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In Chapter 8 five directions of future research are tersely mentioned in order of increasing importance. The investigation of Diophantine relations, such as those reported in Appendix C. The investigation of the behavior in a quantal context of systems described by isochronous Hamiltonians. The investigation of isochronous systems outside the phase space sectors where they behave isochronously. The investigation of the behavior of the isochronous many-body problems treated in Section 5.5, in particular, when the corresponding unmodified system is a realistic molecular dynamics model characterized by a chaotic evolution yielding—in the context of statistical mechanics and thermodynamics—a steadily increasing entropy—which, however, must eventually decrease when the corresponding Ω-modified system returns to its initial state, as entailed by its isochronous character. And, last but by no means least, the investigation of isochronous systems in an applicative context—be it physics, chemistry, biology, medicine, economy, and so on.

This chapter introduces the basic setting: Tonelli Lagrangians and Hamiltonians on a compact manifold. It discusses their main properties and some examples, and provides the opportunity to recall some basic facts on Lagrangian and Hamiltonian dynamics (and on their mutual relation), which will be of fundamental importance in the discussion thereafter.

This chapter explains bosonization, a useful technique for describing the low-energy properties of one-dimensional systems. Bosonization is introduced by presenting a model on which this method is essentially exact. This allows for the derivation of precise formulas that can be used for other models as well. Bosons are mapped with fermions and the effect of the interactions is examined. A one-dimensional system with spin is also discussed, and renormalization equations for sine-Gordon Hamiltonians are presented and analyzed. Furthermore, the generic phase diagram of a one-dimensional chain is obtained.

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials quadratic in the velocities. The important example of Hamiltonians quadratic in the momentum variables is first examined. In the simplest situations discussed in previous chapters, after an explicit integration over momenta p(t) one recovers the usual path integral. More general Hamiltonians are often met, for example, in the quantization of the motion on riemannian manifolds. The analysis is illustrated with the quantization of free motion on a sphere (or hypersphere) S _N-1. A few relevant elements of classical mechanics are considered first.

This chapter constructs the path integral associated with the statistical operator e ^-βH in the case of Hamiltonians of the simple form p2/2m + V (q). The path integral corresponding to a harmonic oscillator coupled to an external, time-dependent force is then calculated. This result allows a perturbative evaluation of path integrals with general analytic potentials. The results are applied to the calculation of the partition function tr e-βH using perturbative and semi-classical methods. The integrand for this class of path integrals defines a positive measure on paths. It is thus natural to introduce the corresponding expectation values, called correlation functions. Moments of such a distribution can be generated by a generating functional, and recovered by functional differentiation. These results can be applied to the determination of the spectrum of a class of Hamiltonians in several approximation schemes.

A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only situations where the Hamiltonian eigenvalues are not degenerate are considered. The power of the path integral formalism is then illustrated beyond simple perturbation theory: it is used to derive a variational principle. It is evaluated, by applying the steepest descent method, in the case of O(N) symmetric Hamiltonians, in the large N limit. Steepest descent calculations of path integrals involve determinants of differential operators that are given a perturbative definition. This chapter shows how to calculate the spectrum of a Hamiltonian in the semi-classical limit or WKB approximation, starting from the semi-classical expansion of the partition function obtained earlier. It is demonstrated that the ground state of simple quantum systems is not degenerate.

This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum system coupled to a magnetic field, and diffusion as described by the Fokker-Planck equation. In both examples, the Hamiltonian contains products of the position and momentum operators. A quantization problem then arises since these operators do not commute, and the correspondence principle is no longer sufficient to determine the quantum Hamiltonian. The order between quantum operators is determined by additional conditions, such as hermiticity or conservation of probabilities. The calculation of the corresponding path integral then suffers from ambiguities, directly related to this quantization problem. The continuum limit is no longer unique, but depends on the limiting process. This chapter also considers a situation where space has a nontrivial topology, in this case a circle, and shows how this influences the calculation of the path integral.

This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.

This chapter summarizes the status of the very active field of research where the strength of the effective Coulombic repulsion among electrons is comparable to the bandwidth of the carriers. It focuses on complex transition-metal oxides, with emphasis on the manganese oxides known as manganites that display the so-called Colossal Magnetoresistance effect. It also describes several other materials such as high-temperature superconductors and multiferroics. In these correlated electron materials the interactions between the electronic spins, their charges and orbitals, and the lattice produce a rich variety of electronic phases and self-organization. The competition and/or cooperation among these correlated electron phases can lead to the emergence of surprising electronic phenomena and also of interesting functionalities via their nonlinear responses to external fields.

Chapter 5 discussed two possible approaches to constructing a relativistically invariant theory of particle scattering. The first attempt — a frontal assault in which one directly writes down for each scattering sector (i.e., with a specified number of incoming and outgoing particles) a manifestly Lorentz-invariant interaction operator containing momentum-dependent Lorentz scalar amplitudes — led to disaster. The resultant theory led to particle interactions which could not be confined to finite regions of space-time. The second attempt, in which the interaction Hamiltonian is written as a spatial integral of a local, Lorentz (ultra-)scalar field, accomplishes the primary goal of producing a Lorentz-invariant set of scattering amplitudes, but its compliance with the clustering principle remains uncertain. This chapter puts this latter requirement into a precise mathematical framework, called second quantization, so that the process of identifying clustering relativistic scattering theories can be simplified and even to some degree automated.

This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on to discuss Aristotelian and Galilean–Newtonian space-times. It explains how Newton’s first law of motion is required to correlate successive copies of space. It discusses the differences between the Newtonian paradigm of motion and the use of Lagrangians in the Calculus of Variations that gives the Euler–Lagrange equations of motion in advanced mechanics. The chapter concludes with a discussion of phase space, Hamiltonians, Poisson brackets, and the role of the Hamiltonian as a generator of translation in time.