Abstract Wiener measure using abelian Yang-Mills action on R4\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 \nonumber \end{equation} using the standard metric on $T\mathbb{R}^4$. When we consider the Lie group $U(1)$, the Lie algebra $\mathfrak{g}$ is isomorphic to $\mathbb{R} \otimes i$, thus $A \wedge A = 0$. For some 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}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space of $\mathfrak{g}$-valued 1-forms, modulo gauge transformations, $\mathcal{A} /\mathcal{G}$, 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, using renormalization techniques found in lattice gauge theory. We will further show that the Area Law formula do not hold in the abelian Yang-Mills theory

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.Comment: 15 page

Positive mass gap of quantum Yang-Mills Fields

We construct a 4-dimensional quantum field theory on a non-separable Hilbert space, dependent on a simple Lie Algebra of a compact Lie group, that satisfies Wightman Axioms. This Hilbert space can be written as a countable sum of Hilbert spaces, each indexed by a non-trivial, inequivalent irreducible representation of its Lie Algebra. In each component Hilbert space, a state is given by a double, a space-like rectangular surface $S$ in $\mathbb{R}^4$ and a measurable section of the Lie Algebra bundle over this surface $S$, represented irreducibly as a matrix. 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 decomposition theorem.Comment: 86 page

An unitary representation of inhomogeneous SL(2,C){\rm SL}(2,\mathbb{C}) using surfaces in R4\mathbb{R}^4

We will construct a 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) surface in $\mathbb{R}^4$, for which 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

Wilson Area Law formula on R4\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 \nonumber \end{equation} using the standard 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 potential.Comment: arXiv admin note: text overlap with arXiv:1701.0152