Now that we've reviewed the math of particle physics and relativity, we can move on to the math of things we can touch, feel, see, hear and smell.
No, we aren't interested in Newtonian Mechanics at this point. We want to show how chaos theory is a bridge from "simple" systems like atoms to "complex" systems like humans, ecologies, and economies.
Just as patterns in the subatomic world give rise to the mathematics of particle physics, patterns in nature and the cosmos give rise to the mathmatics of complex systems.
A universality has been found in patterns beneath the surface of things like the human heart, weather, internet traffic, the human brain, nature's rhythms, and cosmic expansion.
There is a holism and universality not unlike that found in certain subatomic patterns in the world around us.
Historically, mathematicians and scientists have attemped to deal with the nonlinear patterns of complex systems using standard mathematical structures like transformed differential equations and probability theory. The problem is that complex systems involve patterns that ultimately don't cooperate with such standard approaches.
As was the case in quantum mechanics and relativity theory, a new way of thinking is needed if we are really going to make progress in understanding complex systems. The dramatic increase in computing power in recent years has provided opportunities for such new thinking.
State and phase space, perturbation, fractals, attractors, power laws, cascades, symmetries, topologies, multifractals, and phase transitions are examples of notions which have arisen in the search for patterns in complex systems. If the mathematics underlying those patterns happens to relate to the patterns uncovered in the subatomic world, a metaphysical and scientific revolution of unprecedented power will be unleashed.
A mathematics of reality which begins with the fuzzy patterns in the subatomic world and shows how those patterns, even in principle, can be followed up the chain of reality, could lead us to a picture of human and cosmic reality which is holistic, interconnected and transcendent.
Economic activity, political activity, social activity, intellectual activity and spiritual activity could all be viewed from the perspective of universal patterns emerging into structures of truth beauty and justice. Quite a leap in logic, perhaps, but better than most of the other possibilities.
We have discussed in other blogs, and will discuss in future blogs, how such deep perspective can help us deal with the problems of the day. As a highly complex system, the global economy cries out for holisitc analysis. The tools of complexity science and actuarial science are ready to be combined with those of relativity and quantum mechanics to form an analytical structure for public policy analysis in the twenty first century.
Are we up to the challenge?
Yes we are.
Lee
Showing posts with label qcd. Show all posts
Showing posts with label qcd. Show all posts
Monday, February 2, 2009
Friday, January 30, 2009
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 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
Tuesday, January 27, 2009
The Math of Quantum Field Theory
This is the third in a series of blogs on the mathematics of reality.
It is not about the math! It is about getting a feeling for some of the words and methodologies underpinning the scientific advances of recent decades. It is those advances that have hailed the dawn of a new era of understanding.
We met many of the concepts that will be in this blog in the prior one on quantum mechanics. Quantum Field Theory is essentially just an extension of quantum mechanics.
In quantum mechanics we saw that mathematical operators function as observables. The Schrodinger wave equation gave the motion of a particle in one dimension and that structure can be extended to handle multiple particles.
No wave equation we have met so far, however, can be used in both relativity and quantum theory. Attempts to merge realtivity and quantum mechanics were attemped in the equations associated with the names of Klein-Gordon and Dirac, but it was the introduction of fields for wave equations that allowed the theory to truly advance.
In classical mechanics, Lagrangians (Euler-Lagrange equations) are used in deriving equations of motion. In field theory they are used to derive field equations.
A Hamiltonian equation can be derived from the Lagrangian and momentum representations.
Symmetries leave the form of the Lagrangian and equations of motion invariant.
The three fundamental interactions (forces) in the standard model are electromagnetic (em), "weak", and "strong".
Each force is manifest in a "particle". For each interaction there is a field. The generators of the field come from the unitary group which describes the field symmetries.
Elementary particles develop mass in their interaction with a mathmatical formalism called a Higgs Field. While no such field has been found, it is needed for the math of the standard model to describe the fundamental entities in our universe and their interactions. It is often the case that mathematics predicts the existence of an element of reality before our experiments can detect it.
Adding gravity to the standard model has not been accomplished. Relativity and quantum mechanics are mathematically incompatible. String theory, which we will visit in the next blog, is one attempt to create a theory that encompasses all four forces.
In the next blog, on the mathematics of string theory, we will attempt to bring together the philosophical and mathematical implications of discoveries in particle physics.
Lee
It is not about the math! It is about getting a feeling for some of the words and methodologies underpinning the scientific advances of recent decades. It is those advances that have hailed the dawn of a new era of understanding.
