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

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra $\textbf{${\mathfrak g}_{\mathsf u}$}$ that extends $\mathbf{e_9}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn $\textbf{${\mathfrak g}_{\mathsf u}$}$ into a Lie superalgebra $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincar\'e group on $\textbf{$\mathfrak {sg}_{\mathsf u}$}$, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular {\it resonant scattering}. Finally, we complete the model by merging the local $\textbf{$\mathfrak {sg}_{\mathsf u}$}$ into a vertex-type algebra.


Introduction
This is the second of two papers -see also [1] -describing an algebraic model of quantum gravity. In the first paper we have described the basic principles of our model and we have investigated the mathematical structures that may suit our purpose. In particular, we have focused on rank-12 infinite dimensional Kac-Moody, [2], and Borcherds algebras, [3] [4], and we have given physical and mathematical reasons why the latter are preferable.
In our model for the expansion of quantum early Universe, [1] [5], the need for an infinite dimensional Lie algebra stems from the unlimited number of possible 4-momenta, but at each fixed cosmological time the number of generators and roots involved is finite. There is a known algorithm of Lie algebra theory that allows to determine the structure constants among a finite number of generators of a Borcherds algebra [3] [4]. Let us grade the commutators by levels, by saying that the commutators involving n simple roots have level n − 1. A consistent set of structure constants is calculable level by level, and once the structure constants are calculated at level n, they will not be affected by the calculation at any level m > n. There are computer programs that apply this algorithm and give the explicit structure constants level by level, see for instance the package LieRing of GAP, developed by S. Cicalò and W. A. de Graaf [6].
However, for the sake of simplicity, in [1] we have chosen to deal with a simpler Lie algebra, g u , that extends e 8 and e 9 . In the present paper, we will start invertigating a physical model for quantum gravity based on this particular rank-12 algebra g u . We will start by focussing on local aspects of the algebraic model: in Sec. 2, we recall g u , which is then turned into a Lie superalgebra sg u in Sec. 3. Sec. 4 will then discuss interactions, scattering processes and decays, whereas the role of the Poincaré group is analyzed in Sec. 5. Finally, in Secs. 6 and 7 we will define the quantum states, and then we will merge the algebra sg u into a vertex-type algebra, representing the quantum early Universe with its expanding spacetime. 2 The Lie algebra g u We start and consider B + , the Lie subalgebra of the rank-12 Borcherds algebra B 12 introduced in [1] and generated by the Chevalley generators corresponding to positive roots. A further simplification will then give rise to g u , the Lie algebra that acts locally on the quantum state of the Universe [1].
Remark 2.1. Notice that the mass of a particle cannot be arbitrary small, since there is a lower limit, m ≥ 1.
Hence, we extend the possible values of the 4-momentum p := (E, p) by including those with opposite 3-momentump = (E, − p), as explained in [1], so that p = (E p , p x , p y , p z ) , E p ∈ N , p x , p y , p z ∈ Z , p 2 ≤ 0. (2.9) The algebra g u extends the 1+1-dimensional toy model based on e 9 discussed in [1]; it is defined as the algebra generated by x α p and x α+p , such that p 2 ≤ 0, satisfying the following commutation relations: where (·, ·) is the Euclidean scalar product in R 8 , the function ε : is the asymmetry function [2,7,1], and 11) in order to have an antisymmetric algebra. Moreover, for consistency, we require that Remark 2.2. Notice also that p 2 1 , p 2 2 < 0 imply (p 1 + p 2 ) 2 < 0. Thus, there is a subalgebra g + u of g u with the same commutation relations (2.10), but with generators x α+p and x α p such that p 2 < 0 (only massive particles). The proof is in Appendix A.
The algebra g u has a natural 2-grading inherited by that of e 8 , due to the decomposition into the subalgebra d 8 and its Weyl spinor, [1]. The generators x α+p are fermionic (resp. bosonic) if α is fermionic (resp. bosonic), whereas the generators x α p are bosonic, due to the commutation relations (2.10).

The Lie Superalgebra sg u
In order to turn the Lie algebra g u into a Lie superalgebra, we exploit the Grassmann envelope G(g u ) of g u , where g u0 is the boson subalgebra of g u , g u1 its fermionic part, and G 0 , G 1 are the even, odd parts of a Grassmann algebra with infinitely many generators. More precisely, we map each generator X of g u to the generator X ⊗ e x of G(g u ), where e x is even if X is bosonic, odd if X is fermionic, and e x = e y if X = Y . Then the graded Jacobi identity is satisfied, [8], and one obtains, by linearity, a Lie superalgebra, that we denote by sg u .
Let us show this straightforward calculation explicitly. Let X, Y, Z be generators of g u of degree i, j, k ∈ {0, 1} respectively. We remind, from [1], that the generators x α p have degree 0, whereas the generators x α+p have degree α = 0 if α is bosonic and degree α = 1 if α is fermionic. Let [X, Y ] still denote the product of X, Y in g u and X ⊗ e x •Y ⊗ e y the corresponding product in sg u . Then: where J = 0 is the Jacobi identity for g u .
Remark 3.1. The product in the Lie superalgebra sg u is effectively the same as in the Lie algebra g u but its symmetry property is crucial for the elements of the universal enveloping algebra, that appear point by point in the model for the expanding Universe. The universal enveloping algebra is indeed the tensor algebra x, y ∈ sg u , embedded in the tensor algebra, of degree i, j respectively. In particular this makes the fermions comply with the Pauli exclusion principle: x ⊗ x = 0, for x fermionic, whereas the same relation is trivial, 0 = 0, if x is bosonic.
Remark 3.2. The Lie superalgebra sg u does not involve superpartners. The elements are exactly the same as those of the algebra g u . The importance we attribute to this algebra is solely due to the fulfillment of the Pauli exclusion principle.

Remark 3.3.
We also notice that the use of the Grassmann envelope produces zero divisors in the algebra whenever the same fermionic root, with the same momentum, is in two interacting particles.

Interaction graphs
As mentioned in the Introduction of [1], the interactions have a tree structure whose building blocks involve only three particles, and they are expressed by the product in the underlying algebra. The scattering amplitudes are proportional, up to normalization, to the structure constants of the related products. An ordering of the roots has to be a priori set, so that the commutator between two generators is taken according to that order. Quantum interference is obviously independent from the ordering choice.
In this section, we set up a correspondence between graphs and products in the algebra g u spanned by the generators and the procedure can then be trivially extended to sg u by Remark 3.1.
We include the decays among the possible scatterings as resonance interactions, a well known and studied phenomenon in many physical processes, as we now explain.
Suppose that two particles, one with charge α ∈ Φ 8 ∪ {0} and momentum p 1 , the other with charge β ∈ Φ 8 ∪ {0} and momentum p 2 , are present at the same space point and are such that α − β ∈ Φ 8 ∪ {0} and E 1 > E 2 ; then, a decay occurs, with a certain amplitude, producing the outgoing particles of charges (α−β), β with momenta (p 1 − p 2 ) and p 2 respectively, whereas the particle with charge β ∈ Φ 8 ∪{0} and momentum p 2 shifts in space according to the expansion rule, see [1]. The amplitude for the decay is proportional, up to normalization, to the structure constant of the commutator between the outgoing particles (we will comment on this viewpoint on the decays at the end of this section).
The possible situations for an elementary interaction are depicted in the following graphs, Figs. 1 ÷ 3 (the resonant particle is also shown in case of a decay). In the graphs we use wiggly lines for the neutral particles x α p and straight lines for the charged particles x α+p . Red lines indicate outgoing particles and blue lines incoming ones.
We would like to stress that the orientation of the graphs is not significant; these are not Feynmann diagrams, although they resemble them: only the distinction between incoming and outgoing particles matters; it complies with 4-momentum and charge conservation. x α+p 1  x α+p 1 x α+β decay into x α and x β (similarly for x α+p 1 and x β+p 2 interchanged).

The Poincaré group
We refer to section 2.3 of our previous paper [1], in particular we denote by ρ 1 , ρ 2 the roots k 5 − k 6 and k 5 + k 6 respectively.
We have a complex a 1 ⊕ a 1 Lie algebra M generated by x ±ρ 1 , x ±ρ 2 and the corresponding Cartan generators h ρ 1 , h ρ 2 .
The spin subalgebra su(2) spin ∈ M is the compact form of the subalgebra with generators R + := We denote by w the Pauli-Lubanski vector and we classify the g u generators x α+p or x α p with respect to m 2 = −p 2 and w 2 , the two Casimir invariants of the Poincaré group. We use the shorthand notation k := ±k 1 ± k 2 ± k 3 ± k 4 ± k 7 ± k 8 , k e (resp. k o ) when k has an even (resp. odd) number of + signs.
all generators x α+p or x α p such that p 2 = 0 (5.1) Let g e u be the extension of g u that includes all timelike and lightlike momenta (not necessarily with integer component), and let g +e u the subalgebra g e u of massive particles, namely the extension of the subalgebra g + u introduced in Remark 2.2. We regard the following proposition as fundamental for the relativistic behavior of our model. where ad(R) is the adjoint action of the generator R ∈ su(2) spin of the Wigner rotation W (Λ, p). where λ = 0, ± 1 2 , ±1 is the helicity of α and θ is the angle of the SO(2) rotation along the direction of p, analogous to the Wigner rotation of the massive case.

If
The Poincarè group is a subgroup of the automorphism group of g +e u .
Proof. The action P on each generator with a certain mass and spin/helicity acts as the irreducible induced representation, introduced by Wigner, [9]. We only need to prove that it is an automorphism of g +e u , namely that P is nonsingular and preserves the Lie product (2.10). Part of the proof is similar to the classical one, see Lemma 4.3.1 in [10]. The fact that P is non-singular comes from the obvious existence of its inverse transformation. We are left with the proof that P([X, Y ]) = [P(X), P(Y )]. Let us consider in particular P(x α+p ) in (5.2). Since R is an e 8 generator then ad(R) is nilpotent, namely ad(R) r = 0 for some r and and also that ad(R) t = 0 for t ≥ r implies Let α + β ∈ Φ 8 and p 2 1 , p 2 2 < 0. We get: Similarly for the other commutators in (2.10).
The action P can be easily extended to sg u by acting accordingly on the Grassmann variable in order to get the variable associated to the transformed generators of g u .

Initial Quantum State
The initial quantum state of our model of the expanding early Universe is an element of the universal enveloping algebra U sg u of sg u , namely an element of the tensor algebra built on the generators of sg u modulo the relations defining the product in the algebra itself. The initial generators are all in pairs with opposite helicity and opposite 3-momentum, [1], and have a phase or amplitude associated to each of them as a complex coefficient. The interactions and expansions starting from the initial state are such that locally the quantum state is an element of the universal enveloping algebra. Interference plays the crucial role in the quantum behavior of the model, including repulsive versus attractive forces. The quantum nature of gravity appears through the quantum nature of spacetime: at every cosmological instant, a point in space has an amplitude which is the sum of the amplitudes for particles to be at that point.
The initial state has the mean energy of the Universe concentrated on the generators that interact with each other at t = 0. The choice of the initial state is crucial in determining the likelihood for the existence of particles and of an eventual symmetry breaking. It is beyond the scope of this paper to investigate this subject in depth; an algorithm based on the algebra and the expansion rule that we have introduced can be the basis for computer calculations, which should shed some light on the physical consequences of the choice of the initial quantum state.

