Teun van Nuland

Delft University of Technology | TU · Analysis

Given H self-adjoint, V symmetric and relatively H-bounded, and f : R → C satisfying mild conditions, we show that the Gateaux derivative d n dt n f (H + tV)| t=0 exists in the operator norm topology, for every natural n, give a new explicit formula for this derivative in terms of multiple operator integrals, and establish useful perturbation formu...

We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus...

Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a $$\hbox {C}^*$$ C ∗ -algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $$T^*\mathbb {T}^n$$ T...

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space R^n. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ±∞. When φ is more...

We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subje...

Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ+bP-ir)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepa...

Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ + bP − ir)^(−1) for real constants a, b, r =/= 0. This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra R gener...

Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ + bP - i r)^{-1} for real constants a,b,r=/=0. This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra R generate...

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space R^n. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions ϕ that are continuous, nonlinear, and asymptotically linear at ±∞. When ϕ is moreover bou...

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators H and V such that V is bounded and V(H-iI)^{-1} belongs to a Schatten–von Neumann ideal \mathcal{S}^n of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace...

We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative $d$-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order $k$ and in any dimension $d$. Our recursive formula can conveniently produce all...

This thesis consists of two parts, both situated in operator theory, and both motivated by the quest for rigorous quantizations of gauge theories. The first part is based on [Skripka,vN - JST 2022], [van Suijlekom,vN - JNCG 2021], and [van Suijlekom,vN - JHEP 2022], and concerns the spectral action of noncommutative geometry and its perturbative ex...

Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and als...

We present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integra...

A bstract We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability in a generalized sense, while staying within the spectral framework of noncommutative geometry. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and C...

This paper shows how to construct classical and quantum field C*-algebras modeling a U(1)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)^n$$\end{document}-gauge t...

We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern–Simons actions and Yang–Mills actions of all orders. In the odd orders, generalized Chern–Simons forms are integrated against an odd (b,B) -cocycle, whereas, in the even orders, powers of the curvature are integrated against (b,B) -cocycles th...

This paper shows how to construct classical and quantum field C*-algebras modeling a $U(1)^n$-gauge theory in any dimension using a novel approach to lattice gauge theory, while simultaneously constructing a strict deformation quantization between the respective field algebras. The construction starts with quantization maps defined on operator syst...

We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and Chern-Simons forms, for which we show here that the one-loop counterterms are of the sam...

We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern--Simons actions and Yang--Mills actions of all orders. In the odd orders, generalized Chern--Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycl...

We affirmatively settle the question on existence of a real-valued spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $\S^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as i...

Buchholz and Grundling (Comm. Math. Phys., 272, 699--750, 2007) introduced a C$^\ast$-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space, and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $T^*\mathbb{T}^n$ of an $n$-t...

We introduce a novel commutative C*-algebra of functions on a symplectic vector space admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra to the resolvent algebra introduced by Buchholz and Grundling \cite{BG2008}. The associated quantization map is a field-theoretical Weyl quantization compatibl...

We introduce a novel commutative C*-algebra CR(X) of functions on a symplectic vector space (X,σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of CR(X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantizatio...