Tuesday, January 31, 2006

Saturday, January 21, 2006

GEOMETRIC INTERPRETATION OF SCHRÖDINGER'S FORMULA

If we try to take a Geometrical interpretation of Schrödinger's formula, we can see that in this formula there are two questions to solve, one, the minus sign in the formula, and two, the two terms of this, that seems to be rotations in different direction.

We have said that quaternions are very useful like rotation operators, and this, if we apply a quaternion to a vector and his conjugate complex, we have a rotation in one direction.

Thus

q v q* = (q)* v (q* )* = p* v p where p = q*


The identity is a rotation using p, that is the inverse rotation.

So, previous equation indicates us a rotation, and its inverse in negative direction, so we have two rotations in same direction, that is a rotation. This way, there are not efforts, and a free particle that we can use Schrödinger's equation is a redundancy.

If we want to use a formula with efforts, its necessary that it appears two term like this:





But, It exists any formula that has this terms? It's evident that the answer is YES.

We have the equation of continuity:



That has to be equal to zero, due to the condition of continuity.

Now, we can generalize the Schrödinger equation with efforts... next.

R. Aparicio.

Partial and not exactly extracted from "De Natura Visibilium Et Invisibilium". R. Aparicio. Ed. Elaleph.

Thursday, January 19, 2006

GEOMETRIC INTERPRETATION OF ΨΔΨ*

Lets take an area like this:


If we take a volume in an after time, we have:




The volume relative change is:







Now, in 3D, we have that:








The dilatation of volume can be directly relationed with the spacial structure of velocity gradient.





A unit of volume can be TRANSLATED:














Be DEFORMED:


















ROTATED





Or have cutting efforts:

























And







Thus:













This is the angular velocity in Z axis. We can see that when there's a minus sign, it means angular velocity. Same way, we can demostrate that the rest of the two angles is an effort, it means that the formula between parenthesis has a "+" sign.





But, what represents it in Schrödinger's formula?

Answers, next day...

R. Aparicio.

----

References:

- Quaternion and Rotation Sequences, Jack B. Kuipers

- Vectors, tensors and the Basic Equations of Fluid Mechanics, Rutherford Aris.

- Vibraciones y ondas. A. P. French.

- Lectures On Clifford (Geometric) Algebras. Rafal Ablamowicz et. al.

- Mecánica de Fluidos. Victor L. Streeter et. al.

- De Natura Visibilium et Invisibilium. R. Aparicio. Ed. Elaleph.

- Siglo XXI: La física que nos espera. R. Aparicio. Ed. Elaleph.

Wednesday, January 18, 2006

GEOMETRIC INTERPRETATIONS OF THE EVEN SUB GROUP ALGEBRA OF CLIFFORD ALGEBRAS (OR QUATERNIONS).

(vectors in bold type)
A rotation may be represented by a 3x3 matrix, provided that the matrix is orthogonal and has determinant +1. Or alternativately, any such matrix A may be interpreted geometrically as a rotation operator in R3. In order to find the vector w wich is the image of a vector v under such a rotation, we simply represented the vector v by a column matrix whose entries are the components of v, and multiplied it on the left by the rotation matrix. Thus, in matrix form, the rotation is given by the equation w = A v.

The product of a sequence of rotation operators is again a rotation operator, and there are algorithms for finding the axis and the angle of that composite rotation. Quaternions play an important role in an alternative form for a rotation operator, a role that is quite different from the role played by most familiar matrix rotation operator. Further, quaternions may be very efficient for analizing certain situations wich involve rotations in R3. A quaternion can be geometrically interpreted as a rotation in R3.

But there are any questions about the quaternions that it seems like it doesn’t work: How can a quaternion, that is a R4 element, operate on a vector, that is R3? Well, is very obvious that a vector v of R3 can simply be treated as though it were a quaternion (R4) whose real part is zero. A quaternion like this is called pure quaternion. The set of pure quaternions is a subset of the quatenions.

A quaternion somehow represente a rotation and that we may find the image w of some vector v by using the simple product rule w = q v

The product of a quaternion q with a vector v must not only always be defined, but the result must always be a vector.

The simple product q v does not work, and the product vq will not work either, since commuting the factors does not change the real part of this product. This observation leads us to purpose that the desired operator may involve triple or perhaps even higher order products. It may then be possible to insure that the output of the operator will be a vector whenever the input is a vector. Since we wish to operate on vectors using quaternions we will let one of the factors, say p, in a triple quaternion product, be a pure quaternion, representing the vector in question.

If we have two general quaternions, q and r, and a pure quaternion say p, representing a vector, there are six possible products involved these three quaternions:

pqr qrp rpq prq rqp qpr

Quaternions are closed under multiplication, thus, the products qr and rq are simply cuaterniones, and this are not appropriate candidates to be our operator. The appropriate candidates are rpq and qpr. If we make no distinction betewwn the quaternions q and r, our last candidate is the single triple product qpr.

If we let
q=q0+q,
p = 0 + p
and r = r0 + p,

according to the quaternion product the real part of this triple product is:

—r0 (qp) –q0 (pr)– (q X p) • r

Using rules of vector algebra we can rewrite this real part in the form:

—r0 (qp) –q0 (rp) + (q X p) • r


Our operator must be such that the output is a pure quaternion that represents a vector whenever the input is, so it must require that this real part must be zero. To accomplish this, if we suppose that we had r0=q0, this real part may then be rewritten this way:

-q0 (q + r) • p + (q X r ) • p

If this part is zero, its required that r = - q that is

r = r0 + r = q0 – q = q*
q = r*

Thus, there are two triple quaternion products, namely:

qpq* and q*pq

that produce a pure quaternion (a vector) whenever the factor p is a pure quaternion, In terms of a given input vector v we then have two possible triple-product quaternion operators definend by:

w1=q vq* and
w2=q* vq

If we utilize a unit or normalized quaternion (a quaternion with norm 1), we can have:

q= q0 + q

Indeed has norm 1 then

q0^2 + q^2=1

However, since for any angle theta we know that
cos^2 (theta) + sin^2 (theta) = 1

There must be some angle such that:

cos^2 (theta) = q0^2

and

sin^2 (theta) = q^2

Theta must be between –pi and pi.

If we have a unit vector u, we can write the quaternion in terms of this angle:

u=q /q = q / (sin (theta))

Then:

q=q0 + q = cos (theta) + u sin (theta)

In this formula, if the change the angle theta for the angle is negative, we have the conjugate.

This way, we have arrived to an interesting geometric (but restriction) interpretation of geometric propertly of the quaternion product, that is, that a quaternion operator can be a representation of rotations in R3, and much more the product of

qvq*

Now, we will return to the Schrödinger’s formula.

What means ψΔψ*?

References:

- Quaternions and Rotation Sequences, Jack B. Kuipers
- Siglo XXI: La Física que nos espera. R. Aparicio.
- De Natura Visibilium et Invisibilium, R. Aparicio.

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: