We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and ... more We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the 15j symbol, in a particular basis constructed using SU(2) coherent states. This basis makes the geometric interpretation of the variables transparent: they are the vectors normal to the triangles within each tetrahedron. We study the condition under which these states can be considered semiclassical, and we show that the semiclassical ones dominate the evaluation of quantum correlations. Finally, we describe how the constraints reducing BF to gravity can be directly written in terms of the new variables, and how the semiclassicality of the states might improve understanding the correct way to implement the constraints.
In the context of the quest for a holographic formulation of quantum gravity, we investigate the ... more In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3+1 space-time dimensions, the boundary theory lives on the 2+1-dimensional time-like boundary and is supposed to describe the time evolution of the edge modes living on the 2-dimensional boundary of space, i.e. the space-time corner. Focusing on "electric" excitations -- quanta of area -- living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background boundary 2+1-metric. This leads to a deeper conjecture of a correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a SL(2,C) connection, transporting the spinors on the boundary surface and whose SU(2) component would define "magn...
We show that loop gravity can equally well be formulated in in terms of spinorial variables (inst... more We show that loop gravity can equally well be formulated in in terms of spinorial variables (instead of the group variables which are commonly used), which have recently been shown to provide a direct link between spin network states and discrete geometries. This results in a new, unitarily equivalent formulation of the theory on a generalized Bargmann space. Since integrals over the group are exchanged for straightforward integrals over the complex plane we expect this formalism to be useful to efficiently organize practical calculations.
We perform a quantization of the loop gravity phase space purely in terms of spinorial variables,... more We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.
The relation between the 2d Ising partition function and spin network evaluations, reflecting a b... more The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner's 6j-symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j-symbol and Fisher zeros. Using the Ponzano-Regge formula for the asymptotics for the 6j-symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well-known that the 2d intrinsic geometry can be used to parametrize the critical point of the ...
arXiv: General Relativity and Quantum Cosmology, 2014
We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing t... more We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing the phase space $T^*\mathrm{SU}(2)\sim\mathrm{ISO}(3)$ as the Heisenberg double of the Lie group $\mathrm{SO}(3)$ provided with the trivial cocyle. Tackling the issue of accounting for a non-vanishing cosmological constraint $\Lambda \ne 0$ in the canonical framework of 3D loop quantum gravity, $\mathrm{SL}(2,\mathbb{C})$ viewed as the Heisenberg double of $\mathrm{SU}(2)$ provided with a non-trivial cocyle is introduced as a phase space. It is a deformation of the flat phase space $\mathrm{ISO}(3)$ and reproduces the latter in a suitable limit. The $\mathrm{SL}(2,\mathbb{C})$ phase space is then used to build a new, deformed LQG phase space associated to graphs. It can be equipped with a set of Gauss constraints and flatness constraints, which form a first class system and Poisson-generate local 3D rotations and deformed translations. We provide a geometrical interpretation for this latt...
In the context of the quest for a holographic formulation of quantum gravity, we investigate the ... more In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3+1 space-time dimensions, the boundary theory lives on the 2+1-dimensional time-like boundary and is supposed to describe the time evolution of the edge modes living on the 2-dimensional boundary of space, i.e. the spacetime corner. Focusing on "electric" excitations-quanta of area-living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background boundary 2+1metric. This leads to a deeper conjecture of a correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a SL(2, C) connection, transporting the spinors on the boundary surface and whose SU(2) component would define "magnetic" excitations (tangential Ashtekar-Barbero connection), thereby opening the door to writing the loop quantum gravity boundary dynamics as a 2+1-dimensional SL(2, C) gauge theory. Contents I. Loop Quantum Gravity on Space-Time Corners 2 A. Punctures and Spinors on the Boundary 3 B. Spinor dynamics on the 2+1-d time-like boundary 4 C. Boundary Field Dynamics 7 II. Lorentz Connection on the Boundary A. SL(2, C)-holonomies between spinors B. Non-trivial stabilizer and (relative) locality on the boundary C. SL(2, C) boundary theory D. Recovering local SU(2) gauge invariance on the boundary: magnetic excitations Outlook & Conclusion Acknowledgement A. SL(2, C)-holonomy between 3-vectors References
We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N face... more We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N faces. Starting with the Schwinger representation of the su(1, 1) Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the threedimensional Minkowski space R 1,2. Considering N copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with N faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of SU(1, 1)-invariant observables, whose flow by the Poisson bracket generate both area-preserving and area-changing deformations. We further show that the area-preserving observables form a gl N (R) Lie algebra and that they generate a GLN (R) action on Lorentzian polyhedra at fixed total area. That action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a GLN (R) transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as SU(1, 1) intertwiners between unitary SU(1, 1)-representations from the principal continuous series. Those SU(1, 1)-intertwiners are the building blocks of spin network states in loop quantum gravity in 3+1 dimensions for time-like slicing and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.
