All the fundamental laws of physics are now understood in geometrical terms. The classical geometry that plays a fundamental role in our formulation of these laws has been vastly extended by a subject known as noncommutative geometry, my field of specialization in graduate school. However, we still have very little insight into the origins of the geometry of spacetime. This situation has been undergoing a significant evolution in recent years and it now seems possible to understand geometry as an emergent concept. The notion of geometry as an emergent concept is not new, see for example this article for an inspiring discussion and this research article for some recent ideas. I studied such a phenomenon in the context of what is called a “three matrix model.”
In classical physics, physical measurements of the states of a dynamical system are not thought to greatly interfere with the system in any fundamental way. The exact opposite is true in quantum mechanics where the process of taking a measurement “collapses” the probability amplitude of the quantity being measured, forcing the system to assume one of the allowable states. This interference phenomena finds its expression in the famous Heisenberg Uncertainty Principle which says that, even in the most ideal case, the measurement error in one quantity is always inversely proportional to the measurement error in any other. For this reason, the order in which one proceeds with measurements is important. A different outcome may result if a measurement of position is followed by, as opposed to being preceded by, a measurement of speed.
The implications of this are profound. Mathematically, it means that the “algebra” of measurement operations is not commutative. That is, if you first do operation “A” and then follow it by operation “B”, the reverse order may not yield the same result. We say that the “algebra of observables” is noncommutative. This is in stark contrast with classical physics where the observable quantities form a commutative algebra. The ansatz is to think of classical physical systems as being represented by their algebra of observable, and hence measurable quantities. In this setting, physically, that means when regarding a quantum mechanical system, we may regard it as being represented by its noncommutative algebra of observables. This process is called quantization, and the physical structures thought to be represented by that algebra are thought of as a noncommutative geometry.
Matrix models with a background noncommutative geometry have received attention as an alternative setting for the regularization of field theories and as the configurations for structures in string theory. In the model I studied, the situation was quite different. It had no background geometry in the high temperature phase and the geometry itself seemed to “emerge” as the system cooled, much as a Bose condensate or superfluid would emerge as a collective phenomenon at low temperatures. This means that spacetime itself with its geometric structure could possibly have emerged as a consequence of thermodynamic properties of the primordial universe. It is very interesting to wonder about what existence means in the absence of time and geometry.
But that is another discussion.