Loop Quantum Gravity is one of the most promising approaches to quantum gravity. A distinctive feature of this theory is its background independence. No classical background is required in its formulation. This thesis aims to lay the foundations for a possible connection between particle physics's phenomenology and the loop approach to quantum gravity. The Standard Model or quantum field theories rely on a fixed background, so we look for a recipe for identifying states in the kinematical space of Loop Quantum Gravity from which a background metric can be recovered. The first step toward a classical geometry is to consider a spin network state in the kinematical Hilbert space of the theory, peaked on a collection of classical polyhedra. These are many-body states that describe an unentangled collection of quantum polyhedra. We expect all the many-body states that aspire to be candidates for semiclassical geometry to respect an area law. We then use the tensor network representation, which features an area law, to find these states, introducing uniformization operators. These operators, representing geometric constraints on tetrahedra, play the role of Hamiltonians to be minimized using the density matrix renormalization group algorithm on matrix product states. In this thesis, we only focus on the 1-dimensional case, obtaining entangled states of uniform tetrahedra. Future developments in this direction are given by considering higher-dimensional tensor networks associated with spin networks on dual complexes of discretized 3-manifolds.

Loop Quantum Gravity is one of the most promising approaches to quantum gravity. A distinctive feature of this theory is its background independence. No classical background is required in its formulation. This thesis aims to lay the foundations for a possible connection between particle physics's phenomenology and the loop approach to quantum gravity. The Standard Model or quantum field theories rely on a fixed background, so we look for a recipe for identifying states in the kinematical space of Loop Quantum Gravity from which a background metric can be recovered. The first step toward a classical geometry is to consider a spin network state in the kinematical Hilbert space of the theory, peaked on a collection of classical polyhedra. These are many-body states that describe an unentangled collection of quantum polyhedra. We expect all the many-body states that aspire to be candidates for semiclassical geometry to respect an area law. We then use the tensor network representation, which features an area law, to find these states, introducing uniformization operators. These operators, representing geometric constraints on tetrahedra, play the role of Hamiltonians to be minimized using the density matrix renormalization group algorithm on matrix product states. In this thesis, we only focus on the 1-dimensional case, obtaining entangled states of uniform tetrahedra. Future developments in this direction are given by considering higher-dimensional tensor networks associated with spin networks on dual complexes of discretized 3-manifolds.

Quantum Gravity and Tensor Networks

PISANA, ALESSANDRO
2021/2022

Abstract

Loop Quantum Gravity is one of the most promising approaches to quantum gravity. A distinctive feature of this theory is its background independence. No classical background is required in its formulation. This thesis aims to lay the foundations for a possible connection between particle physics's phenomenology and the loop approach to quantum gravity. The Standard Model or quantum field theories rely on a fixed background, so we look for a recipe for identifying states in the kinematical space of Loop Quantum Gravity from which a background metric can be recovered. The first step toward a classical geometry is to consider a spin network state in the kinematical Hilbert space of the theory, peaked on a collection of classical polyhedra. These are many-body states that describe an unentangled collection of quantum polyhedra. We expect all the many-body states that aspire to be candidates for semiclassical geometry to respect an area law. We then use the tensor network representation, which features an area law, to find these states, introducing uniformization operators. These operators, representing geometric constraints on tetrahedra, play the role of Hamiltonians to be minimized using the density matrix renormalization group algorithm on matrix product states. In this thesis, we only focus on the 1-dimensional case, obtaining entangled states of uniform tetrahedra. Future developments in this direction are given by considering higher-dimensional tensor networks associated with spin networks on dual complexes of discretized 3-manifolds.
2021
Quantum Gravity and Tensor Networks
Loop Quantum Gravity is one of the most promising approaches to quantum gravity. A distinctive feature of this theory is its background independence. No classical background is required in its formulation. This thesis aims to lay the foundations for a possible connection between particle physics's phenomenology and the loop approach to quantum gravity. The Standard Model or quantum field theories rely on a fixed background, so we look for a recipe for identifying states in the kinematical space of Loop Quantum Gravity from which a background metric can be recovered. The first step toward a classical geometry is to consider a spin network state in the kinematical Hilbert space of the theory, peaked on a collection of classical polyhedra. These are many-body states that describe an unentangled collection of quantum polyhedra. We expect all the many-body states that aspire to be candidates for semiclassical geometry to respect an area law. We then use the tensor network representation, which features an area law, to find these states, introducing uniformization operators. These operators, representing geometric constraints on tetrahedra, play the role of Hamiltonians to be minimized using the density matrix renormalization group algorithm on matrix product states. In this thesis, we only focus on the 1-dimensional case, obtaining entangled states of uniform tetrahedra. Future developments in this direction are given by considering higher-dimensional tensor networks associated with spin networks on dual complexes of discretized 3-manifolds.
Quantum Polyhedra
Emergence of Space
Spin Networks
Loop Quantum Gravity
Matrix Product State
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12608/34603