# Emergent Space-Time in a Bubble Universe

## Abstract

**:**

## 1. Introduction

#### 1.1. Quantum Theory as a Black Box Model of Reality

**A**, closed in the ultraweak operator topology, with identity, called a W*-algebra [3] since the representation theory of

**A**is then unbiased. I can then interpret any quantum theory (including quantum gravity) as follows: a quantum state $\rho $ is a positive element of the dual space

**A***of

**A**with $\rho \left(I\right)=1$. A representation $\pi $ is a homomorphism from

**A**to a von Neumann subalgebra of the set of all bounded linear operators $\mathcal{B}\left(\mathcal{H}\right)$ acting on a Hilbert space $\mathcal{H}$. The Hilbert space is either finite dimensional or separable. Each quantum state $\rho $ gives rise to a representation in which it is equivalent to a Dirac ket |x> of the Hilbert space with norm one; this is the Gelfand-Naimark-Segal (GNS) construction [1]. An irreducible GNS representation corresponds to $\rho $ being a pure state. Since the state space is a weak*-convex, compact subset of the dual space, it is the closed convex hull of its extreme points These extreme points are the pure states. A finite system observable, such as the total system energy, corresponds to a bounded linear operator acting on this Hilbert space. The subset of states continuous for the ultraweak operator topology on

**A**is the predual ${A}_{*}$. The Gelfand transform mapping applied to an abstract W*-algebra

**A**shows that

**A**is (isomorphic to) the dual space of the linear hull of its predual, whch turns out to be a Banach space. More formally, we have

- An abstract W*-algebra
**A**is a C*-algebra which is (isomorphic to) the dual space of a Banach space [3]; - Observations/measurements of a quantum system such as the total energy in a GNS representation correspond to discrete eigenvalues of the corresponding matrix operator. This implies that the set of observables corresponds to the subset of self-adjoint operators (with real eigenvalues).

#### 1.2. The Big Bang: Everything from Nothing?

#### 1.3. Gravity

## 2. Computational Spin Networks

## 3. The Homology Invariants of Space-Time

#### Homology Invariants of the Network Structure

**X**defined as a finite quiver consisting of a directed graph with a set of nodes n corresponding to entangled spin inputs and directed links between these nodes, with an upper bound of $\frac{n\left(n-1\right)}{2}$ links. Percolation theory [16] applied to this structure implies a critical value of pr denoted ${p}_{c}$ for which the network becomes path connected. The value pr = 1 corresponds to a fully connected graph.

_{c}< pr ≤ 1 we have a range of units of tessellation, including 3-simplexes (tetrahedra) and potentially higher dimensional generalisations. We can use these basic properties to develop the structures, including simplicial complexes, of an Eilenberg singular homology [17,18] associated with a given computational spin network. I cannot in principle resolve the embedding space into minimal subunits of space-time smaller than triangles (2-simplexes) with finite 1- dimensional edges. I define such a topological space as a discrete embedding space. This is consistent with increments of space and time which can be traded off against each other as part of a more fundamental unit; the space-time vector forming the 1-dimensional boundaries of a triangular 2- simplex.

**a**(j) each with varying direction within their local forward light cones, such that the path begins at

**x(0)**and ends at

**x(1)**, with $T\left(a\left(j\right)\right):x\to x+a\left(j\right)$ elements of the relativistic translation subgroup

**T**. The total path is then generated by the ordered product ${\prod}_{j=1}^{j=n}T\left(a\left(j\right)\right)\mathit{x}\left(\mathbf{0}\right)\mathrm{with}\mathrm{the}\mathrm{final}\mathrm{end}\mathrm{point}\mathbf{x}\left(1\right)=\mathit{x}\left(\mathbf{0}\right)+{\sum}_{j=1}^{j=n}a\left(j\right)$. For a fixed initial point

**x(0)**we can identify this path with the finite group product ${\prod}_{j=1}^{j=n}T\left(a\left(j\right)\right)\in \mathit{T}$. If all the increments point in the same direction, and the local light cones have the same orientation, one to another, this gives a straight path in space-time. If, however, the local light cones are rotated one to another, then the path will be piecewise curved. The choice of settings for the light cones constrains the trajectory of the resultant path in the same way that the gravitational field in Einstein’s theory constrains a geodesic path between two space-time events to be of the form

## 4. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Moffat, J.
Emergent Space-Time in a Bubble Universe. *Symmetry* **2021**, *13*, 729.
https://doi.org/10.3390/sym13040729

**AMA Style**

Moffat J.
Emergent Space-Time in a Bubble Universe. *Symmetry*. 2021; 13(4):729.
https://doi.org/10.3390/sym13040729

**Chicago/Turabian Style**

Moffat, James.
2021. "Emergent Space-Time in a Bubble Universe" *Symmetry* 13, no. 4: 729.
https://doi.org/10.3390/sym13040729