Structural Aspects of Quantum Field Theory and Noncommutative Geometry

The following sections are included:

• PREFACE
• Contents

In our first approach to the subject of quantum field theory we begin by neglecting the qualification ‘quantum’, i.e. we restrict ourselves to classical field theory. Simplifying even further, we start with a discussion of free classical fields, of which the electromagnetic field is a main example. It opens the opportunity to introduce relativistic notation and the underlying symmetry, the Lorentz and Poincaré group. Furthermore, one can relate such relativistic fields to the realm of classical mechanics on restricting the system to a finite lattice so that traditional methods are available. Hence the only prerequisites needed are the basic facts of classical electrodynamics…

The Lagrangian formalism for fields can be developed by following the analogy with a system of classical mechanics with a finite number of degrees of freedom. This approach necessitates the introduction of some elements of functional calculus; we only give a poor-man’s-view to that subject.

Up to now we have only introduced a compact 4-dimensional notation, but not discussed its significance. As we shall see, the notation hides fundamental symmetry properties of Minkowski space. The rather formal, albeit basic tools that follow will play an essential rôle in relativistic field theory, to be revealed in the present and the chapter to come.

As we have experienced already (see sec. 1.3 and sec. 3.1), all the basic notions of special relativity follow rather naturally from the theory of relativistic free fields. Now we consider special relativity as a subject of classical physics; a priori, it is then of purely kinematic origin. Indeed, it will be shown below that special relativity can be viewed as the prime example of a theory for which all physical properties of the system are entirely determined by the symmetry group of the underlying space.

One of the most fundamental equations of theoretical physics was created in the year 1928 by Dirac; it is the subject of the present chapter. In retrospect, Dirac’s original motivation for rejecting the Klein-Gordon equation as a consistent quantum mechanical system does not stand critique. For this and other reasons it has become fashionable to avoid Dirac’s approach in obtaining his equation by more sophisticated group theoretical arguments. Here we leave aside the historical motivation and follow, with some omissions, the main line of Dirac’s reasoning. In this context one should also remember that Dirac’s equation had an important predecessor, namely, Pauli’s two-component equation for nonrelativistic electrons.

A discussion of the solutions of the Klein-Gordon and Dirac equation has already be given in our treatment of fields on a lattice. Now we turn to the continuum and give a systematic construction for the solutions of free fields with spin less than or equal to one.

The present chapter mainly centers around the theme of particles and their antiparticles, special emphasis being laid on the discussion of massless neutral fermions.

Having available the basic facts about the symmetries of Minkowski space and the symmetries of fields defined over there, we can now turn to the investigation of conservation laws. They are provided by the so called Noether theorem, according to which conserved quantities are obtained as a sole consequence of the symmetry properties of the corresponding action…

Up to now we have not taken at face one crucial fact, namely, that in the domain of field theory one must also deal with anticommuting variables. They come in through the mere existence of fermions, as explained in sec. 5.6, and so one has to cope with the problem to develop calculational tools for handling such quantities…

The charge conservation law arises as a consequence of the fact that the Lagrangian for a complex field is invariant against the multiplication of the field by a phase factor; such symmetries and nonabelian generalizations thereof are called internal, and thus depend on the detailed structure of the Lagrangian. By contrast, whatever the internal properties of a field theory are, any relativistic field theory is invariant under Poincaré transformations, giving rise to energy-momentum conservation, and to conservation of the relativistic angular-momentum tensor; these universal symmetries, which hold for any type of relativistic field theory, are called external…

In the present chapter we extend the previously developed methods to external symmetries, of which the Poincaré symmetry (and, presumably, its supersymmetric extension) is the prime example. As opposed to internal symmetries, which depend on the specific type of fields and interactions being considered, matter and gauge field theories of whatever a type must respect Poincaré invariance. This entails two types of conservation laws, namely, the conservation of linear and angular momentum. According to the general Weyl strategy, one may thus attempt to gauge these universal external symmetries and look whether they lead to something sensible; this is the approach that we will follow below…

Consider our space-time world, formally being described as Minkowski space; its natural symmetry group is the Poincaré group, as opposed to the Lorentz group, which is only a subgroup of the full symmetry group. In this space there operate fields, governing the physical laws of our world. An example is the Maxwell field, which is responsible for the electromagnetic interaction. In particular, this interaction happens between electrons; but what one must take at face then is the fact that fermions with half-integral spin have wave functions which anticommute, whereas bosons with integral spin have commuting wave functions. Hence, for fermions one gets outside conventional rules of analysis and, we repeat, it took more than a quarter of a century after the perception of Dirac’s equation to invent a calculus, being capable of dealing with anticommuting quantities. We refer to chap. 9, where these matters have already been discussed in some detail…

