måndag 31 augusti 2009

Illusions of Theories of Everything

The ultimate dream of theoretical physicists is a Grand Unified Theory GUT or a Theory Of Everything TOE  as a mathematical equation in the form of a system of partial differential equations, the solutions of which would represent all there is in the World. We find this dream partially realized in specific areas of mechanics and physics identified by a specific system of partial differential equations, such as 
  • fluid mechanics: Navier-Stokes equations 
  • quantum mechanics: Schrödinger's equation
  • celestial mechanics: Newton's equations of motion/gravitation.
One can argue that in a certain sense all of celestial mechanics including the motion of the planets, comets, asteroids et cet in our Solar system, is represented as solutions to Newton's equations of motion/gravitation, that all of quantum mechanics is represented as solutions of Schrödinger's equation, and that all of fluid mechanics is represented as solutions of the Navier-Stokes equations. 

We can thus view Newton's equations, Navier-Stokes equations and Schrödinger's equation as different forms of TOE with Everything representing the totality of a certain part of the World, like a continent of the Earth. 

Newton seemed to be able to describe all of celestial mechanics by his equations of motion and gravitation in his TOE,  which made Newton immensly famous attributed with godlike power. 

Similarly one can argue that all of fluid mechanics can be described by the Navier-Stokes equations, and all of quantum mechanics by Schrödinger's equation as different forms of TOE. 

This can give a physicist in control of a TOE the illusion of superhuman power, but there is a hook: Even if the equations can be written down in a couple of lines, like the Navier-Stokes equations, they can be impossible to solve by analytical mathematics representing solutions in terms of elementary functions such as polynomials and trigonometric functions: Analytical solutions of Newton's equations are known only for the two-body problem of one small body like the Earth orbiting a big body like the Sun. Already three bodies is beyond analytical representation, not to speak of the turbulent solutions of the Navier-Stokes equations. 

If we stop here, a TOE in the form of Navier-Stokes equations instead of a theory of everything, would seem to be rather a theory of nothing. This would be like a jeweler with diamonds still to be captured from the rock. Is a  jeweler without jewels, still a jeweler?

However, one can compute digital solutions of the Navier-Stokes equations using computers, for specific choices of data, and in this way gain insight case by case using a Computational Theory of Everything. Some diamonds thus can be brought to the surface and put into rings to be admired. But we cannot get full insight in one shot. We cannot capture all diamonds in one day in a true TOE.

One can argue that specific knowledge of e.g. fluid mechanics comes form specific computational solutions to the Navier-Stokes equations for specific data, and fluid mechanics is the totality of such specific knowledge. I give examples in my knols on fluid mechanics. See also

So even if a GUT or TOE unifying quantum mechanics and gravitation in the form of one set of equations, e.g. in the form of string theory, the main task of computing and studying specific solutions would remain.

We can make a parallel with Darwin's theory of evolution based on an equation expressing genetic variability + selection by survival of the fittest. Anybody can formulate this equation and the non-trivial part of evolution theory is the study of specific solutions. Still 150 years after Darwin,  Richard Dawkins struggles hard to compute specific solutions of Darwin's equation...

One can even argue that Darwins TOE of evolution as variability + selection, is trivial in the sense that it can be written down in one line, while the determination of specific solutions such as amoebas and human beings is highly non-trivial.

We often hear physicists claim that the atomic electron structure of the periodic table of elements is a consequence of quantum mechanics, but at a closer examination we find the electron structure for atoms with more than one electron, that is all elements except Hydrogen, is unknown as a solution to the Schrödinger equation, see my knols on quantum mechanics.

Likewise, one can argue that the secret of turbulence is hidden as solutions to the Navier-Stokes equations, a secret closed to analytical solution but open to exploration by computation, as well as the n-body problem of celestial mechanics.

The basic differential equations of celestial, fluid and quantum mechanics express basic physical laws of balance or conservation, such as Newton's 2nd law, conservation of mass, momentum and energy. These physical laws are what a blind Nature obeys in its evolution from one moment of time to the next in some form of analog computation, which can be mimicked by digital computation. To evolve according to physical laws does not require any intelligence, just work. In rare cases human intelligence allows shortcuts to analytical solutions, but in general only brute force computation is effective. This is what makes the world go round, whether it is understood by someone or not.

1 kommentar:

  1. Recent research in application of the symmetry group {*1,*2,*3,...*n} to the differential equation for an atom's nuclear emissions of force solved the correlation function for mapping the set of virtual force photons onto the spacetime manifold of the electron cloud region's heat capacity energy fields.
    The atom's RQT (relative quantum topological) data point imaging function is built by combination of the relativistic Einstein-Lorenz transform functions for time, mass, and energy with the workon quantized electromagnetic wave equations for frequency and wavelength. The atom labeled psi (Z) pulsates at the frequency {Nhu=e/h} by cycles of {e=m(c^2)} transformation of nuclear surface mass to forcons with joule values, followed by nuclear force absorption. This radiation process is limited only by spacetime boundaries of {Gravity-Time}, where gravity is the force binding space to psi, forming the GT integral atomic wavefunction. The expression is defined as the series expansion differential of nuclear output rates with quantum symmetry numbers assigned along the progression to give topology to the solutions.
    Next, the correlation function for the manifold of internal heat capacity particle 3D functions condensed due to radial force dilution is extracted; by rearranging the total internal momentum function to the photon gain rule and integrating it for GT limits. This produces a series of 26 topological waveparticle functions of five classes; {+Positron, Workon, Thermon, -Electromagneton, Magnemedon}, each the 3D data image of a type of energy intermedon of the 5/2 kT J internal energy cloud, accounting for all of them.
    Those values intersect the sizes of the fundamental physical constants: h, h-bar, delta, nuclear magneton, beta magneton, k (series). They quantize nuclear dynamics by acting as fulcrum particles. The result is the picoyoctometric, 3D, interactive video atomic model data imaging function, responsive to keyboard input of virtual photon gain events by relativistic, quantized shifts of electron, force, and energy field states and positions.
    Images of the h-bar magnetic energy waveparticle of ~175 picoyoctometers are available online at http://www.symmecon.com with the complete RQT atomic modeling guide titled The Crystalon Door, copyright TXu1-266-788. TCD conforms to the unopposed motion of disclosure in U.S. District (NM) Court of 04/02/2001 titled The Solution to the Equation of Schrodinger.
    (C) 2009, Dale B. Ritter, B.A.