# Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras

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## Abstract

**:**

## 1. Introduction

- there is no classical observable to be quantized: one has to think directly in terms of quantum objects and states of the system;
- there is no spacetime geometry to start with.

## 2. The Lie Algebra ${\mathfrak{g}}_{\mathsf{u}}$

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

## 3. The Lie Superalgebra ${\mathfrak{sg}}_{\mathsf{u}}$

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. Interaction Graphs

## 5. The Poincaré Group

**Proposition**

**2.**

- If ${p}^{2}<0$, fix a transformation ${\Lambda}_{p}$ such that ${\Lambda}_{p}(m,0,0,0)=p$ and let $W(\Lambda ,p):={\Lambda}_{\Lambda p}^{-1}\Lambda {\Lambda}_{p}$ be the Wigner rotation induced by Λ$$\mathcal{P}\left({x}_{\alpha +p}\right)={e}^{ia\xb7\Lambda p}{e}^{\mathrm{ad}\left(R\right)}{x}_{\alpha +\Lambda p}\phantom{\rule{4pt}{0ex}},\mathcal{P}\left({x}_{p}^{\alpha}\right)=\phantom{\rule{4pt}{0ex}}{e}^{ia\xb7\Lambda p}{e}^{\mathrm{ad}\left(R\right)}{x}_{\Lambda p}^{\alpha}$$
- If ${p}^{2}=0$ and ${w}^{2}=0$ the action reduces to$$\mathcal{P}\left({x}_{\alpha +p}\right)={e}^{ia\xb7\Lambda p}{e}^{i\theta (\Lambda )\lambda}{x}_{\alpha +\Lambda p}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\mathcal{P}\left({x}_{p}^{\alpha}\right)={e}^{ia\xb7\Lambda p}{e}^{i\theta (\Lambda )\lambda}{x}_{\Lambda p}^{\alpha}$$where $\lambda =0,\pm \frac{1}{2},\pm 1$ is the helicity of α and θ is the angle of the $SO\left(2\right)$ rotation along the direction of $\overrightarrow{p}$, analogous to the Wigner rotation of the massive case.

**Proof.**

## 6. Initial Quantum State

## 7. Vertex-Type Algebra and Gravitahedra

## 8. Conclusions

- (I)
- spacetime is the outcome of the interactions driven by an infinite-dimensional Lie superalgebra ${\mathfrak{sg}}_{\mathsf{u}}$; it is discrete, finite and expanding;
- (II)
- the algebra ${\mathfrak{sg}}_{\mathsf{u}}$ incorporates 4-momentum and charge conservation; it involves fermions and bosons, with fermions fulfilling the Pauli exclusion principle;
- (III)
- ${\mathfrak{sg}}_{\mathsf{u}}$ is a Lie superalgebra without any supersymmetry forcing the existence of superpartners for the particles of the Standard Model;
- (IV)
- every particle has positive energy and it is either timelike or lightlike;
- (V)
- the initial state is an element of the universal enveloping algebra of ${\mathfrak{sg}}_{\mathsf{u}}$;
- (VI)
- the interactions are local, and the whole algebraic structure is a vertex-type algebra, due to a mechanism for the expansion of space (in fact, an expansion of matter and radiation);
- (VII)
- the emerging spacetime inherits the quantum nature of the interactions, hence Quantum Gravity is an expression for quantum spacetime—in particular, there is no spin-2 particle;
- (VIII)
- the Poincaré group has a natural action on the local algebra;
- (IX)
- once an initial state is fixed, the model can be viewed as an algorithm for explicit computer calculations of physical quantities, like scattering amplitudes, density matrix, partition function, mean energy, von Neumann entropy, etc.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proposition**

