Invariants in Quantum Geometry

Lim, Adrian P. C.

doi:10.1016/s0034-4877(21)00013-6

Cited by 3 publications

(16 citation statements)

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“…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]. Thus, we now have a consistent way of quantizing physical observables into operators, which we will summarize the results in Section 7.…”

Section: Spin Networkmentioning

confidence: 53%

“…In [31], we defined the hyperlinking number of a hyperlink, and showed how one can compute Wilson Loop observables from it. In [32], we showed that the hyperlinking number is equal to the linking number of the projection of a hyperlink to form a planar graph, up to ±1, provided we define a time-ordering (Definition 4.5) for the component loops. Under time-like isotopy (Definition 4.3) and time ordering (Definition 5.6), the hyperlinking number of a hyperlink will be an invariant.…”

Section: Spin Networkmentioning

confidence: 99%

“…In [31] or [32], we defined the hyperlinking number sk(l u , L) between an oriented time-like loop l u and an oriented time-like hyperlink L, computed from χ(L, L). Suppose the matter hyperlink…”

Section: Hooft Inmentioning

confidence: 99%

“…Even though the hyperlinking number is well-defined for a pair of time-like hyperlinks {L, L}, tangled to form a time-like hyperlink χ(L, L), it is however not an invariant under the equivalence relation given in Definition 4.3. To obtain an equivalence invariant, it is necessary to time-order the loops, as was done in [32]. Definition 5.6 (Time-ordered pair of hyperlinks) Suppose we have a time-like hyperlink, denoted as χ(L, L), consisting of 2 non-empty sets of timelike hyperlink,…”

Section: Hooft Inmentioning

confidence: 99%

“…Suppose we have a matter loop l and a geometric loop l, both tangled together to form a time-like hyperlink χ(l, l). Suppose that l < l. In [32], we showed that the hyperlinking number sk(l, l) will be equal to 3 × lk(π 0 (l), π 0 (l)); we will add a negative sign in front if l > l.…”

Section: Hooft Inmentioning

confidence: 99%

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Loop representation of Quantum Gravity

Lim¹

2021

Preprint

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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.

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