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This chapter gives an overview of compounds with incommensurate crystal structures. Superspace is an excellent tool for crystal chemical considerations, as it is demonstrated for selected classes of compounds with incommensurate crystal structures as well as for compounds with periodic structures. The relations between individual compounds of a homologous series can be better understood by employing a unified superspace group and a unified structure model for all compounds in such a series. Relations between commensurately and incommensurately modulated structures of the different phases of A₂BX₄ type compounds are made explicit by the superspace approach. So-called t-plots are introduced as a versatile method for the crystal chemical analysis of incommensurately modulated structures and composite crystals. They are used to analyse the stability of the incommensurate structures of the chemical elements, and to compute the atomic valences in aperiodic crystals.

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.

This chapter introduces the superspace description of the crystal structures of incommensurate composite crystals, and the characterization of their symmetry by superspace groups. The treatment parallels that of incommensurately modulated crystals in most aspects. The particular features of composite crystals are highlighted with respect to the diffraction and structure in superspace, superspace groups, the structure factor of Bragg reflections, and t-plots.

This chapter develops the relation between crystal structures of aperiodic crystals and superstructures. Superstructures are described as commensurately modulated structures, and it is shown that superspace methods can be applied to this particular kind of periodic crystals. Alternatively, superstructures are obtained as the commensurate approximation to incommensurately modulated crystals and composite crystals. Relations are derived between modulation functions and superspace groups of the modulated-structure description, and atomic coordinates and supercell space groups of the superstructure description.

This chapter discusses the development of SUSY Lagrangians. Topics covered include the free Wess-Zumino model, commutators of SUSY transformations, an extension of the auxiliary field formalism to the gauge interactions of massless vector multiplets, SUSY gauge theories, and superspace. Exercises are provided at the end of the chapter.

Superspace is an extension of space-time which includes new coordinates with unusual algebraic rules. The nature of superspace is quantum mechanical, because it requires the concept of particle spin. This chapter describes the meaning of superspace and supersymmetry, and shows how these ideas can provide a solution to the naturalness problem. The impact of supersymmetry in the quest for unification is also discussed, from its role in superstring theory to the result of gauge coupling unification. Finally, the chapter describes how experiments at the LHC could discover supersymmetry.

This chapter, although dealing entirely with classical physics, prepares the ground for the following chapters by developing in full detail the Hamiltonian, or canonical, formulation of general relativity, also called 3+1 decomposition. This is achieved by a decomposition of four-dimenensional spacetime into a foliation of spacelike hypersurfaces. Special attention is devoted to open spaces and the structure of the configuration space. The canonical formalism is presented for metric, connection, and loop variables.

The motivation for supersymmetry. The algebra, the superspace, and the representations. Field theory models and the non-renormalisation theorems. Spontaneous and explicit breaking of super-symmetry. The generalisation of the Montonen–Olive duality conjecture in supersymmetric theories. The remarkable properties of extended supersymmetric theories. A brief discussion of twisted supersymmetry in connection with topological field theories. Attempts to build a supersymmetric extention of the standard model and its experimental consequences. The property of gauge supersymmetry to include general relativity and the supergravity models.

In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

First a general description of the concept of crystalline structures is presented with some historical background information. The classical approach of periodic structures is presented along with the important topic of symmetry and its role characterizing physical properties. The limitations of the classical model are then introduced in view of the new experimental observations discovered since the 1970s. New forms of crystalline structures including incommensurately modulated and composite structures are presented along with quasicrystalline structures (quasicrystals). The necessity to extend the theory of space group symmetry is then discussed and the concept of superspace symmetry is introduced in order to describe these new forms of matters.

This chapter first introduces the mathematical concept of aperiodic and quasiperiodic functions, which will form the theoretical basis of the superspace description of the new recently discovered forms of matter. They are divided in three groups, namely modulated phases, composites, and quasicrystals. It is shown how the atomic structures and their symmetry can be characterized and described by the new concept. The classification of superspace groups is introduced along with some examples. For quasicrystals, the notion of approximants is also introduced for a better understanding of their structures. Finally, alternatives for the descriptions of the new materials are presented along with scaling symmetries. Magnetic systems and time-reversal symmetry are also introduced.

This chapter discusses the X-ray and neutron diffraction methods used to study the atomic structures of aperiodic crystals, addressing indexing diffraction patterns, superspace, ab initio methods, the structure factor of incommensurate structures; and diffuse scattering. The structure solution methods based on the dual space refinements are described, as they are very often applied for the resolution of aperiodic crystal structures. Modulation functions which are used for the refinement of modulated structures and composite structures are presented and illustrated with examples of structure models covering a large spectrum of structures from organic to inorganic compounds, including metals, alloys, and minerals. For a better understanding of the concept of quasicrystalline structures, one-dimensional structure examples are presented first. Further examples of quasicrystals, including decagonal quasicrystals and icosahedral quasicrystals, are analysed in terms of increasing shells of a selected number of polyhedra. The notion of the approximant is compared with classical forms of structures.

This is a unique chapter that discusses classical path integrals in both configuration space and phase space. It examines both Lagrangian and Hamiltonian formulations before qualitatively discussing some interesting features of gauge fixing. This formulation is then linked to superspace and Grassmann variables for a fermionic field theory. The chapter then shows that the corresponding operatorial formulation is none other than the Koopman–von Neumann theory. In parallel to quantum theory, the classical propagator or the transition amplitude between two classical states is given exactly by the phase space partition function. The functional Dirac delta is discussed, and the chapter closes by briefly mentioning Faddeev–Popov ghosts, which were introduced earlier in the chapter.