Natura Visibilium et Invisibilium

2006-12-25T02:48:00.000-08:00

2006-01-31T14:40:00.000-08:00

2006-01-21T09:42:00.000-08:00

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.

2006-01-19T12:25:00.000-08:00

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.

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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.

2006-01-18T12:38:00.000-08:00

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 (q • p) –q0 (p • r)– (q X p) • r

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

—r0 (q • p) –q0 (r • p) + (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.

2006-01-15T10:49:00.000-08:00

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:

http://www.elaleph.com/libros_buscar.cfm?str_autor=Aparicio%20S%E1nchez%20Rafael&btn_buscar=1

2006-01-13T06:31:00.000-08:00

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.

2006-01-13T06:00:00.000-08:00

DE NATURA VISIBILIUM ET INVISIBILIUM

Approximately ten years ago, the students of Industrial Engineering of The Universidad Politécnica of Valencia decided to do a joke to the novice students. The courses of Industrial Engineering where six, and much students passed much more than 13 years to end this. This way, students and teachers were easily confounded because his ages and beards were similar. The innocent but amusing joke was to take a veteran student, to disguise to him with a white dressing gown like a teacher, and distribute they (teachers)around the classroom. The false teacher, seriously, started the lesson indicating that it was going to do a verification of the base level of the students around their general knowledge, and asked some questions extremely difficult for a beginner student. The false teacher, asked accurately to the real ones about concepts that scared the students like "rotational of a divergence" or "applied laplacians to vectorial functions" or "hermitic or hilbert spaces", that evidently they not have studied last years. One time the students were really scared, asked to a one of they to go to the blackboard, conscious that the level was going to be very high.

And then, the "teacher" asked to him:

— Please, could you say to me the correct definition of a “vector”.

— Eeehh... an arrow... two points

—the student doubted.

— Go down, to the receptionist, and say to him that give to you a box of vectors. The student, rapidly go out for a "box of vectors". But evidently, the vectors don't exist in real world, and by the way, when he go out, the real teachers, the veteran students and any who know this, laughed strongly.

The vectors are representative models, same way that much of our structures and theories are models that we use to explain ourselves the natural world and the reality. Every scientific theory starts on the reality, it is used thought a theoretical model and after is used in real world. Simply resumed, this is the essence of the scientific method. The problem arises and grows when the models that we use to represent reality are every time much complex. From birthing of the nature philosophers, denominated "physics", to practically finals of XIX century, the models of the world have had some revolution, but were comprehensible. For example, there where models based in plane Earth, that after where changed for another's with an spherical Earth, the center of the universe. Much more before, Erastostenes, librarian of the Alexandrian library, calculated the diameter of the Earth with a much more approximation that Cristobal Colon used. Later where much changes, like Earth wasn't the center of the universe, and Sun was. If this is scary, it's really certain that this steps allows humans to eliminate the problems about the interpretation of reality, because it's a much more approximation theory and model of the reality. It means that every abstraction of the reality, every model that humans construct often is conceptually much complex, but at the same time lacks existent paradoxes.

At different times have been immutable references, like the plane Earth, or immobile, or the regularity of the stars and the Sun, and recently the constancy of the velocity of light in vacuum, his wave particle characteristic an the theories that are related to this like especial relativity and general relativity and quantum mechanics. In the work of finding this constant references thought humans could establish his points of view, it arrives at a moment that "space" and "time" are redefined, with Einstein, like a union, a interval that was the last reference of one model. High velocities introduce a new elements of anxiety for de human beings, that needed a new step to security. This security became partially with the Einstein's theories.

But Dalí known it and used it. Confusion is the previous step to a jump of conscience to a higher level that allows to learn from two opposite points of view of the same natural phenomenon. When you see to the “Torero Alucinógeno” of Dalí, the first impression is disturbing, confusing. If you are the sufficiently patient, the first impression is changed, and the brain starts to work more higher, and it looks for an order in the apparent chaos, or the the message that is hidden in it. And is this way that a little miracle occurs in the spectator mind: a new way to see allows us to follow the way that explains us the painter biography.

Same way, an eclipse could have been confusing to the ancient babilonic observers, because something regular like sun and its revolutions was repently changed. Only when they could constant that it was not permanently, they could interpretate this not definitive phenomenon. Maybe gods became infuriated, maybe the Sun remained slept that morning... but really what it happened was that there was a new phenomenon, and a new knowledge of the world. Tales of Mileto gained fame of wise because he had a table and predicted what year this was going to happen. Stars were stable, but sometime, strange things passed

Faraday, Maxwel, Hertz ideas were a giant steps through the unification of extrange phenomenon like electricity and electromagnetism that later were the motor of great changes in technology at beginnings of XX century. Einstein, Young, Broglie, Heisenberg, Planck, etc. broke any "absolute references". But when the reference tree to house was disappeared, only one thing could help man to go to the correct way: human thinking. It could be that things that are created through imagination and philosophy can be completed through it.

Natura abhorret a vacua

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Extracted of "Siglo XXI: La Física que nos espera", Cap. 1, de R. Aparicio.