# Iterant Algebra

## Abstract

**:**

## 1. Introduction

## 2. Iterants

## 3. Iterants and Discrete Processes

**Discrete Calculus and the Temporal Shift Operator.**If we have a discrete time series $X,{X}^{\prime},{X}^{\prime \prime},\cdots ,$ then it is convenient to define an operator J so that ${X}^{t}J=J{X}^{t+1}$, and it is this temporal shift operator that can be used to correlate discrete calculus for the time-series. For example, we can define a discrete derivative D by the equation

## 4. Matrix Algebra via Iterants

**Remark**

**1.**

#### 4.1. Iterants of Arbirtarily High Period and General Matrix Algebras

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Example 1.**

- 1.
- Consider the cyclic group of order three.$${C}_{3}=\{1,S,{S}^{2}\}$$$$\left(\right)open="("\; close=")">\begin{array}{ccc}1& S& {S}^{2}\\ S& {S}^{2}& 1\\ {S}^{2}& 1& S\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{ccc}1& S& {S}^{2}\\ {S}^{2}& 1& S\\ S& {S}^{2}& 1\end{array}$$
- 2.
- Consider the symmetric group on six letters,$${S}_{6}=\{1,R,{R}^{2},F,RF,{R}^{2}F\},$$$$\left(\right)open="("\; close=")">\begin{array}{cccccc}1& R& {R}^{2}& F& RF& {R}^{2}F\\ R& {R}^{2}& 1& RF& {R}^{2}F& F\\ {R}^{2}& 1& R& {R}^{2}F& F& RF\\ F& {R}^{2}F& RF& 1& {R}^{2}& R\\ RF& F& {R}^{2}F& R& 1& {R}^{2}\\ {R}^{2}F& RF& F& {R}^{2}& R& 1\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{cccccc}1& R& {R}^{2}& F& RF& {R}^{2}F\\ {R}^{2}& 1& R& {R}^{2}F& F& RF\\ R& {R}^{2}& 1& RF& {R}^{2}F& F\\ F& {R}^{2}F& RF& 1& {R}^{2}& R\\ RF& F& {R}^{2}F& R& 1& {R}^{2}\\ {R}^{2}F& RF& F& {R}^{2}& R& 1\end{array}$$$$R=\left(\right)open="("\; close=")">\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}$$
- 3.
- Consider the group $G={C}_{2}\times {C}_{2},$ the “Klein 4-Group”. Take $G=\{1,A,B,C\}$ where ${A}^{2}={B}^{2}={C}^{2}=1$, $AB=BA=C.$ G has the multiplication table, which is also its G-Table for $\mathbb{V}ec{t}_{4}(G,\mathbb{F}):$$$\left(\right)open="("\; close=")">\begin{array}{cccc}1& A& B& C\\ A& 1& C& B\\ B& C& 1& A\\ C& B& A& 1\end{array}$$$$\alpha =[1,-1,-1,1],\beta =[1,1,-1,-1],\gamma =[1,-1,1,-1].$$$$I=\alpha A,J=\beta B,K=\gamma C.$$$${I}^{2}={J}^{2}={K}^{2}=IJK=-1,IJ=K,JI=-K.$$
- 4.
- Since complex numbers commute with one another, we could consider iterants whose values are in the complex numbers. This is just like considering matrices whose entries are complex numbers. Thus, we shall allow a version of i that commutes with the iterant shift operator $\eta .$ Let this commuting i be denoted by $\iota .$ Then, we are assuming that$${\iota}^{2}=-1,$$$$\eta \iota =\iota \eta ,$$$${\eta}^{2}=+1.$$$$I=\iota \u03f5,$$$$J=\u03f5\eta ,$$$$K=\iota \eta .$$$${I}^{2}=\iota \u03f5\iota \u03f5=\iota \iota \u03f5\u03f5=(-1)(+1)=-1,$$$${J}^{2}=\u03f5\eta \u03f5\eta =\u03f5(-\u03f5)\eta \eta =-1,$$$${K}^{2}=\iota \eta \iota \eta =\iota \iota \eta \eta =-1,$$$$IJK=\iota \u03f5\u03f5\eta \iota \eta =\iota 1\iota \eta \eta =\iota \iota =-1.$$$${I}^{2}={J}^{2}={K}^{2}=IJK=-1.$$
- 5.
- Similarly,$$H=[a,b]+[c+d\iota ,c-d\iota ]\eta =\left(\right)open="("\; close=")">\begin{array}{cc}a& c+d\iota \\ c-d\iota & b\end{array}$$If in the above Hermitian matrix form, we take $a=T+X,b=T-X,c=Y,d=Z,$ then we obtain an iterant and/or matrix representation for a point in Minkowski spacetime:$$H=[T+X,T-X]+[Y+Z\iota ,Y-Z\iota ]\eta =\left(\right)open="("\; close=")">\begin{array}{cc}T+X& Y+Z\iota \\ Y-Z\iota & T-X\end{array}$$$$Det(H)={T}^{2}-{X}^{2}-{Y}^{2}-{Z}^{2}.$$
- 6.
- Hamilton’s Quaternions are generated by iterants, as discussed above, and we can express them purely algebraicially by writing the corresponding permutations as shown below:$$I=[+1,-1,-1,+1]s,$$$$J=[+1,+1,-1,-1]l,$$$$K=[+1,-1,+1,-1]t,$$$$s=(12)(34),$$$$l=(13)(24),$$$$t=(14)(23).$$One can verify that$${I}^{2}={J}^{2}={K}^{2}=IJK=-1.$$
- 7.
- In all these examples, we can interpret the iterants as short hand for matrix algebra based on permutation matrices, or as indicators of discrete processes. The discrete processes become more complex in proportion to the complexity of the groups used in the construction. We began with processes of order two, then considered cyclic groups of arbitrary order, then the symmetric group ${S}_{3}$ in relation to $6\times 6$ matrices, and the Klein 4-Group in relation to the quaternions. In the case of the quaternions, we know that this structure is intimately related to rotations of three- and four-dimensional space and many other geometric themes.

