arXiv subjects

Maximilian Hanusch

Publications and source records attributed to Maximilian Hanusch.

16 recordsLinked to original sources

arXiv2023

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups

We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor's sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker-Campbell-Dynkin-Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.

math.FA↗

arXiv2023

A $\mathcal{C}^k$-Seeley-Extension-Theorem for Bastiani's Differential Calculus

We generalize a classical extension result by Seeley in the context of Bastiani's differential calculus to infinite dimensions. The construction follows Seeley's original approach, but is significantly more involved as not only $C^k$-maps (for $k\in \mathbb{N}\cup\{\infty\}$) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certain compatibility (and continuity) conditions. Various applications are discussed as well.

math.FA↗

arXiv2022

Decompositions of Analytic 1-Manifolds

In a previous article, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is proven under the assumption that the action is non-contractive (which is less restrictive than regular and separately analytic).

math.DG↗

arXiv2022

Symmetries of Analytic Curves

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves, each of them discretely generated by the symmetry group (i.e., each such curve naturally decomposes into countably many symmetry free subcurves that are mutually and uniquely related by the Lie group action). We additionally extend the classification result to the analytic 1-submanifold case. Specifically, we show that an analytic 1-submanifold is either free or (exponential, i.e.) analytically diffeomorphic (via the exponential map) to the unit circle or an interval. The corresponding decomposition results in the free case are outlined in this paper, but proven in a separate one.

math.DG↗

arXiv2022

Regularity of Lie Groups

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $μ$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $μ$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ it is differentiable at zero $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey$\hspace{1pt}/ \hspace{1pt}$integral$\hspace{1pt}$ complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.

math.FA↗

arXiv2020

The Strong Trotter Property for Locally $μ$-convex Lie Groups

We show that an infinite dimensional Lie group in Milnor's sense has the strong Trotter property if it is locally $μ$-convex. This is a continuity condition imposed on the Lie group multiplication that generalizes the triangle inequality for locally convex vector spaces, and is equivalent to $C^0$-continuity of the evolution map on its domain. In particular, the result proven in this paper significantly extends the respective result obtained by Glöckner in the context of measurable regularity.

math.FA↗

arXiv2020

The Regularity Problem for Lie Groups with Asymptotic Estimate Lie Algebras

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that $\mathrm{evol}$ is $C^\infty$-continuous on $\mathrm{D}\cap C^\infty([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is $C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore show that $G$ is k-confined for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$ if $G$ is constricted. (The latter condition is slightly less restrictive than to be asymptotic estimate.) Results obtained in a previous paper then imply that an asymptotic estimate Lie group $G$ is $C^\infty$-regular if and only if it is Mackey complete, locally $μ$-convex, and has Mackey complete Lie algebra - In this case, $G$ is $C^k$-regular for each $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ (with ``smoothness restrictions'' for $k\equiv\mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially complete with integral complete Lie algebra.

math.FA↗

arXiv2019

Differentiability of the Evolution Map and Mackey Continuity

We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$-semiregular Lie group $G$ (for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$) admits a particular kind of sequentially continuity $-$ called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially-, and the Mackey continuous context. We furthermore conclude that if the Lie algebra of $G$ is a Fréchet space, then $G$ is $C^k$-semiregular (for $k\in \mathbb{N}\sqcup\{\infty\}$) if and only if $G$ is $C^k$-regular.

math.FA↗

arXiv2018

Uniqueness of the Representation in Homogeneous Isotropic LQC

We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux $^*$-algebra that is invariant under residual diffeomorphisms $-$ both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to have a well-defined action on the quantum algebra, we have let them act on the fiducial cell as well as on the dynamical variables, thereby recovering covariance. Consistency with Ashtekar and Campiglia in the Bianchi I case is also shown.

gr-qc↗

arXiv2016

Kinematical uniqueness of homogeneous isotropic LQC

In a paper by Ashtekar and Campiglia, invariance under volume preserving residual diffeomorphisms has been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). In this paper, we use invariance under all residual diffeomorphisms to single out the standard kinematical Hilbert space of homogeneous isotropic LQC for both the standard configuration space $\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one $\mathbb{R} \sqcup \mathbb{R}_{\mathrm{Bohr}}$. We first determine the scale invariant Radon measures on these spaces, and then show that the Haar measure on $\mathbb{R}_{\mathrm{Bohr}}$ is the only such measure for which the momentum operator is hermitian w.r.t. the corresponding inner product. In particular, the measure is forced to be identically zero on $\mathbb{R}$ in the Fleischhack case, so that for both approaches, the standard kinematical LQC-Hilbert space is singled out.

