Talk:Quaternion

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There's something messed up with this page, starting after the equation in the history subsection. Using Firefox, everything after that point is rendered in a much smaller font.--Starwed 20:26, 2 May 2005 (UTC)

I wrote a short introduction so that non-mathematicians stumbling across this article, or curious folks wondering what quaternions are, would have something to grasp before being plunged into inaccessible terminology. I think this is important for math articles if they are to serve a purpose other than preaching to the converted (so to speak).

However, that doesn't mean that my introduction is particularly good - I swiped part of it from the History subsection. Any good historians/sociologists of math out there who can improve it? - DavidWBrooks 20:17, 1 Jun 2004 (UTC)

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[edit] Correction to quaternion argument arg(p) and logarithm ln(p)

The quaternion argument arg(p) was listed as:

\arg(p) = \arccos\left(\frac{\operatorname{Scalar}(p)}{|p|^2}\right)

The norm should not be squared however, so it should be:

\arg(p) = \arccos\left(\frac{\operatorname{Scalar}(p)}{|p|}\right)

The quaternion logarithm ln(p) was listed as:

\ln(p) = \ln(|p|) + \sgn(p)\arg(p)

The sign should be taken from the vector part only not the whole quaterion

\ln(p) = \ln(|p|) + \sgn(\vec{u})\arg(p)

This follow from the exponential notation of quaternions:

p = r\exp([0, \vec{u}]\phi) = r\cos(\phi) + r\sin(\phi)\vec{u} \land |\vec{u}| = 1

Hkuiper 17:34, 30 Oct 2004 (UTC)

[edit] ijk = -1 does not imply associativity

The article says the equation ijk = -1 implies the quaternions are associative. Actually it assumes it, but does not imply it.

John Baez

The statement is now removed. -- Fropuff 01:57, 2005 May 2 (UTC)

[edit] Quaternion multiplication typo?

I think there's a typo in quaternion multiplication. I haven't figured out how to get the images working here, but I hope I can make myself clear with my own ascii notation.

The article defines two quaternions

q = a + u_

p = t + v_ [the underscores denote the vector part]

The article gives the proper form for

pq = at - u_.v_ + av_ + tu_ + v_ x u_

but not for qp,

qp = at - u_.v_ + au_ + tv_ - v_ x u_

which should be

qp = at - u_.v_ + av_ + tu_ - v_ x u_

A look at the source for the section says that both the img and alt versions are faulty.

Quaternion multiplication
pq :

The usual non-commutative multiplication between two quaternions is termed the Grassmann product. This product has been described briefly above. The complete form is described below:

<img class='tex' src="/math/c8439d2102f59aa407da93e2da708782.png" alt="pq = at - \vec{u}\cdot\vec{v} + a\vec{v} + t\vec{u} + \vec{v}\times\vec{u}" />
pq = (at − bx − cy − dz) + (bt + ax + cz − dy)i + (ct + ay + dx − bz)j + (dt + az + by − cx)k

Due to the non-commutative nature of the quaternion multiplication, pq is not equivalent to qp. The Grassmann product is useful to describe many other algebraic functions. The vector portion of the multiplication of qp follows:

<img class='tex' src="/math/b3cff5a7d4f6c36c9cb09d393887c3c9.png" alt="qp = at - \vec{u}\cdot\vec{v} + a\vec{u} + t\vec{v} - \vec{v}\times\vec{u}" />

[edit] Quaternions and quantum theory

Small particles do not obey the laws of classical logic. Instead these particles obey the axioms of a weaker logic, the quantum logic. Classical logic has the structure of an orthocomplementary modular lattice. Quantum logic has the structure of an orthocomplementary weakly modular lattice. Classical logic is isomorphic with the Venn diagrams. Venn diagrams are often represented by a series of overlying circles. The structure of quantum logic is far more complex. There exists a mathematical representation of this structure in the form of the subspaces of a Hilbert space. A Hilbert space is a collection of analytical functions. A countable but still infinite number of mutually independent functions can span this space. The space can be defined over the real numbers, over the complex numbers and maximally over the quaternions. Linear operators that work on these functions have eigen-values that belong to eigen-functions of these operators. The eigen-functions of such operators span the Hilbert space. The eigen-values are the things that appear to us as our physical world. Group theory applied to these operators reveals the basic formula’s of quantum theory. When quaternions are used as the applied number system, then the quaternionic quantum theory results.

With quaternions, concepts like spin and parity become a trivial interpretation. When the multiplication rule for quaternions is written as:

pq = at ± (u,v) + av + tu ± uxv

then the sign of the second and the last term can be taken at will, while still a proper multiplication rule holds. (u,v) stands for the inner product of the vectorial parts. u x v stands for the outer product of the vectorial parts. In physical sense the first sign choice directly relates to the physical feature: parity. The second sign choice relates to spin.

The real part of the physical quaterion can often be interpreted as time, while the vector part represents the three dimensional space we live in. However if you take a quaternionic Fourier transform of that world, you end up in another view of that world where time is replaced by energy and the location vector space is replaced by the impulse space. (Impulse has a close relation with force).

(source: Jauch (1968) Quaternion Quantum Mechanics)

The Fourier transform based relation between the two views causes the Heisenberg’s uncertainty relation. You cannot precisely know time and energy of a particle and you cannot know precisely the location and the impulse of elementary particles. An individual in time-location space has a huge extension in energy-impulse space and vice versa. A rather vague object in time-space will be rather compact in energy-impulse space. This may give a physics based support to the fact that sensitive people can use the associative capabilities of their brains to collect information from ‘the other view’ and interpret it so that it becomes a meaning in a wider time-location frame.

J.A.J. van Leunen


[edit] Rodrigues

I am a novice when it comes to quaternions and am struggling to read Altmann's book on rotations quaternions and double groups. Interestingly this book is quoted in the article but there is no mention of Rodrigues who -if I must believe Altmann- has come up with a more rigorous version of quaternions than Hamilton. Can anyone comment to that?

af:Gebruiker:Jcwf

[edit] About a reference

I'd like to note that (currently fourth) external reference ("Doing Physics with Quaternions") points to a website of the same title, which is unfortunately nothing but one wild mess of gibberish and nonsense. [The author does not appear to ever have heard of vector bundles and gauge theories when he states thats "quaternion multiplication is reminiscent of spin", for instance.] There is nothing encyclopedic about it, and (not mentioning the potential harm that website could have on the innocent laymen, or on the laughing muscles of visitors with some mathematical knowledge) the reference should be deleted.

[edit] Division over reals

The article states that only the real, complex, and quaternion numbers form a finite-dimensional associative division algebras over the field of real numbers. There should be *four* division algebras - we need to add octonion numbers.

[edit] error in quaternion inverse

I was writing some code using this page as a guide. My code didn't work. It took a long time to track down the problem.

The definition of the inverse is wrong. It shows a grassman product in the denominator. Uses same notation as earlier for grassman product.

But this should be instead the dot or inner product.

[edit] Reorganization

I have noticed that a lot of the information in this article is repeated in more than one place, and the logical coherence from start to finish isn't tight. This is not uncommon when many editors contribute. I want to make known that I started to reorganize it to be more concise and better structured. Usually, when I do this, I rarely add/delete/change much. Usually it is just a matter of changing the outline (section titles, etc.) and moving content appropriately.

On a side note, my apologies for inadvertently marking my first reorganizational edit as a minor edit; it was clearly not. Baccyak4H 20:35, 14 November 2006 (UTC)

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