Up to now we have mostly been concerned with conceptual aspects of general relativity, including the extension to spinors and supergravity. In this final chapter one of the main applications of conventional general relativity is considered, cosmology, which describes the time evolution of the universe. Due to a breakthrough in observational cosmology in recent years, this subject has undergone a revolutionary development so that the variety of available cosmological models is severely restricted…

Having covered the foundations of classical free and interacting field theories in the preceding first two parts, we can now address our main theme, i.e. the quantum theory of fields. In the present third part we will exclusively concern ourselves with the operator approach…

Before entering the discussion on a scattering theory for quantum fields, we must explain some formal aspects of quantum mechanical perturbation theory; the present chapter is entirely devoted to these matters. In the first section, time-dependent perturbation theory is covered and the basic properties of the scattering matrix are derived. These results are made use of in the second section to solve some basic questions of stationary state perturbation theory. In particular, the Gell-Mann & Low theorem is dealt with; but we give a modified version of the original approach that contains more information. The added virtue of this modification consists in the fact that it entails, as a corollary, Goldstone’s theorem. However, in contrast to Goldstone’s original proof, we avoid the use of quantum field theoretic methods so that the formula for the energy shift also holds in the realm of one-particle quantum mechanics. Then we return to scattering theory in the third section and develop Green’s function techniques that culminate in the Lippmann-Schwinger equation. We continue by introducing a refinement of the interaction picture in the fourth section, the in and out picture. This will enable us to perform the transition to quantum fields and to prove the Gell-Mann & Low formula; we conclude by deriving the reduction formulae of Lehmann, Symanzik and Zimmermann.

We will we be concerned with the theory of quantized interacting fields in this chapter. In particular, our interest will focus on quantum electrodynamics, usually abbreviated as QED, which is treated by perturbative methods. In this first approach to an interacting quantized system, we use the interaction picture so that only the first section of the previous chapter is actually needed. We also restrict ourselves to lowest nontrivial order, the so called tree graph approximation…

The original strategy to approach quantum mechanical problems proceeds by trying to solve Schrödinger’s partial differential equation exactly. But this direct attack mostly is impracticable since least of all problems are solvable exactly; if not, then one resorts to perturbative methods. Another strategy, initiated by Dirac (1933) and elaborated upon by Feynman (1948), is the path integral technique; actually, such a kind of functional integrals was studied already earlier by the mathematician Wiener in the context of Brownian motion. Since then Feynman’s approach (as displayed in Feynman and Hibbs (1965)) has been clarified in some essential aspects and been considerably simplified. But up to the time of writing it is still not accepted by mathematicians; what we hope to convey, however, is the message that the path integral is better behaved than its reputation.

The harmonic oscillator is of fundamental importance in many areas, one particular example being quantum field theory, where we have seen that the normal mode expansion of the Klein-Gordon field yields decoupled harmonic oscillators.

Time ordered products of Heisenberg operators have already appeared repeatedly, mostly in the field theoretic context; of course, they also play a significant role in ordinary quantum mechanics, and this is the topic to be addressed below.

In the present chapter, the path integral representation of the generating function is treated by perturbative techniques. We begin with the derivation of the perturbation series for the path integral; passing to imaginary time or inverse temperature, the ground state can then be obtained as a low temperature limit. This result is used to calculate the perturbative expansion for the ground state energy of the quartic anharmonic oscillator.

The present chapter is devoted to nonperturbative methods within the path integral approach to quantum mechanics. Our access comprises the analogue of the conventional Wentzel-Kramers-Brillouin (WKB) approximation for the conventional Schrödinger theory, and its generalization with a gyroscopic term in the Lagrangian action; the latter case has always been a subtle affair, with no conclusive answer being available in the literature. The solution given below affords an alternative to Schwinger’s derivation of the Heisenberg-Euler effective action, as we shall show in a later chapter. A further section is devoted to the problem to define a path integral for a particle in an external gravitational field. It is addressed by means of heat kernel techniques. Following de Witt (1965), we calculate the first three coefficients of the heat kernel expansion; they will be made use of at various places. We conclude with an essay on the role of functional determinants in calculating partition functions.

