Our experiments indicate an intersection of superstring theory and the gravitation and guage symmetry problems as the point at which the Delta function is resolved and an inflationary constant can be established.





M. Blagojevic Gravitation and Gauge Symmetries

Synopsis



I. Space, time and gravitation

In this chapter we give an overview of some aspects of the structure of space, time and gravitation, that are important for the understanding of gravitation as a gauge theory. These aspects include: - development of the principle of relativity from classical mechanics and electrodynamics, and its influence on the structure of space and time; - formulation of the principle of equivalence, and introduction of gravitation and the corresponding geometry of curved space. The purpose of this exposition is to illuminate those properties of space, time and gravitation that have had an important role in the development of general relativity (GR), and still have an influence on various attempts to build alternative approaches to gravity.

II. Spacetime symmetries

In physical processes at law energies the gravitational field does not have a significant role, since the gravitational interaction is extremely weak. The structure of spacetime without gravity is determined by Einstein's relativity principle. The equivalence of inertial reference frames is expressed by the Poincare symmetry in M4.

In physical processes at high energies, where dimensional constants become practically negligible, one expects the appearance of scale invariance, so that the relevant symmetry goes over into Weyl symmetry, or even higher, conformal symmetry. The conformal symmetry is a broken symmetry of basic physical interactions. Of particular interest for string theory is the conformal symmetry in two dimensions.

Having in mind the importance of these spacetime symmetries in particle physics, we give a review of those properties of Poincare and conformal symmetries that are of interest for their localization and construction of the related gravitational theories. Understanding gravity as a theory based on local spacetime symmetries represents an important step towards the unification of all fundamental interactions.

III. Poincare gauge theory

It is well known that the existence and interaction of certain fields, such as the electromagnetic field, can be closely related to invariance properties of the theory. On the other hand, it is much less known that Einstein's GR is invariant under local Poincare transformations. This property is based on the principle of equivalence, and gives a rich physical content to the concept of local, or gauge symmetry. Instead of thinking of the local Poincare symmetry as being derived from the principle of equivalence, the whole idea can be reversed, in accordance with the usual philosophy of gauge theories. When the gravitational field is absent, it has become clear from a host of experiments that the underlying symmetry is given by the Poincare group. If we now want to make a physical theory invariant under local Poincare transformations, it is necessary to introduce new, compensating fields, which, actually, represent the gravitational interaction.

Localization of Poincare symmetry leads to the Poincare gauge theory of gravity (PGT), that contains GR as a special case. In PGT one can naturally couple gravity not only to the energy-momentum, but also to the spin of matter fields. The exposition of PGT is followed by the corresponding geometric interpretation, leading to Riemann-Cartan space. Then, we study a simple but important case of Einstein-Cartan theory, and give a short account of some basic dynamical features of the new gravitational theory. IV. Weyl gauge theory

The principle of gauge invariance was invoked for the first time in Weyl's unified theory of gravity and electromagnetism (1918). The geometry of Weyl's theory represents an extension of Riemannian structure of spacetime in GR. Weyl spacetime is based on the idea of invariance under the local change of the unit of length (gauge), which is realized by introducing an additional compensating field. Weyl tried to interpret this field as the electromagnetic potential, but, as further development showed, that was not possible. Passing through a process of change, the idea evolved into a new symmetry principle related to the local change of phase, the principle that underlies modern understanding of gauge theories of fundamental interactions.

Weyl's old idea gained new strength after some interesting discoveries in particle physics, in the sixties. Experimental results on deep inelastic electron-nucleon scattering showed that scattering amplitudes behaved as if all masses were negligible, and focused our attention on physical theories that are scale invariant. Localization of this symmetry brings us back to the old Weyl theory, which now describes not an electromagnetic but a gravitational interaction. Weyl's idea can be realized in two complementary ways: - as a gauge theory, based on the Weyl group W(1,3), and - as a geometric theory, obtained by extending Riemannian structure with local scale invariance. The first approach gives a new meaning to the old geometric construction.

V. Hamiltonian dynamics

All attempts to quantize the theory of gravity encounter serious difficulties. In order to find a solution to this problem, it seems to be useful to reconsider the fundamental principles of classical dynamics. In this context, the principles of Hamiltonian dynamics are seen to be of great importance not only for a basic understanding of the classical theory, but also for its quantization.

Theories of basic physical interactions, such as the electroweak theory or GR, are theories with gauge symmetries. In the Hamiltonian formalism such dynamical systems are characterized by the presence of constraints. The Hamiltonian formulation results in a clear picture of physical degrees of freedom and gauge symmetries, and enables a thorough understanding of the whole constrained dynamical structure. In the first part of this chapter we present the basic ideas of Dirac's Hamiltonian formalism, and develop a systematic approach to the construction of gauge generators.

