arXiv subjects

Matthias Schötz

Publications and source records attributed to Matthias Schötz.

16 recordsLinked to original sources

arXiv2026

The bulk spectral gap is certifiable from above but uncomputable from below

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. Here, we therefore consider the bulk spectral gap problem, which concerns only the intrinsic excitations of the infinite system, independently of any boundary. We first give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk: no algorithm can decide whether the bulk gap is positive, already for translationally invariant nearest-neighbour chains of fixed local dimension. The bulk spectral gap is thus certifiable from above but uncomputable from below. More precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. Finally, we apply the upper-bounding algorithm to the spin-1/2 kagome Heisenberg antiferromagnet to produce the first nontrivial certified upper bounds on its bulk gap.

quant-ph↗

arXiv2026

The Effective Lasserre's Perturbative Positivstellensatz

We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of $p$ augmented by weighted polynomial tails of the form $\sum_{n=0}^N (x\cdot x)^n/(n!)^t$ for $0 < t < 1$. Our main result provides an explicit quantitative bound on the truncation order $N$ required to achieve an $\varepsilon$-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that $N$ grows polynomially in $1/\varepsilon$, with rate $N = O((\|p\|/\varepsilon)^{1/(1-t)})$.

math.OC↗

arXiv2024

Rings of almost everywhere defined functions

The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean and localizable. Here we assume that the positive cone of $R$ is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on $X$ is one that is defined on a dense open subset of $X$. A partially ordered commutative ring $R$ is archimedean if the underlying additive partially ordered abelian group is archimedean, and $R$ is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the $σ$-bounded case, lattice-ordered commutative rings ($f$-rings), partially ordered fields, and commutative operator algebras.

math.RA↗

arXiv2024

Real Nullstellensatz for 2-step nilpotent Lie algebras

We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra $\mathbb R[x_1, \dots, x_d]$ we consider the universal enveloping *-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical *-involution). Evaluation at points of $\mathbb R^d$ is then generalized to evaluation through integrable *-representations, which in this case are equivalent to filtered *-algebra morphisms from the universal enveloping *-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such *-algebra morphisms as the real ideals of the universal enveloping *-algebra.

math.AG↗

arXiv2024

Associativity and Commutativity of Partially Ordered Rings

Consider a commutative monoid $(M,+,0)$ and a biadditive binary operation $μ\colon M \times M \to M$. We will show that under some additional general assumptions, the operation $μ$ is automatically both associative and commutative. The main additional assumption is localizability of $μ$, which essentially means that a certain canonical order on $M$ is compatible with adjoining some multiplicative inverses of elements of $M$. As an application we show that a division ring $\mathbb F$ is commutative provided that for all $a \in \mathbb F$ there exists a natural number $k$ such that $a-k$ is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings (``extended'' in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e.~for (extended) $f$-rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem.

math.RA↗

arXiv2023

Symmetry Reduction of States I

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra $g$. The key idea advocated for in this article is that the ``correct'' notion of positivity on a *-algebra $A$ is not necessarily the algebraic one, for which positive elements are sums of Hermitian squares $a^*a$ with $a \in A$, but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on $A$ thus depends on this choice of positivity on $A$, and the notion of positivity on the reduced algebra $A_{red}$ should be such that states on $A_{red}$ are obtained as reductions of certain states on $A$. We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold $M$, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a $U(1)$-action.

math-ph↗

arXiv2022

Positivstellensätze for Semirings

In this paper we develop a number of results and notions concerning Positivstellensätze for semirings (preprimes) of commutative unital real algebras. First we reduce the Archimedean Positivstellensatz for semirings to the corresponding result for quadratic modules. Various applications of the Archimedean Positivstellensatz for semirings are investigated. A general Positivstellensatz with denominators is proved for filtered algebras with semirings. As an application we derive a denominator-free Positivstellensatz for the cylindrical extension of an algebra with Archimedean semiring. A large number of illustrating examples are given.

math.AG↗

arXiv2022

Gelfand-Naimark Theorems for Ordered *-Algebras

The classical Gelfand--Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for $σ$-bounded closed ordered *-algebras a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand--Naimark representation theorems to classes of *-algebras larger than C*-algebras, and which especially contain *-algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $σ$-bounded, then the order of V is induced by the extremal positive linear functionals on V.

math.OA↗

arXiv2022

Symmetry Reduction of States II: A non-commutative Positivstellensatz for CPn

We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n}, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give an algebraic description of the real-valued U(1)-invariant polynomials that are positive in certain *-representations on Hilbert spaces of holomorphic sections of line bundles over CP^n. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive elements. As an application, all *-representations of the quantization of the polynomial *-algebra on CP^n, obtained e.g. through phase space reduction or Berezin--Toeplitz quantization, are determined.

math.QA↗

arXiv2020

Universal Continuous Calculus for Su*-Algebras

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e. "strictly" positive elements are invertible, and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for C*-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.

math.FA↗

arXiv2020

Equivalence of Order and Algebraic Properties in Ordered *-Algebras

The aim of this article is to describe a class of *-algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered *-algebras (*-algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C*-norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered *-algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su*-algebras as uniformly complete Archimedean ordered *-algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su*-algebras generalize some important properties of C^*-algebras to algebras of unbounded operators. Similarly, they generalize some properties of Phi-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered *-algebras. As an example, Su*-algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as *-algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system.

math.OA↗

arXiv2020

Wick Rotations in Deformation Quantization

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from $\mathbb{C}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space $\mathbb{CP}^n$ and the complex hyperbolic disc $\mathbb{D}^n$. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of "polynomial" functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking e.g. the star products on $\mathbb{CP}^n$ and $\mathbb{D}^n$. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the *-involution of pointwise complex conjugation.

math.QA↗

arXiv2019

Stone-Weierstraß Theorems for Riesz Ideals of Continuous Functions

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone-Weierstraß-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.

math.FA↗

arXiv2018

On Characters and Pure States of *-Algebras

It is easy to see that every character (i.e. unital *-homomorphism to the complex numbers) of a commutative unital associative *-algebra is a pure state (i.e. extreme point in the convex set of all normalized positive linear functionals). This article gives sufficient conditions for the converse to be true as well. In order to formulate these results together with similar ones, e.g. for locally convex *-algebras, the notion of an abstract O*-algebra (unital associative *-algebra with an order defined by positive linear functionals) is introduced. Many concepts and intermediary results discussed here also apply to the non-commutative case.

math.FA↗

arXiv2018

A Convergent Star Product on the Poincaré Disc

On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra.

math.CV↗

arXiv2017

Convergent Star Products for Projective Limits of Hilbert Spaces

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.

math.QA↗