User:Quantim/Fun With Quaternions

(Extracted from http://wiki.mindcloud.org/wiki/Quantim/Fun_With_Quaternions ;-D)

A Quaternion is a mathematical construct invented by Sir Hamilton in the 1800s, before vectors. Basically he wanted what we would today call a 'vector space' that is capable of direct division - something which can't be done with either vectors or spacetime 4-D metrics. They are essentially a 4-D vector thing with one scalar component & a 3-D vector component IN THE ONE THING! This is very useful because you no longer have to worry about vector or scalar 'spaces' as they both can be expressed as a quaternion.

Contents

Quaternions Basics Explained

Hamilton considered the idea of 3 orthogonal axes but wasn't sure how to get there. The answer lies in imaginary numbers. We "all" know that:
i2 = − 1
which gives rise to the imaginary axis and numbers like
z = x + yi
which is mapped to a 2-D plane, one imaginary axis & one real axis. Hamilton extended this idea to three imaginary axes while keeping the one real axis. He did this by introducing other square roots of one, i.e
i2 = j2 = k2 = − 1 so that each of i,j,k is a unit vector for an imaginary axis, with 1 being the unit vector for the real axis. In order to make these imaginary axes "behave" the following relations are used:
ij = − ji = k
jk = − kj = i
ki = − ik = j
so that these unit vector imaginary number things do not obey the seemingly intuitive commutative law for addition i.e. a+b=b+a. At the time Hamilton introduced these ideas to the public, non-commutation was a mere curiosity and not at all relevant to any real physics. However, in this modern age of quantum freakin theory and all the 'commutation' and 'anti-commutation' relations flying at YOUR FACE from everywhere, this concept doesn't seem too dificult to grasp. In fact, upon inspecting them, i was immediately able to realise that, "these i,j,k things don't commute... they ANTI-COMMUTE!" so yay me for remember what that meant.

anyhow, with these 3 imaginary unit vectors, along with the real unit vector (it is just 1 but we will denote it u, which is no biggy since vectors write their 3 real vectors, which are all exactly one, to be i, j, k... INVOKING direction via dot product rather than it coming naturally as with these three things... but i digress) WITH the 3 imaginary vectors and one real one, we define a quaternion Q as:
Q = q_{0}\hat u + q_{1}\hat i + q_{2}\hat j + q_{3}\hat k

well that looks like nothing but a 4-D vector. big deal. thing is with quaternions, there is no dot product or cross product to remember. You just multiply them out directly... er like a dot product. there's a page away from the mindcloud i'll link to at some stage which describes the exact multiplication algorithm thing - i've done it by hand, how boring. some points:

One assumes that the ability to divide directly will greatly simplify the mathematics. This means you could just divide an operator away rather than having to do pesky inverse relations e.g. integrating away a derivative. However, i haven't gotten that far in researching with them yet.

Quaternions as Scalar-and-Vector

It is interesting to see what happens when we tentatively separate the quaternion into scalar and vector components, like
\begin{align} Q &= q_{0}\hat u + q_{1}\hat i + q_{2}\hat j + q_{3}\hat k\\ &= (q_{0}\hat u) + (q_{1}\hat i + q_{2}\hat j + q_{3}\hat k)\\ &= (\phi,\vec{A}) \end{align}
where we use φ to denote a scalar potential & \vec{A} to denote the associated vector field. There is something important about this coupling of the scalar potential to a vector field - in vector notation they are inherently separate even though in the case of electromagnetism they are aspects of the same thing. Multiplication of two arbitrary quaternions becomes really freakin easy, compared to doing all the ugly symbol matching of expanding brackets with 4 terms each. observe!
(a,\vec{B})(c,\vec{D}) = (ac - \vec{B}\cdot\vec{D} , a\vec{D} + \vec{B}c + \vec{B}\times\vec{D} )

i'm sure that would have made more sense had i explained multiplication properly, or at the very least linked to it. I think it's neat that quaternions contain both dot and cross products of the vector components in a single multiplication operation. Quaternions are essentially bigger than vectors cos they can do more.

Quant!M's Quaternionic Musi(c)ing

Quant!M has found it useful to define an operator \Box as:
\begin{align} \Box &= \frac{\partial}{\partial t} \hat u + \frac{\partial}{\partial x}\hat i + \frac{\partial}{\partial y}\hat j + \frac{\partial}{\partial z}\hat k\\ &= (\frac{\partial}{\partial t} ,\nabla) \end{align}

which quantim thinks of as the change operator. It is similar to the D'Alembert operator used in space-time 4-vectors but that applies each partial derivative twice. Quant!M has been applying this operator to arbitrary quaternions:
\begin{align} \Box Q &= (\frac{\partial}{\partial t} ,\ \nabla) (\phi,\vec{A})\\ &= (\frac{\partial\phi}{\partial t} - \nabla\cdot\vec{A},\ \frac{\partial\vec{A}}{\partial t} + \nabla\phi + \nabla\times\vec{A}) \end{align}
which aint all that neat but if we equate \Box Q = 0 then we have end up with 2 equations for 2 unknowns:
\frac{\partial\phi}{\partial t} = \nabla\cdot\vec{A}, and
\frac{\partial\vec{A}}{\partial t} = - \nabla\phi - \nabla\times\vec{A}
which superficially look a hell of a lot like maxwell's original equations. Means i'm on the right track i guess. Really i want to relate this to gravity and process physics. Something physicists have done when deriving vector fields & forces is assume the existence of a scalar potential φ - with units of "Energy per unit ___" where in the blank spot you put charge or mass, depending on whether it is electromagnetism or gravity respectively. In gravity the acceleration is related to this potential by:
\vec{g} = -\nabla\phi
but also, by definition
\vec{g} = \frac{d\vec{v}}{dt}

which made quantim think a bit - if these quantities of velocity and potential are related through \vec{g} then is it possible to arrive at a relation between the two quantities directly, without invoking acceleration due to gravity?

I considered the quaternion \Phi = (\phi, \vec{v}) and applied the change operator \Box:
\Box\Phi = (\frac{\partial\phi}{\partial t} - \nabla\cdot\vec{v},\ \frac{\partial\vec{v}}{\partial t} + \nabla\phi + \nabla\times\vec{v})
leading to:
\frac{\partial\phi}{\partial t} = \nabla\cdot\vec{v}, and
\frac{\partial\vec{v}}{\partial t} = - \nabla\phi - \nabla\times\vec{v}

if we look at the latter of these two equations, if \nabla\times\vec{v} = 0 which means there is no vorticity in a flowing vector field with velocity \vec{v} then indeed we have:
\frac{\partial\vec{v}}{\partial t} = - \nabla\phi = \vec{g}

which is mathematically pretty nice. Quant!M is currently musing on what a quaternion with a scalar component and an associated velocity (vector) field component could mean physically. He's thinking it may precisely define gravity in a 3-space flowing worldview. I'm wondering when general relativity will pop out. meh.

Further Research

Quant!M is aware of a single quaternion equation that contains all of maxwell's equations, at least as the textbooks define them (history tells a different story). He is wondering if he may arrive at a similar result for the new 3-space theory of gravity if he plays with the convective derivative rather than just the partial time thing. yeah.

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