In the second part of the chapter we present the Hamiltonian analysis of general PGT. It leads to a simple form of the gravitational Hamiltonian, representing a generalization of the canonical ADM Hamiltonian from GR, and enables a clear understanding of the interrelation between dynamical and geometric aspects of the theory. As an important example we analyze Einstein-Cartan theory without matter fields. This analysis represents a basis for a simple transition to Ashtekar's formulation of GR, in which encouraging results concerning the quantization of gravity are obtained.

VI. Symmetries and conservation laws

The existence of gauge symmetries is related to the presence of arbitrary multipliers in the total Hamiltonian, i.e. to the presence of first class constraints. The old question about the nature of constraints and the form of gauge generators has been resolved by Castellani, who developed an algorithm for constructing all the canonical gauge generators (1982). In this chapter we study the structure of Poincare gauge generators in PGT. An essential step in the procedure is the determination of the algebra of first class constraints.

A continuous symmetry of the action leads, via Noether's theorem, to a differentially conserved current. The conservation of the corresponding charge, which is an integral quantity, can be proven only under certain assumptions about asymptotic behavior of the basic dynamical variables. A clear and consistent picture of gravitational energy as a conserved quantity emerges only after the role and importance of boundary conditions has been fully recognized.

Assuming that the symmetry of PGT in the asymptotic region is the global Poincare symmetry, one can easily find the corresponding generators. Since the generators act on dynamical variables via the Poisson brackets, they should have well defined functional derivatives. A careful analysis of this requirement leads to the appearance of certain surface terms in the correct expressions for the generators. These terms determine conserved quantities, such as the energy, momentum and angular momentum of the gravitational system.

VII. Gravity in flat spacetime

In the framework of Riemannian geometry Einstein's GR is expressed as a theory in which gravitational phenomena are related to the geometry of spacetime. Notable analogies between electromagnetic and gravitational interactions inspired many attempts to unify these two theories. Today we know that any program of geometric unification must be necessarily more general, since the world of fundamental interactions contains much more than just electromagnetism and gravity.

There is, however, another approach to unification, which is based on the idea that gravity can be described as a relativistic quantum field theory in flat spacetime, like all the other fundamental interactions. In this approach, developed by Feynman in the sixties, the gravitational force is explained by the virtual exchange of a particle called the graviton.

In field theories long range forces are produced by the exchange of massless particles, as in electrodynamics. In order to determine the spin of the graviton, we analyze different possibilities and compare them with experimental data. Such considerations lead to the conclusion that the graviton is most successfully described as a massless tensor field of spin 2. If the graviton is coupled to the energy-momentum tensor of matter, the resulting theory of gravity - predicts slightly incorrect value for the precession of Mercury's orbit, and - yields field equations which are not consistent. Thus, the best description of gravity in flat spacetime is given by the massless spin 2 field, but its interaction with matter is not consistent.

VIII. Nonlinear effects in gravity

A closer inspection of the massless spin 2 field theory shows that its inconsistency comes from the fact that the graviton itself has an energy-momentum, which has to be included into a complete theory. The related correction of the theory leads to nonlinear effects: the energy-momentum of matter is a source of the gravitational field, whose energy-momentum becomes a source of the new field, etc. Can this nonlinear correction account for the small discrepancy in the precession of Mercury's perihelion? Surprisingly enough, the answer is yes. Moreover, the ``simplest" solution to the nonlinear self-coupling problem leads to a theory which is identical to Einstein's GR. This result gives an unexpected geometric interpretation to the field-theoretic approach.

In order to express the essential features of this problem and its resolution in the most simple manner, we first consider an analogous problem in Yang-Mills theory. Then, we go over to the gravitational field by studying first the scalar, and then the tensor theory. Particular attention is devoted to the first order formalism, which significantly simplifies the whole method.

IX. Supersymmetry and supergravity

One of the central goals of elementary particle physics at the end of this century is the unification of gravity with the other interactions, within a consistent quantum theory.

Supersymmetry is a symmetry that relates bosons and fermions, in a way which is consistent with basic principles of relativistic quantum field theory. It is characterized by both commutation and anticommutation relations between the symmetry generators, in contrast to the standard Lie group structure. Early investigations of quantum properties of supersymmetric theories led to an impressive result: some well known perturbative divergences stemming from bosons and fermions ``canceled" just because of supersymmetry. In this chapter we introduce the simple supersymmetric extension of the Poincare algebra, and consider two examples of supersymmetric field theories: the free Wess-Zumino model and supersymmetric electrodynamics. Then, we discuss representations on states and fields, and illustrate the construction of interacting field theories by studying the case of the Wess-Zumino model.

Since the concept of gauge invariance has been established as the basis for our understanding of particle physics, it is natural to elevate the idea of supersymmetry to the level of gauge symmetry, introducing thus the gravitational interaction into the world of supersymmetry. Quantum supergravity is found to be more finite than ordinary GR. To what extent these result might be extended to a satisfying quantum theory of gravity remains a question for the future. In the second part of this chapter we turn our attention to (classical) supergravity: we discuss its linearized and complete formulation, find the form of the related gauge algebras, and clarify the role of auxiliary fields.