There is another quantization method, the importance of which can not be overestimated, called coherent or holomorphic quantization (see Faddeev (1976)). It is adapted to Hamiltonians being built from operators of harmonic oscillator type. In particular, it is the only method that is available for the treatment of fermions by means of path integral techniques. As it appears, it seems worthwhile to stress that coherent states are not restricted to quadratic Hamiltonians only, though they were first applied to that case. Originally, this method is due to Schrödinger and in the sequel it was elaborated by Bargmann, Fock, Segal and others.

We aim at modelling a fermionic analogue of N bosonic momentum operators ${\widehat{p}}_i$ and generalized coordinate operators ${\widehat{q}}^j$, denoted by ${\widehat{\zeta }}_i$ and ${\widehat{\eta }}^j$ in what follows. Though such fermionic degrees of freedom are treated in close parallel to the bosonic case, the outcome will be that they describe unphysical fermions because negative norm states get involved…

In the preceding chapter we have derived the path integral for bosonic and fermionic quantum mechanical systems. There we only had to cope with a finite number of degrees of freedom. What we attempt in the present part is to generalize the above results to the quantum field theoretic situation with an uncountably infinite number of degrees of freedom. This is achieved on considering the system on a spatial lattice; afterwards the lattice constant is sent to zero. Hence, also in the quantum field theoretic context the path integral will only be defined through the continuum limit of its discrete version.

The present chapter is devoted to the path integral treatment of a scalar field theory. This lends itself to a perturbative analysis. But already in the first nontrivial order of perturbation theory one encounters divergent expressions that must be regularized - which is the subject of renormalization theory. These matters are discussed in some detail for the so called ϕ⁴-theory. Amongst the variety of possible regularization schemes the method of dimensional regularization is chosen; it has some definite advantages over earlier techniques. We conclude with the derivation of a nonperturbative result, the Coleman-Weinberg effective action.

We have already studied the interaction between photons and electrons in the operator formalism; but there only tree graphs were considered. Now we turn to the path integral treatment and take it as a starting point for the investigation of the regularization and renormalization of the electromagnetic interaction. Afterwards a discussion on the structure of the vacuum in quantum electrodynamics is given, including the Casimir effect and the Euler-Heisenberg effective field theory.

For an abelian gauge theory we have seen that gauge invariance requires additional considerations in order to arrive at a consistent path integral treatment. For a nonabelian gauge theory the analysis will turn out to be even more sophisticated. We here are content with the description of the so called Faddeev-Popov ‘trick’, which enables us to finally write down the path integral for a Yang-Mills theory and to discuss its renormalization. But this approach is to be considered as rather intuitive since, as always, a ‘trick’ calls for a deeper foundation; a discussion relying on more basic principles is rather intricate and is postponed at a later stage.

Consider a quantum mechanical system of N identical particles, being subject to an interparticle potential. The N-particle Schrődinger equation then is

As we have already noticed, the Dirac-Feynman path integral approach makes it conspicuous in which sense quantum mechanics is directly related to classical mechanics; this relation is most obvious in the Hamiltonian version of the functional integral. We shall exhibit in the present chapter that the same remark applies to quantum statistical mechanics, which can efficiently be treated by means of functional integral techniques. This will be demonstrated for both the canonical ensemble, where the Feynman path integral provides the naturally adapted frame, and the grand canonical ensemble, for which the coherent state path integral offers the ideal vehicle to investigate the properties of the partition function for systems with variable particle number. Beyond this, the functional integral approach will turn out to have definite advantages over the conventional operator treatment using Green’s function techniques.

As explained in the preceding chapter, the Dirac-Feynman type of path integral is naturally adapted to the canonical ensemble. We now turn to the partition function of the grand canonical ensemble; in that case it is the coherent state path integral which is ideally suited to discuss its properties. In the literature, this theme is not as adequately dealt with as it deserves. An early reference on this subject is Casher, Lurié & Revzen (1968); among the rare more advanced treatments being available there are the notable monographs of Popov (1987) and Negele & Orland (1988). The present approach supersedes the traditional operator approach via Green’s functions techniques, the latter mainly being due to Abrikosov and Gorkov (see Abrikosov, Gorkov & Dzyaloshinskii (1963)); their work was recently honoured with the Nobel prize. As we shall demonstrate on the examples of Bose-Einstein condensation and superconductivity in two separate chapters to follow, path integral methods in the holomorphic representation will turn out to have some definite advantages over conventional operator techniques in many body theory.

