Relationship between causal multiway systems and spin foams in loop quantum gravity
Geeth Chandra Ongole
Baylor University
Geeth Chandra Ongole
Baylor University
Wolfram model hypergraph could have a reasonable interpretation as a spin network corresponding to the quantum state of a gravitational field on a given spacelike hypersurface. The time-evolved description of such a hypergraph (via a causal graph) possibly corresponds to a spin foam.
Introduction
Introduction
Loop quantum gravity(LQG) is born out of many attempts at quantizing gravity. LQG falls under a broad class of many approaches to quantum gravity called canonical quantum gravity(QG). Canonical QG theories are very different by construction and are background independent, unlike string theory. Following an approach laid out by Dirac,
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a theory manifestly covariant in its action is quantized by initially casting the theory into a canonical form by extracting its Hamiltonian.
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Promoting the phase space variables of the theory as operators and finding commutation relations is the key.
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Using the operators to rewrite the constraints that build the Hamiltonian and further identifying the Hilbert space is followed which usually is not straightforward.
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Finally, the equations of the theory are given by the Schrodinger equation written in operator form.
This usually is the recipe for formulating such a theory.
LQG is obtained by starting with the Einstein-Hilbert action and casting it into a Hamiltonian form by using the Ashtekar variables(in the form of triads/vierbiens) which is the major difference from its predecessors, where ADM variables were used. These triads being spineless(background independent) by nature help in part for constructing a background independent theory. Surprisingly, the quantized Hamiltonian which is a sum of constraint equations that are individually zero on the phase space, govern the dynamics of the system. These constraint equations are usually termed as vector and scalar(Hamiltonian) constraints; vector constraints generate the diffeomorphic invariance of the spatial hypersurface and the scalar constraints store the time reparametrization invariance of the theory. LQG also has another constraint because of the extra freedom facilitated by the triads, this is usually referred to as the Gauss constraint. LQG has many advantages over other canonical QG theories, quantum mechanical counterparts of geometrical definitions such as area and volume are some of them.
This usually is the recipe for formulating such a theory.
LQG is obtained by starting with the Einstein-Hilbert action and casting it into a Hamiltonian form by using the Ashtekar variables(in the form of triads/vierbiens) which is the major difference from its predecessors, where ADM variables were used. These triads being spineless(background independent) by nature help in part for constructing a background independent theory. Surprisingly, the quantized Hamiltonian which is a sum of constraint equations that are individually zero on the phase space, govern the dynamics of the system. These constraint equations are usually termed as vector and scalar(Hamiltonian) constraints; vector constraints generate the diffeomorphic invariance of the spatial hypersurface and the scalar constraints store the time reparametrization invariance of the theory. LQG also has another constraint because of the extra freedom facilitated by the triads, this is usually referred to as the Gauss constraint. LQG has many advantages over other canonical QG theories, quantum mechanical counterparts of geometrical definitions such as area and volume are some of them.
Spin Foams
Spin Foams
There is a different approach, using the path integral formulation for obtaining an LQG-like theory, often referred to as the covariant LQG. Such a theory enjoys the advantages of General Relativity, such as the general covariance and some features from canonical LQG. A correlation between the quantum states of such a theory and spin networks was found. They have been realised as the eigenstates of the quantum area operator. Spin networks are graphs with edges labelled by the representation of a group(referred to as a spin), usually SU(2) for 3d-gravity and the nodes given by the intertwining operators. A spin foam is a spin network evolved in time. Another interpretation is, a spin foam is a time ordered arrangement of spin network states. Transition amplitudes(or loosely the inner product) between two states can be further calculated from these representations.
One of the constraints which composes the Hamiltonian, scalar constraint is of importance here. Scalar constraint when acts on a spin network create new links. A series of such actions give a spin foam. Mathematically, a spin foam is a 2-complex with the faces labelled by the representation of a group(referred to as spin) and the edges given by intertwining operators(after the action of scalar constraint). After the action of the above mentioned scalar constraint on a node, the spin of the new link formed is given by the tensor product of the edges supporting it.