In canonical quantum gravity, the presence of spatial boundaries naturally leads to boundary quan... more In canonical quantum gravity, the presence of spatial boundaries naturally leads to boundary quantum states, representing quantum boundary conditions for the bulk fields. As a consequence, quantum states of the bulk geometry need to be upgraded to wave-functions valued in the boundary Hilbert space: the bulk becomes quantum operator acting on boundary states. We apply this to loop quantum gravity and describe spin networks with 2d boundary as wave-functions mapping bulk holonomies to spin states on the boundary. This sets the bulk-boundary relation in a clear mathematical framework, which allows to define the boundary density matrix induced by a bulk spin network states after tracing out the bulk degrees of freedom. We ask the question of the bulk reconstruction and prove a boundary-to-bulk universal reconstruction procedure, to be understood as a purification of the mixed boundary state into a pure bulk state. We further perform a first investigation in the algebraic structure of i...
arXiv: General Relativity and Quantum Cosmology, 2003
We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravi... more We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravity states and how causality can be consistently implemented in it.
arXiv: General Relativity and Quantum Cosmology, 2006
We review the canonical analysis of the Palatini action without going to the time gauge as in the... more We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.
Considering that a position measurement can effectively involve a momentum-dependent shift and re... more Considering that a position measurement can effectively involve a momentum-dependent shift and rescaling of the "true" space-time coordinates, we construct a set of effective space-time coordinates which are naturally non-commutative. They lead to a minimum length and are shown to be related to Snyder's coordinates and the five-dimensional formulation of Deformed Special Relativity. This effective approach then provides a natural physical interpretation for both the extra fifth dimension and the deformed momenta appearing in this context.
arXiv: General Relativity and Quantum Cosmology, 2006
After a brief review of spin networks and their interpretation as wave functions for the (space) ... more After a brief review of spin networks and their interpretation as wave functions for the (space) geometry, we discuss the renormalisation of the area operator in loop quantum gravity. In such a background independent framework, we propose to probe the structure of a surface through the analysis of the coarse-graining and renormalisation flow(s) of its area. We further introduce a procedure to coarse-grain spin network states and we quantitatively study the decrease in the number of degrees of freedom during this process. Finally, we use these coarse-graining tools to define the correlation and entanglement between parts of a spin network and discuss their potential interpretation as a natural measure of distance in such a state of quantum geometry.
The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory ... more The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory are analyzed and used to constrain the form of the effective action.
Quantum states of geometry in loop quantum gravity are defined as spin networks, which are graph ... more Quantum states of geometry in loop quantum gravity are defined as spin networks, which are graph dressed with SU(2) representations. A spin network edge carries a half-integer spin, representing basic quanta of area, and the standard framework imposes an area matching constraint along the edge: it carries the same spin at its source and target vertices. In the context of coarse-graining, or equivalently of the definition of spin networks as projective limits of graphs, it appears natural to introduce excitations of curvature along the edges. An edge is then treated similarly to a propagator living on the links of Feynman diagrams in quantum field theory: curvature excitations create little loops -- tadpoles -- which renormalize it. This relaxes the area matching condition, with different spins at both ends of the edge. We show that this is equivalent to combining the usual SU(2) holonomy along the edge with a Lorentz boost into SL(2,C) group elements living on the spin network edges...
We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled ... more We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemeterizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the su(1, 1) Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical non-vanishing minimum volume. Since the Hamiltonian is an element in the su(1, 1) Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a time-like irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the min...
