arXiv subjects

Hanno Hammer

Publications and source records attributed to Hanno Hammer.

12 recordsLinked to original sources

arXiv2006

Semigroup extensions of isometry groups of compactified spacetimes

We investigate the possibility of semigroup extensions of the isometry group of an identification space, in particular, of a compactified spacetime arising from an identification map $p: \RR^n_t \to \RR^n_t / Γ$, where $\RR^n_t$ is a flat pseudo-Euclidean covering space and $Γ$ is a discrete group of primitive lattice translations on this space. We show that the conditions under which such an extension is possible are related to the index of the metric on the subvector space spanned by the lattice vectors: If this restricted metric is Euclidean, no extensions are possible. Furthermore, we provide an explicit example of a semigroup extension of the isometry group of the identification space obtained by compactifying a Lorentzian spacetime over a lattice which contains a lightlike basis vector. The extension of the isometry group is shown to be isomorphic to the semigroup $(\ZZ^{\times},\cdot)$, i.e. the set of nonzero integers with multiplication as composition and 1 as unit element. A theorem is proven which illustrates that such an extension is obstructed whenever the metric on the covering spacetime is Euclidean.

hep-th↗

arXiv2006

Determination of the characteristic directions of lossless linear optical elements

We show that the problem of finding the primary and secondary characteristic directions of a linear lossless optical element can be reformulated in terms of an eigenvalue problem related to the unimodular factor of the transfer matrix of the optical device. This formulation makes any actual computation of the characteristic directions amenable to pre-implemented numerical routines, thereby facilitating the decomposition of the transfer matrix into equivalent linear retarders and rotators according to the related Poincare equivalence theorem. The method is expected to be useful whenever the inverse problem of reconstruction of the internal state of a transparent medium from optical data obtained by tomographical methods is an issue.

physics.optics↗

arXiv2005

Compactification along Lightlike Lattices

Spacetimes obtained by dimensional reduction along lattices containing a lightlike direction can admit semigroup extensions of their isometry groups. We show by concrete examples that such a semigroup can exhibit a natural order, which in turn implies the existence of preferred coordinate charts on the underlying space. Specifically, for spacetimes which are products of an external Minkowski space with an internal two-dimensional Lorentzian space, where one of the lightlike directions has a compact size, the preferred charts consist of "infinite-momentum" frames on the internal space. This implies that fields viewed from this preferred frame acquire extreme values; in particular, some of the off-diagonal components of the higher-dimensional metric, which may be regarded as gauge potentials for a field theory on the external Minkowski factor, vanish. This raises the possibility of regarding known gauge theories as part of more extended field multiplets which have been reduced in size since they are perceived from within an extreme frame. In the case of an external 4-dimensional Minkowski spacetime times a two-dimensional Lorentzian cylinder, the field content as seen in the preferred frame is that of a five-dimensional Kaluza-Klein theory, where the electrodynamic potentials Am may depend, in addition to the external spacetime coordinates, on a fifth coordinate along a lightlike direction. The fact that the metric along this direction is zero obstructs the generation of field equations from the Ricci tensor of the overall metric.

hep-th↗

arXiv2004

Collective states in highly symmetric atomic configurations, and single-photon traps

Abbreviated Abstract: We study correlated states in a circular and linear-chain configuration of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behaviour is mediated by exchange of transverse photons between the atoms. For a circular configuration of atoms the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. In this case, the radiationless energy eigenstates carry a $\mathbb{Z}_N$ quantum number $p=0,1, ..., N$ which is analogous to the angular momentum quantum number $l= 0, 1, ...$, carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s-states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing $\mathbb{Z}_N$ quantum number $p$. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of $λ/2$ emerges both in the linear-chain and the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jump-like "critical" behaviour for next-neighbour distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configuration exhibit exponential photon trapping once the next-neighbour distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for {\it single-photon traps}.

quant-ph↗

arXiv2004

Reconstruction of spatially inhomogeneous dielectric tensors via optical tomography

A method to reconstruct weakly anisotropic inhomogeneous dielectric tensors inside a transparent medium is proposed. The mathematical theory of Integral Geometry is cast into a workable framework which allows the full determination of dielectric tensor fields by scalar Radon inversions of the polarization transformation data obtained from six planar tomographic scanning cycles. Furthermore, a careful derivation of the usual equations of integrated photoelasticity in terms of heuristic length scales of the material inhomogeneity and anisotropy is provided, making the paper a self-contained account about the reconstruction of arbitrary three-dimensional, weakly anisotropic dielectric tensor fields.

physics.optics↗

arXiv2004

Collective excitations in circular atomic configurations, and single-photon traps

Correlated excitations in a plane circular configuration of identical atoms with parallel dipole moments are investigated. The collective energy eigenstates, their level shifts and decay rates are computed utilizing a decomposition of the atomic state space into carrier spaces for the irreducible representations of the symmetry group $\ZZ_N$ of the circle. It is shown that the index $p$ of these representations can be used as a quantum number analogously to the orbital angular momentum quantum number $l$ in hydrogen-like systems. Just as the hydrogen s-states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the atoms in the circle can occupy the central atom only for vanishing $\ZZ_N$ quantum number $p$. If a central atom is present, the $p=0$ state splits into two and shows level-crossing at certain radii; in the regions between these radii, damped Rabi oscillations between two "extreme" $p=0$ configurations occur. The physical mechanisms behind super- and subradiance at a given radius and the divergence of the level shifts at small interatomic distances are discussed. It is shown that, beyond a certain critical number of atoms in the circle, the lifetime of the maximally subradiant state increases exponentially with the number of atoms in the configuration, making the system a natural candidate for a {\it single-photon trap}.

