[I changed the title of this blog to prepend Proto to vector space since a vector space is defined over a mathematical field.  While I explored the possibility that the positive real numbers might make up a mathematical field, that effort did not pan out.  In a subsequent blog, I will show how many properties of a vector space are shared by the positive real numbers over the group Z2.]

My research quest is to understand the nature of quaternions and how much of their structure is reflected in physics.  This blog will tell the tale of how I fooled myself three different ways on this journey.  My resolution of these particular problems struck me by surprise.  Ontology is a fancy word for why.  Hopefully I can show that physics makes the odd math reasonable.

A Fool for History

A quaternion as defined in the King James version of the Bible (Acts 12:4, whatever that means), is four Roman soldiers.  Four biblical quaternions were sent to bring Saint Peter to be crucified, a bit of history I missed in my church going days.  Quaternions and crucifixion go together, how ironic.  I inherited a bit of numerology about quaternions, that they should involve four numbers.  Before blogging at Science 2.0, I created a clay and pipe cleaner model of quaternion multiplication:

My second blog here on March 14, 2011 featured a discussion of this graph.  I thought this graph represented all the ways to form quaternion products.

Eleven months later, David Halliday pointed out that the model was not up to the task:
It may help you to recognize that the picture on the left is not a representation of Q8—or any group, for that matter.

If you actually want to make a model of Q8 you will need eight (8) balls (nodes).  Using the vertices of a cube, for the eight elements of Q8, where you have two concentric tetrahedra inscribed within the cube (in the two possible orientations for such), you will have connections through the major diagonal (multiplication by -1), connections through the diagonals of the faces of the cube, and connections along the edges of the cube (and loops at each vertex for multiplication by the identity).
No matter how many times I re-read the critique, it always sounded right.  My toy model was too small to do the job.

It took one month and \$225 to construct the upgrade:

Another five revisions were required before I could say that I was confident that the cube was a physically accurate representation of the quaternion group Q8.  I got the message about quaternions and eight.  Not really.

A Fool Falls in the Woods Without Making a Sound

I announce the new and improved model for quaternion multiplication in March, 2012 , a year after blog #2.  This line was in the first paragraph:
A few months ago, David Halliday and I started talking about the finite group Q8, which over the real numbers becomes the quaternions.
This sounded reasonable to my ear.  The real numbers are the simplest mathematical field I am aware of where one can do calculus.  The basis elements of Q8 are easy enough to list:

$(e_0, -e_0, e_1, -e_1, e_2, -e_2, e_3, -e_3)$

Here the e's are general.  To be honest, I gravitate to (1, -1, i, -i, j, -j, k, -k).  Consider the basis vector e0 over the real numbers.  It can represent everything that the basis vector -e0 can over the real numbers, with no exceptions.  If I demand that the basis vectors are linearly independent, I am obligated to avoid the real numbers.  Just to check I was using the correct lingo, I looked it up:
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent.
The family of 8 vectors is linearly dependent over the real numbers, but linearly independent over the positive real numbers.

To be a vector, one must be able to add them and multiply by a scalar.  Adding them up is straightforward.  What is questionable is multiplying by a negative scalar.  Since every plus becomes a minus and visa versa, this should be fine.  Just a bit odd.

The New Idea Fool

I felt a need to work over the positive real numbers and zero so a vector with a basis of +e0 would be different from one with a basis of -e0.  The positive real numbers (without 0) with the multiplication operator form a group:
1. One member times another always creates a third (2*3=6).
2. There is an identity element (1).
3. Every member has an inverse (8 * ⅛ = 1).
4. Products are associative ( 2 * (3 * 4) = (2 * 3) * 4)
The problem is with the addition operator over the positive real numbers and zero.  It is manifestly obvious that the additive inverse does not exist.  All the additive inverses are in the set of negative real numbers.

Many people give up after spotting one manifestly obvious problem.  This is where the fun begins for me.  Is there any way to make things work?  Multiplication as repetitive addition was suppose to be the difficult case.  My initial tries in the comments section failed to pass the associative property needed by groups.

