When we choose a quaternion basis {1, i, j, k}, there are
and
.
Then a quaternion q is denoted by
Equation (1) can also be written as the form of dual complexes
when we introduce a scalar function 
and a vector function
, Equation (1) can be also recorded as a scalar-vector construction of quaternion
And when we record Equation (1.4) as
it is a Cayley-Dickson construction [
So, there exist scalar-vector branching and CayleyDickson branching in quaternion and we can call the branches a Hamilton representation of a quaternion, which guides us into the branch world.
Meanwhile, there is multi-vector Mk (k = 0, 1, 2, 3, 4), where Mk is a multi-vector of grade k. k = 0 is scalar, k = 1 vector, k = 2 bivector, k = 3 pseudovector and k = 4 pseudoscalar [3-5] as
in which Ψ = φ – iθ constructs a complex wave function of matter, while A = V – iU forms a complex vector function of matter particles in space-time and F maintains a bivector as interaction. Equation (5) means that matter combines wave function and vector function with their interaction, which is an image fitting the duality of waveparticle, within the concept of combining mass and energy as matter.
The conjuncture of M is
M can be divided into two parts, even M, M+, as left M, ML, and odd M,
, as right M, MR:
The reversion of multi-vector M, denoted by
, can be defined as
In order to describe the branching process, quaternion and multi-vector mathematics are suitable. When we combine quaternionic algebra, geometric algebra and calculus, the mathematical structure for branch dynamics will become apparent.
Suppose R, C, and H denote respectively real, complex, and quaternion fields. For a complex 
with
, record 
and
, called the real part and imaginary part respectively. And for a quaternion 
with
, record 
and
, called the real or scalar part and vector or pure quaternion part respectively.
While a quaternion (q) can be expressed by the scalar-vector construction and the Cayley-Dickson construction, there is a conjugation of q
with 
and norm
. Then we have
and
where 
and P constructs 3-dimentional Euclidean vector space.
Using Pauli matrices and Dirac spinors [6-8], we know that a quaternion can split into 2 × 2 complex matrices by substitution 
[
are the standard complex Pauli matrices as follows (together with unit matrix
)
and Dirac matrices become
in which
.
So, a quaternion has an equivalent representation which we can call the Pauli representation as
We see that Equation (15) is the same as the conjugation of q, Equation (10), as
, which means that the Pauli representation and Hamilton representation become conjugations of each other. The algebraic structure shows that the Hamilton representation and Pauli representation exist naturally for a quaternion, which constructs a conjugation pair.
In the Hamilton representation, there are a CayleyDickson branch, a Grassman branch and a Euclidian branch, in which the Cayley-Dickson branch is produced by the multiplication of quaternion G1 and quaternion G2 with the form of the dual complex function form as
The Grassman branch produced by the multiplication of quaternion G1 and quaternion G2 with the form of scalar-vector representation as left branch
and a similar Euclidian branch as right branch
Noncommutative associative quaternion algebra provides rich algebraic branch structures, which establishes the foundations of branching. And a similar structure could be broadened to octonion (both quaternion algebra and octonion algebra belong to Clifford algebra), if we abandoned the associative property and then got only the alternative algebraic structure.
For multi-vector M, the frame basis can be unified in geometric algebra, spanned by
And for multi-vectors M and N, the geometric product is defined as
where. means inner product and 
outer product.
, 
(20)
,
(24)
= A and Ψ = φ – iθ and A are linked by following equation
and
, one knows dynamical mechanism.