X. Kaluza-Klein theory

Very soon after the discovery and experimental verification of GR, Kaluza (1921) proposed that the four--dimensional spacetime be supplemented with a fifth dimension, in order to give a unified account of the gravitational and electromagnetic interactions, the only two basic interactions known at the time. While Kaluza studied only classical structure of the five-dimensional gravity, Klein (1926) gave the first analysis of the compatibility of this theory with quantum mechanics. Much later, in the sixties, these investigations were generalized to higher number of dimensions, giving rise to a unification of gravity with non-Abelian gauge theories.

In the papers of Kaluza and Klein it is not clear whether the fifth dimension should be taken seriously, or merely as a useful mathematical device necessary to obtain a unified four-dimensional theory, whereupon its physical meaning is completely lost. We begin this chapter with an account of Kaluza's approach to this question, which relies on the so-called cylinder condition, expressing the assumption that all geometric (and physical) objects are independent of the fifth coordinate. The meaning of this assumption is explained by using the modern concept of spontaneous compactification: the fifth coordinate is curled up into a tiny circle whose radius is of the order of Planck length, so that it can not be seen at energies well bellow the Planck energy. Then, we give a systematic description of the five-dimensional theory, including the structure of the ground state, the construction of the effective four-dimensional theory, dynamical properties of the massless sector and matter fields. Finally, we go over to the non-Abelian generalization, and discuss some specific mechanisms of spontaneous compactification.

Significant advances in our understanding of fundamental interactions in the seventies brought a revival of interest for Kaluza's ideas. Since the electroweak and strong interactions are successfully described as gauge theories, the Kaluza-Klein approach may serve as a framework for studying the unification with gravity. In spite of many advances, the realization of these ideas has been accompanied by many difficulties, so that even today, almost eight decades after its birth, there is no realistic Kaluza-Klein theory. Nevertheless, physicists continue to study this approach, having a feeling that there must be at least a grain of truth in these fascinating ideas.

XI. String theory

String theory is the quantum theory of fundamental interactions in which the basic constituents are one-dimensional, rather than point-like objects. Since one mode of oscillation of the (closed) string is a massless, spin 2 state that can be identified with the graviton, string theory necessarily contains gravity. A significant success of string theory in a consistent treatment of quantum gravity in the eighties led to a renewed interest for identifying the underlying symmetry principles of strings, and for the construction of the related covariant field theory. Although in principle one can discover all the relevant properties of a theory by using a given gauge, this may be difficult in practice. In the covariant formulation string theory can be seen as a natural extension of many concepts that have become particularly important to modern physics. The construction of the covariant theory is only a first step in establishing a clear picture of the related geometric structure. Our understanding of nonperturbative semiclassical phenomena and spontaneous symmetry breaking has been developed in the gauge-invariant framework, and these properties play an essential role in the effective reduction of string theory to four dimensions. One also expects that quantum properties of the theory are more transparent in a covariant formulation.

The subject of this chapter is to present an introduction to the covariant field theory of free bosonic strings, which will cover main features of the classical theory. Our exposition is based on the gauge invariant Hamiltonian formalism, which clearly relates the two-dimensional reparametrization invariance, given in the form of Virasoro conditions, with gauge invariances in field theory. The classical action for the bosonic string is introduced in analogy with the related point-particle action. We describe the oscillator formalism for both open and closed strings, and obtain the classical Virasoro algebra as the algebra of first class constraints. After introducing the first quantization of the theory, we derive the quantum Virasoro algebra, and discuss the structure of the string states in terms of the usual component fields of different spin. Finally, we turn our attention to gauge symmetries and the basic principles underlying the construction of the action for the free bosonic string field theory. The physical content of the theory is clarified by identifying electromagnetism and gravity as massless sectors of open and closed string field theories, respectively.

The bosonic string is used to illustrate the common features of all string models, without to many technical complications. Interacting models and supersymmetric formulations can be described in a similar way, but are technically more involved.

Appendix

The Appendix consists of twelve separate sections (A-L), which are in different relations to the main text.

- Technical appendices J and L (Dirac spinors, Fourier expansion) are indispensable for the exposition in chapters IX and XI.

- Appendices A, H and I (internal local symmetries, Lorentz group, Poincare group) are very useful for the exposition in chapters III (A) and IX (H, I).

- Appendices C, D, E, F and G (de Sitter gauge theory, the scalar-tensor theory, Ashtekar's formulation of GR, constraint algebra and gauge symmetries, and covariance, spin and interaction of massless particles) are supplements to the main exposition, and may be studied according to the reader's choice.

- The material in appendices B and K (differentiable manifolds, symmetry groups and manifolds) is not necessary for the main exposition. It gives a deeper mathematical foundation for the geometric considerations in chapters III, IV and X.