitable structure. The O-theory was developed to seek a reformulation of quantum theory which does not depend on classical time. In this theory, right from the word go, quantum systems do not live in 4D spacetime, even at low energies. They live in a higher dimensional space, which contains 4d spacetime as a subset, and where the extra dimensions are complex. The absolute magnitude of the scale of these extra dimensions is microscopic, but not Planck length. It is of the order of the short range of the electroweak and strong interactions. These extra dimensions, accessible only to quantum systems, play the role of the wormhole of Susskind and Maldacena. But the wormhole like feature has not been invoked in an ad hoc manner; it is a part of the theory, and indicates the modification of the spacetime of special relativity that Einstein hinted at.

None of this is inconsistent with quantum theory as we know it. Quantum dynamics can be written on octonionic space, or to an excellent approximation, on classical 4D spacetime. When we do the latter, without realising that this is only an approximation to the former, we are confronted with the non-locality puzzle (to which we then seek ad hoc solutions), and confronted with so many free parameters in the standard model of particle physics. We seem to be solving quantum problems one at a time: non-locality, BH information loss, origin of matter-antimatter asymmetry, etc. with the solution to one problem having no bearing on the solution to another problem. However we need to realise that there are so many difficulties at the interface of quantum theory, relativity, standard model, and cosmology, that at this juncture we need to address the core foundational problems of quantum theory and spacetime, and come up with a theory from which gravitation, and quantum theory, are emergent. The rest then is likely to take care of itself.

When the dynamical variables do no commute, the underlying coordinate geometry must also be non-commutative.

The Einstein hole argument shows that for the points of spacetime to be distinguishable from each other, the spacetime has to be overlaid by a metric. The metric at a point then acts as a flag, a marker so to say, labelling the point.

In quantum theory as currently formulated, there is assumed to exist a background spacetime and a universe dominated by classical bodies and fields. This permits a metric which serves as marker.

However, when there are no classical objects around, the metric undergoes quantum fluctuations [because it is being produced by quantum sources] and hence can no longer serve as a marker. The distinguishability of spacetime points is lost.

We see that in conventional quantisation, we necessarily keep the universe dominantly classical, and quantise a negligible fraction of it. What should we do if we want to quantise everything that is classical, in one go?

Consider Newtonian mechanics, or special relativity as a starting point. Raise every dynamical degree of freedom (configuration variables and their corresponding canonical momenta) to the status of matrices. Do not impose Heisenberg quantum commutation relations [q,p]=i\hbar by hand (they will emerge). Replace the four dimensional spacetime coordinate geometry by a non-commuting coordinate system. The Einstein hole argument no longer applies.

We have a matrix-valued polynomial Lagrangian. It's trace defines the trace Lagrangian to be used in the action principle. Use a new absolute time to describe evolution. The Lagrangian dynamics so defined is a pre-quantum theory, from which quantum theory emerges. In a sense this pre-quantum theory can be called true quantisation, a no-holds barred purist quantisation wherein all classical elements are removed. What we call quantum theory is a stop-gap, a half-way home, which serves very well phenomenologically, but leaves a lot unexplained as well.

The non-commutative geometry dictates what we mean by elementary particles, and what properties they have. It also dictates what form the Lagrangian takes. In Newton's mechanics the simplest description of the universe is as a collection of colliding point particles, and the Lagrangian for each one of them is simply it's kinetic energy. So we raise the position variable of each point particle to a (Grassmann number valued) matrix, and define its velocity as the time rate of change of its matrix-valued position vector in the non-commutative space. Then we can define the kinetic energy as the trace Lagrangian - in place of the mass of the particle there is a length scale, which is the only parameter associated with the particle. The universe consists of such colliding matrix-particles.

Because the entries in the matrices are Grassmann numbers, there is a natural place for bosonic and fermionic degrees of freedom, and for particles and forces. And there seems to be a possibility that we can derive our observed universe from the above pre-quantum, pre-spacetime dynamics, because the observed symmetries possibly coincide with those of the underlying non-commutative geometry.

