One of the best weapons in science is analogy. We apply the template of existing known processes in discovering/understanding new processes. Nature seems to somehow agree with us. Wave motion is one example of such a concept. In mathematics and physics, the Laplace operator is a differential operator used in modeling of many different kinds of wave propagation. It is used in formulating equations for acoustics, fluid dynamics heat flow, forming the Helmholtz equation, all the major equations in electrostatics, Electro-Magnetism, and in representing the kinetic energy term of the SchrÃ¶dinger equation in Quantum Theory.

Harmonic motion is one other such concept - it has been applied successfully, over and over again in various (intially, to an untrained eye atleast) unrelated fields like mass-spring arrangements, a molecule inside a solid, an electron stuck in an atom, a car stuck in a ditch being rocked out, a pendulum and the earth in its orbit. (source: http://www.slideshare.net/makadelhi/applications-of-shm)

Capacitors and Inductors are also such a concept. While capacitors alone interact as point objects, inductive objects (consisting of a spinning object in suspension about an orthogonal axis) behave as objects with a finite extension. Thus, viewing all interactions as being either capacitive or inductive can become a generalized technique that sorts the spatially extended objects (ie field elements) from point objects in both Electro-Magnetism (EM) and Iner-Gravitation (lets say, IG). This analogy raises spin/rotation to a unique postion of being the progenitor of both effects via capacitive and inductive suspension. By giving us the ability to distinguish between capacitive and inductive interactions, this analogy can also help harmonize IG with EM by solving the problem of renormalization when combining gravitons with strong and electroweak interactions. The following section explores one way to resolve this current problem in Physics.

All this means we can guarantee a unified structure to natural laws that would still look very familiar, but with a twist (almost literally) of the third derivative. In merging Mechanics and Gravitation with EM, we open the door to the Unified Field Theory. Lets not forget that Gravitation has long been the wedge that kept the whole structure from beautifully fitting together. In addition, it will give us the key to building ships that can cross space and make green transportation a reality.

In order to incorporate gravity into the Unified Field Theory framework, we have to work to replace curved spacetime as in general relativity with a situation where the gravitational interaction is mediated by gravitons. However, attempts to replace general relativity (GR) with gravitons have run into serious theoretical difficulties at high energies (processes with energies close to or above the Planck scale) because of infinities arising due to quantum effects (in technical terms, gravitation is nonrenormalizable).

Even just trying to combine the graviton with the strong and electroweak interactions runs into fundamental difficulties which boil down to the non-renormalizability of the results. The incompatibility of GR and quantum mechanics (QM) is another current problem in physics. Both these problems involve mechanics/gravitation's relationship with the remaining three forces.

In physics, although in principle we can predict the behavior of matter by keeping track of each atom, it is often more practical to treat matter as a continuum and then taking the continuum limit. Newton for example considered that air could be modeled as a lattice of mass points. He assumed the simplest possible lattice – equal masses spaced equally along the direction of propagation.

Laplace used this conception of air to successfully calculate the speed of sound. In fact, the entire set of Newton's Laws of Motion as well as wave theory itself can be deduced by taking the continuum limit of this simple lattice. If you wish to see the derivation, it is available here:

[http://en.wikibooks.org/wiki/General_Mechanics/The_Continuum_Limit]

In Quantum Field Theory (QED), renormalization refers to a collection of techniques used to take a continuum limit of space and time.

(source: http://en.wikipedia.org/wiki/Renormalization)

*When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill defined. In order to define them, the continuum limit has to be taken carefully.*

Renormalization determines the relationship between parameters in the theory, when the parameters describing large distance scales differ from the parameters describing small distances.

end quote

Renormalization determines the relationship between parameters in the theory, when the parameters describing large distance scales differ from the parameters describing small distances.

However renormalization when gravitons and strong or electroweak interactions are combined is hindered by the problem of infinities. The problem of infinities is an old one dating back to the 19th and early 20th century. Back then, it arose in the application of classical electrodynamics to point particles. This first version of the problem was solved by QED (as will be discussed immediately below) and the second version of this problem that has arisen with respect to Gravitons is currently unsolved and is an open problem.

To put it simply,

(source: http://en.wikipedia.org/wiki/Renormalization)

quote

*the mass of a charged particle should include the mass-energy involved in its electrostatic field. Assume that the particle is a charged spherical shell of radius re. The energy in the field is*

end quote

m

_{em}= q

^{2}/8*P*r

_{e}

where

m

_{em}= electron mass

Now normally, this classical electrodynamics formula performs well for all electromagnetic interactions for which quantum mechanics is not relevant. However, notice what happens when the radius falls to zero. The energy becomes infinite when re is zero. This directly implies that the point particle would be infinitely massive and could never be moved - an absurd conclusion.

Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative calculations many integrals were divergent.

Further,(source:http://en.wikipedia.org/wiki/Renormalization)

*when calculating the electromagnetic interactions of charged particles, it is tempting to ignore the back-reaction of a particle's own field on itself. But this back reaction is necessary to explain the friction on charged particles when they emit radiation. If the electron is assumed to be a point, the value of the back-reaction diverges, for the same reason that the mass diverges, because the field is inverse-square.*

end quote

## No comments :

## Post a Comment