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Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space
General Relativity and Quantum Cosmology · Submitted 30 Apr 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the su(1,1) representation label and interpreted as the minimally resolved geometric scale.
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We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1,1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1,1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra su(1,1). We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the su(1,1) representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the su(1,1) representation label sets a lower cut-off for the loop quantum gravity spins (= U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the su(1,1) representation label and interpreted as the minimally resolved geometric scale.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. | contribution 0.97 | 0.96 | 0.29 | asserted 1.00 |
| s1 | A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1,1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. | method 0.86 | 0.63 | 0.06 | asserted 1.00 |
| s2 | The SU(1,1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra su(1,1). | method 0.98 | 0.79 | 0.03 | asserted 1.00 |
| s3 | We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. | method 0.72 | 0.93 | 0.06 | asserted 1.00 |
| s4 | This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. | result 0.57 | 0.91 | 0.15 | asserted 1.00 |
| s5 | As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. | result 0.52 | 0.71 | 0.16 | asserted 0.98 |
| s6 | We explicitly discuss how the su(1,1) representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the su(1,1) representation label sets a lower cut-off for the loop quantum gravity spins (= U(1) representation labels in LQC). | contribution 0.42 | 0.82 | 0.10 | asserted 1.00 |
| s7 | Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the su(1,1) representation label and interpreted as the minimally resolved geometric scale. | result 0.56 | 0.94 | 0.60 | asserted 1.00 |
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Original abstract
We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1,1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1,1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra su(1,1). We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the su(1,1) representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the su(1,1) representation label sets a lower cut-off for the loop quantum gravity spins (= U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the su(1,1) representation label and interpreted as the minimally resolved geometric scale.
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References
How the literature uses it
9 arXiv papers cite this in 39 sentences
Papers citing it · 28 sentences without a confident label
Builds on it 1 paper
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The main content of the present paper will be to compute the analogue of [ref] in a coherent state path integral, i.e. to extract the renormlised Hamiltonian, canonical action, and path integral measure directly from a coherent state path integral based on the group quantisation given in section [ref] or equivalently the canonical theory in [ref] sketched in this section.
Path integral renormalisation in loop quantum cosmology · Renormalised Hamiltonian from canonical quantisation · 0.97
Uses its method or tool 1 paper
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Given that we will choose the Hamiltonian of the system to be the linear combination of generators in [ref], we know that the time evolution operator \(-i H t∈SU(1,1)\), so that the entire time evolution of the coherent states can be computed by this very simple procedure [ref] , leading to [equation]
Path integral renormalisation in loop quantum cosmology · Coherent states · 0.92
Uses its data or values 1 paper
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The error made this way is very sensitive to taking the quantum numbers even slightly too large, i.e. if we peak the bounce volume on $2 v_m$ instead of $v_m$, we already overestimate the critical density by $87%$ as opposed to $100%$ at infinite bounce volume[To obtain these numbers, we simply evaluate the functions plotted in figure 1 of [ref] at $α≈0.268$.]. * While the model system considered in this paper as well as [ref] is very simple and not claimed to be an accurate reflection of our universe, it serves the purpose of illustrating the importance of renormalisation.
Path integral renormalisation in loop quantum cosmology · Discussion · 0.88
Compares with it 1 paper
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An interesting observation which contrasts the canonical [ref] and path integral quantisations [ref] slightly is that the canonical Hamiltonian operator [ref] does not have any small volume corrections for $j=1/2$.
Path integral renormalisation in loop quantum cosmology · Discussion · 1.00
Cites it as background 4 papers
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A more detailed review is given in [ref] .
Path integral renormalisation in loop quantum cosmology · Group quantisation, coarse graining map, and renormalised operators · 0.95
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This looks very much like a very naïve renormalisation procedure and in future work it would be interesting to relate this to previous work in the context of coarse graining and renormalisation in loop quantum cosmology [ref] .