quant-ph↗

arXiv2004

Characteristic Parameters in Integrated Photoelasticity: An Application of Poincare's Equivalence Theorem

The Poincare Equivalence Theorem states that any optical element which contains no absorbing components can be replaced by an equivalent optical model which consists of one linear retarder and one rotator only, both of which are uniquely determined. This has many useful applications in the field of Optics of Polarized Light. In particular, it arises naturally in attempts to reconstruct spatially varying refractive tensors or dielectric tensors from measurements of the change of state of polarization of light beams passing through the medium, a field which is known as Tensor Tomography. A special case is Photoelasticity, where the internal stress of a transparent material may be reconstructed from knowledge of the local optical tensors by using the stress-optical laws. - We present a rigorous approach to the Poincare Equivalence Theorem by explicitly proving a matrix decomposition theorem, from which the Poincare Equivalence Theorem follows as a corollary. To make the paper self-contained we supplement a brief account of the Jones matrix formalism, at least as far as linear retarders and rotators are concerned. We point out the connection between the parameters of the Poincare-equivalent model to previously introduced notions of the Characteristic Parameters of an optical model in the engineering literature. Finally, we briefly illustrate how characteristic parameters and Poincare-equivalent models naturally arise in Photoelasticity.

physics.optics↗

arXiv2003

Tree Structures: A Variational Approach to Shannon--Wiener Information

Entanglement measures based on a logarithmic functional form naturally emerge in any attempt to quantify the degree of entanglement in the state of a multipartite quantum system. These measures can be regarded as generalizations of the classical Shannon-Wiener information of a probability distribution into the quantum regime. In the present work we introduce a previously unknown approach to the Shannon-Wiener information which provides an intuitive interpretation for its functional form as well as putting all entanglement measures with a similar structure into a new context: By formalizing the process of information gaining in a set-theoretical language we arrive at a mathematical structure which we call ''tree structures'' over a given set. On each tree structure, a tree function can be defined, reflecting the degree of splitting and branching in the given tree. We show in detail that the minimization of the tree function on, possibly constrained, sets of tree structures renders the functional form of the Shannon-Wiener information. This finding demonstrates that entropy-like information measures may themselves be understood as the result of a minimization process on a more general underlying mathematical structure, thus providing an entirely new interpretational framework to entropy-like measures of information and entanglement. We suggest three natural axioms for defining tree structures, which turn out to be related to the axioms describing neighbourhood topologies on a topological space. The same minimization that renders the functional form of the Shannon-Wiener information from the tree function then assigns a preferred topology to the underlying set, hinting at a deep relation between entropy-like measures and neighbourhood topologies.

hep-th↗

arXiv2002

Orthogonality relations for triple modes at dielectric boundary surfaces

We work out the orthogonality relations for the set of Carniglia-Mandel triple modes which provide a set of normal modes for the source-free electromagnetic field in a background consisting of a passive dielectric half-space and the vacuum, respectively. Due to the inherent computational complexity of the problem, an efficient strategy to accomplish this task is desirable, which is presented in the paper. Furthermore, we provide all main steps for the various proofs pertaining to different combinations of triple modes in the orthogonality integral.

quant-ph↗

arXiv1999

Spacetime Topology, Conserved Charges, and the Splitting of Classical Multiplets

We examine consequences of non-trivial topology in background spacetimes and super-spacetimes: It is shown how the semi-invariance of a brane Lagrangian under supertranslations gives rise to topological extensions of the Noether charge algebra carried by D-branes. We investigate how isometry groups of lightlike compactified spacetimes admit an extension to a semigroup. It is shown how the splitting of classical multiplets caused by multiply-connectedness of a phase space can be described in the framework of Symplectic Geometry, based on the concept of symplectic covering spaces and local moment maps.

math-ph↗

arXiv1999

Local moment maps and the splitting of classical multiplets

We generalize the concept of global moment maps to local moment maps, whose different branches are labelled by the elements of the fundamental group of the underlying symplectic manifold. These branches can be smoothly glued together by employing fundamental-group-valued \u Cech cocycles on the phase space. In the course of this work we prove a couple of theorems on the liftability of group actions to symplectic covering spaces, and examine the possible extensions of the original group by the fundamental group of the quotient phase space. It it shown how the splitting of multiplets, this being a consequence of the multiply-connectedness of the quotient phase space, can be described by identification maps on a space of multiplets derived from a symplectic universal covering manifold. The states that are identified in this process are related by certain integrals over non-contractible loops in the quotient phase space.

math-ph↗

arXiv1997

Topological Extensions of Noether Charge Algebras carried by D-p-branes

We derive the fully extended supersymmetry algebra carried by D-branes in a massless type IIA superspace vacuum. We find that the extended algebra contains not only topological charges that probe the presence of compact spacetime dimensions but also pieces that measure non-trivial configurations of the gauge field on the worldvolume of the brane. Furthermore there are terms that measure the coupling of the non-triviality of the worldvolume regarded as a U(1)-bundle of the gauge field to possible compact spacetime dimensions. In particular, the extended algebra carried by the D-2-brane can contain the charge of a Dirac monopole of the gauge field. In the course of this work we derive a set of generalized Gamma-matrix identities that include the ones presently known for the IIA case. In the first part of the paper we give an introduction to the basic notions of Noether current algebras and charge algebras; furthermore we find a Theorem that describes in a general context how the presence of a gauge field on the worldvolume of an embedded object transforming under the symmetry group on the target space alters the algebra of the Noether charges, which otherwise would be the same as the algebra of the symmetry group. This is a phenomenon recently found by Sorokin and Townsend in the case of the M-5-brane, but here we show that it holds quite generally, and in particular also in the case of D-branes.

hep-th↗