We met many of the concepts that will be in this blog in the prior one on quantum mechanics. Quantum Field Theory is essentially just an extension of quantum mechanics.
In quantum mechanics we saw that mathematical operators function as observables. The Schrodinger wave equation gave the motion of a particle in one dimension and that structure can be extended to handle multiple particles.
No wave equation we have met so far, however, can be used in both relativity and quantum theory. Attempts to merge realtivity and quantum mechanics were attemped in the equations associated with the names of Klein-Gordon and Dirac, but it was the introduction of fields for wave equations that allowed the theory to truly advance.
The Dirac equation is like the Schrodinger with a modified Hamiltonian. Dirac fields satisfy the Klein-Gordon equation and relativistic relations among energy, mass and momentum.
Solutions to the Dirac equation can be found in momentum space using a Fourier expansion.
The Klein-Gordon equation emerges from a substitution of the quantum mechanical operators for energy and momentum into the Einstein special relativity energy, mass, and momentum relations. By viewing the equation in the context of a field rather than a particle, creation and annihilation operators are allowed to arise.
In classical mechanics, Lagrangians (Euler-Lagrange equations) are used in deriving equations of motion. In field theory they are used to derive field equations.
A Hamiltonian equation can be derived from the Lagrangian and momentum representations.
Symmetries leave the form of the Lagrangian and equations of motion invariant.
Path integrals allow for calculation of amplitudes of quantuim transitions. In classical mechanics, the path is deterministic. In quantum mechanics , there is no trajectory per se, only path integrals.
In classical physics, there were two components, objects and the fields which linked them. Quantum physics tells us that particles are mere manifestations of fields.
Like many findings in quantum physics, this has deep philosophical implications. At the core of reality is not a thing, but a connection. Everything is connected, and elements have no meaning in isolation.
In quantum theory, predictions involve calculating probability amplitudes. The S-Matrix is one of the tools used. Feynman Rules allow for fairly straightforward calculation of amplitude/probabilities. Feynman Diagrams show particle actions (scattering, decay...).
The mathematics of group theory formalizes symmetry structures. Unitary groups (U1) are important because unitary tansformations leave probabilities for transitions among states unaffected. Unitary operators commute with Hamiltonians.
SU(2) symmetries are important in electroweak interaction work, and SU(3) symmetries are important in strong force/quark study (QCD).
The standard model includes QED and QCD. Quantum Electrodynamics (QED) works with electromagnetic interactions. Electromagnetic forces arise from photon exchangeof electrons in an em field. Feynman Diagrams illustrate QED processes.
QED explains the behavior of charged particles in an em field within the framework of quantum theory. Chemistry rests on the behavior of the electrons surrounding the nucleus of the atom.
The number of particles which had been detected by physicists by the mid twentieth century was large, not unlike the number of elements which Mendeleev was able to classify for chemistry.
In the 1960's, Murray Gell-Mann and Israel Ne'eman were able to classify particles. into patterns based on qualities called charge, spin and strangeness.
Initially the particles were grouped into eight patterns, and the name given to the system was the Eightfold Way. Particles in the nucleus, neutrons and protons, were actually found to be made up of more fundamental entities called quarks.
After a good deal of mathematical gymnastics, involving symmetries, Yang-Mills equations, Abelian Theory, renormalization, and the Higgs Mechanism, we came up with Quantum Chromodynamuics(QCD).
In QED. gauge invariance (U1) involves a Lagrangian for the em field, a Dirac Lagrangian, and an interaction term.
The combination of QED and QCD gives rise to the standard model of physics which describes the entire particle world, the nature and interactions of fundamental particles. Remember that in particle physics, particles have wave as well as particle characteristics.
The three fundamental interactions (forces) in the standard model are electromagnetic (em), "weak", and "strong".
Each force is manifest in a "particle". For each interaction there is a field. The generators of the field come from the unitary group which describes the field symmetries.
Elementary particles develop mass in their interaction with a mathmatical formalism called a Higgs Field. While no such field has been found, it is needed for the math of the standard model to describe the fundamental entities in our universe and their interactions. It is often the case that mathematics predicts the existence of an element of reality before our experiments can detect it.
Adding gravity to the standard model has not been accomplished. Relativity and quantum mechanics are mathematically incompatible. String theory, which we will visit in the next blog, is one attempt to create a theory that encompasses all four forces.
In the next blog, on the mathematics of string theory, we will attempt to bring together the philosophical and mathematical implications of discoveries in particle physics.
Lee
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