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# [WSS21] Quantum Gravity in (2+1) Dimensions and Multiway System

Posted 13 days ago

This project is an attempt to reconcile the multiway paradigm and loop quantum gravity. In particular, we focused on the deformed (2+1)
Introduction The current paradigm describing the action of fundamental forces (strong nuclear, weak nuclear and electromagnetism) is based on the formalism of Yang-Mills theories in what is called the Standard Model of fundamental interactions. Gravity, described by General Relativity, has been excluded from this paradigm. However, there is nothing inconsistent between quantum theory and general relativity. The latter is the only consistent theory for an interacting spin-2 particle, and it is possible to treat it as an effective field theory . The problem with this view is that of not considering the formal aspects of quantum field theories. The axiomatic approach to quantum field theory provides for axioms (called Wightman axioms for the Minkowski case and Osterwalder - Schrader for the Euclidean one) it is possible to reconstruct the interacting fields from the correlation functions of free fields. All the axioms in question foresee using a Minkowski background metric, and it is not clear how to reconstruct the fields in the case of a general metric, as expected by general relativity. Furthermore, general relativity, treated as a field theory, is non-renormalizable: it is not perturbative for energies above the Planck mass. The attempt to construct a non-perturbative quantum theory of gravity, consistent from a mathematical point of view and background-independent, takes the name of Canonical Quantum General Relativity, often abbreviated to Quantum Gravity. In its path-integral formulation, this is called Loop Quantum Gravity, where the “loops” indicate the variables introduced by Rovelli and Smolin.
Space from Combinatorics In trying to construct an independent background theory, it is necessary to give up an a priori notion of causality. This notion is then recovered from the local relationship between the objects of which the theory is built. Objects of this type in Loop Quantum gravity are spin networks whose causal evolution constitutes spin foams. In this sense, space is a derived notion that emerges from combinatorial principles. The combinatorics then force us to give away the concept of continuous spacetime and replace it with a series of relationships between discrete objects and combine them to obtain a large scale limit, the usual physical laws.
Quantized Geometry and Loop Quantum Gravity Loop Quantum Gravity is a theory for quantum gravity in which one tries to encode the effect of quantum superposition of geometry into a framework that is well defined conceptually and technically . The empirical inputs come from Gravity and the Quantum Theory, and the resulting theory is consistent with the Standard Model and has no ultraviolet divergences . However, it is not a theory of unification. Loop Quantum Gravity omits Supersymmetry, but there are several attempts to incorporate the gauge/gravity duality (for example, in the 3D case assuming as boundary field theory the 2D Ising model on a lattice) . Finally, there is no breaking of the Lorentz invariance at the Plank scale . In this work we focus on the (2+1)
Spin Networks Spin networks were first introduced by Sir. Roger Penrose in an attempt to recover continuous geometry from combinatorial laws. They are a set of vertices and edges on top of which some numbers are placed. These numbers were chosen to be units of angular momentum. In the context of Loop Quantum Gravity, the spin networks lose the connection to the original Penrose idea, and they turn out to be generalized eigenvectors of the area and volume operator. These two operators form complete commuting observables. Thus they are both diagonalized by the spin network basis. To construct a spin network in (2+1) dimensional gravity, we consider a simple example where the foliated spacetime is such that each spatial hypersurface has a cube’s topology. The idea is to truncate the number of the degree of freedom of this hypersurface by choosing a triangulation. Out[]= The triangulation form a graph whose links are the edges of the triangles. We extract the graph from the mesh, obtaining the Regge discretization. Out[]= The discretized space provides us with a truncation of the Hilbert space of the quantized geometry. The idea is to take the dual graph and assign spin numbers (or colourings) to it. Then a gauge-invariant theory is obtained if the colour is allowed by triangular inequalities, which in the dual triangulation takes the form of the Clebsch-Gordon conditions for the