When is quantum gravity necessary?

It is true that at the Planck energy scale, the action for the gravitational field becomes of the order \hbar, and hence quantisation of gravity becomes essential.

However, while this is a sufficient condition for a non-classical treatment of gravitation, it is not necessary. Magnitude of the action of the gravitational field is not the only criterion for deciding whether spacetime should be treated non-classically. The point structure of the spacetime manifold also needs to be attended to.

Even at low energies, if every matter source is quantum and in a superposition of position states, the produced gravitational field will be in a superposition of different classical gravitational fields. At a given spacetime point, the gravitational field does not take a unique classical value. This implies that the point structure of the underlying spacetime manifold is no longer operationally defined, and is hence lost. This is happening even though the action of the gravitational field is much larger than \hbar. As a consequence, it becomes necessary to reformulate quantum theory without making reference to classical spacetime.

Our conventional formulation of quantum theory takes for granted that the universe is dominated by classical objects; so that it has a definite classical metric. This allows a spacetime with a definite point structure to be assumed. But this is an approximate situation.

The standard model of particle physics has so many undetermined free parameters precisely because it is formulated on a spacetime with definite point structure. This is an approximate description. When this approximation is removed, and spacetime replaced by non-commutative spacetime [at low energies] we begin to see evidence that the parameters of the standard model are not free.

There are two aspects to quantum gravity. One is the quantum nature of the metric. The other is the point structure of spacetime. The former becomes significant only at Planck energy scale, and is not relevant to low energy particle physics. However, the latter - the classical point structure of spacetime - is already lost at low energies if all matter sources are quantum in nature. The spacetime is non-classical even though gravity is weak. This is relativistic weak quantum gravity, and its relevance for the standard model of particle physics is non-trivial. This fact has been overlooked thus far.

Where does an electron live?

The belief that a quantum particle such as the electron lives in spacetime is the last vestige of a mechanistic Newtonian view of the universe. In which space is absolute, time is absolute, and objects are embedded in space and time.

Special relativity took away the independent absolute status of space and of time, and gave us instead a four dimensional spacetime, in which classical material bodies and fields live. The geometry of spacetime is absolute and flat.

General relativity taught us that the geometry of spacetime is not absolute, but is determined by the classical bodies and fields which live in it. In turn, the geometry tells matter how to move. Also, it is a consequence of the Einstein hole argument that it is not possible to give an operational meaning to the point structure of the spacetime manifold, unless there overlies on it a non-trivial classical metric.

Quantum theory tells us that microscopic objects do not move on classical trajectories. However, the theory does retain classical spacetime, with its point structure. This latter retention however contradicts the Einstein hole argument. Because the metric undergoes quantum fluctuations, and is no longer classical. Therefore a quantum particle such as the electron cannot be said to live in a classical spacetime, except in an approximate roundabout sense.

Rather, a quantum particle lives in a complex space whose geometric symmetries are a union of the internal symmetries of the standard model (electroweak and color) and the Lorentz symmetry of the standard model. In such a space, the electron evolves dynamically, in a newly introduced absolute time, which is a feature of this complex space, not possessed by classical geometry.

When the elementary particles living in such a space form large scale entangled systems, they undergo a quantum-to-classical transition and are confined to 4D Lorentz invariant spacetime, which, because of the classical nature of the bodies which dominate it, acquires the classical geometry given by Einstein's general relativity.

A quantum particle such as the electron, which has not undergone the quantum-to-classical transition, continues to live in the original complex space. We incorrectly assume that the electron also lives in the emergent classical spacetime. However, this classical 4D spacetime is the domain only of classical objects and fields. Not of quantum systems.

