The Math of String Theory

If there is to be a "Theory of Everything" for the forces of nature, string theory at the moment seems to have a good chance of playing a major role. Everything that happens in the physical universe may be ultimately represented in a single, elegant equation.

The mathematics of string theory gives hope to the dream of a resolution of some of the problems in the current model of particle physics. By adding dimensions to the framework, string theory provides new avenues to mathematical consistency.

Of course, if some version of string theory holds , as was the case with quantum mechanics, our world view will have to be altered dramatically. In addition to matter and energy, space and time themselves will cease to have their commonsense characteristics.

On a conceptual level, string theory uses one dimensional strings rather than zero dimensional particles in developing the formulas. The way these strings "vibrate" determines what type of particle emerges, and ultimately, what kind of universe emerges.

Just as the strings of a piano have resonant frequencies, so do the vibrating strings of subatomic reality. Whereas the strings of musical instruments produce notes, string vibrations produce particles.

The forces of nature also emerge from string vibration patterns. As a result, all matter and forces known become notes on a universal string.

Particle properties are a manifestatinof resonant pattterns of string vibrations as well.

Stretching the metaphor, there is a school of thought that this framework can be brought to bear on the more complex systems which have emerged in our universe, including life and consciousness.

While chaos theory and complexity science show new laws come into play as system compexity increases, there is a belief that those new laws may ultimately be found to be consistent with the deeper mathematics of string theory.

It takes a while to digest even the conceptual, much less the mathematical, framework of string theory. If right, it posits a multidimensional labyrinth of interconnected structures mysteriously intertwined with our notions of space and time.

In conventional physics, events occur in four dimensions. They can be undersood in the context of four factors.

In quantum mechanics, the microscopic properties of space become dissimilar to the analogous properties in classical physics. The notion of a violent quantum foam versus the smooth surface of relativity causes and results in mathematical incompatiblity between the theories.

The standard model of particle physics posits a world of pointlike objects. String/Superstring Theory unveils a cosmos emerging from a microscopic landscape of strings whose vibrations emerge as symphony of reality.

Everything in the universe emerges from vibrating strings.

When strings and their vibrations are fundamental (rather than point particles), the incompatibility of microscopic spatial structures between relaivity and quantum physics vanishes.

Supersymmery is part of string theory. Fermions and bosons are paired, symmetric, in a supersymmetric universe, and the mathematics becomes more elegant as dimensions are added.

Fundamental properties of the universe depend on the number and structure of dimensions in which the fundamental strings vibrate.

In supersymmetry, Hamiltonians commute with supercharges and Lagrangians are transformed to produce mixed bosonic and fermionic fields. Dirac and Pauli matrices are utilized in the representation, and a supersymmetric Lagrangian emerges.

A particular class of six dimensional spaces known as Calabi-Yau spaces have the geometric struture required to meet the mathematical properties of the theory

The mathematics of string theory is important in its own right. Even if it is found that string theory doesn't fully describe our universe, the underlying mathematics has blazed new trails of understanding.

As Robert Pirsig said in Zen and the Art of Motorcycle Maintenance,"...the stream of national consciousness moves faster now, and is broader, but it seems to run less deep...Some channel deepening is called for."

There are a number of cosmic and subatomic implications of the math of string theory. These will be explored throughout my blogs.

Like a Beethoven Symphony, or Mozart Concerto, our world is a labyrinth of elements which emerges in a much more beautiful form than could be predicted from an assemblage of the parts.

The solution of a wave equation can be written in terms of a superposition, a Fourier expansion. The symmetries of the Lagrangian, the energy momentum tensor, and the Hamiltonian which governs time evolution of the worldsheet are elements of the classical analysis of strings.

The quantum extension of classical theory involves covariant quantization and Virasoro constraints. At this point there are 26 space-time dimensions.

Conformal field theory is used in peturbative string theory. A conformal transformation maps a region of the complex plane onto a more workable space.

In T-duality, open strings with Neumann boundary conditions are transformed to those with Dirichlet boundaries.

Spacetime supersymmetry involves superfields, superspaces, Grassman coordinates, and Green- Schwarz formalism.

In conclusion, string theory develops a mathematical framework by which to model the essence of all subatomoic activity. It shows that at the deepest level, reality may be interconnected and multidimensional. This has profound philosophical implcations.

This concludes our blogs on the mathematics of subatomic interactions. (whew!)

The next blogs in Mathematical Universe will deal with the mathematics of larger systems, up to and including the global economic system.

Lee