SU(2) SU(2) SU(2) q 2r q = Exp[2*Pi*I/r] It was shown that deforming the gauge group account for adding the cosmological constant to the Einstein - Hilbert action. In this quantum gravity deformed version, the maximum spin allowed is finite and depends on the deformation parameter through r, so that the cosmological constant provides an infrared cutoff of the geometry of the universe. Going back to the triangulation, we choose as a simple (and computationally lighter) parameter r=3 Domain[r_] := Table[i/2, {i, 0, r - 2}]; Domain[3] Out[]= 0, 1 2 Then gauge conditions are imposed. In what follows, we have reported all the possible colours of the triangulation that respect the triangular inequalities. Notice that, even in the case of two allowed spins, the number of combinations is high. Out[]//Short= ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,49,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Once the spin number is assigned, we can extract the dual graph. By taking a random coloured cube from the list, we will provide the relative spin network. Out[]= , The dual graph is constructed, placing a node in each triangle, connecting them following triangles. The spin assignment to a segment of the triangles is then inherited by the link crossing that segment. For our example, we obtain the following spin network. Out[]=
Other Examples The procedure we implemented is general, and it is possible to change the topology or the colouring number. However, the labelling and triangulation are computationally costly: increasing the allowed colouring and the triangulation results in a prolonged evaluation time. ◼ Here we labelled the dual triangulation of a triangulated sphere, imposing the maximum spin be 1/2, reporting two random allowed colourings. Out[]= , ◼ The increasing of the colouring can be seen in the tetrahedra example. Here we raised r from 3 to 4. Out[]= 1 .r=3 Out[]= 2 .r=4 Out[]= 3 .r=5 Out[]//Short= ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,145,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Dynamical Evolution What we succeeded to obtain are the deformed q-nets for a 3-manifold. The relative spin foams are the evolution in time of the initial spin network. If we start with a generic triangulation, we can write an ultra-local model for the Hamiltonian constrain, supposing that the Hamiltonian add new tetrahedra each time on the top of the triangulation. The new spin network formed by including the dual of this new tetrahedra then should satisfy the Clebsch-Gordon condition (to be gauge invariant). This translates into the following rules 1 .The first rule is the one that attaches the new tetrahedra to the original q-net. So, for example, if the original network has the form secondinitial = {UndirectedEdge[z, y, 1/2], UndirectedEdge[y, x, 1/2], UndirectedEdge[y, a, 1/2], UndirectedEdge[a, b, 1/2], UndirectedEdge[a, c, 1/2]}; Graph[secondinitial, EdgeLabels -> "EdgeTag", VertexLabels -> Automatic, ImageSize -> Small] Out[]= creates new vertices attached in the following way Out[]= 2 .The second rule instead changes the graph’s connectivity by replacing the two internal vertices (the two trivalent) with the other two, such that the spin current changes in the following way. Out[]=
The Multiway System As a final step, we are ready to implement the evolution into a Multiway System paradigm. By going back to the cube, we obtain the following tree graph. Each branch of this graph represents the quantum superposition of geometry. As the evolution number grows, a mini-universe example with spin-1/2 cutoff is reproduced. multiwayRule = Association["StateEvolutionFunction" -> (Join[applyRule1[#1], applyRule2[#1]] & ), "StateEquivalenceFunction" -> SameQ, "StateEventFunction" -> Identity, "EventDecompositionFunction" -> Identity, "EventApplicationFunction" -> Identity, "SystemType" -> "None", "EventSelectionFunction" -> Identity]; ResourceFunction["MultiwaySystem"][multiwayRule, {initSpinNetwork}, 5, "StatesGraphStructure"] Out[]= As an example, we report one branch, applying only Rule 1 after 3 evolutions. Out[]//Short= , , , , , , , , , , , , , , , , , , , , The final picture to have in mind is that the spacetime is built out of quantum tetrahedra; those faces are coloured that appears and disappears, changing colour in a fuzzy and furious way. As an artistic representation, here is a pictorial view of the process. Out[]=
Transition Amplitudes Suppose to have two graphs, one in the future and one in the past. The Hamiltonian constrain acts only on nodes, and we want to write an amplitude between the two graphs. The deformation of the SU(2) 2r In[]:= Q[n_, r_] := Sin[Pi*(n/r)]/Sin[Pi/r]; FQ[n_, t_] := Product[Q[m, t], {m, 1, n}]; Δ[a_, b_, c_, |