Quantized Curvature in Loop Quantum Gravity

Lim, Adrian P. C.

doi:10.1016/s0034-4877(19)30007-2

Cited by 4 publications

(8 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The path integrals involving area, volume and curvature, were all defined and computed in [31], [34] and [35] respectively. Each of these path integrals can be explicitly computed using topological invariants defined in [32], hence consistent with the view point that geometric notions should play a central role in LQG, as stated in [8].…”

Section: Spin Networkmentioning

confidence: 99%

“…In [35], we quantized the curvature of S into an operator FS using the following path integral expression (indexed by a parameter κ)…”

Section: Definementioning

confidence: 99%

“…The main theorem in [35] says that the limit of this sequence of path integrals in Expression 7.3 as κ goes to infinity, is computed from the linking number between L and the surface S, denoted as lk(L, S). See [32].…”

Section: Definementioning

confidence: 99%

See 2 more Smart Citations

Loop representation of Quantum Gravity

Lim¹

2021

Preprint

View full text Add to dashboard Cite

A hyperlink is a finite set of non-intersecting simple closed curves in R 4 ≡ R × R 3 , each curve is either a matter or geometric loop. We consider an equivalence class of such hyperlinks, up to time-like isotopy, preserving time-ordering. Using an equivalence class and after coloring each matter component loop with an irreducible representation of su(2) × su(2), we can define its Wilson Loop observable using an Einstein-Hilbert action, which is now thought of as a functional acting on the set containing equivalence classes of hyperlink.Construct a vector space using these functionals, which we now term as quantum states. To make it into a Hilbert space, we need to define a counting probability measure on the space containing equivalence classes of hyperlinks.In our previous work, we defined area, volume and curvature operators, corresponding to given geometric objects like surface and a compact solid spatial region. These operators act on the quantum states and by deliberate construction of the Hilbert space, are self-adjoint and possibly unbounded operators.Using these operators and Einstein's field equations, we can proceed to construct a quantized stress operator and also a Hamiltonian constraint operator for the quantum system. We will also use the area operator to derive the Bekenstein entropy of a black hole.In the concluding section, we will explain how Loop Quantum Gravity predicts the existence of gravitons, implies causality and locality in quantum gravity, and formulate the principle of equivalence mathematically in its framework.

show abstract

Section: Spin Networkmentioning

confidence: 99%

“…In [35], we quantized the curvature of S into an operator FS using the following path integral expression (indexed by a parameter κ)…”

Section: Definementioning

confidence: 99%

See 1 more Smart Citation

Loop representation of Quantum Gravity

Lim¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Quantized curvature now becomes an invariant under an equivalence relation, computed using the linking number between a surface and a hyperlink. See [7].…”

Section: Definition 33 (Confinement Number)mentioning

confidence: 99%

“…Another geometrical object that appears in Loop Quantum Gravity would be a compact surface, with or without boundary. As seen in [6] , [7] and [8], the surface plays an important role in the quantization of area and curvature.…”

mentioning

confidence: 99%

Invariants in Quantum Geometry

Lim¹

2017

Preprint

View full text Add to dashboard Cite

In our earlier work, we used Einstein-Hilbert path integrals to quantize gravity in loop quantum gravity. The observables are actually area and curvature of a surface and volume of a three dimensional manifold, which are quantized into operators acting on quantum states. The quantum states are defined using a set of non-intersecting loops in R × R 3 .A successful quantum theory of gravity in R × R 3 should be invariant under the diffeomorphism group. This means that the eigenvalues of the quantized operators should yield invariants for ambient isotopic classes of surfaces and three dimensional manifold. In this article, we would like to discuss some of these invariants which appear in loop quantum gravity. We will also define and discuss an equivalence class of submanifolds to be considered in loop quantum gravity.

show abstract

Loop Representation of Quantum Gravity

Lim

2023

Ann. Henri Poincaré

View full text Add to dashboard Cite

Quantized Curvature in Loop Quantum Gravity

Cited by 4 publications

References 16 publications

Loop representation of Quantum Gravity

Loop representation of Quantum Gravity

Invariants in Quantum Geometry

Loop Representation of Quantum Gravity

Contact Info

Product

Resources

About