This raised a question for me: what does the associative property do?  In evaluating any expression, there can be several orders of operation.  For example:

$(a + b) + c$

means a adds to b first, then sum that with c.  While

$a + (b + c)$

indicates that b adds to c first, then sum that with a.  The associative property says these must be equal:

$(a + b) + c = a + (b + c)$

This certainly is a nice property, one I have used since my first days with algebra.  Yet, is it necessary?  Imagine doing algebra where parentheses were not allowed, ever.  Could you proceed?  It can be done with a rule that one works left to right.  Banishing parentheses makes associative and distributive laws moot.  It is a tautology that:

$a + b + c = a + b + c$

but a question remains open if:

$a + b + c =?= b + c + a$

With addition, both sides are the same.  But addition will not have any additive inverses in the set of positive real numbers.

This is where I turned to physics in order to step outside the box.  A photon not only is a particle, but it is its own antiparticle.  Could I cook up a scheme where a number was its own additive inverse?  That could be done by treating addition, the "+", more abstractly.  It is just a binary operator after all.  To George Orwell's slogans: "WAR IS PEACE", "FREEDOM IS SLAVERY", and "IGNORANCE IS STRENGTH", I add my own: "PLUS IS MINUS".  With the Orwellian plus, every number is its Orwellian additive inverse.

The Orwellian plus requires one more component to do its job: take the magnitude of the result.  Since I think of the positive real numbers and zero as magnitudes, taking a magnitude appears to be a reasonable thing to do.  Taking the magnitude defines the set.  Let's do a few calculations to see we are on the same page:

8 "+" 8 = 0
8 "+" 10 = 10 "+" 8 = 2
8 "+" 3 "+" 1 = 8 "+" 1 "+" 3 = 4
3 "+" 1 "+" 8 = 6

Do the positive real numbers and zero with the Orwellian plus form a group?  The answer is no because the associative property is not there (4 does not equal 6 in the above calculation, so the order does matter for the Orwellian plus).  What of the other three properties?
1. One element "plus" another always makes a third positive real number or zero.
2. There is an additive identity element, zero.
3. Every element has an additive inverse, itself.
There might be a bit of jargon to label this three out of four situation, but let me not muddy the waters: the positive real number and zero with the Orwellian plus is not a group because it is not associative.  Can one still do important things like taking a derivative? There are slight changes in how to write things out:

$f[q + {\delta}] + f[q] * \delta^{-1} \lim_{\delta \rightarrow 0} = f'$

This definition should generate the expected results over the positive reals and zero.

WHY an 8D Vector Space?

My first reaction to try and embrace the eight-ness of quaternions was: WTF???  For years I had talked about spacetime, three dimensions for space, one for time, the total is four.  I claim to suffer from Minkowski vision.  I'l n'y a pas espace sans temps, pas de temps sans espace is a bit of French I see every morning.

A few blogs ago, I admitted complete and utter defeat on an issue as happens to me.  It was about time, that it is a vector having a positive definite value called the magnitude and a direction.  One direction is positive, the other negative.  How can one interpret those two directions?

The past is negative.  The future is positive.  The past is not the future.  More than that, no elements of the past can mix with the future.  The past is separate from the future.  More technically, the past is orthogonal to the future. [CORRECTION: I should have written, the past is linearly independent of the future for an 8D vector space over the positive real numbers and zero.  This blog only deals with the addition operator and multiplication by a scalar.] Calculus describes the past's relationship to the future where change can only happen now.

Left is not right.  Left is a mirror reflection of right.  Mirror reflections never meet.  That separation of mirror reflections is built into the basis vectors of 8D spacetime.

Up is not down.  One can again use mirrors, but there is always a separation between the two.  Likewise, near is not far.

As I worked through the details of the 8D spacetime vector proposal, I saw the logic of the Orwellian plus.  I interpret the origin as the home of the observer.  Go out in time or in space from the origin.  What can be Orwellian added is something that is closer to the origin.  One ends up describing things that are closer to home.

Anyone wishing to discuss dot or cross products will have to wait until next week.  The rules are all there in the quaternion group Q8 over the positive real numbers and zero.

Doug

Snarky Puzzle

What property of physics makes the 8D spacetime vector space sound unreasonable?  Let Lord Kelvin be your guide.

Next Monday/Tuesday: The 8D Multiplication Table
[Update: That will be July 9.  Dance camp and a death in the family are the cause of the delay.]