User:RogueTeddy/sandbox/String Theory (mathematical underpinnings)

In this article, the mathematics of string theory is discussed. Broadly speaking, the central idea of string theory is to extend conventional physics, which deals with particles, or point-like objects in 0-categories, to the consideration of 'strings', or arrows in 1-categories.

Higher order string theories correspond to n-arrows in n-categories, but for the intents and purposes of most modern string theories, low dimensional higher categories (up to but not exceeding 3-categories) suffice. In this way, String Theory can be thought of as an extension of classical theories of physics (Quantum Mechanics, General Relativity, Quantum Field Theory, Quantum Electrodynamics, etc) from 0-categories to higher dimensional categories.

Another way of thinking of this is extending from physical spaces, to the consideration of function spaces, where functions become the fundamental objects.

Still another way of thinking of this is in terms of the cybernetics or higher cybernetics of a system.

A common misconception is that string theory is about 1-dimensional physical objects, called strings. This is a product of miscommunication and projection of abstractions from 1-category theory to terms in which can be commonly digested for the purposes of scientific journalism and popularisation. However, 'strings' are better thought of as functions.

Example
Let {x_1, x_2, ..., x_n} be coordinates for a chart within a differentiable manifold. Then the maps {f_ij: x_i \mapsto x_j} form a chart for the function space.

For an even simpler example, suppose that the differentiable manifold is R^n. Then this chart is global.

Particles follow geodesics within a differentiable manifold, if it is endowed with an additional structure that gives it a distance function, such as a (pseudo)-Riemannian metric (should this be pseudo, if we are doing physics, we consider such with index 1 over R^4).

So a 0-particle follows a 0-geodesic. Let a 1-particle, or a 0-string, follow a 1-geodesic. For 1-geodesics to make sense, we need a distance function defined over the function space (locally) R^n -> R^n. There are several different choices we can make of appropriate exotic differentiable structures. These form different geometries with different 1-geodesics. Therefore the paths that objects in function space will follow depend on the choice of exotic differentiable structure.

Exotic differentiable structures can be built in certain ways. Generically, we are interested in different generalised Levi-Civita Connections, induced by operators acting on 2-tuples of pre-geometric structures of the form b(t, u) = \delta(\sigma(t) - u), where \sigma is a Riemannian metric:


 * Induced by action *(b_1, b_2) := b_1 * b_2
 * Induced by action ^(b_1, b_2) := b_1 ^ b_2
 * Induced by action \circ(b_1, b_2) := b_1 \circ b_2

These form different string theories.

M-theory as an umbrella for 2-categorical theories
M-theory concerns 2-cybernetics, or the cybernetics of cybernetics. There are various flavours of cybernetic theories, which can be brought wholly under the umbrella of a meta-cybernetic theory, which is built out of objects from various types of function-function spaces, or 2-categories.

Broadly speaking, if one considers extending to 2-categories, one has five choices of operator, inducing 5 types of 2-strings:


 * \circ (composition)
 * \circ^(2) (tetrated composition)
 * \star (multiplication)
 * \wedge (exponentiation)
 * \wedge^{(2)} (tetration)

Dynamics are obtained by applying the Cramer-Rao inequality in each instance to deduce the natural geometric invariant induced by the corresponding exotic geometric structures defined on function-function spaces.

M-theory then is the study of structures that bring all of these objects under one umbrella. It has interesting connections with Surgery theory (including cobordisms, etc), and the theory of Jet Bundles.

Increased complexity: cybernetics
The prior considerations assume that one is dealing with exotic geometric structures that rely on tensors that have rank a power of 2. However, it is also natural to study cybernetic classes of structures as well (where tensors have rank a power of 3) or meta-cybernetic structures (where tensors have rank a power of 5); as well as mixed theories (wherein one uses tensors that have rank that has as prime factors 2, 3, and 5). This makes things a bit more complicated.

History
There are multiple ways of constructing cybernetic theories, or types of function spaces.