arXiv subjects

Rodrigo Vargas Le-Bert

Publications and source records attributed to Rodrigo Vargas Le-Bert.

8 recordsLinked to original sources

arXiv2020

Renormalization flow fixed points for higher-dimensional abelian gauge fields

A connection modulo gauge symmetry on the trivial principal bundle $M\times G$ is a morphism from the loop group of $M$ into $G$. Thus, considering only loops around the 2-cells of a distinguished family of progressively refined cellular structures on $M$, the observable algebra $A$ of an abelian gauge field can be presented as an inductive limit of quotients of polynomial algebras. In that context, it turns out that the state $μ_λ:A\rightarrow\mathbb{C}$ of the Yang-Mills field on the sphere can be written $μ_λ= μ_0\mathrm{e}^{λL}$ with $λ$ an interaction strength parameter, $L:A\rightarrow A$ an explicit second-order partial differential operator and $μ_0$ the state of an almost surely flat connection. Extrapolating, we provide analogous states for the case of abelian gauge fields on $\mathbb{R}^d$.

math-ph↗

arXiv2018

Field Theory Done Right

An effective formalism for white noise analysis, conceptually equivalent to Wilsonian renormalization theory, is introduced. Space-time gets represented by a boolean lattice of coarse regions, energy scales become space-time partitions by lattice regions, and observables are elements of a projective limit with connecting maps given by partial integration of high-energy degrees of freedom. The framework allows for a seamless generalization of the Wick product and the $\mathcal S$-transform to essentially arbitrary Lévy noises, and we provide a tool to make explicit calculations in several cases of interest, including Gauss, Poisson and Gamma noises (we shall thereby encounter pretty familiar polynomials, like falling factorials and Hermite polynomials). Armed with this, we turn to constructive quantum field theory. We adopt an Euclidean approach and introduce a sufficient condition for reflection positivity, based on our $\mathcal S$-transform, enabling us to construct non-trivial quantum fields by simply specifying compatible families of effective connected $n$-point functions. We exemplify this by producing a field with quartic interaction in dimension $d\leq 8$. Its connected $n$-point functions vanish except for the propagator and the connected $4$-point function, which is that of the $ϕ^4$ field up to order $\hbar$. This model satisfies all the physical requirements of a non-trivial quantum field theory.

math-ph↗

arXiv2017

Wick Ordering and Kinetic Energy Renormalization for Lévy White Noise Fields

Let $S=\mathbb{T}^d$ be a torus and $μ$ the probability distribution of a Lévy white noise field $x:S\rightarrow\mathbb{R}$. Using projective limit measures we address the problem of making sense of $\mathrm{e}^{-T(x)}$, where $T(x) = \int_S \lvert\nabla x(s)\rvert^2 \mathrm{d}s$ is the kinetic energy, as a function in $L^1(μ)$. We start by making sense of $T(x)$ itself as a sort of distribution, which is achieved by a generalization of Wick ordering. Then we specify to the case of a $Γ$ field, finding that Wick ordering does not eliminate all divergences. Higher order renormalization would be required, but the model seems to be non-renormalizable.

math-ph↗

arXiv2015

Towards an Analytic Theory of Stochastic and Quantum Fields

We propose a method for the rigorous construction of physically relevant functional measures. In shaping it we get several conceptual insights, which can perhaps be summarized by the following statement: the renormalized interaction Lagrangian should be the generator of a flow on a space of asymptotically free cylinder functional measures with density given, in the case of Boson fields with polynomial self-interaction, by a generalized form of the Appell polynomials.

math-ph↗

arXiv2015

Cylinder Measures in Renormalization Theory

The standard approach to renormalization relies, technically, on the asymptotic perturbation of Gaussian measures embodied in Feynman diagram theory. From a mathematical standpoint this is not good enough, because thereby solving the renormalization problem does not immediately traduce into having a rigorous construction of the corresponding theory. Here, we start developing an approach to renormalization based on cylinder measures. After explaining how renormalization can be mathematically better understood in terms of them, we argue that a renormalization problem can, under certain hypothesis, be reduced to that of the corresponding strongly coupled theory, and obtain a family of solutions for the case of scalar bosons whose interaction Lagrangian does not contain derivatives. As a further application, we produce an explicit, formal expression for the cylinder measure of a local field with effectively quartic interaction at any given, fixed scale, in arbitrary dimension.

math-ph↗

arXiv2012

On the geometry underlying a real Lie algebra representation

Let $G$ be a real Lie group with Lie algebra $\mathfrak g$. Given a unitary representation $π$ of $G$, one obtains by differentiation a representation $dπ$ of $\mathfrak g$ by unbounded, skew-adjoint operators. Representations of $\mathfrak g$ admitting such a description are called \emph{integrable,} and they can be geometrically seen as the action of $\mathfrak g$ by derivations on the algebra of representative functions $g\mapsto<ξ,π(g)η>$, which are naturally defined on the homogeneous space $M=G/\kerπ$. In other words, integrable representations of a real Lie algebra can always be seen as realizations of that algebra by vector fields on a homogeneous manifold. Here we show how to use the coproduct of the universal enveloping algebra of $\mathfrak g$ to generalize this to representations which are not necessarily integrable. The geometry now playing the role of $M$ is a locally homogeneous space. This provides the basis for a geometric approach to integrability questions regarding Lie algebra representations.

math.RT↗

arXiv2012

On Complex Manifolds and Observable Schemes

We work out the construction of a Stein manifold from a commutative Arens-Michael algebra, under assumptions that are mild enough for the process to be useful in practice. Then, we do the passage to arbitrary complex manifolds by proposing a suitable notion of scheme. We do this in the abstract language of spectral functors, in view of its potential usefulness in non-commutative geometry.

math.CV↗

arXiv2011

Integrable representations of involutive algebras and Ore localization

Let $\mathcal A$ be a unital algebra equipped with an involution $(\cdot)^\dagger$, and suppose that the multiplicative set $\mathcal S\subseteq \mathcal A$ generated by the elements of the form $1 + a^\dagger a$ satisfies the Ore condition. We prove that: (i) Cyclic representations of $\mathcal A$ admit an integrable extension (acting on a possibly larger Hilbert space), and (ii) Integrable representations of $\mathcal A$ are in bijection with representations of the Ore localization $\mathcal A\mathcal S^{-1}$ (which we prove to be an involutive algebra). This second result is a limited converse to a theorem by Inoue asserting that representations of symmetric involutive algebras are integrable.

math.OA↗