gr-qc↗

arXiv2016

Invariant Connections in Loop Quantum Gravity

Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the antihomomorphisms $Θ\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $Φ\colon G\times \mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ whose translation maps $Φ_g$ are continuous; whereby continuities of $Θ$ and $Φ$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $ϕ\colon G \times X\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if $ϕ_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.

math-ph↗

arXiv2016

Invariant Connections and Symmetry Reduction in Loop Quantum Gravity

The intention of this thesis is to provide general tools and concepts that allow to perform a mathematically substantiated symmetry reduction in (quantum) gauge field theories. Here, the main focus is on the framework of loop quantum gravity (LQG), where we concentrate on the reduction of the quantum configuration space, and the construction of a normalized Radon measures on the reduced one. More precisely, we introduce a new way to symmetry reduce the LQG-configuration space directly on the quantum level, and then show that this always leads to a (strictly) larger reduced space than quantizing the classical configuration space of invariant connections (traditional approach). We prove a general classification theorem for such invariant connections, which we then use to calculate the classical configuration space for the homogeneous and the spherically symmetric case. Here, the backbone of the introduced reduction concept is a lifting result for group actions on sets to spectra of $C^*$-subalgebras of the bounded functions thereon; and as a further application of this, we single out the standard kinematical Hilbert space of homogeneous isotropic loop quantum cosmology by means of the same invariance condition for both the standard configuration space $\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one $\mathbb{R}\sqcup\mathbb{R}_{\mathrm{Bohr}}$. Along the way, symmetries of embedded analytic curves under a given analytic Lie group action are investigated, and a first classification result is proven for the case that the action is proper or pointwise proper and transitive, and only admits normal stabilizers.

math-ph↗

arXiv2015

Uniqueness of Measures in Loop Quantum Cosmology

In a paper of Ashtekar and Campiglia, residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations w.r.t. the extended $\mathbb{R}$-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space $\mathbb{R}_\mathrm{Bohr}$ as well as on the Fleischhack one. Thus, in both situation, the same condition singles out the standard kinematical Hilbert space of LQC.

math-ph↗

arXiv2015

Projective Structures in Loop Quantum Cosmology

Projective structures have successfully been used for the construction of measures in the framework of loop quantum gravity. In the present paper, we establish such structures for the configuration space $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$, recently introduced in the context of homogeneous isotropic loop quantum cosmology. In contrast to the traditional space $\mathbb{R}_{\mathbb{Bohr}}$, the first one is canonically embedded into the quantum configuration space of the full theory. In particular, for the embedding of states into a corresponding symmetric sector of loop quantum gravity, this is advantageous. However, in contrast to the traditional space, there is no Haar measure on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$ defining a canonical kinematical $L^2$-Hilbert space on which operators can be represented. The introduced projective structures allow to construct a family of natural measures on $\mathbb{R}\sqcup \mathbb{R}_{\mathbb{Bohr}}$ whose corresponding $L^2$-Hilbert spaces we finally investigate.

math-ph↗

arXiv2015

A Characterization of Invariant Connections

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $ψ\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where $G$ admits only one orbit type. Along the way, we give applications to loop quantum gravity.

math-ph↗

arXiv2010

Hochschild-Kohomologien von Observablenalgebren in der Klassischen Feldtheorie

This is my diploma thesis in german language. In the context of formal deformation theorie of assoziative observables in classical field theory I consider the symmetric algebra S(V) on an arbitrary-dimensional R- or C-vectorspace V as a prototype of comprehensive observables algebras in quantum field theory. In this framework I calculate the Hochschild cohomologies of S(V) with values in S(V)-S(V)-bimodules M. In the case that V is a locally convex vectorspace I compute the continuous Hochschild cohomologies for the (with help of the pi-tensor product) locally convex topologised symmetric Algebra on V and likewise for the completion Hol(V) of S(V) if V is in addition a Hausdorff space and M is complete. For all this cases and in the situation of symmetric bimodules M I prove generalized Hochschild-Kostant-Rosenberg theorems by use of explicite chain maps. Furthermore I have found useful statements about the differential Hochschild cohomologies in the case that M is a bimodule whose right modul multiplication can be written as a sum of the left modul multiplication and higher differential terms.

math-ph↗