Vertex-type algebra and Gravitahedra
Space expansion leads to an enrichment of the algebra. The locality of interactions suggests to embed the algebra in a vertex-type operator algebra, in which the generators of sg u act as vertex operators on a discrete space that is being built up, step by step, by sg u driven interactions.
The tree structure of the interactions allows for a description of scattering amplitudes in terms of associahedra or permutahedra, [11]- [18], with structure constants attached to each vertex; see Fig. 4 for the interaction of four particles, producing the associahedron K 4 . A vertex is interpreted as an interaction with universal time flowing from top to bottom in the trees of Fig. 4. However, if one includes the gravitational effect of space expansion, one should describe the interactions through permutahedra P n−1 rather than associahedra; see Fig. 5 for the interaction of four particles, producing the permutahedron P 3 . The two trees in Fig. 5   A complete graphical description of the interactions, including the spacetime effects, hence gravity, can be quite complicated and needs a deep study. A research program with this goal has initiated, and the name gravitahedra has been coined for the polytopes that will eventually, and hopefully, describe such interactions.
The fact that locally the quantum state is an element of the universal enveloping algebra means that we can assign to it labels q of space Q, which are triples of rational numbers, due to the expansion by p/E, where E, p x , p y , p z are integers, [1]. The vertex-type algebra is therefore the algebra U sg u (Q), whose relations have been extended in order to include the commutation of elements with different space-labels.

Conclusion
In the pair of papers given by [1] and the present paper, we have presented an intrinsically quantum and relativistic theory of the creation of spacetime starting from a quantum state as cosmological boundary condition, which we conceive to play a key role in any fundamental theory of Quantum Gravity. We have discussed the general framework of a workable model, based on a rank-12 infinite dimensional Lie superalgebra, which can be applied to the quantum era of the first cosmic evolution. Our model can accommodate the degrees of freedom of the particles we know, without superpartners, namely spin-1 2 fermions and spin-0 and spin-1 bosons obeying the proper statistics.
The quantum nature of gravity is intrinsically unobservable, because observation implies the destruction of the entanglement and the collapse of the wavefunction. Thus, the deal in Quantum Gravity is the following : the intrinsically quantum and relativistic description of an intrinsically unobservable regime should be made consistent with the existence of a macroscopic observer, and thus of a (semi)classical observational symmetry, emerging in the theormodynamical/macroscopic limit in which the entanglement becomes irrelevant. Our model tackles this crucial issue of Quantum Gravity, and solves it with elegance : indeed, the Poincaré group emerges from both the "spin" sector (e 8 ) and the kinematical sector (complementary of e 8 in g u ) of the Lie superalgebra sg u . Besides the absence of superpartners and the implementation of the Pauli exclusion principle, the emergence of the Poincaré group is a crucial feature of our model. We should stress that, of course, the Poincaré group can be defined only in the thermodynamical limit in which the observer can be consistently decoupled from the evolutive dynamics of the Universe, given in toto by sg u . Especially in an early Universe, the back-reaction of the observer on the object of the observation should be relevant, and thus the abstraction of a decoupled and distinct observer is not totally consistent during the early stages of the Universe.
Many physical properties have still to be verified and/or fulfilled, like the proton decay, the confinement of quarks, the attractive nature of gravity on the large scale. The general framework of the model leaves however a great freedom of choice, and this is as a benefit for those who believe this is a promising approach and wish to explore it.
There is much left for future work, to start with the definition of a particular quantum initial state allowing to perform some preliminary computer calculations that may give an idea of how the model effectively works. In particular, the density matrix, von Neumann entropy, mean energy, scattering amplitudes can be explicitly calculated according to our model.
We end this series of two papers by recapitulating what we consider the main physical features of our approach: I) spacetime is the outcome of the interactions driven by an infinite-dimensional Lie superalgebra sg u ; it is discrete, finite and expanding; II) the algebra sg u incorporates 4-momentum and charge conservation; it involves fermions and bosons, with fermions fulfilling the Pauli exclusion principle; III) sg 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 sg 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..

A Appendix
We prove Proposition 2.3.
The algebra g u with relations (2.10), (2.11), (2.12) is obviously infinite dimensional, and its product is antisymmetric. We only need to prove that it fulfills the Jacobi identity.