arXiv subjects

Albert Much

Publications and source records attributed to Albert Much.

At least 37 records · Page 2Linked to original sources

arXiv2023

Non-commutative Geometry from Perturbative Quantum Gravity in de Sitter spacetime

We show that a non-commutative structure arises naturally from perturbative quantum gravity in a de Sitter background metric. Our work builds on recent advances in the construction of observables in highly symmetric background spacetimes [Brunetti et al., JHEP 08, 032 (2016); Fröb and Lima, Class. Quant. Grav. 35, 095010 (2018)], where the dynamical coordinates that are needed in the relational approach were established for such backgrounds to all orders in perturbation theory. We show that these dynamical coordinates that describe events in the perturbed spacetime are naturally non-commuting, and determine their commutator to leading order in the Planck length. Our result generalizes the causal non-commutative structure that was found using the same approach in Minkowski space [Fröb, Much and Papadopoulos, Phys. Rev. D 107, 064041 (2023)].

gr-qc↗

arXiv2023

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations (CCR) which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

math-ph↗

arXiv2023

Non-commutative Geometry from Perturbative Quantum Gravity

Trying to connect a fundamentally non-commutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are non-commutative geometrical effects already present in the regime where perturbative quantum gravity provides a predictive framework? Moreover, is it necessary to introduce non-commutativity by hand, or does it arise through quantum-gravitational effects? We show that the first question can be answered in the affirmative, and the second one in the negative: perturbative quantum gravity predicts non-commutativity at the Planck scale, once one clarifies the structure of observables in the quantum theory.

gr-qc↗

arXiv2023

Non-commutative coordinates from quantum gravity

Local observables in (perturbative) quantum gravity are notoriously hard to define, since the gauge symmetry of gravity -- diffeomorphisms -- moves points on the manifold. In particular, this is a problem for backgrounds of high symmetry such as Minkowski space or de Sitter spacetime which describes the early inflationary phase of our universe. Only recently this obstacle has been overcome, and a field-dependent coordinate system has been constructed to all orders in perturbation theory, in which observables are fully gauge-invariant. We show that these field-dependent coordinates are non-commutative, and compute their commutator to second order in the Planck length. This provides the first systematic derivation of non-commutativity that arises due to quantum gravity effects.

gr-qc↗

arXiv2022

An approximate local modular quantum energy inequality in general quantum field theory

For every local quantum field theory on a static, globally hyperbolic spacetime of arbitrary dimension, assuming the Reeh-Schlieder property, local preparability of states, and the existence of an energy density as operator-valued distribution, we prove an approximate quantum energy inequality for a dense set of vector states. The quantum field theory is given by a net of von Neumann algebras of observables, and the energy density is assumed to fulfill polynomial energy bounds and to locally generate the time translations. While being approximate in the sense that it is controlled by a small parameter that depends on the respective state vector, the derived lower bound on the expectation value of the spacetime averaged energy density has a universal structure. In particular, the bound is directly related to the Tomita-Takesaki modular operators associated to the local von Neumann algebras. This reveals general, model-independent features of quantum energy inequalities for a large class of quantum field theories on static spacetimes.

math-ph↗

arXiv2022

A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes

The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.

math-ph↗

arXiv2022

Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes

We develop a rigorous method to parametrize complex structures for Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of globally hyperbolic spacetimes. In particular, we consider static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a huge class of spacetimes we prove that the differential equation for the complex structure is given by the Gelfand-Dikki equation.

math-ph↗

arXiv2021

On Global Hyperbolicity of Spacetimes: Topology Meets Functional Analysis

This chapter is an up-to-date account of results on globally hyperbolic spacetimes, and serves several purposes. We begin with the exposition of results from a foundational level, where the main tools are order theory and general topology, we continue with results of a more geometric nature, and we conclude with results that are related to current research in theoretical physics. In each case, we list a number of open questions and formulate, for a class of spacetimes, an interesting connection between global hyperbolicity of a manifold and the geodesic completeness of its corresponding space-like surfaces. This connection is substantial for the proof of essential self-adjointness of a class of pseudo differential operators, that stem from relativistic quantum field theory.

math.DG↗

arXiv2021

Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

math-ph↗

arXiv2019

Stationary Spacetimes and Self-Adjointness in Klein-Gordon Theory

We consider the problem of essential self-adjointness of the spatial part of the Klein-Gordon operator in stationary spacetimes. This operator is shown to be a Laplace-Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential selfadjointness is proven assuming smoothness of the metric components and semi-boundedness of the potential. This extends a recent result for static spacetimes to the stationary case. Furthermore, we generalize the results to certain non-globally hyperbolic spacetimes.