## 5. Schrödinger’s Equation

#### 5.1. Brownian Walks and the Diffusion Equation

#### 5.2. An Iterant Intepretation of Schrödinger’s Equation

**Replacing i by An Iterant.**Now, however, suppose that we replace i by ${(-1)}^{n(t)}$ at time step $t=n(t)\tau $ where $n(t)$ is a non-negative integer. Instead of writing

**Remark**

**3.**

## 6. The Framed Braid Group and the Sundance Bilson-Thompson Model for Elementary Particles

## 7. Iterants and the Standard Model

**Iterant Formulation of the $su(3)$ Lie Algebra.**We now have the specific iterant formulas

## 8. Iterants, Braiding and the Sundance Bilson-Thompson Model for Fermions

## 9. Clifford Algebra, Majorana Fermions and Braiding

## 10. The Dirac Equation and Majorana Fermions

**A Plane Wave Solution to the Dirac Equation.**Note that

#### 10.1. U and ${U}^{\u2020}$ as Creation and Annihilation Operators

#### 10.2. Iterant Formulation of the Dirac Equation

#### 10.3. Writing in the Full Dirac Algebra

#### 10.4. Majorana Fermions in the Sense of Majorana

**A solution to the Majorana–Dirac Equation.**Let $\rho (x,t)={e}^{(p\u2022x-Et)}.$ Note that $\rho $ is a a real-valued function. Let

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Sundance Bilson-Thompson Framed Braid Fermions (“(3)” under the labels for the up and down quarks and antiquarks represent the fact that there are three permutations of charge placement giving the three colours).

**Figure 6.**Representation of $\mu \to {\nu}_{\mu}+{W}_{-}\to {\nu}_{\mu}+\overline{{\nu}_{e}}+{e}^{-}.$

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Kauffman, L.H.
Iterant Algebra. *Entropy* **2017**, *19*, 347.
https://doi.org/10.3390/e19070347

**AMA Style**

Kauffman LH.
Iterant Algebra. *Entropy*. 2017; 19(7):347.
https://doi.org/10.3390/e19070347

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Kauffman, Louis H.
2017. "Iterant Algebra" *Entropy* 19, no. 7: 347.
https://doi.org/10.3390/e19070347