We have studied the effects of spontaneous symmetry breaking already earlier (see sec. 10.3) in the context of the standard model of particle physics, at increasing level of complexity; beginning with a system with a discrete ℤ₂-symmetry, we turned to an abelian U(1)-symmetry, which was then gauged, and afterwards generalized to the nonabelian case. However, the discussion given there was entirely classical. We now attempt the treatment of the quantum mechanical situation; this is done in the present chapter for a nonrelativistic bosonic system on the example of Bose-Einstein condensation, which may serve as the paradigm for the spontaneous breakdown of a rigid U(1)-symmetry. In the next chapter, the generalization to a spontaneously broken gauged local U(1)-symmetry will then be performed on the example of a fermionic superconducting system.

The following sections are included:

• Introduction
• Effective Action

In the preceding two chapters we have treated spontaneous symmetry breaking of nonrelativistic field theories at low temperatures. We now turn to relativistic fields and begin this investigation with the relativistic ideal gas, for both bosons and fermions. Afterwards we study spontaneous symmetry breaking on the example of a self interacting scalar field theory at nonzero temperature. Such an investigation is motivated by the known fact that heating of a superconductor destroys superconductivity. As was first pointed out by Kirzhnits & Linde (1972), in the relativistic case something similar happens since the spontaneously broken symmetry is restored at high temperatures; subsequently this effect was confirmed by Dolan & Jackiw (1974) and Weinberg (1974) by quantitative arguments. This phenomenon has important applications in cosmology since it says that at an early stage of the evolution of the universe the original symmetry of the unified theory was intact and the weak and strong interactions were long ranged, as the electromagnetic interaction.

As we have learned up to now, functional methods can rather effectively be utilized in analyzing complicated thermodynamical systems, such as superfluids and superconductors. We conclude the present part with a discussion of a third example, the quantum Hall system; one reason is that it builds a bridge to more recent developments, which will form the content of the second volume…

The following sections are included:

• PREFACE
• Contents

Classical mechanics is an old science, and one could argue that this branch of theoretical physics is so antique and so thoroughly understood that one may be content with the standard textbook knowledge. This subject, however, has undergone a rapid development in the second half of the century just passed, with far reaching results. Furthermore, some deep problems in quantum field theory require a detailed knowledge of these achievements. It is for this and reasons to be explained that we enter into the study of such matters in the present and chapters to come.

In the nineteenth century conservation laws still had a rather mysterious status in science; only in the twentieth century they were unburdened from this reputation. In particular, in the first quarter of the century just having passed, stimulated by the insight delivered by the theory special relativity, it became clear that the classical conservation laws are deeply related to the symmetries of space-time which, in nonrelativistic mechanics, are given by the Galilei group. It is the purpose of the present chapter to describe the connection between symmetries and conservation laws; these developments are tied to the names E. Noether and (her protector) D. Hilbert.

It is a known fact (Bertrand’s theorem) that the only rotationally invariant central potentials in three dimensions, for which the orbits of bound motion close on themselves, are given by the harmonic oscillator potential V ∼ r² and the Kepler potential V ∼ 1/r. We investigate the classical and quantum treatment of these two familiar examples; they belong to the rare cases which are exactly soluble. For both of these one knows from the standard analysis of the Schrödinger equation that the energy eigenvalues are degenerate, and in order to describe this curious fact, the degeneracy is said to be accidental ; remarkably, the ground state wave function is unique…

Up to now we have mostly restricted our considerations to conventional systems; for these, the Lagrangian is strictly invariant under the given symmetries. We now weaken this requirement slightly and extend our investigation to those systems, which are only almost invariant, i.e. the (point) transformations alter the Lagrangian by a total time derivative. As we shall see, this apparently innocuous modification entails drastic changes in the properties of the system since the symmetries generally become anomalous, i.e. the original commutators get modified by an additional contribution. One thus enters a subject, which in mathematics is called cohomology theory; it will turn out to be an indispensable tool to reach a proper understanding of a both simple and basic quantum mechanical system, the Landau problem, which in sec. 34.2 we have already touched upon…

