Rules for multiplying quaternion exponentials

doug  <sweetser@world.com>

Introduction
New method for multiplying quaternion exponentials
Testing the proposal
Implications

Introduction

Multiplying two exponentials is at the heart of modern analysis, whether one works with Fourier transforms or Lie groups.  Given a Lie algebra of a Lie group in a sufficiently small area the identity, the product of two exponentials can be defined using the Campbell-Hausdorff formula:

\(Exp[X] Exp[y]\  = \ 
      \((X\  + \ Y)\)\  + \ \(1\/2\)[X, \ Y] \((X\  + \ Y)\)\ 
 + \ 
          \(1\/12\) \((\([\([X, \ Y]\), Y]\)\  - \ \([\([X, \ Y]\), X]\))\) 
            \((X\  + \ Y)\)\  + \ ...\)

This formula is not easy to use, and is only applicable in a small area around unity. Quaternion analysis that relies on this formula would be very limited.

New method for multiplying quaternion exponentials

I have discovered (perhaps for the first time) a simpler and general way to express the product of two quaternion exponentials as the sum of two components.  The product of two quaternions splits into a commuting and an anti-commuting part.  The rules for multiplying commuting quaternions are identical to those for complex numbers.  The anticommuting part needs to be purely imaginary.  The Grassman product (q q') of two quaternion exponentials and the Euclidean product (q* q') should both have these properties.  Together these define the needs for the product of two quaternion exponentials.

\(Let\ q = \ \(Exp[X]\ \ \ q'\  = \ Exp[Y]\)
	\)q q'  = 
       {q,  q'}/2  +  
        Abs[q,  q']/2 Exp[(π/2) (([q,  q'])/Abs)[q,  q']] \(	where\ \ \ {q, \ q'}\  = \ 
      \(q\ q'\  + \ q\^\('*\)\ \ \(q\^*\)\ \ \ and\ \ [q, \ q']\  = \ 
        q\ q'\  - \ q\^\('*\)\ \ \(q\^*\)\)\ 
\)\(\(q\^*\) q'\  = \ same\ as\ above\)\(	where\ \ \ {q, \ q'}\  = \ 
      \(\(q\^*\)\ q'\  + \ q\^\('*\)\ \ q\ \ \ and\ \ [q, \ q']\  = \ 
        \(q\^*\)\ q'\  - \ q\^\('*\)\ \ q\)\ \ \)

Changing the commutator changes the type of product of quaternion exponentials.  Andrew Millard made the suggestion for the Grassman product that unifies these  approaches nicely.  This may have an interesting philosophical impact, but for now, let's just check the math :-)

Testing the proposal

Define a few Mathematica functions to test these proposals.

\(expProduct[comm_, \ anti_]\  := 
	
      anti/2\  + \ abs[comm]\ qexp[π\/2\ comm\ /\ abs[comm]]/2\)

\(commGrassman[q1_, \ q2_]\  := 
	
      q1\ .q2\  - \ Transpose[q2].Transpose[q1]\)

\(antiGrassman[q1_, \ q2_]\  := 
	
      q1.q2 + Transpose[q2].Transpose[q1]\)

\(commEuclid[q1_, \ q2_]\  := 
	
      Transpose[q1]\ .q2\  - \ Transpose[q2].q1\)

\(antiEuclid[q1_, \ q2_]\  := 
	
      Transpose[q1]\ .q2\  + \ Transpose[q2].q1\)

\(A\  = \ q[2, 1, 3, 4]; 
B\  = \ q[1, 2, 3, 5]; \)

Without the new tools, find the products.

N[(qexp[A].qexp[B]).{1, 0, 0, 0}]

N[(Transpose[qexp[A]].qexp[B]).{1, 0, 0, 0}]

Test the proposal.

\(N[expProduct[commGrassman[qexp[A], qexp[B]], 

          													antiGrassman[qexp[A], qexp[B]]]].{1, 0, 
        0, 0}\)

\(N[expProduct[commEuclid[qexp[A], qexp[B]], 
													
          antiEuclid[qexp[A], qexp[B]]]].{1, 0, 0, 0}\)

It works!

Implications

In terms of mathematics, a few new identities for multiplying quaternion exponentials were established.  These might be part of an effort to do Fourier analysis with quaternions.  That analysis would be applicable to both Grassman and Euclidean products, since work with the former can be transformed into the latter by changing the commutator relations.  This may have the greatest impact for physics.  Quantum mechanics has grown to very high levels of sophistication due in part to the norm being definite in a Hilbert space.  General relativity relies on the indefinite Minkowski metric, and that leads to technical problems with non-compact groups.  Doing physics with quaternions does not even use metrics! It may become transparent to bounce between Hilbert spaces where the scalar of squaring a quaternion is a norm, and spacetime where the scalar of squaring a quaternion is an interval. Prove something in quantum mechanics, then apply it to spacetime.  Could be cool.