arXiv subjects

Adrian P. C. Lim

Publications and source records attributed to Adrian P. C. Lim.

15 recordsLinked to original sources

arXiv2026

From Loop Quantum Gravity to a Theory of Everything

Witten described how a path integral quantization of Wilson Loop observables will define Jones polynomial type of link invariants, using the Chern-Simons gauge theory in $\mathbb{R}^3$. In this gauge theory, a compact Lie group ${\rm G}$, together with a representation of its Lie Algebra $\mathfrak{g}$, describe the symmetry group and fundamental forces acting on the particles respectively. However, it appears that this theory might be part of a bigger theory. We will incorporate this theory into the Einstein-Hilbert theory, which when reformulated and quantized using a ${\rm SU}(2) \times {\rm SU}(2)$ gauge group, gives us a quantized theory of gravity in $\mathbb{R}^4$. In this theory, we can quantize area, volume and curvature into quantum operators. By using both the Chern-Simons and Einstein-Hilbert action, we will write down a path integral expression, and compute the Wilson Loop observable for a time-like hyperlink in $\mathbb{R}^4$, each component loop is coloured with a representation for the Lie Algebra $\mathfrak{g} \times [\mathfrak{su}(2) \times \mathfrak{su}(2)]$, unifying the fundamental forces with gravity. This Wilson Loop observable can be computed using link diagrams, and it can be written as a state model, satisfying a Homfly-type skein relations. We will show that the Wilson Loop observable remains an eigenstate for the quantum operators corresponding to spin curvature, but it is not an eigenstate for the area and volume quantized operators, unless the representation for $\mathfrak{g}$ is trivial. This implies that in the Planck scale where quantum gravity is important, we see that all the particles are indistinguishable, hence the fundamental forces disappear and only interaction between matter and space-time remains.

gr-qc↗

arXiv2025

Abstract Wiener measure using abelian Yang-Mills action on $\mathbb{R}^4$

Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group. For a $\mathfrak{g}$-valued 1-form $A$, consider the Yang-Mills action \begin{equation} S_{\rm YM}(A) = \int_{\mathbb{R}^4} \left|dA + A \wedge A \right|^2\ dω\nonumber \end{equation} using the Euclidean metric on $T\mathbb{R}^4$. When we consider the Lie group ${\rm U}(1)$, the Lie algebra $\mathfrak{g}$ is isomorphic to $\mathbb{R} \otimes i$, thus $A \wedge A = 0$. For a simple closed loop $C$, we want to make sense of the following path integral, \begin{equation} \frac{1}{Z}\ \int_{A \in \mathcal{A} /\mathcal{G}} \exp \left[ \int_{C} A\right] e^{-\frac{1}{2}\int_{\mathbb{R}^4}|dA|^2\ dω}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space $\mathcal{A} /\mathcal{G}$ containing $\mathfrak{g}$-valued 1-forms modulo gauge transformations, and $Z$ is some partition function. We will construct an Abstract Wiener space for which we can define the above Yang-Mills path integral rigorously, applying renormalization techniques found in lattice gauge theory. We will further show that the Area Law formula does not hold in the abelian Yang-Mills theory.

math.PR↗

arXiv2025

Wilson Area Law formula on $\mathbb{R}^4$

Let $\mathfrak{g}$ be the Lie Algebra of a compact semi-simple gauge group. For a $\mathfrak{g}$-valued 1-form $A$, consider the Yang-Mills action \begin{equation} S_{\rm YM}(A) = \int_{\mathbb{R}^4} \left|dA + A \wedge A \right|^2\ dω, \nonumber \end{equation} using the Euclidean metric on $T\mathbb{R}^4$. We want to make sense of the following path integral, \begin{equation} {\rm Tr}\ \int_{A \in \mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}} \exp \left[ c\int_{S} dA\right] e^{-\frac{1}{2}S_{\rm YM}(A)}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space of $\mathfrak{g}$-valued 1-forms, modulo gauge transformations $\mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}$. Here, $S$ is some compact flat rectangular surface. Using an Abstract Wiener space, we can define a Yang-Mills path integral rigorously, for a compact semi-simple gauge group. Subsequently, we will then derive the Wilson area law formula from the definition, using renormalization techniques and asymptotic freedom. One of the most important applications of the Area Law formula will be to explain why the potential measured between a quark and antiquark is a linear function of its distance.

math-ph↗

arXiv2025

Positive mass gap of quantum Yang-Mills Fields

We construct a 4-dimensional quantum field theory on a Hilbert space, dependent on a simple Lie Algebra of a compact Lie group, that satisfies Wightman's axioms. This Hilbert space can be written as a countable sum of non-separable Hilbert spaces, each indexed by a non-trivial, inequivalent irreducible representation of the Lie Algebra. In each component Hilbert space, a state is given by a triple, a space-like rectangular surface $S$ in $\mathbb{R}^4$, a measurable section of the Lie Algebra bundle over this surface $S$, represented irreducibly as a matrix, and a Minkowski frame. The inner product is associated with the area of the surface $S$. In our previous work, we constructed a Yang-Mills measure for a compact semi-simple gauge group. We will use a Yang-Mills path integral to quantize the momentum and energy in this theory. During the quantization process, renormalization techniques and asymptotic freedom will be used. Each component Hilbert space is the eigenspace for the momentum operator and Hamiltonian, and the corresponding Hamiltonian eigenvalue is given by the quadratic Casimir operator. The eigenvalue of the corresponding momentum operator will be shown to be strictly less than the eigenvalue of the Hamiltonian, hence showing the existence of a positive mass gap in each component Hilbert space. We will further show that the infimum of the set containing positive mass gaps, each indexed by an irreducible representation, is strictly positive. In the last section, we will show how the positive mass gap will imply the Clustering Theorem.