Constrained systems of the anholonomic type constitute a notoriously difficult and vast subject, up to the present day. A systematic treatment was initiated by P. Dirac, who was mainly interested in conventional general relativity, which is a constrained system of intriguing complexity. Our attempt is more modest; we focus on abelian and nonabelian gauge theories of the Maxwell and Yang-Mills type, which are difficult enough to cope with. Also, we do not want to repeat the entire liturgy and restrict the investigation to Hamiltonian systems so that there is no need to introduce what are called primary and secondary constraints. In the present chapter, we collect the main results on classical reduction being scattered in the mathematical literature (Abraham and Marsden (1978), Guillemin and Sternberg (1984), Woodhouse (1992), and Marsden and Ratiu (1999); the physical aspects are covered in the monographs of Sundermeyer (1982), and Henneaux and Teitelboim (1992)). As will be seen, the classical reduction is rather perfectly understood (Marsden and Weinstein (1974)). After reduction, however, one ends up with a system, the phase space of which will no longer be flat; hence, one is deep down in the intricacies of quantizing systems with a nontrivial metric.

In the last chapter no mention at all has been made of a Hamiltonian, which governs the time evolution or dynamics in phase space; this is the topic we address now. Prime examples of constrained systems are Maxwell’s and Yang-Mills theory. However, this property cannot be seen in the Lagrangian formalism since the time-component of the gauge field appears quadratically in L so that its role as a Lagrangian multiplier is not manifest. The standard approach to convert the quadratic into a linear dependence is to use a (Palatini type) first order formalism. However, such an approach is unnecessary complicated and does not take into account that anholonomic constraints are notoriously difficult to cope with in a Lagrangian treatment. Instead, one must treat such systems in the Hamiltonian framework; only then the arguments can considerably be simplified and also become transparent.

It is shown that Faddeev’s path integral formula for a system with first class constraints hides an important symmetry, uncovered by Becchi, Rouet and Stora, and independently by Tyutin. The BRS invariance was originally detected in Yang-Mills theory (see chap. 27). But as will be seen, the invariance is also present in the finite dimensional context, and here we content ourselves with a discussion of the simpler situation since the underlying structure can more clearly be revealed in this case.

One of the main problems of quantization since its perception consists in associating to a given phase space function a uniquely determined operator. The difficulties arise from ordering ambiguities because, in general, one has many possibilities to replace a product of the momenta and coordinates by a quantum mechanical counterpart…

There is more about the Weyl approach than meets the eyes from the material covered in the preceding chapter. The additional insight needed is the connection with symplectic invariance, which is the topic we now turn to.

The basic principle underlying Weyl quantization is of group theoretical nature, namely, it relies on the Heisenberg-Weyl group, which is a central extension of the abelian group of translations of phase space; in essence, it is this fact that gives rise to the construction of the Weyl operator, and as a consequence of coherent states. So the idea comes to mind to use other Lie groups in the attempt to obtain a generalization of the Weyl approach. In order to probe such ideas, the simplest example being available is the group SU(2), which is intimately related to the spin of a particle. This will lead us to the concept of geometric quantization (see Woodhouse (1992), and further references given there) since one needs a strategy to work out the generalization of the Weyl operator for an arbitrary semisimple Lie group…

It is shown that the Weyl formalism for fermions (Berezin (1966)) can be developed in close analogy to the bosonic case, and we shall see that the results turn out to be rather similar. Though there are also some essential differences, nevertheless, this is a really amazing fact.

Consider a classical system of finite or (even non denumerable) infinite dimension, and assume that it is invariant against some symmetry transformations; according to Noether’s theorem, these symmetries give rise to conserved quantities. If such a system is subjected to quantization, however, it may happen that the classical symmetries do not survive the quantization process, i.e. they are broken at the quantum level; in such a situation one speaks of an anomaly. We have already discussed a related phenomenon in the quantum mechanical context with a finite number of degrees of freedom; in the present chapter we study the field theoretic situation and establish the connection of anomalies with a deep result of mathematics, the celebrated Atiyah-Singer index theorems.

