Saturday, March 03, 2012

Electromagnetism, fluid mechanics, and the mathematical universe

A theory in modern mathematical physics consists of a mathematical formalism, and a set of rules linking parts of the mathematical structure to parts of the physical world. Observational and measurement data is thereby embedded into the theoretical formalism as an empirical substructure.

In terms of the philosophy of science, if you think that physics captures the nature of the world which exists beyond the empirical data, then you're a realist, whilst if you think that the theory is simply a means for organising the empirical data, and generating reliable predictions, then you're an empiricist.

One of the problems which realists face is distinguishing between physical and non-physical mathematical structure; some parts of the mathematical formalism appear to be convenient, but surplus to the physical structure.

An interesting illustration of these issues can be found by comparing classical electromagnetism with fluid mechanics. These two fields represent very different aspects of the physical world, but use the same branch of mathematics to do so.

For example, in electromagnetism, there is a vector field called the magnetic field B, and this field is defined to be the result of applying a differential operator called the curl, to a vector potential A:

Now, in fluid mechanics, there is a vector field called the vorticity, and this field is defined to be the result of applying a differential operator called the curl, to the velocity vector field:

What's interesting about this is that in electromagnetism some people regard the object from which the curl is taken, the vector potential A, to be surplus mathematical structure, whilst in fluid mechanics, there are people who regard the object obtained by taking the curl, the vorticity, to be the surplus mathematical structure:

"In a wholly classical context, electromagnetism acts on charged particles only through the electromagnetic field...the electromagnetic potential has no independent manifestations, and seems best regarded as an element of 'surplus mathematical structure'," (Richard Healey, Gauging what's real: The conceptual foundations of contemporary gauge theories, OUP, 2007,p21).

"Vorticity, the quantification of the strength of such vortices, is not actually physics—vorticity is a purely mathematical definition. Indeed, vorticity is constructed from the velocity gradients described above—which are physics: the amount velocity changes over a given distance," (p46, Introductory lectures on turbulence: Physics, Mathematics and Modeling, J. M. McDonough).

Whether McDonough's assertion is accurate is something of a moot point; one could make a decent counter-argument for saying that vorticity is an objective, physical pattern to be found in velocity fields. For the sake of argument, however, let us assume that McDonough is correct.

Now, the theories in which these structures are embedded are non-isomorphic: electromagnetic fields must satisfy the Maxwell equations, while the velocity/vorticity fields must satisfy the Navier-Stokes equations. The electromagnetic and velocity/vorticity fields, then, differ not only in the way they are mapped to the physical world, but also by virtue of the overall structures in which they are embedded.

Max Tegmark has argued that it should be possible to infer the interpretation of a theory from its intrinsic mathematical structure: "Suppose we were given mathematical equations that completely describe the physical world, including us, but with no hints about how to interpret them...the only way in which familiar physical notions and interpretations...can emerge are as implicit properties of the structure itself that reveal themselves during the mathematical investigation," (The Mathematical Universe, Tegmark 2008,p5).

Perhaps, then, electromagnetism and fluid mechanics would be a good test-bed for this hypothesis. If you can show, from the intrinsic mathematical structures alone, why the magnetic vector potential A is surplus mathematical structure in electromagnetism, whilst the vorticity is surplus mathematical structure in fluid mechanics, without resorting to physical interpretation, then Tegmark has a viable hypothesis.