Quantization and the Resolvent Algebra

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 compatible with the work of Binz, Honegger and Rieckers \cite{BHR}. We also define a Berezin-type quantization map on the whole C*-algebra, which continuously and bijectively maps it onto the resolvent algebra. This C*-algebra, generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure and applicability of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum

Noncompact uniform universal approximation

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb R^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all continuous activation functions $\varphi\neq0$ with asymptotically linear behaviour at $\pm\infty$. When $\varphi$ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let $\overline{\mathcal{N}_\varphi^l(\mathbb R^n)}$ denote the vector space of functions that are uniformly approximable by neural networks with $l$ hidden layers and $n$ inputs. For all $n$ and all $l\geq2$, $\overline{\mathcal{N}_\varphi^l(\mathbb R^n)}$ turns out to be an algebra under the pointwise product. If the left limit of $\varphi$ differs from its right limit (for instance, when $\varphi$ is sigmoidal) the algebra $\overline{\mathcal{N}_\varphi^l(\mathbb R^n)}$ ($l\geq2$) is independent of $\varphi$ and $l$, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of $\varphi$ equals its right limit, $\overline{\mathcal{N}_\varphi^l(\mathbb R^n)}$ ($l\geq1$) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of $l\geq1$, whereas in the former case $\overline{\mathcal{N}_\varphi^2(\mathbb R^n)}$ is strictly bigger than $\overline{\mathcal{N}_\varphi^1(\mathbb R^n)}$.Comment: 9 pages, 2 figure

C*-algebraic results in the search for quantum gauge fields

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 expansions. We prove the existence of a higher-order spectral shift function under the relative Schatten class assumption, give a converging series expansion of the spectral action in terms of Chern--Simons and Yang--Mills forms, and show one-loop renormalizability of the spectral action in a generalized sense. The second part is based on [Stienstra,vN 2020] and [vN - LMP 2022] and concerns a non-perturbative approach to quantum gauge theory by means of Hamiltonian lattice gauge theory and strict quantization. We construct C*-algebras of U(1)^n-gauge observables on the lattice, show that they are conserved under the relevant time evolutions, construct continuum limit C*-algebras, and show that the result constitutes a strict deformation quantization.Comment: PhD thesis. 149 pages, 3 figure

The basic resolvents of position and momentum operators form a total set in the resolvent algebra

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 generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results obtain for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.Comment: 8 pages, no figure

One-loop corrections to the spectral action

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 Chern-Simons forms. In the spirit of random noncommutative geometries, we consider the path integral over matrix fluctuations around a fixed noncommutative gauge background and show that the corresponding one-loop counterterms are of the same form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.Comment: 15 pages; minor corrections mad

Cyclic cocycles in the spectral action

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 that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr$(f(D+V))$ in powers of $V=\pi_D(A)$, but unlike the Taylor expansion we expand in increasing order of the forms in $A$. This extends [Connes--Chamseddine 2006], which computes only the scale-invariant part of the spectral action, works in dimension at most 4, and assumes the vanishing tadpole hypothesis. In our situation, we obtain a truly infinite odd $(b,B)$-cocycle. The analysis involved draws from recent results in multiple operator integration, which also allows us to give conditions under which this cocycle is entire, and under which our expansion is absolutely convergent. As a consequence of our expansion and of the gauge invariance of the spectral action, we show that the odd $(b,B)$-cocycle pairs trivially with $K_1$.Comment: In this new version (v2) we have corrected the statement and proof of Lemma 5. The subsequent results are unaffecte

L'Ecologisme en Allemagne et en France : deux modes différents de construction d'un nouvel acteur politique

SummaryThe N-terminally truncated variant of photoactive yellow protein (Δ25-PYP) undergoes a very similar photocycle as the corresponding wild-type protein (WT-PYP), although the lifetime of its light-illuminated (pB) state is much longer. This has allowed determination of the structure of both its dark- (pG) as well as its pB-state in solution by nuclear magnetic resonance (NMR) spectroscopy. The pG structure shows a well-defined fold, similar to WT-PYP and the X-ray structure of the pG state of Δ25-PYP. In the long-lived photocycle intermediate pB, the central β sheet is still intact, as well as a small part of one α helix. The remainder of pB is unfolded and highly flexible, as evidenced by results from proton-deuterium exchange and NMR relaxation studies. Thus, the partially unfolded nature of the presumed signaling state of PYP in solution, as suggested previously, has now been structurally demonstrated

Fire Promotes Pollinator Visitation: Implications for Ameliorating Declines of Pollination Services

Pollinators serve critical roles for the functioning of terrestrial ecosystems, and have an estimated annual value of over $150 billion for global agriculture. Mounting evidence from agricultural systems reveals that pollinators are declining in many regions of the world, and with a lack of information on whether pollinator communities in natural systems are following similar trends, identifying factors which support pollinator visitation and services are important for ameliorating the effects of the current global pollinator crisis. We investigated how fire affects resource structure and how that variation influences floral pollinator communities by comparing burn versus control treatments in a southeastern USA old-field system. We hypothesized and found a positive relationship between fire and plant density of a native forb, Verbesina alternifolia, as well as a significant difference in floral visitation of V. alternifolia between burn and control treatments. V. alternifolia density was 44% greater and floral visitation was 54% greater in burned treatments relative to control sites. When the density of V. alternifolia was experimentally reduced in the burn sites to equivalent densities observed in control sites, floral visitation in burned sites declined to rates found in control sites. Our results indicate that plant density is a proximal mechanism by which an imposed fire regime can indirectly impact floral visitation, suggesting its usefulness as a tool for management of pollination services. Although concerns surround the negative impacts of management, indirect positive effects may provide an important direction to explore for managing future ecological and conservation issues. Studies examining the interaction among resource concentration, plant apparency, and how fire affects the evolutionary consequences of altered patterns of floral visitation are overdue. DOI: 10.1371/journal.pone.007985