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.
