# The Geometry of Noncommutative Spacetimes

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Noncommutative Spacetime—An Operational Approach

## 3. Noncommutative Geometry à la Connes

**Definition**

**1.**

- $\mathcal{A}$—a dense ${}^{*}$-subalgebra of a ${C}^{*}$-algebra $\overline{\mathcal{A}}$;
- $\mathcal{H}$—a separable Hilbert space with a faithful representation $\pi :\mathcal{A}\to \mathcal{B}(\mathcal{H})$ via bounded operators;
- $\mathcal{D}$—an unbounded self-adjoint operator on $\mathcal{H}$ with a compact resolvent.

**Example**

**1.**

- ${\mathcal{A}}_{\mathcal{M}}={C}^{\infty}(\mathcal{M})$—the algebra of smooth functions on $\mathcal{M}$,
- ${\mathcal{H}}_{\mathcal{M}}={L}^{2}(\mathcal{M},S)$—the space of square summable sections of the spinor bundle S over $\mathcal{M}$,
- $\mathcal{D}\phantom{\rule{-6.94443pt}{0ex}}/\phantom{\rule{0.166667em}{0ex}}=-i{\gamma}^{\mu}{\nabla}_{\mu}^{S}$—the (curved) Dirac operator associated with S,

**Example**

**2.**

**Example**

**3.**

## 4. Causality in Noncommutative Spacetimes

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

## 5. The Foundations of Quantum Field Theory Revisited

## Acknowledgments

## Conflicts of Interest

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^{1.}In general, $\mathcal{A}$ does not need to contain a unit. However, it always contains an approximate unit, which can be utilised to rigorously express the normalisation requirement [29].^{3.}A concrete model of a noncommutative spacetime with nonlocal events, but a rigid causal structure was developed in [63].^{4.}Such a viewpoint leads to the so-called ‘zigzag picture of the electron’ [71, Section 25.2].

**Figure 1.**The causal structure of a two-sheeted 2-dim Minkowski spacetime. The shaded region illustrates the causal future of the event $(p,-)$.

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