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New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling
General Relativity and Quantum Cosmology · Submitted 18 May 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group.
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We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group. The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already: First one performs a canonical analysis of Lorentzian General Relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in the first paper of this series to obtain a connection theory with SO(D+1) as the internal gauge group subject to additional constraints. The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. A quantisation of the Hamiltonian constraint is provided.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group. | contribution 0.99 | 0.95 | 0.37 | asserted 1.00 |
| s1 | The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already: First one performs a canonical analysis of Lorentzian General Relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in the first paper of this series to obtain a connection theory with SO(D+1) as the internal gauge group subject to additional constraints. | method 1.00 | 0.69 | 0.11 | asserted 1.00 |
| s2 | The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. | interpretation 0.41 | 0.35 | 0.05 | asserted 1.00 |
| s3 | A quantisation of the Hamiltonian constraint is provided. | contribution 0.73 | 0.90 | 0.26 | asserted 1.00 |
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The paper
Original abstract
We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group. The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already: First one performs a canonical analysis of Lorentzian General Relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in the first paper of this series to obtain a connection theory with SO(D+1) as the internal gauge group subject to additional constraints. The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. A quantisation of the Hamiltonian constraint is provided.
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References
How the literature uses it
23 arXiv papers cite this in 36 sentences
Papers citing it · 31 sentences without a confident label
Builds on it 1 paper
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The results are shown to extend to coupling of fermionic matter treated in [ref] .
Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector · Linear Simplicity Constraints · 0.85
Uses its method or tool 1 paper
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The Hilbert space for the bulk theory in higher dimension can be constructed in the approach proposed by Bodendorfer and his collaborators [ref] .
The Entropy of Higher Dimensional Nonrotating Isolated Horizons from Loop Quantum Gravity · The entropy of isolated horizon · 0.96
Cites it as background 3 papers
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First attempts to add spacelike extra dimensions to LQG were made in Refs. [ref] .
Compactified Spacelike Extra Dimension & Brane-Higgs Field · Introduction · 0.90
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Matter couplings in LQG have been extensively explored (see e.g. [ref] ).
Fermions on Quantum Geometry and Resolution of Doubling Problem · Introduction · 0.98
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Next to the Ashtekar-Barbero variables, there exists another set of canonical variables, in terms of which loop quantum gravity can be constructed in $D+1$ dimensions for $D ≥2$ [ref] .
Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition · Higher dimensions · 0.88
What it cites · 53 sentences without a confident label
Builds on 3 papers
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The reason why the procedure introduced in [ref] generalises nicely to Dirac fermions is the strong resemblance of the Clifford algebras, which differ only by factors of $i$ for different signatures and the Euclidean signature of the internal gauge group which ensures that $Σ^IJ$ is a Hermitian matrix the Euclidean case.
New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis · Phase Space Extension · 0.83
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The reason why the procedure introduced in [ref] generalises nicely to Dirac fermions is the strong resemblance of the Clifford algebras, which differ only by factors of $i$ for different signatures and the Euclidean signature of the internal gauge group which ensures that $Σ^IJ$ is a Hermitian matrix the Euclidean case.
New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis · Phase Space Extension · 0.83
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In this paper we extended the new connection formulation of General Relativity in all spacetime dimensions $D+1≥3$ [ref] to arbitrary gauge boson (for compact structure groups) and standard fermionic couplings.
New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory · Conclusion · 0.99
Uses the method or tool of 8 papers
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The explicit quantisation follows the (extended) toolkit of [ref] : * Choose a triangulation $T(γ, ϵ)$ of the spatial slice $σ$ adapted to the graph $γ$. * Use classical identities in order to express the Hamiltonian constraint in terms of connections $A_aIJ$, volumes $V(x, ϵ)$ and Euclidean Hamiltonian constraints $ℋ_E(x, ϵ)$. * Replace all phase space variables by their corresponding regulated quantities. * Instead of the the integration $∫_σd^Dx$, put a sum $1/D! ∑_v ∈V(γ)$ over all the vertices $v$ of the graph $γ$. * For every spatial $ϵ$-symbol, put a sum $2^D/E(v) ∑_v(Δ)=v$ over all $D$-simplices having $v$ as a vertex.
New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory · Implementation of the Hamiltonian Constraint Operator · 0.99
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After having obtained the fermionic contributions to the classical constraints we quantise them using standard methods [ref] and using the representation [ref] .
QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories · Introduction · 0.99
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After having obtained the fermionic contributions to the classical constraints we quantise them using standard methods [ref] and using the representation [ref] .
Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories · Introduction · 0.99
Cites as background 8 papers
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The quantisation of fermionic degrees of freedom was described in detail in [ref] , which we assume the reader to be familiar with.
QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories · Implementation of the Hamiltonian Constraint Operator · 0.83
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The quantisation of fermionic degrees of freedom was described in detail in [ref] , which we assume the reader to be familiar with.
Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories · Implementation of the Hamiltonian Constraint Operator · 0.83
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We refer to our companion papers [ref] for details.
On the Implementation of the Canonical Quantum Simplicity Constraint · Outline of the Quantum Simplicity Constraints · 0.88
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