In recent years, there has been an increasing interest to provide supermechanics with a solid geometrical base. Some of the difficulties are right at the beginning: there is a general consent about the configuration space, which is taken to be a supermanifold, but there is no general agreement on what the velocity phase space should be. Naturally, the candidate for that should be a generalization of the tangent bundle in the context of supergeometry; unfortunately there are several different (and reasonable) notions that could serve that purpose. The tangent supermanifold introduced by Ibort and Marín-Solano in [8] seems to be the right candidate from the point of view of a physicist since a good deal of supermechanics can rigorously be developed; nevertheless, the main disadvantage of the tangent supermanifold is that it is not a bundle in any sense, which forces a local coordinate approach that does not give too much insight, or a purely algebraic approach, in practice difficult to handle; besides, in classical mechanics one usually takes advantage of the fact that vector fields, for instance, are, after all, sections of a bundle. In this sense, the tangent superbundle introduced by Sánchez–Valenzuela [10,11,2] seems to be quite appropriate, from a theoretical point of view, since there is a one–to–one correpondence between sections of this superbundle and supervector fields regarded as superderivations. Unfortunately, the tangent superbundle is too big, its dimension is (2m + n, 2n + m) if the dimension of the configuration space is (m,n).

Abstract

Introduction

Four-dimensional Einstein-Maxwell theory admits remarkable static multiblack-hole metrics in which the electrostatic repulsion exactly balances the gravitational attraction [1, 2, 3]. These solutions have the following important properties.

1. (i) each black hole contained in the solutions is of the extreme Reissner-Nordstrom type, i.e., has its mass equal to its charge;

2. (ii) the relative positions of the individual black holes are free parameters that can be specified independly of their masses (the exact balance between the electromagnetic repulsion and the gravitational attraction holds for any configuration); and

3. (ii) all the multi-black hole solutions have some exact supersymmetries [4, 5, 6]; actually, they are the only black hole solutions with this property.

Recently, three-dimensional metrics describing single black holes have been found in the context of the Einstein theory with a negative cosmological constant [7]. Even though the single (2+1)-black hole solutions occur in a much simpler setting, they share global causal properties quite similar to their (3+1)-parents [8].

The purpose of this note is to investigate whether one can construct static multi-black-hole solutions in 2+1 dimensions. We show that this is not possible, in spite of the fact that multiparticle solutions are known to exist for (2+1)-Einstein theory with zero cosmological constant [9].

Abstract

Introduction

In the absence of a viable quantum description of the gravitational field attention is sometimes directed to the so called “mini-superspace” models [1] in which all but a small number of degrees of freedom of the gravitational field are suppressed and the dynamics is reduced from field theory to quantum mechanics. Such a programme is fraught with both conceptual and technical difficulties. It ignores many effects that may be of relevance in determining a viable quantum description. One restricts to quantum states describing highly symmetric geometries that possess a preferred class of spacelike foliations that may be used to order temporal phenomena in a classical spacetime. Even within this restricted framework there is no preferred way to effect a quantisation of Einstein's equations of motion tensorial nature of these equations gives rise to a constrained canonical system and there is no known criterion that singles out a particular mapping from the classical constraints to a set of quantum operators on a Hilbert space.

Motivation

The background for the investigations to be discussed below is provided byrecent studies of canonical quantum gravity in 3+1 dimensions. However, related issues arise in other physical theories that can be formulated on spaces of connections and are invariant under a corresponding group of gauge transformations. We will be interested in the so-called loop approach to the quantization of gravity, and will have a closer look at the (2+1)-dimensional theory, in the hope of gaining further understanding of the general approach.

Let us summarize in a nutshell the ideas that have gone into proposals for a quantization program for general relativity.

1. * The starting point is Ashtekar's reformulation of 3+1-dimensional Hamiltonian gravity in terms of Yang-Mills variables, namely, an sl(2, (ℂ)-valued pair (A, E) of a connection one-form and its conjugate momentum [1].

2. * On the phase space spanned by these variables, define Wilson loop variables T0(γ) = TrP exp ∫γA, and momentum-dependent generalizations which together form a closed Poisson bracket algebra of loop functions.

3. * Next, “quantize” this classical structure by finding representations of the Wilson loop algebra on spaces of wave functions that are themselves labelled by spatial loops γ.

4. * Rewrite the Hamiltonian in terms of loop variables, regularize it appropriately and look for solutions of the Wheeler-DeWitt equation, i.e. wave functions that are annihilated by the quantum Hamiltonian operator.

The general principle of operation of a scanning tunneling microscope (STM) – and related scanning probe microscopies (SPM) as well – is surprisingly simple. In STM a bias voltage is applied between a sharp metal tip and a conducting sample to be investigated (metal or doped semiconductor). After bringing tip and sample surface within a separation of only a few Ångström units (1 Ångström unit (Å) = 0.1 nanometer (nm) = 10−10 m), a tunneling current can flow due to the quantum mechanical tunneling effect before ‘mechanical point contact’ between tip and sample is reached. The tunneling current can be used to probe physical properties locally at the sample surface as well as to control the separation between tip and sample surface. The distance control based on tunneling is very sensitive to small changes in separation between the two electrodes because the tunneling current is strongly (exponentially) dependent on this separation, as we will see later (section 1.2). By scanning the tip over the sample surface while keeping the tunneling current constant by means of a feedback loop, we can follow the surface contours with the tip which – to a first approximation – will remain at constant distance from the sample surface. By monitoring the vertical position z of the tip as a function of the lateral position (x, y), we can get a three-dimensional image z(x,y) of the sample surface.

With the increasing degree of miniaturization in microelectronics, we unquestionably become confronted with structures of matter on a scale below 100 nm. The transition from microtechnology (lateral dimensions of 0.1–100 μm) to nanotechnology (lateral dimensions of 0.1–100 nm) requires the ability to fabricate smaller structures as well as the exploration and application of new physical phenomena occurring on the refined scale which becomes comparable to the characteristic lengths associated with the elementary processes in physics (e.g. the electron mean free path). In particular, entering the nanotechnology age involves the following tasks:

1. nanopositioning and nanocontrol of processes,

2. nanoprecision machining,

3. finding natural supersmooth surfaces and/or creating them,

4. fabrication of nanometer-scale structures,

5. analysis of nanometer-scale structures,

6. understanding the physical properties of matter on a nanometer scale,

7. development of nanometer-scale devices,

8. finding appropriate architectures for nanometer-scale structures, and

9. linking the ‘nanoscopic’ to the macroscopic world.

With the invention of STM and related scanning probe methods, we have been equipped in good time with the appropriate tools to attack most of these tasks, and there can be no doubt that STM-based technology and nanotechnology will interact closely in the coming decades.

1. STM-based technology has enabled us to control the position and motion of arbitrarily small objects down to a sub-Ångström unit scale (section 1.10).

2. STM and AFM can nowadays routinely be used to control the roughness of surfaces from a millimeter down to the atomic scale.

3. […]

STM and related scanning probe microscopies (SPM) have become important experimental techniques in chemistry, particularly solid state chemistry, as well as in solid state physics. There exist two major fields where SPM has already made significant contributions.

1. Chemical reactions at the solid–vacuum interface, i.e. surface reactions, which are in the focus of surface science research where the border between surface chemistry and surface physics (section 4.1) has become blurred.

2. Chemical reactions at the solid–liquid interface, particularly those initiated by electrochemical processes.

In the following, we will concentrate on some of the achievements of STM and related SPM techniques towards an atomic-level understanding of chemical reaction processes occurring at the solid–vacuum interface (section 5.1) and the solid–liquid interface (section 5.2).

Surface reactions

STM, or generally SPM, can contribute in various ways to a detailed investigation of surface chemical reactions.

1. Characterization of the atomic and electronic structure of the as-prepared surface before the initiation of chemical reactions is important to identify the variety of inequivalent surface sites and their structural as well as electronic characteristics.

2. In the initial stage of the reaction process, site-specific modifications of the substrate surface and chemisorbed species have to be characterized. In particular, a correlation between the local reactivity and the atomic and electronic properties of individual surface sites has to be established. As the chemical reaction proceeds, it is also important to study how the reaction at one particular surface site affects the local electronic structure of neighboring sites with possible influences on their local reactivity. This way, detailed information about the mechanism of the surface chemical reaction, from the initial nucleation to the growth of reacted surface regions, can be obtained.

3. […]

The application of scanning probe microscopy (SPM) to organic material, from small molecules to supramolecular assemblies, is a challenging task, which requires to address the following key issues.

1. Suitable substrates have to be found exhibiting a surface roughness considerably less than the size of the molecular species to be deposited in order to allow clear distinction between substrate and molecular features in SPM images. For SPM studies under ambient conditions, substrate surfaces must be chemically inert. The applicability of STM additionally requires electrically conducting substrates.

2. Deposited molecular species have to be immobilized in some way to allow stable SPM imaging.

3. Interpretation of STM images of molecular species requires profound understanding of their electronic structure and transport properties. In addition, the elastic properties must be known for interpretation of STM as well as AFM results.

A variety of different substrates has already been tried for SPM studies of molecules and biological specimens. At first sight, layered materials (section 4.1.3) seem to be particularly attractive because they usually provide large, atomically flat terraces after sample cleavage. In addition, the surfaces of graphite or transition metal dichalcogenides (TMD), for instance, are relatively inert, which is favorable for SPM studies in ambient air. On the other hand, the binding strength of molecules to these surfaces tends to be low in the absence of surface defects.

Nanometrology

Nanometrology is defined as the science of measuring the dimensions of objects or object features to uncertainties of lnm or less. The demand for nanometrology comes together with advances in integrated circuit technology where uncertainty requirements, e.g. in mask alignment, will soon approach the length scale lnm (Teague, 1992). The achievement of atomic-resolution real-space imaging of single-crystal surfaces by SPM has opened up novel opportunities in the field of nanometrology.

1. The highly-ordered atomic lattice of a single-crystal surface can serve as a reference against which the position and motion of an object can be measured and controlled. This idea has triggered the development of a dual tunnel-unit STM (Fig. 7.1), where one tunnel unit is used to provide a crystal reference for the second unit (Kawakatsu and Higuchi, 1990; Kawakatsu et al., 1991). To obtain a reference lattice over technologically relevant areas, there is a strong need for single-crystal surfaces being atomically flat over extended surface regions without the presence of steps or dislocations.

2. Piezoelectric crystals have proved to allow highly accurate and repeatable motion down to a subatomic length scale. This is clearly demonstrated by SPM images showing regular two-dimensional crystal lattices with measured corrugation amplitudes below 0.1 Å.

However, several problems have to be solved before successful application of SPM in nanometrology can be achieved.

1. Tip instabilities associated with switching of either the vertical or lateral position of the tip with respect to the substrate must be eliminated by preparation of highly stable tips with reliable long-term performance.

2. […]