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.