Vector versus quaternion products

Subject: vector versus quaternion products
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/25
Message-Id: <5m83ou$84f@agate.berkeley.edu>
Newsgroups: sci.physics.research

Hello:

Descartes developed analytic geometry, where position is
represented by coordinates that can be used in equations. Hamilton's
quaternions provided the first means of working with products.
Today's vector products are a watered-down version of quaternion
multiplication. I think it is dangerous that so much of physics is
based on such denuded land. In this post, I hope to substantiate
these abrasive comments :-)

Hamilton had to make a daring leap to 4 dimensions to find the next
division algebra after complex numbers. He had no notion what to
do with this collection of 4 numbers, so the first thing he did was to
drop one of them. He worked with "pure quaternions":

0 + x i + y j + z k

Multiply a pair together:

(0 + x1 i + y1 j + z1 k)(0 + x2 i + y2 j + z2 k)

= - x1 x2 - y1 y2 - z1 z2 + (y1 z2 - z1 y2) i
(x1 z2 - z1 x2) j
(x1 y2 - y1 x2) k

Hamilton called the first three terms the scalar product and the latter
six the vector product (yes, he coined these ubiquitous terms). Pure
quaternions are not closed under multiplication, i.e., they don't
recreate another pure quaternion. That strikes me as a mathematical
gremlin that should come with a parental warning label. Gremlins
cause mischief. For example, the real numbers are not closed under
the square root operation, and that gives rise to complex numbers.
The square root gremlin is famous. The pure quaternion product
gremlin is stealthy, which makes it far more dangerous.

As physicists, we need not live in denial of quaternions with 4
dimensions. Instead, we should openly embrace them. What is an
event in spacetime but scalar time and vector x, y, and z? What is
the electromagnetic potential but scalar phi and vector Ax, Ay, and
Az? What is a particle's momenergy (a Wheelerism) but scalar E and
vector px, py, and pz?

As John Baez has pointed out, much of physics involves symmetry. A
basic algebraic form of symmetry is even and odd, how exchange of
order in a function will not change a sign (even) or will flip the sign
(odd). Why not build this symmetry into the foundations of
everyday mathematics used in physics? With quaternions, this is
easy to do.

Consider the Grassman outer product:

(q1 q2 - q2 q1)/2

= 0 + (y1 z2 - z1 y2) i
(x1 z2 - z1 x2) j
(x1 y2 - y1 x2) k

This is better known as the cross product. If q1 and q2 change
places, then the sign will change

(q1 q2 - q2 q1)/2 = - (q2 q1 - q1 q2)/2

This is obvious based on this definition, instead of a convenient
accident of algebra.

Yin/yang, even/odd, Grassman inner product/ Grassman outer
product. What is the Grassman inner product?

(q1 q2 + q2 q1)/2

= t1 t2 - x1 x2 - y1 y2 - z1 z2
+ (t1 x2 + x1 t2) i
+ (t1 y2 + y1 t2) j
+ (t1 z2 + z1 t2) k

If q1 and q2 trade places, the sign will not change:

(q1 q2 + q2 q1)/2 = (q2 q1 + q1 q2)/2

This is obvious based on this definition. The scalar terms is the one
generated by two 4-vectors using the Minkowski metric (sans
Minkowski), the ruler of special relativity. Unfortunately, by using
the Minkowski metric, the vector information is tossed away, a high
crime in physics.

The dot product is even. Define the conjugate of a quaternion just as
for complex numbers: q* = t - x i - y j - z k. The Euclidean inner
product (a term I coined because it can generate the square of the
Euclidean norm) is

(q1* q2 + q2* q1)/2

= t1 t2 + x1 x2 + y1 y2 + z1 z2
+ 0 i + 0 j + 0 k

If q1 and q2 trade places, the sign will not change. Quaternions are
closed under the operation of the Euclidean inner product because
one starts with quaternions and it ends with a quaternion. It expands
the notion of a dot product by inviting in scalar terms into a vector
function in a natural way.

Yin/yang, even/odd, Euclidean inner product/Euclidean outer
product. What is the Euclidean outer product?

(q1* q2 - q2* q1)/2

= 0 + (t1 x2 - x1 t2 - y1 z2 + z1 y2) i
(t1 y2 - y1 t2 - x1 z2 + z1 x2) j
(t1 z2 - z1 t2 - x1 y2 + y1 x2) k

If q1 and q2 trade places, the sign flips. I'm not certain what role
this plays in physics, but I have a prime suspect. Angular
momentum is a strange and wonderful part of our world which
depends on the Grassman outer product of quaternions (or the cross
product in vector-speak). Spin is even stranger and more
mysterious, related to angular mo but different. The Euclidean outer
product is related enough to angular momentum (via the negative of
the cross product) yet different enough (via the odd product
involving time) to _be_ spin.

To summarize, compare vector products to quaternion products. The
vector cross product is identical to the Grassman outer product. The
Minkowski product of two 4-vectors is the scalar portion of the
Grassman inner product but omits the symmetric vector portion. The
scalar dot product of two vectors is contained in the Euclidean inner
product but omits the scalar-scalar term. And there is no obvious
counterpart to the Euclidean outer product. Quaternion products
offer more information starting with the same initial data.

If you like watered-down, lukewarm coffee, stick to vector products.
It's safe, and it works with the right plug-ins, like the Minkowski
metric and spin. Me, I'm going to guzzle scalding cappuccino to see if
I can get enough of a buzz to see spin.


Doug Sweetser
http://world.std.com/~sweetser

ps. I don't actually drink coffee.
I do hope someday to write a JAVA applet that
will feature a peep show of spin 1/2 particles.