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.

Where does a quantum particle such as an electron live? Is E_6 the sought for symmetry group of unification?

Where does a quantum particle such as an electron live?

No, it does not live in spacetime, not 4D nor a higher dimension like 10D.

A quantum system even at low energies is an `atom of spacetime-matter' described by a pre-quantum pre-spacetime matrix valued Lagrangian dynamics. The Lagrangian has a certain symmetry group, say/perhaps E_6, and this is where the electron lives. It lives in E6. Now E6 is not a spacetime symmetry, neither is it an internal SU(n) symmetry. Beautifully, it is a unification of both, quite befitting an atom of space-time-matter, which is what an electron is.

How then does our classical 4D spacetime emerge? When a large number of quantum degrees of freedom living in E6 get entangled, the quantum to classical transition takes place. These emergent classical matter degrees of freedom get confined to a 4D spacetime [a subgroup of E6] and their localisation defines and gives rise to a 4D classical spacetime. This is compactification without compactification.

Quantum systems continue to live in E_6. We can describe their dynamics, from the vantage point of the emergent classical 4D spacetime, in the conventional language of quantum field theory.

Hence one must never try to compactify the extra dimensions - in a sense these extra dimensions are complex dimensions; they are needed to describe the internal symmetries.

Is E_6 the sought for symmetry group of unification?

We made three copies of Left-Right symmetric fermions using three copies of complex split bioctonions. These are also three copies of the Clifford algebra Cl(7), each made from two copies of Cl(6). This suggests the complexified exceptional Jordan algebra [whose automorphism group is E_6] of 3x3 matrices with octonionic entries.

The following sub-group structure is suggested:

(i) SU(3)_color X SU(2)_L x U(1) : color-electro-weak

8x3 = 24 LH fermions and 12 SM bosons

(ii) SU(3)_grav X SU(2)_R x U(1) : pre-gravitation

8x3 = 24 RH fermions and 12 bosons (8 gravi-gluons, 1 dark photon, two Lorentz bosons, one Higgs boson).

(iii) SO(1,3) ~ SL(2,C) is 6 dimensional, shared by particles in (i) and (ii). The Higgs mediates between LH and RH fermions.

Do the numbers add up?

24 + 24 = 48 fermions (including 3 RH sterile neutrinos)

12 + 12 = 24 gauge bosons

6 generators for SO(1,3) Lorentz group of 4D spacetime

These add up to 48 + 24 + 6 = 78

Now, E_6 is 78 dimensional. So is this a good indication?

By computing the eigenvalues of the exceptional Jordan algebra for three fermion generations, and in conjunction with the Lagrangian for a pre-quantum pre-spacetime dynamics we derived the low energy fine structure constant, and mass ratios for charged fermions.

Critically entangled fermions form macroscopic classical objects and descend to 4D spacetime with Lorentz symmetry.

Quantum systems [entanglement is sub-critical] live in E_6, i.e. partly in 4D spacetime and partly beyond in the complex internal dimensions.

Perhaps these are good signs ... ?

[There is no dark matter nor a non-vanishing cosmological constant in this theory. The emergent spacetime geometry shows evidence in favour of MOND and RMOND, and a long-range modification of GR playing the role of dark energy.]

The tension between quantum mechanics and relativity, and what it signifies for the standard model of particle physics?

The principle of quantum linear superposition is well-tested for elementary particles such as electrons. The theory of special relativity is also well tested. However, the two are not consistent with each other, except in an approximate sense.

We realise this when we ask the question: what kind of spacetime geometry and gravitation is produced by an electron? It is tempting to think of the produced gravitation as a very tiny (quantum) perturbation of the flat spacetime of special relativity. However this cannot be correct, precisely because of the validity of the principle of quantum linear superposition.

Imagine for simplicity that the electron state is a superposition of two localised position states A and B. Had the electron been at A, it will produce a perturbation of flat space time, with the perturbation peaked at A. Similarly if it had been at B, the produced perturbation would have been peaked at B. If we now consider a point X in space, the field there will be a superposition of the fields due to the location A and the location B. Such a field is not a perturbation of flat spacetime ! If we want to write down the Schrodinger equation for a test particle, then what time parameter to use at location X? That determined by the clock rate fixed by location A or by location B? There is ambiguity. The concept of a classical spacetime is lost [even in a perturbative sense] the moment we consider the implications of quantum superposition for spacetime geometry. In order to describe quantum spacetime - the one produced by an electron - we must entirely give up on the flat spacetime of relativity, even at low energies.

But quantum field theory on a flat spacetime works extremely well. We are able to construct the highly successful standard model of particle physics, assuming the flat spacetime of relativity, and assuming Lorentz invariance. Does that not imply that flat spacetime is an excellent approximation in the limit of low gravity? No, it does not imply that. There are 26 free parameters in the standard model, which have to be put in by hand, after measuring them experimentally. Could it be that these parameters get determined uniquely, and are not arbitrary, if we describe the standard model, not on flat spacetime, but on that spacetime which elementary quantum particles produce. Gravity is tiny, but it is not a perturbation of flat spacetime. What then is it a perturbation of?

The gravity is a perturbation of a [non-commutative] spinor spacetime, from which the flat spacetime of relativity is derived as an approximation. When we describe the standard model on this spinor spacetime, we find evidence that the parameters of the standard model are not free, but take values as measured in experiments. Such a noncommutative spinor spacetime is compatible with the quantum superposition principle. We could say that when we take the square root of the Klein-Gordon equation to arrive at the Dirac equation, we should also take the square root of flat spacetime so as to arrive at the noncommutative spinor spacetime. Then we write down the Dirac equation on this spinor spacetime. Right away we find that electric charge is quantised [0, 1/3, 2/3, 1 : neutrino, down quark, up quark, electron]. And that the low energy fine structure constant is 1/137. It cannot be any other value.