**A1.**

**Proposition**

**A2.**

- (a)
- At least one of $X,Y,Z$ is of type-0
- (a1)
- If $X,Y,Z$ are all of the type-0 then Jacobi holds trivially.
- (a2)
- If $X={x}_{{p}_{1}}^{\alpha},Y={x}_{{p}_{2}}^{\beta}$ are of type 0 and $Z={x}_{\gamma +{p}_{3}}$ is of type 1 then ${J}_{1}=0$, ${J}_{2}=(\alpha ,\gamma )(\beta ,\gamma ){x}_{\gamma +{p}_{1}+{p}_{2}+{p}_{3}}$ and ${J}_{3}=-(\alpha ,\gamma )(\beta ,\gamma ){x}_{\gamma +{p}_{1}+{p}_{2}+{p}_{3}}$, hence $J=0$.
- (a3)
- If $X={x}_{{p}_{1}}^{\alpha}$ is of type 0 and $Y={x}_{\beta +{p}_{2}},Z={x}_{\gamma +{p}_{3}}$ are of type 1, then ${J}_{1}=(\alpha ,\beta )[{x}_{\beta +{p}_{1}+{p}_{2}},{x}_{\gamma +{p}_{3}}]$, ${J}_{2}=(\alpha ,\gamma )[{x}_{\beta +{p}_{2}},{x}_{\gamma +{p}_{1}+{p}_{3}}]$ and ${J}_{3}=-[{x}_{{p}_{1}}^{\alpha},[{x}_{\beta +{p}_{2}}$, ${x}_{\gamma +{p}_{3}}\left]\right]$. We have 3 cases:
- (a3.i)
- $\beta +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ then ${J}_{1}={J}_{2}={J}_{3}=0$;
- (a3.ii)
- $\beta +\gamma \in {\Phi}_{8}$ then ${J}_{3}=-(\alpha ,\beta +\gamma )\u03f5(\beta ,\gamma ){x}_{\beta +\gamma +{p}_{1}+{p}_{2}+{p}_{3}}=-({J}_{1}+{J}_{2})$;
- (a3.iii)
- $\beta +\gamma =0$ then ${J}_{1}=-(\alpha ,\beta ){x}_{{p}_{1}+{p}_{2}+{p}_{3}}^{\beta}$, ${J}_{2}=(\alpha ,\beta ){x}_{{p}_{1}+{p}_{2}+{p}_{3}}^{\beta}$ and ${J}_{3}=0$, hence $J=0$.

