Friday, January 13, 2006

ABOUT THE IMPORTANCE OF CLIFFORD ALGEBRAS IN PHYSICS (II)


One of the most important questions that a Physics Scientific must to confront is de diversity of notations to model the Physical Reality, thus, we have so much quantity of ways to reflect a physical magnitude, like can see in this graphic:







Usually, we utilize the wave function in quantum mechanics, due to the Schrödinger equation (and others). In this case, the wave function is in a sub group that is the sub group of complex numbers. But if we take a look to Dirac or Pauli formulation, there's anything that doesn't work: Dirac needs a special wave function with 4 homogeneous components, and the solution isn't exactly like Schrödingers' one. And Pauli utilizes strange matrix that are not exactly complex numbers. It's much more simpler include these ones in a group much more adecuate: the Even Sub Group of Cliford Algebras' .

The classical Schrödinger formulation gives an interesting result, that is the equation of continuity of current of probability:




But, what's the meaning of this formulation of current of probability? really it seems like if there's anything that doesn't work correctly. And it is this way. If we remember, the differential operator of velocity is:






Now, let's take a look to the previous formulation: the operator velocity is the result of apply this differential operator to the wave function, and this is the h constant, the mass, and the nabla operator. Thus,



Lets think now about this. If we take the operator and apply it, due that it isn't a quaternionic number, it must be a vector. But, what's the wave function here? and, what's the meaning of this formula?

To understand it, we must start to familiarize with an interesting aplication of the quaternions, one application that, if you take a vector and multiply it (quaternionic multiplication) by a wave function expressed in quaternions of module one, and his conjugate like this:


...then we are utilizing "rotations", but an special way of rotations: wave hipercomplex function rotations, part of the Clifford Algebras previously exposed, concretely the Even Sub Group of Clifford Algebras.

Next post we well talk about rotations and quaternion-rotations, and what are the interpretations in GA of the Schrödinger equation.

R. Aparicio.

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References

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

(2) Lectures on Clifford (Geometric) Algebras and Applications, Rafal Ablamowicz & Garret Sobczyk.
(3) Siglo XXI: La física que nos espera. R. Aparicio.
(4) De Natura Visibilium et Invisibilium. R. Aparicio.

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