One of the constraints which composes the Hamiltonian, scalar constraint is of importance here. Scalar constraint when acts on a spin network create new links. A series of such actions give a spin foam. Mathematically, a spin foam is a 2-complex with the faces labelled by the representation of a group(referred to as spin) and the edges given by intertwining operators(after the action of scalar constraint). After the action of the above mentioned scalar constraint on a node, the spin of the new link formed is given by the tensor product of the edges supporting it.
Spin Foams in Wolfram Model
Spin Foams in Wolfram Model
Mathematically spin networks are constructed by typically populating tetrahedrons edge to edge for a 3d-gravity model(2+1) which combine 3 spins at a vertex. For constructing a spacetime(3+1) from a spin foam, pentagons need to be populated edge to edge and 10 spins are to be combined at a vertex. The time evolution of such a spin network gives a spin foam. As mentioned earlier, the edges carry a spin label implicitly carrying a group representation. The partition function(also referred to as weight) is given by the Wigner-6j symbol for 2+1-gravity and 15j symbol for 3+1 gravity. The Wigner-6j symbol is given by.It can alternatively be calculated by the SixJSymbol. The partition function(weight) of this 2+1 model is given by .The following discussion will be limited to 2+1 gravity which is inspired by the Ponzano-Regge model and later improved over the years.Wolfram Model(WM) environment by construction is background(co-ordinate) independent which is ideal for such an implementation and is also demanded by the theory(LQG). The idea is to construct a hypergraph using a WM. A given rule should populate tetrahedrons to build a spin network. The spins have to be carefully assigned to the edges of the hypergraph and the 6j-symbol calculated. This facilitates in further calculating the partition function using the above formula. There are various approaches in calculating the total partition function of the spin network which allows for further calculations on the spin network. The construction can be made more fundamental by interpreting the scalar, vector and Gauss constraints in the context of WM physics. These constraints which govern the diffeomorphism and time reparametrization are not entirely understood in the conventional LQG setting and such an interpretation could indeed help in a better understanding. Evolution in WM context could be given by a multiway causal graph giving a spin foam. Causal invariance interpreted as conformal invariance seen in General relativity needs to be verified here.Finding a rule that generates a hypergraph as desired(tetrahedral populated edge to edge) was a major difficulty in the implementation. A rule similar to the one which knits a grid(like here) is required which is rather counter-intuitive. A rule needs to be carefully devised to obtain such a hypergraph which will need various permutations and combinations and a proper choice of initial conditions. One way to get around this difficulty is to emulate the process followed in constructing such surfaces like here. One obvious but rather tedious job is to construct a tetrahedron and join them edge to edge one by one which also helps in assigning the spins to the edges. This seemingly obvious method also proves to be not so straightforward. Other lattice objects could be used to populate a spin network. End results of such spin foams can be compared to the conventional LQG theories to find better alternatives to the lattice objects. One of the major problems in LQG theories is the convergence to Einstein’s equations in the classical limit, such problems have potential solutions in the WM physics context. Implementation of Spin foams in WM physics indeed has great potential for further exploration and hopefully could solve many major problems that plague LQG.
Keywords
Keywords
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Loop quantum gravity (LQG)
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Spin networks
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Spin foams
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Quantum gravity
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Canonical quantum gravity
Acknowledgement
Acknowledgement
Mentor: Cameron Beetar
I am thankful for the support, guidance and help provided by Xerxes D. Arsiwalla, Hatem Elshatlawy, José Manuel and Max Piskunov directly or indirectly. My mentor Cameron Beetar had constantly been supervising me and motivated me to get a better understanding of the topic. I also thank Stephen Wolfram and Jonathan Gorard for organising this winter school and bringing together people from various places with interesting perspectives in physics.
References
References
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Perez, Alejandro. “Introduction to loop quantum gravity and spin foams.” arXiv preprint gr-qc/0409061 (2004).
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Nicolai, Hermann, and Kasper Peeters. “Loop and spin foam quantum gravity: A brief guide for beginners.” Approaches to fundamental physics. Springer, Berlin, Heidelberg, 2007. 151-184.
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Wolfram, Stephen. “A Class of Models with the Potential to Represent Fundamental Physics.” arXiv (2020): arXiv-2004.