- (b)
- None of $X,Y,Z$ is of type-0. Let $X={x}_{\alpha +{p}_{1}},Y={x}_{\beta +{p}_{2}},Z={x}_{\gamma +{p}_{3}}$ be all of type 1. For any two roots of ${\Phi}_{8}$, say $\alpha ,\beta $ without loss of generality, we have three cases:
- (b1)
- $\alpha +\beta \notin {\Phi}_{8}\cup \left\{0\right\}$:
- (b1.i)
- if both $\alpha +\gamma ,\beta +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ then $J=0$ trivially;
- (b1.ii)
- if $\beta +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ and $\alpha +\gamma \in {\Phi}_{8}\cup \left\{0\right\}$ then ${J}_{1}={J}_{3}=0$. Since both $(\alpha ,\beta ),(\beta ,\gamma )\in \{0,1,2\}$ then $(\alpha +\gamma ,\beta )\ge 0$ hence if $\alpha +\gamma \in {\Phi}_{8}$, then $\alpha +\beta +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ and ${J}_{2}=0$. On the other hand if $\alpha =-\gamma $ then ${J}_{2}=[{x}_{{p}_{1}+{p}_{3}}^{\alpha},{x}_{\beta +{p}_{2}}]=(\alpha ,\beta ){x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}$. But $(\beta ,\gamma )=-(\beta ,\alpha )$ and $(\alpha ,\beta ),(\beta ,\gamma )\in \{0,1,2\}$ imply $(\alpha ,\beta )=0$ hence $J=0$;
- (b1.iii)
- if $\beta +\gamma \in {\Phi}_{8}$ and $\alpha +\gamma \in {\Phi}_{8}$ then ${J}_{2}=\u03f5(\gamma ,\alpha )[{x}_{\alpha +\gamma +{p}_{1}+{p}_{3}},{x}_{\beta +{p}_{2}}]$ and ${J}_{3}=\u03f5(\beta ,\gamma )[{x}_{\beta +\gamma +{p}_{2}+{p}_{3}},{x}_{\alpha +{p}_{1}}]$. If $\alpha +\beta +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ then ${J}_{2}={J}_{3}=0$ hence $J=0$. If $\alpha +\beta +\gamma \in {\Phi}_{8}$ then ${J}_{2}+{J}_{3}=\u03f5\left(\gamma ,\alpha )\left(\u03f5\right(\gamma ,\beta )\u03f5\right(\alpha ,\beta )+\u03f5\left(\beta ,\gamma )\u03f5\right(\beta ,\alpha \left)\right){x}_{\alpha +\beta +\gamma +{p}_{1}+{p}_{2}+{p}_{3}}$. Since $2=(\alpha +\beta +\gamma ,\alpha +\beta +\gamma )=6+2(\alpha ,\beta )+2(\beta ,\gamma )+2(\alpha ,\gamma )=2+2(\alpha ,\beta )$, we get $(\alpha ,\beta )=0$ and, from Proposition A2, $\u03f5(\alpha ,\beta )=\u03f5(\beta ,\alpha )$ and $\u03f5(\gamma ,\beta )=-\u03f5(\beta ,\gamma )$, implying ${J}_{2}+{J}_{3}=0$ and $J=0$. Finally if $\alpha +\beta +\gamma =0$ then $(\alpha ,\beta )=(\alpha ,-\alpha -\gamma )=-2+1=-1$ and $\alpha +\beta $ would be a root, contradicting the hypothesis.
- (b1.iv)
- if $\beta +\gamma \in {\Phi}_{8}$ and $\alpha +\gamma =0$ then ${J}_{2}=(\alpha ,\beta ){x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}$ and ${J}_{3}=\u03f5(\beta ,\alpha \left)\u03f5\right(\beta -\alpha ,\alpha ){x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}=-{x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}$. But $(\alpha ,\beta )=-(\gamma ,\beta )=1$ hence ${J}_{2}+{J}_{3}=0$ and $J=0$.
- (b1.v)
- If $\beta +\gamma \in {\Phi}_{8}$ and $\alpha +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ then ${J}_{2}=0$ and $(\beta ,\alpha )\ge 0$, $(\gamma ,\alpha )\ge 0$ imply $(\beta +\gamma ,\alpha )\ge 0$ hence $\beta +\gamma +\alpha \notin {\Phi}_{8}\cup \left\{0\right\}$ therefore ${J}_{3}=\u03f5(\beta ,\gamma )[{x}_{\beta +\gamma +{p}_{2}+{p}_{3}},{x}_{\alpha +{p}_{1}}]=0$ and $J=0$.
- (b1.vi)
- If $\beta +\gamma =0$ and $\alpha +\gamma \in {\Phi}_{8}$ then${J}_{2}=-\u03f5(\alpha ,\beta \left)\u03f5\right(\alpha -\beta ,\beta ){x}_{\alpha +{p}_{1}+{p}_{2}+{p}_{3}}={x}_{\alpha +{p}_{1}+{p}_{2}+{p}_{3}}$ and${J}_{3}=-[{x}_{{p}_{2}+{p}_{3}}^{\beta},{x}_{\alpha +{p}_{1}}]=-{x}_{\alpha +{p}_{1}+{p}_{2}+{p}_{3}}$, being $(\alpha ,\beta )=-(\alpha ,\gamma )=1$, implying $J=0$.
- (b1.vii)
- If $\beta +\gamma =0$ and $\alpha +\gamma =0$ then ${J}_{2}=[{x}_{{p}_{1}+{p}_{3}}^{\beta},{x}_{\beta +{p}_{2}}]=2{x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}$ and ${J}_{3}=-[{x}_{{p}_{2}+{p}_{3}}^{\beta},{x}_{\beta +{p}_{1}}]=-2{x}_{\beta +{p}_{1}+{p}_{2}+{p}_{3}}$ and $J=0$.
- (b1.viii)
- If $\beta +\gamma =0$ and $\alpha +\gamma \notin {\Phi}_{8}\cup \left\{0\right\}$ then ${J}_{2}=0$; $(\alpha ,\beta )=0$ since $(\alpha ,\beta )\ge 0$ and $(\alpha ,\gamma )=-(\alpha ,\beta )\ge 0$, therefore ${J}_{3}=-[{x}_{{p}_{2}+{p}_{3}}^{\beta},{x}_{\alpha +{p}_{1}}]=0$ and $J=0$.From now on $\alpha +\beta ,\alpha +\gamma ,\beta +\gamma \in {\Phi}_{8}\cup \left\{0\right\}$.

