# Non-Commutative Worlds and Classical Constraints

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## Abstract

**:**

## 1. Introduction

## 2. Time Series and Discrete Physics

**We adjust the discrete derivative.**We shall add an operator J that in this context accomplishes the time shift:

**Discrete observations do not commute.**Let A and B denote quantities that we wish to observe in the discrete system. Let $AB$ denote the result of first observing B and then observing $A.$ The result of this definition is that a successive observation of the form $x\left(Dx\right)$ is distinct from an observation of the form $\left(Dx\right)x.$ In the first case, we first observe the velocity at time t, and then x is measured at $t+\Delta t$. In the second case, we measure x at t and then measure the velocity.

## 3. Review of Non-Commutative Worlds

**Discrete Derivatives are Replaced by Commutators.**There is a lot of motivation for replacing derivatives by commutators. If $f\left(x\right)$ denotes (say) a function of a real variable $x,$ and $\tilde{f}\left(x\right)=f(x+h)$ for a fixed increment $h,$ define the discrete derivative $Df$ by the formula $Df=(\tilde{f}-f)/h,$ and find that the Leibniz rule is not satisfied. One has the basic formula for the discrete derivative of a product:

**Advanced Calculus and Hamiltonian Mechanics or Quantum Mechanics in a Non-Commutative World.**In $\mathcal{A}$, there are as many derivations as there are elements of the algebra, and these derivations behave quite wildly with respect to one another. If one takes the concept of curvature as the non-commutation of derivations, then $\mathcal{A}$ is a highly curved world indeed. Within $\mathcal{A}$, one can build a tame world of derivations that mimics the behaviour of flat coordinates in Euclidean space. The description of the structure of $\mathcal{A}$ with respect to these flat coordinates contains many of the equations and patterns of mathematical physics.

**Hamilton’s Equations are Part of the Mathematical Structure of Non-Commutative Advanced Calculus.**

**The Simplest Time Series Leads to the Diffusion Constant and Heisenberg’s Commuator.**Consider a time series $\{Q,{Q}^{\prime},{Q}^{\u2033},\phantom{\rule{0.166667em}{0ex}}\cdots \}$ with commuting scalar values. Let

- Let $\dot{Q}Q$ denote the sequence: observe Q, then obtain $\dot{Q}.$
- Let $Q\dot{Q}$ denote the sequence: obtain $\dot{Q}$, then observe $Q.$

**Schroedinger’s Equation is Discrete.**Here is how the Heisenberg form of Schroedinger’s equation fits in this context. Let $J=(1-\frac{i}{\hslash}H\Delta t).$ Then, $\nabla \psi =[\psi ,J/\Delta t],$ and we calculate

**Dynamical Equations Generalize Gauge Theory and Curvature.**One can take the general dynamical equation in the form

**Non-Commutative Electromagnetism and Gauge Theory.**One can use this context to revisit the Feynman-Dyson derivation [16,17] of electromagnetism from commutator equations, showing that most of the derivation is independent of any choice of commutators, but highly dependent upon the choice of definitions of the derivatives involved. Without any assumptions about initial commutator equations, but taking the right (in some sense simplest) definitions of the derivatives one obtains a significant generalization of the result of Feynman-Dyson. We give this derivation in [18] and in [13,14,15] using diagrammatic algebra to clarify the structure. In this section, we use X to denote the position vector rather than $Q,$ as above, and the partial derivatives $\{{\partial}_{1},{\partial}_{2},{\partial}_{3}\}$ are each covariant derivatives represented by commutators with $\dot{{X}_{1}},\dot{{X}_{2}},\dot{{X}_{2}}$, respectively.

**Theorem**

**1.**

- $\ddot{X}=E+\dot{X}\times B,$
- $\nabla \u2022B=0,$
- ${\partial}_{t}B+\nabla \times E=B\times B,$
- ${\partial}_{t}E-\nabla \times B=({\partial}_{t}^{2}-{\nabla}^{2})\dot{X}.$

**Remark**

**1.**

## 4. Constraints—Classical Physics and General Relativity

**The First Constraint.**The first constraint is the equation

**The Second Constraint.**The second constraint is the symmetrized analog of the second temporal derivative:

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**A Relationship with General Relativity.**We choose a non-commutative metric representative ${g}^{ij}$ in the non-commutative world with an inverse ${g}_{ij}$ so that ${g}^{ij}={g}^{ji},{g}_{ij}={g}_{ji},$ and ${g}^{ik}{g}_{kj}={\delta}_{j}^{i}.$ We can use the quadratic Hamiltonian $H=\frac{1}{4}({g}^{ij}{P}^{i}{P}^{j}+{P}^{i}{P}^{j}{g}^{ij})$ as previously discussed, but we simplify the calculations below by taking $H=\frac{1}{2}\left({g}^{ij}{P}^{i}{P}^{j}\right).$ No essential difference ensues in the results. We assume that the ${g}^{ij}$ commute with the coordinate representatives ${Q}^{k}$ so that $[{g}^{ij},{Q}^{k}]=0$ for all choices of $i,j,k$ and similarly for the ${g}_{ij}.$ We take ${P}^{i}$ and ${Q}^{j}$ as described at the beginning of this section. It is then an easy calculation to verify that

**Lemma**

**1.**

**Proof.**

**Remark**

**3.**

## 5. The Kilmister Equation

- ${R}_{abcd}={R}_{cdab}={R}_{dcba},$
- ${R}_{abcd}=-{R}_{bacd}=-{R}_{abdc}.$

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Definition**

**1.**

**Remark**

**4.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**5.**

**Theorem**

**4.**

## 6. On the Algebra of Constraints

#### Algebra of Constraints

- $\dot{\theta}=\theta h,$
- $\ddot{\theta}=\theta {h}^{2}+\theta \dot{h}.$

- $\left\{\dot{\Theta}\right\}=\{\Theta H\}=\frac{1}{2}(\Theta H+H\Theta ),$
- $\left\{\ddot{\Theta}\right\}=\{\Theta {H}^{2}\}+\{\Theta \dot{H}\}=\frac{1}{3}(\Theta {H}^{2}+H\Theta H+{H}^{2}\Theta )+\frac{1}{2}(\Theta \dot{H}+\dot{H}\Theta ).$

**Proposition**

**1.**

**Proof.**

**The Third Constraint.**We now go on to an analysis of the third constraint. The third constraint consists in the two equations

- $\left\{\stackrel{\u20db}{\Theta}\right\}=\{\Theta {H}^{3}\}+3\{\Theta H\dot{H}\}+\{\Theta \ddot{H}\},$
- $\left\{\stackrel{\u20db}{\Theta}\right\}={\left\{\ddot{\Theta}\right\}}^{\u2022}$, where$${\left\{\ddot{\Theta}\right\}}^{\u2022}=\left\{\{\Theta H\}{H}^{2}\right\}+2\{\Theta H\dot{H}\}+\left\{\{\Theta H\}\dot{H}\right\}+\{\Theta \ddot{H}\}.$$

**Proposition**

**2.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Einstein’s Equations and the Bianchi Identity

**Bianchi Identity and Jacobi Identity.**Now lets turn to the context of non-commutative worlds. We have infinitely many possible convariant derivatives, all of the form

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