We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and ... more We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the 15j symbol, in a particular basis constructed using SU(2) coherent states. This basis makes the geometric interpretation of the variables transparent: they are the vectors normal to the triangles within each tetrahedron. We study the condition under which these states can be considered semiclassical, and we show that the semiclassical ones dominate the evaluation of quantum correlations. Finally, we describe how the constraints reducing BF to gravity can be directly written in terms of the new variables, and how the semiclassicality of the states might improve understanding the correct way to implement the constraints.
In the context of the quest for a holographic formulation of quantum gravity, we investigate the ... more In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3+1 space-time dimensions, the boundary theory lives on the 2+1-dimensional time-like boundary and is supposed to describe the time evolution of the edge modes living on the 2-dimensional boundary of space, i.e. the space-time corner. Focusing on "electric" excitations -- quanta of area -- living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background boundary 2+1-metric. This leads to a deeper conjecture of a correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a SL(2,C) connection, transporting the spinors on the boundary surface and whose SU(2) component would define "magn...
We show that loop gravity can equally well be formulated in in terms of spinorial variables (inst... more We show that loop gravity can equally well be formulated in in terms of spinorial variables (instead of the group variables which are commonly used), which have recently been shown to provide a direct link between spin network states and discrete geometries. This results in a new, unitarily equivalent formulation of the theory on a generalized Bargmann space. Since integrals over the group are exchanged for straightforward integrals over the complex plane we expect this formalism to be useful to efficiently organize practical calculations.
We perform a quantization of the loop gravity phase space purely in terms of spinorial variables,... more We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.
The relation between the 2d Ising partition function and spin network evaluations, reflecting a b... more The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner's 6j-symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j-symbol and Fisher zeros. Using the Ponzano-Regge formula for the asymptotics for the 6j-symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well-known that the 2d intrinsic geometry can be used to parametrize the critical point of the ...
arXiv: General Relativity and Quantum Cosmology, 2014
We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing t... more We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing the phase space $T^*\mathrm{SU}(2)\sim\mathrm{ISO}(3)$ as the Heisenberg double of the Lie group $\mathrm{SO}(3)$ provided with the trivial cocyle. Tackling the issue of accounting for a non-vanishing cosmological constraint $\Lambda \ne 0$ in the canonical framework of 3D loop quantum gravity, $\mathrm{SL}(2,\mathbb{C})$ viewed as the Heisenberg double of $\mathrm{SU}(2)$ provided with a non-trivial cocyle is introduced as a phase space. It is a deformation of the flat phase space $\mathrm{ISO}(3)$ and reproduces the latter in a suitable limit. The $\mathrm{SL}(2,\mathbb{C})$ phase space is then used to build a new, deformed LQG phase space associated to graphs. It can be equipped with a set of Gauss constraints and flatness constraints, which form a first class system and Poisson-generate local 3D rotations and deformed translations. We provide a geometrical interpretation for this latt...
In the context of the quest for a holographic formulation of quantum gravity, we investigate the ... more In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3+1 space-time dimensions, the boundary theory lives on the 2+1-dimensional time-like boundary and is supposed to describe the time evolution of the edge modes living on the 2-dimensional boundary of space, i.e. the spacetime corner. Focusing on "electric" excitations-quanta of area-living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background boundary 2+1metric. This leads to a deeper conjecture of a correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a SL(2, C) connection, transporting the spinors on the boundary surface and whose SU(2) component would define "magnetic" excitations (tangential Ashtekar-Barbero connection), thereby opening the door to writing the loop quantum gravity boundary dynamics as a 2+1-dimensional SL(2, C) gauge theory. Contents I. Loop Quantum Gravity on Space-Time Corners 2 A. Punctures and Spinors on the Boundary 3 B. Spinor dynamics on the 2+1-d time-like boundary 4 C. Boundary Field Dynamics 7 II. Lorentz Connection on the Boundary A. SL(2, C)-holonomies between spinors B. Non-trivial stabilizer and (relative) locality on the boundary C. SL(2, C) boundary theory D. Recovering local SU(2) gauge invariance on the boundary: magnetic excitations Outlook & Conclusion Acknowledgement A. SL(2, C)-holonomy between 3-vectors References
We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N face... more We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N faces. Starting with the Schwinger representation of the su(1, 1) Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the threedimensional Minkowski space R 1,2. Considering N copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with N faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of SU(1, 1)-invariant observables, whose flow by the Poisson bracket generate both area-preserving and area-changing deformations. We further show that the area-preserving observables form a gl N (R) Lie algebra and that they generate a GLN (R) action on Lorentzian polyhedra at fixed total area. That action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a GLN (R) transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as SU(1, 1) intertwiners between unitary SU(1, 1)-representations from the principal continuous series. Those SU(1, 1)-intertwiners are the building blocks of spin network states in loop quantum gravity in 3+1 dimensions for time-like slicing and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.