- (b2)
- $\alpha +\beta \in {\Phi}_{8}$:
- (b2.i)
- If $\alpha +\gamma ,\beta +\gamma \in {\Phi}_{8}$ then $(\alpha +\beta +\gamma ,\alpha +\beta +\gamma )=0$ hence $\alpha +\beta +\gamma =0$. Then ${J}_{1}=-\u03f5(\alpha ,\beta ){x}_{{p}_{1}+{p}_{2}+{p}_{3}}^{\alpha +\beta}$, ${J}_{2}=\u03f5(\alpha +\beta ,\alpha ){x}_{{p}_{1}+{p}_{2}+{p}_{3}}^{\beta}$, ${J}_{3}=\u03f5(\beta ,\alpha +\beta ){x}_{{p}_{1}+{p}_{2}+{p}_{3}}^{\alpha}$. Since $\u03f5\left(\alpha +\beta ,\alpha )=\u03f5\right(\beta ,\alpha +\beta )=\u03f5(\alpha ,\beta )$, ${x}_{p}^{\alpha +\beta}={x}_{p}^{\alpha}+{x}_{p}^{\beta}$, see (13), we get $J=0$.
- (b2.ii)
- If $\alpha +\gamma \in {\Phi}_{8}$ and $\beta +\gamma =0$ then $\alpha -\beta \in {\Phi}_{8}$ which is impossible.
- (b2.iii)
- If $\alpha +\gamma =0$ and $\beta +\gamma \in {\Phi}_{8}$ then $\beta -\alpha \in {\Phi}_{8}$ which is impossible.
- (b2.iv)
- If $\alpha +\gamma =0$ and $\beta +\gamma =0$ then $\alpha =\beta $ which is impossible.

- (b3)
- $\alpha +\beta =0$:
- (b3.i)
- If $\alpha +\gamma \in {\Phi}_{8}$ then $\beta +\gamma =-\alpha +\gamma \notin {\Phi}_{8}$; we can only have $\beta +\gamma =0$ implying $\alpha =\gamma $, that contradicts $\alpha +\gamma \in {\Phi}_{8}$.
- (b3.ii)
- If $\beta +\gamma \in {\Phi}_{8}$ then $\alpha +\gamma =-\beta +\gamma \notin {\Phi}_{8}$; we can only have $\alpha +\gamma =0$ implying $-\alpha =\beta =\gamma $ that contradicts $\beta +\gamma \in {\Phi}_{8}$.
- (b3.iii)
- If both $\alpha +\gamma =0$ and $\beta +\gamma =0$ then $\alpha =\beta $ which contradicts $\alpha +\beta =0$.

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**Figure 1.**$[{x}_{{p}_{1}}^{\alpha},{x}_{\beta +{p}_{2}}]=(\alpha ,\beta ){x}_{\beta +{p}_{1}+{p}_{2}}$; (

**a**): ${x}^{\alpha}$ absorption by ${x}_{\beta}$; (

**b**): ${x}^{\alpha}$ emission by ${x}_{\beta}$ (similarly for ${x}_{{p}_{1}}^{\alpha}$ and ${x}_{\beta +{p}_{2}}$ interchanged).

**Figure 2.**$[{x}_{\alpha +{p}_{1}},{x}_{-\alpha +{p}_{2}}]=-{x}_{{p}_{1}+{p}_{2}}^{\alpha}$; (

**a**): ${x}_{\alpha}-{x}_{-\alpha}$ annihilation; (

**b**): pair creation (similarly for ${x}_{\alpha +{p}_{1}}$ and ${x}_{-\alpha +{p}_{2}}$ interchanged).

**Figure 3.**$[{x}_{\alpha +{p}_{1}},{x}_{\beta +{p}_{2}}]=\epsilon (\alpha ,\beta ){x}_{\alpha +\beta +{p}_{1}+{p}_{2}}$; (

**a**): ${x}_{\alpha}-{x}_{\beta}$ scattering; (

**b**): ${x}_{\alpha +\beta}$ decay into ${x}_{\alpha}$ and ${x}_{\beta}$ (similarly for ${x}_{\alpha +{p}_{1}}$ and ${x}_{\beta +{p}_{2}}$ interchanged).

**Figure 4.**Associahedron ${\mathbb{K}}_{4}$. Adjacent vertices $\left(st\right)u\to s\left(tu\right)$, for sub-words $s,t,u$.

**Figure 5.**(

**a**): whose; (

**b**): renders a magnification, shows the permutahedron ${\mathbb{P}}_{3}$, pertaining to the interaction of 4 particles.

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Truini, P.; Marrani, A.; Rios, M.; Irwin, K.
Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras. *Symmetry* **2021**, *13*, 2289.
https://doi.org/10.3390/sym13122289

**AMA Style**

Truini P, Marrani A, Rios M, Irwin K.
Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras. *Symmetry*. 2021; 13(12):2289.
https://doi.org/10.3390/sym13122289

**Chicago/Turabian Style**

Truini, Piero, Alessio Marrani, Michael Rios, and Klee Irwin.
2021. "Space, Matter and Interactions in a Quantum Early Universe. Part II: Superalgebras and Vertex Algebras" *Symmetry* 13, no. 12: 2289.
https://doi.org/10.3390/sym13122289