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.

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