Sunday, January 15, 2006

ABOUT THE IMPORTANCE OF CLIFFORD ALGEBRAS ON PHYSICS

THE EVEN SUBALGEBRA CL3 and the Center sub algebra of Clifford Algebras are a very interesting mathematical tool to use in today Physics. There's an interesting recompilation of articles in reference (1), and an interesting recompilation of articles on Clifford (Geometric) Algebras on (2). Usually, we use the vectorial field to representate us the world, but the dispersion of systems and geometries only produces problems. The vectorial field its used for example for the study of forces in classical physics, thus, the velocity or the acceleration are vectors:



The vectors are imbibed in the Clifford Algebras (or Geometrical Algebras) through the utilization of quaternions, and making that his three imaginary components (i, j, k) were the orthonormal basis of a vectorial space (the pure quaternions). And more, the scalar part, could be utilized to represent the fourth element, this that is related with changes in time (like time were a scalar-real and vectors quaternionic-imaginary.

Thus, we can take a particular acceleration that is a part of the GA:


The Clifford Algebras can be divided in 2 subgroups. The center sub algebra, that is isomorphic to the complex numbers, and the even subalgebra, that is isomorphic to the quaternions:



Usually, quaternions are represented with this letter:

The quaternion group or the even algebra on Clifford Algebras is isomorphic if we take the bivectors and associate them to the complex numbers:



Like we have said before, if we know that vectorial field can be represented like the even sub algebra of Clifford Algebras, the acceleration could be a quaternion, and this, one multivector (a sum of k-blades, like scalars, bi-vectors, tri-vectors, pseudoscalars...).


Clifford Algebras and much more, his even sub algebra group, is going to be very important to symbolize the elements of a new theory that utilises them to reformulate any parts of the physics, but not to delete anything, but to generalize it. Anybody could ask, with a very part of reason: why need we this mathematical tool? Simply to do the same that others, but separating scalar parts to the vectorial ones, but having influence ones to others.

If we take a typical formulation of the second Newton Law, we can represent it this way:


Or much extensive, this ways (in vectors and tensors representation):




If we take a look on this, we can see that there exists another formulation more general, like the equation of transport, usually utilized in fluid mechanics:


One part of this is the continuity equation, that is:



Now lets wait a moment. If we utilize quaternionic operators, we can utilize the nabla differential operator that is:

Can be generalized this way:


If we take a look on it, we can see that a quaternion is really a sum of a scalar and a vector. It's a sum like the imaginary ones, that have his independent parts, bust they "interact" when we calculate with they.


If we take a look on quantum mechanics, we can see that there exists an operator, the operator "quantity of moment".


In three axes:






And its "time part":


This can be simply represented with the quaternion:


Or much simply:


Now, we propose a "little but a great" change. The wave function, we utilize not the complex numbers (the center part of Clifford Algebras), but utilize the even sub algebra group:


In this representation, we have this:


... and it can be represented by a special quaternion, one that have the module one:




Now, to continue this trip, we need another tool of GA. The "geometric product". This is:








In 2D isomorphic space, and the very important triple product of GA:


... that has two terms that are, one, a representation of the variation in different directions of a square (like a deformation):


... and the other, a representation of the variation of they in the same, like a rotation:


The question now is this, Could we find an expression of the wave function in even sub algebras, that were a much great generalization that existent in complex numbers?

This question is for another day... now, I'm going to eat.

R. Aparicio.

------ References

(1) http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf

(2) Lectures on Clifford (Geometric) Algebras and Applications, Rafal Ablamowicz & Garret Sobczyk.
(3) Author's books:

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