Hilbert Book Model Project/Quaternions

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Quaternionions

Number systems

Number systems exist in many forms. They differ in their arithmetic capabilities.[1]

Algorithms exist that construct higher dimensional number systems from lower dimensional number systems. In this way, the dimension increases with a factor two. These procedures start from the real numbers and produce complex numbers, quaternions, octonions, sedions, and higher dimensional numbers.

The arithmetic capabilities may increase with the dimension of the number system until that dimension reaches four and after that limit, the arithmetic capabilities start to decrease.

Hilbert spaces can only cope with number systems that are division rings. In a division ring, every non-zero member owns a unique inverse.

Bi-quaternions use complex-number-based coefficients, rather than real-number-based coefficients. The bi-quaternions do not form a division ring.

Natural numbers form the simplest number system. The positive numbers and the integers are extensions of the natural number system.

Rational numbers consist of groups of fractions that have the same value. All rational numbers can be labeled by a natural number. This makes the rational number system countable.

The real number system adds all limits of convergent series of rational numbers. This addition destroys the countability of the real number system. The real number system is a continuum.

The natural number system, the positive number system, the integer number system and the rational number system have infinitely many elements. Cantor introduced the notion of cardinality. The cardinality is indicated by natural numbers, but the cardinality of the full set of natural numbers is indicated by cardinality ${\displaystyle \aleph _{0}}$,

The cardinality of the real numbers, the complex numbers, the quaternions, the octonions and the sedions equals ${\displaystyle \aleph _{1}}$. Also hiher cardinalities exist.

The Hilbert Book Model applies number systems that can serve its base model, which is based on a combination of an infinite dimensional separable Hilbert space and its unique non-separable companion that embeds its separable partner. The model selects the most versatile division ring. The quaternionic numbers system contains real number systems and complex number systems as subsets.

Versions

Depending on their dimension, number systems exist in many versions that differ in their ordering symmetry. Applying a Cartesian coordinate system followed by a polar coordinate system can achieve this ordering symmetry. The Hilbert Book Model exploits the fact that a quaternionic Hilbert space can harbor multiple versions of quaternionic number systems that serve as parameter spaces that each own their private ordering symmetry and that can float on top of a background parameter space. The difference between the ordering symmetry of a floating platform and the ordering symmetry of the background parameter space determines the symmetry flavor of the platform.

Quaternion arithmetic

Quaternions exist of a one-dimensional real part, and a three-dimensional imaginary part that can be represented by a real number valued scalar and a three-dimensional vector that has real number valued coefficients. In this way, the quaternionic number system represents a four-dimensional vector space that features a Euclidean structure.

Here we represent a quaternion ${\displaystyle q}$ by a real part ${\displaystyle q_{r}}$ and a spatial vector part ${\displaystyle {\vec {q}}}$.

${\displaystyle q\,{\overset {\underset {\mathrm {def} }{}}{=}}\,q_{r}+{\vec {q}}}$

(1)

The quaternionic conjugate ${\displaystyle q^{*}}$ is

${\displaystyle q^{*}\,{\overset {\underset {\mathrm {def} }{}}{=}}\,q_{r}-{\vec {q}}}$

(2)

Summation is commutative and associative

${\displaystyle a+b=b+a}$

(3)

${\displaystyle (a+b)+c=a+(b+c)}$

(4)

Multiplication follows from

${\displaystyle a\,b=(a_{r}+{\vec {a}})(b_{r}+{\vec {b}})=a_{r}\,b_{r}-\langle {\vec {a}},{\vec {b}}\rangle +a_{r}\,{\vec {b}}+b_{r}\,{\vec {a}}\ {\color {Red}\pm }\ {\vec {a}}\times {\vec {b}}}$

(5)

${\displaystyle \langle {\vec {a}},{\vec {b}}\rangle }$ is the inner vector product and ${\displaystyle {\vec {a}}\times {\vec {b}}}$ is the external vector product.

${\displaystyle {\color {Red}\pm }}$ indicates that depending on the ordering symmetry, the quaternionic number system exists in right-handed and in left-handed versions.

A right-handed quaternion cannot multiply with a left-handed quaternion.

${\displaystyle (a\,b)^{*}=b^{*}\,a^{*}}$

(6)

The norm ${\displaystyle |q|}$ equals

${\displaystyle |q|={\sqrt {q\,q^{*}}}={\sqrt {q_{r}q_{r}+\langle {\vec {q}},{\vec {q}}\rangle }}}$

(7)

${\displaystyle q^{-1}={\frac {q^{*}}{|q|^{2}}}}$

(8)

Phase

The phase ${\displaystyle q_{\varphi }}$ in radians of quaternion ${\displaystyle q}$ follows from

${\displaystyle q=|q|\exp {\biggl (}q_{\varphi }\,{\frac {\vec {q}}{|{\vec {q}}|}}{\biggr )}}$

(9)

${\displaystyle {\frac {\vec {q}}{|{\vec {q}}|}}}$ is the spatial direction of ${\displaystyle q}$.

Quaternionic rotation

In multiplication, quaternions do not commute. Thus, in general, ${\displaystyle a\,b/a\neq b}$. In this multiplication, the imaginary part of ${\displaystyle b}$ that is perpendicular to the imaginary part of ${\displaystyle a}$ is rotated over an angle that is twice the complex phase ${\displaystyle \varphi }$ of ${\displaystyle a}$.

This graph means that if ${\displaystyle \varphi =\pi /4}$, then the rotation ${\displaystyle a\,b/a}$ shifts ${\displaystyle b_{\perp }}$to another dimension. This fact puts quaternions for which the size of the real part equals the size of the imaginary part in a special category. They can switch states of tri-state systems. In addition, they can switch the color charge of quarks. This means that in such pairs, they behave as gluons.

Reflection

Each quaternion ${\displaystyle c}$ can be written as a product of two complex numbers ${\displaystyle a}$ and ${\displaystyle b}$ of which the imaginary base vectors are perpendicular

${\displaystyle c=(a_{0}+a_{1}{\vec {i}})(b_{0}+b_{1}{\vec {j}})=c_{0}+c_{1}{\vec {i}}+c_{2}{\vec {j}}+c_{3}{\vec {k}}=a_{0}b_{0}+a_{1}b_{0}{\vec {i}}+a_{0}b_{1}{\vec {j}}\pm a_{1}b_{1}{\vec {k}};\ {\vec {i}}{\vec {j}}=\pm {\vec {k}}}$

Rotating with a pair of ${\displaystyle {\vec {k}}}$ vectors will invert quaternion ${\displaystyle c}$ in the direction of ${\displaystyle {\vec {k}}}$.

${\displaystyle {\vec {k}}\ c/{\vec {k}}=c_{0}-c_{1}{\vec {i}}-c_{2}{\vec {j}}+c_{3}{\vec {k}}}$