math-ph↗

arXiv2019

Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type

A general formula is calculated for the connection of a central metric w.r.t.\ a noncommutative spacetime of Lie-algebraic type. This is done by using the framework of linear connections on central bi-modules. The general formula is further on used to calculate the corresponding Riemann tensor and prove the corresponding Bianchi identities and certain symmetries that are essential to obtain a symmetric and divergenceless Einstein Tensor. In particular, the obtained Einstein Tensor is not equivalent to the sum of the noncommutative Riemann tensor and scalar, as in the commutative case, but in addition a traceless term appears.

math-ph↗

arXiv2018

A New Algorithm for the Inverse of Matrices with Noncommuting Entries

By using the quasi-determinant the construction of Gel'fand et al. leads to the inverse of a matrix with noncommuting entries. In this work we offer a new method that is more suitable for physical purposes and motivated by deformation quantization, where our constructed algorithm emulates the commutative case and in addition gives corrections coming from the noncommutativity of the entries. Furthermore, we provide an equivalence of the introduced algorithm and the construction via quasi-determinants.

math-ph↗

arXiv2017

Curving Flat Space-Time by Deformation Quantization?

We use a deformed differential structure to obtain a curved metric by a deformation quantization of the flat space-time. In particular, by setting the deformation parameters to be equal to physical constants we obtain the Friedmann-Robertson-Walker (FRW) model for inflation and a deformed version of the FRW space-time. By calculating classical Einstein-equations for the extended space-time we obtain non-trivial solutions. Moreover, in this framework we obtain the Moyal-Weyl, i.e. a constant non-commutative space-time, by a consistency condition.

math-ph↗

arXiv2017

Gravity from Quantum Spacetime by Twisted Deformation of the Quantum Poincaré Group

We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from Quantum Gravity. Specifically we consider a two-parameter class of twisted Poincaré algebras, from which Lie-algebraic noncommutativities of the translations are derived as well as associative star-products, deformed Riemannian geometries, Lie-algebraic twisted Minkowski spaces and quantum effects that arise as noncommutativities. Starting from a universal differential algebra of forms based on the above mentioned Lie-algebraic noncommutativities of the translations, we construct the noncommutative differential forms and Inner and Outer derivations, which are the noncommutative equivalents of the vector fields in the case of commutative differential geometry. Having established the essentials of this formalism we construct a bimodule, required to be central under the action of the Inner derivations in order to have well defined contractions and from where the algebraic dependence of its coefficients is derived. This again then defines the noncommutative equivalent of the geometrical line-element in commutative differential geometry. We stress, however, that even though the components of the twisted metric are by construction symmetric in their algebra valuation, this is not so for their inverse and thus to construct it we made use of Gel'fand's theory of quasi-determinants, which is conceptually straightforward but computationally becoming quite complicate beyond an algebra of 3 generators. The consequences of the noncommutativity of the Lie-algebra twisted geometry are further discussed.

math-ph↗

arXiv2017

A Poincaré Covariant Noncommutative Spacetime?

We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the \textbf{whole} Poincaré group.

math-ph↗

arXiv2016

Massless QFT and the Newton-Wigner Operator

In this work, the second-quantized version of the spatial-coordinate operator, known as the Newton-Wigner-Pryce operator, is explicitly given w.r.t. the massless scalar field. Moreover, transformations of the conformal group are calculated on eigenfunctions of this operator in order to investigate the covariance group w.r.t. probability amplitudes of localizing particles.

math-ph↗

arXiv2016

Remarks on QFT in the Coordinate Space

Generators of the Poincaré group, for a free massive scalar field, are usually expressed in the momentum space. In this work we perform a transformation of these generators into the coordinate space. This (spatial)-position space is spanned by eigenvectors of the Newton-Wigner-Pryce operator. The motivation is a deeper understanding of the commutative spatial coordinate space in QFT, in order to investigate the non-commutative version thereof.

math-ph↗

arXiv2016

Learning a Deep Model for Human Action Recognition from Novel Viewpoints

Recognizing human actions from unknown and unseen (novel) views is a challenging problem. We propose a Robust Non-Linear Knowledge Transfer Model (R-NKTM) for human action recognition from novel views. The proposed R-NKTM is a deep fully-connected neural network that transfers knowledge of human actions from any unknown view to a shared high-level virtual view by finding a non-linear virtual path that connects the views. The R-NKTM is learned from dense trajectories of synthetic 3D human models fitted to real motion capture data and generalizes to real videos of human actions. The strength of our technique is that we learn a single R-NKTM for all actions and all viewpoints for knowledge transfer of any real human action video without the need for re-training or fine-tuning the model. Thus, R-NKTM can efficiently scale to incorporate new action classes. R-NKTM is learned with dummy labels and does not require knowledge of the camera viewpoint at any stage. Experiments on three benchmark cross-view human action datasets show that our method outperforms existing state-of-the-art.

cs.CV↗