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

The Mathematics of Relativity

This blog is meant for readers with a modicum of interest in the mathemtics of reality.

While no mathematical background is necessary to enjoy this blog, mathematical interest is.

This particular blog will discuss the mathematics of relativity. Later blogs will discuss the mathematics of quantum mechanics, quantum field theory, string theory, complexity science, economics, finance and actuarial science.

Descriptions like that which follows can be found in many relativity publications. In this blog we will be following the formalism laid out in Relativity Demystified.


The mathematical structure of relativity is as straightforward as it is complex. Einstein's special and general theories of relativity change the way we think about our world. Many phenomena of common sense fall within the mathmatics of relativity.

The basic idea flows from a series of mathematical relationships. In 1865, physicist James Clerk Maxwell published his equations which posited that electromagnetic (em) waves exist, and move at the speed of light. Electromagnetic fields can be shown to satisfy wave equations using simple vector calculus.

This is important, because wave equations are used extensively in physics to model physical events. It is important in the context of relativity because it can be shown that em waves (light) always travel at the same speed. It is the constancy of the speed of light in all frames of reference which leads to many of the unique elements of relativity theory.

In Newtonian physics, distances and times are fixed. When it was discovered that waves don't need a medium in which to travel, a mathematical tool called a Lorentz Transformation was applied to explain the results.

Newton used two equations to describe gravity. The first describes the path of a particle through space. The second describes how mass acts as a source of gravity.

Einstein's equations have similar form. In general relativity, the energy-momentum tensor acts as a source of gravitation, and relates to the curvature of spacetime via the Einstein equation.

Symmetries lead to conservation laws. In relativity, Killing vectors can be used to tease out symmetries. The motions of particles and light waves in spacetime involves Lagrangian methods which we will deal with in the math of quantum mechanics blog.

In terms of these blogs, it is important to note that understanding the mathematics of relativity is not just an abstract exercise. Harmonious Universe posits that a metaphysics based on mathematics and science is at least a possibility.

From the mathematics of relativity, particle physics, chaos theory, complexity science and actuarial science, patterns can be seen which point to the possibility that all of reality can be understood at a deep level. The evolution of the universe itself is mathematiclly derivable.

These ideas can be followed without understanding the math, but understanding the math enhances the insights.

Getting back to the math of relativity, now that we have a basic framework, we can begin to build the mathematical structure. The study of relativity involves what most physics does, the analysis of events.

In physics, events occur in a framework. That framework usually involves space, time, matter, and energy. Elements of matter and energy interact in space and time and reality emerges from these interactions.

Galilean Transformations were used to study how phenomena appear to different observers in Newtonian Mechanics. In Relativity Theory, Lorentz Transformations apply.

Among the tools used in developing some of the key elements of Relativity theory are, vectors, tensors, metrics, and manifolds.

Vectors are simply symbols which represent magnitude and direction. Tensors map from vectors to the real numbers. Metrics describe three dimensional space. Manifolds are continuous spaces of points that can be curved globally, but locally are Euclidean.

Many tools of mathematics have as their primary role the simplification of calculation and analysis. Transformations often accomplish this quite directly.

One example is a gauge transformation which is quite prominent in quantum theory, and is used in relativity to tease out wave equations.

In relativity, much of the theory is nonlinear, but can be worked on using linear methods which are much more manageable.

Lie derivatives, Killing vectors, Riemann tensors, Ricci tensors, Cartan Equations, and Einstein's field equations are all in the relativity tool box.

The most famous equation of all, e=mc**2 came out of these elements

From relativity theory came a new vision of the world and a new way of living.

Einstein's relativity theories changed the landscape of the human enterprise. Combined with advances in particle physics, it changed metaphysics itself.

Lee

The New Structure of Mathematics

Our blogs will now become somewhat specialized.

Huffington Post will be a general site...Harmony.

This blog, Mathematical Universe, will deal with the underlying mathematics of Harmonious Universe...truth.

Quantum Universe will deal with the underlying science...beauty.

The Art of Hardball will deal with the underlying politics and economics...justice.

The new structure of mathematics will build up from the deep structures of pure and particle mathematics through the cobbled structures of economics, finance and actuarial science.

The unifying structures will be those of complexity science. At it s most basic level, complexity science is the application of the most advanced computer and analytical structures to the most advanced problems of society.

It is the structure which emerges when the best scientists apply their deepest wisdom to the deep problems they attack.

No one will have to go back to school to follow this blog. A major point of the new structure of mathematics is that it be coherent to a general audience.

That doesn't mean sacrificing analytical robustness. It merely means organizing and explaining the tools better.

The new structure itself naturally starts with pure mathmatics. This involves mathematical structures that don't necessarily relate to anything we observe.

It should be noted that much of what is originally considered pure mathematics is ultimately found to apply to real problems. Similarly, much applied math (ie. that being developed in string theory) is often found to be pure.

The first level of mathematics beyond pure math is that being used in particle physics. One doesn't have to be able to manipulate Lagrangians, Hamiltonians. Hermitians, Laguerre's, Legendre's, Fouriers, Path Integrals, Hilbert Spaces, Lie Groups, Topologies, or Tensors to know that Schrodinger Waves, Dirac Seas, Feynman Diagrams, Pauli Exclusions, Planck Lengths, Heisenberg Uncertainties, and Maxwell's Equations help us understand how fundamental particles waves and fields become atoms, molecules and us.

The mathematics of particle physics points us to the mathematics of chemistry which in turn points us to biology. Neurology and psychology are on the same trajectory . Now society itself is in view mathematically.

This is where complexity science really shines. Along with some of the structures of chaos theory (fractals, universality...), complexity science is cobbling together a mathematics of everything.

Skeptics will say that humanity and its institutions are too complicated to approach mathmatically. There is some wisdom in that skepticism if not taken to far.

Remember that if we don't use the best math, we will use an inferior one, like the math behind the models which have missed all the recent global meltdowns.

What we are saying is that complexity science gives us a conceptual framework by which to combine the best math and computer science being used in the labs and universities with that being used in business and government. By considering connections among econometric, actuarial and financial models with those of Quantum Mechanics, String Theory and Relativity we will find better ways of connecting mathematics and our realities.

That will put us in a better position to anticipate problems and resolve them so as to optimize human welfare.

Is that not a worthy goal?

The journey begins.

Lee