math-ph↗

arXiv2023

An unitary representation of inhomogeneous ${\rm SL}(2,\mathbb{C})$ using surfaces in $\mathbb{R}^4$

We will construct a non-separable Hilbert space for which the inhomogeneous ${\rm SL}(2,\mathbb{C})$ acts on it unitarily. Each vector in this Hilbert space is described by a (rectangular) space-like surface in $\mathbb{R}^4$, for which a frame consisting of a time-like vector and a space-like vector, and a vector field is defined on it. The inner product on this Hilbert space is defined via a surface integral, which is associated with the area of the surface.

math-ph↗

arXiv2021

Loop representation of Quantum Gravity

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R}^4 \equiv \mathbb{R} \times \mathbb{R}^3$, 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 $\mathfrak{su}(2) \times \mathfrak{su}(2)$, 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.

gr-qc↗

arXiv2020

Invariants in Quantum Geometry

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or `linked' with each other, and they are closely associated with the linking number between links in $\mathbb{R}^3$. Because we distinguish the time-axis from spatial subspace in $\mathbb{R}^4$, we see that these equivalence relations, will also imply causality.

math.GT↗

arXiv2018

Quantized Curvature in Loop Quantum Gravity

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R} \times \mathbb{R}^3$. The Einstein-Hilbert action $S(e,ω)$ is defined on the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$, which are the dynamical variables in General Relativity. Define a functional $F_S(ω)$, by integrating the curvature $dω+ ω\wedge ω$ over the surface $S$, which is $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued. We integrate $F_S(ω)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $ω$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the quantized curvature can be computed from the linking number between $L$ and $S$.

math-ph↗

arXiv2017

Path Integral Quantization of Volume

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $R$ be a compact set inside $\mathbf{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $V_R(e)$ as the volume of $R$, for a given choice of $e$. The Einstein-Hilbert action $S(e,ω)$ is defined on $e$ and $ω$. We will quantize the volume of $R$ by integrating $V_R(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $R$, and the exponential of the Einstein-Hilbert action, over the space of vierbein $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the volume operator can be computed by counting the number of half-twists in the projected hyperlink, which lie inside $R$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the volume operator gives the total kinetic energy, which comes from translational and angular momentum.

math.PR↗

arXiv2017

Area Operator in Loop Quantum Gravity

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,ω)$ is defined on $e$ and $ω$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $ω$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$.

math-ph↗

arXiv2017

Einstein-Hilbert Path Integrals and Chern-Simons Integrals

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied earlier in our previous work on the Chern-Simons path integrals in $\mathbb{R}^3$. We will show that the Wilson Loop observable can be computed from a link diagram of a hyperlink, projected on a plane. Only crossings in the diagram will contribute to the path integral. Furthermore, we will show that it is invariant under an equivalence relation defined on the set of hyperlinks.

math.DG↗

arXiv2017

Einstein-Hilbert Path Integrals in $\mathbb{R}^4$

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. The Einstein-Hilbert action $S(e,ω)$ is defined using $e$ and $ω$. We will define a path integral $I$ by integrating a functional $H(e,ω)$ against a holonomy operator of a hyperlink $L$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $ω$. Three different types of functional will be considered for $H$, namely area of a surface, volume of a region and the curvature of a surface $S$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will derive and write these infinite dimensional path integrals $I$ as the limit of a sequence of Chern-Simons integrals.

math.PR↗

arXiv2010

Asymptotic estimates on the time derivative of entropy on a Riemannian manifold

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hyperbolic space shows that the time derivative of the entropy is asymptotically bounded by two positive constants.

math.DG↗

arXiv2010

A note on Gaussian correlation inequalities for nonsymmetric sets

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $γ_d(A\cap B)\geq γ_d(A)γ_d(B)$, where $γ_d$ is the standard Gaussian measure on $\mathbb{R}^d$. This generalizes Proposition 1 in [Arch. Rational Mech. Anal. 161 (2002), 257--269].

math.PR↗

arXiv2006

Path Integrals on a Compact Manifold with Non-negative Curvature

A typical path integral on a manifold, $M$ is an informal expression of the form \frac{1}{Z}\int_{σ\in H(M)} f(σ) e^{-E(σ)}\mathcal{D}σ, \nonumber where $H(M)$ is a Hilbert manifold of paths with energy $E(σ) < \infty$, $f$ is a real valued function on $H(M)$, $\mathcal{D}σ$ is a \textquotedblleft Lebesgue measure \textquotedblright and $Z$ is a normalization constant. For a compact Riemannian manifold $M$, we wish to interpret $\mathcal{D}σ$ as a Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$, equipped with its natural $G^{1}$ metric. Given an equally spaced partition, ${\mathcal{P}}$ of $[0,1],$ let $H_{\mathcal{P}%}(M)$ be the finite dimensional Riemannian submanifold of $H(M) $ consisting of piecewise geodesic paths adapted to $\mathcal{P.}$ Under certain curvature restrictions on $M,$ it is shown that \[ \frac{1}{Z_{\mathcal{P}}}e^{-{1/2}E(σ)}dVol_{H_{\mathcal{P}}% }(σ)\toρ(σ)dν(σ)\text{as}\mathrm{mesh}% ({\mathcal{P}})\to0, \] where $Z_{\mathcal{P}}$ is a \textquotedblleft normalization\textquotedblright constant, $E:H(M) \to\lbrack0,\infty)$ is the energy functional, $Vol_{H_{\mathcal{P}%}}$ is the Riemannian volume measure on $H_{\mathcal{P}}(M) ,$ $ν$ is Wiener measure on continuous paths in $M,$ and $ρ$ is a certain density determined by the curvature tensor of $M.$

math.PR↗