Elementary particles, physical space, and the octonions

When we write down the Schrodinger equation, evolution is described by a classical time parameter. Suppose there were to be no classical objects in the universe at low energies, can we continue to use the classical time parameter in the Schrodinger equation? Low energy by itself does not imply classical behaviour. After all, a free electron at low energies is not classical; it is quantum. If every physical system in the universe were to be low energy and quantum (i.e. microscopic), will we still have classical spacetime? Probably not. Because the points of spacetime will themselves be forced into superposition, removing their commutative, classical nature. Can we describe this new situation by non-commuting numbers, such as the octonions? Perhaps we can: octonions might be able to describe the non-classical spacetime, as well as the elementary particles themselves, when classical spacetime ceases to exist.

Octonions were discovered by the Irish mathematician John Graves, in 1843, within months of the discovery of quaternions by his friend Rowan Hamilton. Along with real numbers and complex numbers, the quaternions and octonions are the only two other division algebras. i.e. number systems in which the four operations of addition, subtraction, multiplication and division are possible.

We are of course very familiar with real numbers. Amongst other things, they describe the four dimensional spacetime of our universe, R^4, having a Lorentzian signature. Complex numbers play an important role in quantum mechanics. They describe the quantum state. Quaternions, though not much appreciated as such, are very much always `around' in our physics. Hamilton invented them as a generalisation to complex numbers, to describe rotations in three spatial dimensions. Quaternions are equivalent to vectors in three dimensions: the dot product and cross product of two vectors in 3D is related to the product of two quaternions. In fact, after the quaternions were discovered, they led to a vectors vs. quaternions war: which of the two should be used in coordinate geometry? The undergraduate mathematics syllabus then used to have a full course on the quaternions. But eventually vectors won and quaternions were mostly forgotten. However, they are related to the Pauli matrices and also to the Lorentz algebra of 4D spacetime. In fact, complex quaternions can serve as coordinates in a physical space whose symmetry group is the Lorentz group. Note that if the symmetry group of a space is the Lorentz group, it does not necessarily follow that the space is 4D Minkowski flat spacetime R^4. It can be the quaternionic space instead.

Nature uses real numbers, complex numbers and quaternions. Why does it not use the last of the division algebras, the octonions? Or maybe it does?

So we appreciate that real numbers, complex numbers, and the quaternions can be used to define space. Is physical space, the one in which physical objects like stars and electrons live, necessarily to be described by real numbers? Perhaps not. Can quaternions and octonions describe physical space [a part of physical reality of nature] in which elementary particles such as quarks, electrons and neutrinos live? Perhaps yes.

A collection of quantum systems, each having an action of order \hbar, does not give rise to/live in, 4D Minkowski real-valued spacetime, even at low energies. This is a quantum gravitational situation, and spacetime is non-classical. Perhaps the non-classical space is octonionic. This is where a free electron in flight might be living, until it is absorbed by a classical measuring apparatus. When that happens, the absorbed electron, along with the measuring apparatus of which it is now a part, becomes a part of the classical universe of macroscopic bodies embedded in 4D classical Minkowski spacetime.

There is nothing wrong in proposing that a free electron in flight lives in a complex-valued space, such as the one described by the octonions. In such a space the 4D Lorentz symmetry is unified with the internal gauge symmetries SU(3)_c X SU(2)_L x U1)_Y of the standard model. These latter complex valued spaces, labeled also by the octonions, become a part of physical reality, along with 4D Lorentzian spacetime.

We are conditioned to believe that everything must live in 4D classical spacetime; even the elementary particles of quantum theory. But this is not so. Quantum particles live in a space which is labeled not by the real numbers, but by the other division algebras.

What then of the time parameter in the Schrodinger equation? It is replaced by a new absolute notion of time, the Connes time, which is an essential feature of a non-commutative geometry, such the geometry of the space of octonions. Sometimes these are called quantum Riemannian geometries. There is no concept of absolute Connes time in the Riemannian geometry of 4D classical spacetime.