In the present section we investigate the integrated form of anomalies and associated induced effective actions. They provide examples of quantum field theories which, apart from their physical relevance, have important impact as well on mathematical disciplines…

In the present part we begin with a new subject; it was founded by A. Connes and is called noncommutative geometry. As a motivation, recall from our treatment of Weyl quantization that it gives rise to the construction of a new product on the space of phase space functions with the pointwise product. This is the noncommutative Groenewold-Moyal product, and it is in this way that the Groenewold-van Howe theorem, which states that classical and quantum mechanics at the level of Dirac’s naïve or ideal quantization rules are incompatible, is circumvented. Hence, it seems likely to replace the conventional commutative algebra of functions over a manifold by a noncommutative algebra. Astonishingly enough, on this substitute a differential and integral calculus in rather close analogy to the commutative case can be developed, and one is even able to introduce the concept of a connexion; these topics will be the subject of the present chapter…

The purely algebraic notions of noncommutative geometry are now brought together with analytical methods. Of course, the main source for this subject still is the book of Connes (1994), and the further book by Connes and Marcolli (2008), in which the more recent developments are covered. Again we have profited from the treatments of Landi (1997), and Gracia-Bondia, Varilly and Figueroa (2001); for a detailed list of references, the reader is referred to the latter monograph. In order to fix the notation, some basic facts of functional analysis and C*-algebras are collected in an appendix to this chapter.

Having available all the above rather subtle prerequisites, we are now able to address the main topic, the standard model of particle physics; it will turn out that noncommutative geometry has tremendous impact on this theme…

The present chapter is devoted to the subject of noncommutative quantum field theory. Such theories became popular after the discovery that they arise as certain limits of string theory (Connes, Douglas & Schwarz (1998), Schomerus (1999), and Seiberg & Witten (1999)). Here we only give a rather limited account; for further information, in particular the connection with string theory, the reader is referred to the reviews of Douglas & Nekrasov (2001) and Szabo (2003).

The following sections are included:

• Dark Matter and Torsion
  • Aspects of torsion
  • Dark matter and torsion physics
• Randall-Sundrum model

In the present part we address a new theme, the subject of quantum groups. They may be viewed as quantum deformations of the classical Lie groups; but we shall only work at the level of Lie algebras, and so the heading should better read ‘quantum symmetries’ or ‘quantum algebras’. Such constructs figure in various branches of theoretical physics, as there are spin chains, conformal quantum field theory, and the like. We shall need these deformations, among others, in order to fill a gap having been left open up to now, namely, to reach a proper understanding of what we call the Landau problem. Beyond the fact that it constitutes one of the most fundamental systems of quantum mechanics, it will also be seen to be a paradigmatic example of a physical system in which the beautiful architecture of a quantum group is realized…

The following sections are included:

• Almost Cocommutative Hopf Algebras
• Quasitriangular Hopf Algebras
• Ribbon Hopf Algebras
• Matrix Realizations of the Universal $\mathcal{R}$-Operator and Artin’s Braid Group
• Quasitriangular Hopf Algebras and ∗-Structures

The present chapter is devoted to the main example of a quantum group, the ‘quantum deformation’ of the Lie algebra of the group SU₂, denoted U_q(su₂), where q is a complex parameter. For a physicist, the Lie algebra su₂ is well known from elementary quantum mechanics. Here, however, a modification is investigated, which also has its origin in quantum mechanics, but which will turn out to be of even more profound relevance than the Lie algebra su₂ itself. Hence, quantum mechanics requires more general symmetries than the conventional Lie group symmetries we are used to…

In this last part noncommutative geometry and quantum symmetries are brought together. As we will show, the prime example of a noncommutative geometry, the noncommutative torus ${\mathbb{T}}^d{}_{\theta }$, plays the role of an auxiliary algebra for the quantum algebra U_q(su_d) with q = exp2πiθ and rational θ, in the sense that the generators of U_q(su_d) can be expressed in terms of those of the noncommutative torus…

Finally, we return to the topic of the fractional quantum Hall effect. In sec. 34.12 we have been able to derive Laughlin’s trial wave function as an exact ground state; the basic ingredient was to subject the bulk electrons to an abelian Chern-Simons gauge field in order to ensure that they follow braided paths…

The theta functions of Jacobi and Riemann form a class of special functions, being of importance in both mathematics and physics. In mathematics, for example, one main subject is algebraic geometry; in physics, we have already seen applications on several occasions (see the secs. 38.4 and 34.7). The deeper reason for the relevance of this class of functions rest on the fact that its construction relies on the Heisenberg-Weyl group, the symmetry group of quantum mechanics. This insight, being due to Cartier (Cartier (1966)), is extensively made use of in the work of Mumford (Mumford (1983) and (1991))…

The following section is included:

• Index