The Physical Relevance of the Fiducial Cell in Loop Quantum Cosmology · Conclusions · 0.92
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(See also [ref] for appearances of this algebra or related versions in quantum cosmology.) In these variables, the Hamiltonian [equation] is linear (again, up to the absolute value).
Tunneling dynamics in cosmological bounce models · Harmonic systems · 0.90
What it cites · 51 sentences without a confident label
Contests 5 papers
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The same criticism applies to computations within full loop quantum gravity (LQG) [ref] or related models [ref] , where one acts with the fundamental non-renormalised Hamiltonian on coarse quantum states, as well as to spin foam calculations in the same spirit, such as [ref] .
Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory · Conclusion · contest 0.72
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The same criticism applies to computations within full loop quantum gravity (LQG) [ref] or related models [ref] , where one acts with the fundamental non-renormalised Hamiltonian on coarse quantum states, as well as to spin foam calculations in the same spirit, such as [ref] .
Quantum Reduced Loop Gravity: Semiclassical limit · Conclusion · contest 0.72
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The same criticism applies to computations within full loop quantum gravity (LQG) [ref] or related models [ref] , where one acts with the fundamental non-renormalised Hamiltonian on coarse quantum states, as well as to spin foam calculations in the same spirit, such as [ref] .
A quantum reduction to Bianchi I models in loop quantum gravity · Conclusion · contest 0.72
Builds on 4 papers
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Suitable candidate regularisations that generalise [ref] to such a one-parameter family were obtained in [ref] as j_z = v/2λ, k_±= √(v^2-v_m^2)/2 λ e^±2 i λb , from which [ref] can be easily verified.
Group theoretical Quantization of Isotropic Loop Cosmology · CVH algebra and regularisation · 0.92
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Suitable candidate regularisations that generalise [ref] to such a one-parameter family were obtained in [ref] as j_z = v/2λ, k_±= √(v^2-v_m^2)/2 λ e^±2 i λb , from which [ref] can be easily verified.
The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology · CVH algebra and regularisation · 0.92
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While [ref] started from a classical Poisson algebra with this property, we are going to construct such operators directly on the loop quantum cosmology (LQC) Hilbert space in this paper.
Coarse graining as a representation change · Introduction · 0.93
Uses the method or tool of 3 papers
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The reason for doing so in this paper is purely technical: it gives us access to techniques using Perelomov coherent states for the Lie group SU$(1,1)$ and, as a consequence, allows us to compute analytically a non-trivial renormalisation group flow by applying the results of [ref] .
Coarse graining as a representation change · CVH algebra and regularisation · 0.95
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Using the techniques of [ref] and the present paper, it is possible to perform an analytic computation of $ρ_b$ for arbitrary $v_b$, in particular also for the limiting case $v_b →v_gap$, where the bounce volume (in a single cell) agrees with the volume gap.
Group theoretical Quantization of Isotropic Loop Cosmology · Renormalised vs non-renormalised dynamics · 0.83
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Using the techniques of [ref] and the present paper, it is possible to perform an analytic computation of $ρ_b$ for arbitrary $v_b$, in particular also for the limiting case $v_b →v_gap$, where the bounce volume (in a single cell) agrees with the volume gap.
The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology · Renormalised vs non-renormalised dynamics · 0.83
Cites as background 26 papers
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Section [ref] reviews [ref] and the associated main idea of using the $𝔰𝔲(1,1)$ structure of the CVH algebra for coarse graining.
Coarse graining as a representation change · Introduction · 0.84
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Seminal papers about using an $𝔰𝔲(1,1)$ structure in LQC include [ref] .
Dynamical coherent states and physical solutions of quantum cosmological bounces · Quantising the CVH algebra and coarse graining · 1.00
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Seminal papers about using an $𝔰𝔲(1,1)$ structure in LQC include [ref] .
Dynamics for a 2-vertex Quantum Gravity Model · Quantising the CVH algebra and coarse graining · 1.00
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