Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” October 4, 2007Posted by dorigo in Blogroll, internet, mathematics, physics, science.
Marni D. Sheppeard is a theoretical physicist in New Zealand. After 20 years of stints in experimental condensed matter physics, biomedical engineering, lattice QCD, electronics manufacture, programming, teaching, mountaineering and the real world, she has finally found some time to work on her main interest, quantum gravity. You’ll also find her serving excellent lattes in central Christchurch.
IS CATEGORY THEORY USEFUL ?
In the past, physics has made great progress with the mathematics of classical symmetries. For example, motion of a body in a plane can be decomposed into straight line translations and rotations.
The collection of all possible rotations is described by a circle in the plane, marked with a reference point which represents no rotation. Other points on the circle represent rotations by an angle corresponding to the angle between the given point and the reference point. Any rotation has a reverse operation, namely a rotation by the same angle in the opposite direction, which equals the rotation that moves from the original point around the remainder of the circle. That is, a pair of rotations is represented by a pair of arrows which make up a circle. These arrows combine to form a single arrow around the whole circle. Note that this is the same physical operation as the original marked reference point. Two shorter arrows could also be combined into a single rotation.
The idea of combining arrows in this way is what Category Theory is about. By definition, a category is just a collection of points along with arrows between them which can be combined to form new arrows. But points don’t have to represent actual points in a classical space. A point might represent a whole space, and the arrows ways of mapping one space into another. By studying these arrows we have a way to look at properties of spaces that doesn’t involve drawing complicated pictures in higher dimensions, because a space is represented by a single point and arrows are only one dimensional. But the concept of category is even more general than this. Points might represent sets, and the arrows functions between sets. Now much of 20th century mathematics is built upon the properties of sets, and here there is the possibility of using functions and sets together. But there seems to be a snag: surely the collection of points and arrows in any category must form sets! And it is true that the basic axioms for a category state the existence of such sets. However, by being a little bit more sneaky we can describe the category of all sets and then ask ourselves about other categories that might replace this one.
Why on earth would we want to do this in physics? First, observe that for rotations in the plane there is only one point, the marked reference. All classical symmetries, in any dimension, correspond to categories with only one point. In the quantum world, however, we secretly play with categories with more than one point. For example, an atom emits light only at certain frequencies determined by state transitions for its electrons. A transition may be pictured as an arrow between states. Two consecutive transitions combine to give a photon of frequency equal to the sum of its component transitions. This looks similar to the planar rotations, except that each state now gives a point in the category.
Another category associated to quantum physics takes points to be spaces of states. This category has a logical aspect, similar to the category of sets but also with glaring differences. Whereas elements of sets obey the rule that they either exist or do not exist, quantum matter only takes on this feature when it is observed. The logic of quantum mechanics is built from operators (projections) that take a space of states and pick out a specific choice of state. Such an operator, as an arrow in the quantum category, has the feature that doing it twice is the same as doing it once, because once a state is chosen the second arrow will just select the state again. This type of operation appears in many places in mathematics. In category theory, a map that selects the point at the start of an arrow is such an operator, because after the point is selected the second iteration of the map just reselects the point, which is viewed as an arrow from the point to itself that does nothing. Category theory is unique in the way it combines algebra, geometry and logic.
But the important feature of a quantum particle that is really not understood is its mass. To derive mass numbers correctly would require some knowledge about Quantum Gravity, the theory of quantized mass. Only the photon has zero mass and travels at the local speed of light relative to observers made of massive particles. In the late 1960s, Sir Roger Penrose considered spacetime events as collections of light rays incident on a point from the celestial sphere. Mathematically, this amounts to thinking of the sphere as a category of sets of light rays, with its own non-classical logic. This is still a one dimensional category. Categories in higher dimensions, with higher dimensional arrows, are currently been investigated as a language for gravity.
In three dimensions a basic arrow may be replaced by a cube. We imagine that the cube contains arrows on the faces and one arrow in the interior. An analogue of a projection operator is a map that sends the whole cube to one half of its boundary. Is it possible that particle masses could be derived from the simple algebra associated to such operators? Carl Brannen has been studying projection operator algebras associated to cubes. He has shown that the electron, muon and tau masses are described by a simple three-by-three array of numbers, as are the three neutrino masses. Current work is extending this analysis to baryons and mesons.