In canonical quantum gravity, the presence of spatial boundaries naturally leads to boundary quan... more In canonical quantum gravity, the presence of spatial boundaries naturally leads to boundary quantum states, representing quantum boundary conditions for the bulk fields. As a consequence, quantum states of the bulk geometry need to be upgraded to wave-functions valued in the boundary Hilbert space: the bulk becomes quantum operator acting on boundary states. We apply this to loop quantum gravity and describe spin networks with 2d boundary as wave-functions mapping bulk holonomies to spin states on the boundary. This sets the bulk-boundary relation in a clear mathematical framework, which allows to define the boundary density matrix induced by a bulk spin network states after tracing out the bulk degrees of freedom. We ask the question of the bulk reconstruction and prove a boundary-to-bulk universal reconstruction procedure, to be understood as a purification of the mixed boundary state into a pure bulk state. We further perform a first investigation in the algebraic structure of i...
arXiv: General Relativity and Quantum Cosmology, 2003
We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravi... more We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravity states and how causality can be consistently implemented in it.
arXiv: General Relativity and Quantum Cosmology, 2006
We review the canonical analysis of the Palatini action without going to the time gauge as in the... more We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.
Considering that a position measurement can effectively involve a momentum-dependent shift and re... more Considering that a position measurement can effectively involve a momentum-dependent shift and rescaling of the "true" space-time coordinates, we construct a set of effective space-time coordinates which are naturally non-commutative. They lead to a minimum length and are shown to be related to Snyder's coordinates and the five-dimensional formulation of Deformed Special Relativity. This effective approach then provides a natural physical interpretation for both the extra fifth dimension and the deformed momenta appearing in this context.
arXiv: General Relativity and Quantum Cosmology, 2006
After a brief review of spin networks and their interpretation as wave functions for the (space) ... more After a brief review of spin networks and their interpretation as wave functions for the (space) geometry, we discuss the renormalisation of the area operator in loop quantum gravity. In such a background independent framework, we propose to probe the structure of a surface through the analysis of the coarse-graining and renormalisation flow(s) of its area. We further introduce a procedure to coarse-grain spin network states and we quantitatively study the decrease in the number of degrees of freedom during this process. Finally, we use these coarse-graining tools to define the correlation and entanglement between parts of a spin network and discuss their potential interpretation as a natural measure of distance in such a state of quantum geometry.
The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory ... more The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory are analyzed and used to constrain the form of the effective action.
Quantum states of geometry in loop quantum gravity are defined as spin networks, which are graph ... more Quantum states of geometry in loop quantum gravity are defined as spin networks, which are graph dressed with SU(2) representations. A spin network edge carries a half-integer spin, representing basic quanta of area, and the standard framework imposes an area matching constraint along the edge: it carries the same spin at its source and target vertices. In the context of coarse-graining, or equivalently of the definition of spin networks as projective limits of graphs, it appears natural to introduce excitations of curvature along the edges. An edge is then treated similarly to a propagator living on the links of Feynman diagrams in quantum field theory: curvature excitations create little loops -- tadpoles -- which renormalize it. This relaxes the area matching condition, with different spins at both ends of the edge. We show that this is equivalent to combining the usual SU(2) holonomy along the edge with a Lorentz boost into SL(2,C) group elements living on the spin network edges...
We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled ... more We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemeterizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the su(1, 1) Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical non-vanishing minimum volume. Since the Hamiltonian is an element in the su(1, 1) Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a time-like irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the min...
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