Quantum mechanics, as for example in the case of a non-relativistic particle, can be treated in either of two ways. One can work with the differential-equation form of the theory, by studying the Schrödinger equation. Alternatively, one can study the Feynman path integral, which gives the integral form of the Schrödinger differential approach. The Feynman path integral has the advantage of incorporating the boundary conditions on the particle, for example that the particle is at spatial position xa at an initial time ta, and at position xb at final time tb. The path integral leads naturally to a semi-classical expansion of the quantum amplitude, valid asymptotically as the action of the classical solution of the equations of motion becomes large compared to Planck's constant ħ.

One moves from quantum mechanics to quantum gravity by replacing the spatial argument x of the wave function by the three-dimensional spatial geometry hij(x). A typical quantum amplitude is then the amplitude to go from an initial three-geometry hijI to a final geometry hijF, specified (say) on identical three-surfaces ΣI, ΣF. To complete the description in the asymptotically flat case, one needs to specify asymptotic parameters such as the time T between the two surfaces, measured at spatial infinity. To make the classical boundary-value problem elliptic and (one hopes) well-posed, one rotates to imaginary time –iT. The Feynman path integral would again give a semi-classical expansion of the quantum amplitude, were it not for the infinities present in the loop amplitudes.

Introduction

Before embarking on the full theory of N = 1 supergravity in the following chapters, it is necessary to review some of what is known about quantum cosmology based on general relativity, possibly coupled to spin-0 or spin-1/2 (non-supersymmetric) matter. The ideas presented in this chapter, based to a considerable extent but not exclusively on Hamiltonian methods, will recur throughout the book. Perhaps the main underlying idea is that there is an analogy between the classical dynamics of a point particle with position x and that of a three-geometry hij(x). The theory of point-particle dynamics, when written in parametrized form [Kuchař 1981] and cast into Hamiltonian form, and the theory of general relativity, again in Hamiltonian form, bear a strong resemblance. In the Hamiltonian form of general relativity, hij(x) can be taken to be the ‘coordinate’ variable, corresponding to x in particle dynamics. In section 2.2, for parametrized particle dynamics, it is shown following [Kuchař 1981] how a constraint arises classically in the Hamiltonian theory, which, when quantized, gives the appropriate Schrödinger or wave equation for the quantum wave function ψ(x, t). As described in subsequent sections, the quantization of the analogous constraint in general relativity gives the Wheeler–DeWitt equation [DeWitt 1967, Wheeler 1968], a second-order functional differential equation for the wave function Ψ[hij(x)], which contains all the information in quantum gravity, if only one could solve and interpret it.

The Hamiltonian form of general relativity is derived from the Einstein–Hilbert Lagrangian in section 2.3.

The application of canonical methods to gravity has a long history [De-Witt 1967]. In [Dirac 1950] a general Hamiltonian approach was presented, which allowed for the presence of constraints in a theory, due to the momenta not being independent functions of the velocities. In particular, this occurs in general relativity, because of the underlying coordinate invariance of gravity. The general approach above was applied to general relativity in [Dirac 1958a,b, 1959] and further described in [Dirac 1965]. It was seen that there are four constraints, usually written ℋi(i = 1,2,3) and ℋ⊥, associated with the freedom to make coordinate transformations in the spatial and normal directions relative to a hypersurface t = const. in the Hamiltonian decomposition. Classically, these four constraints must vanish for allowed initial data. In the quantum theory, as will be seen in chapter 2, these constraints become operators on physically allowed states Ψ, which must obey ℋiΨ = 0, ℋ⊥Ψ = 0. Here, in the simplest representation, Ψ is a functional of the spatial metric hij(x). It was shown in [Higgs 1958, 1959] that the constraints ℋiΨ = 0 precisely describe the invariance of the wave function under spatial coordinate transformations. The Hamiltonian formulation of gravity was also studied by [Arnowitt et al. 1962], who provided the standard definition of the mass or energy M of a spacetime, as measured at spatial infinity.

Quasars, which can be a thousand times brighter than an ordinary galaxy, are the most distant objects observable in the Universe. How quasars produce the luminosity of 1013 suns in a volume the size of the solar system continues to be a major question in astronomy. Distant quasars are very rare objects whose study has been blocked by their scarcity. Recent technical advances, however, have opened new paths for their discovery. Forty quasars with redshifts greater than 4 have been found since 1986. Redshift 4 corresponds to a light travel time of more than 10 billion years. As a result, we are now able to probe the epoch shortly after the Big Bang when quasars may have first formed and to study the universe when it was less than a tenth its present age.

Quasars were one of the main discoveries thirty years ago that revolutionized astronomy. While they and the black holes thought to occur in their centers have become household words today, quasars are as enigmatic in many ways as they were when first discovered. Whatever their nature, they offer us views of the Universe never before seen, especially at distances far beyond what astronomers of the previous generation expected to see. In this chapter I wish to review briefly their history, how extraordinary their properties are, and how they serve as probes of the Universe to nearly as far as the visible horizon.

This book originated as a symposium at the American Association for the Advancement of Science annual meeting in San Francisco in 1989. The topic, The Farthest Things in the Universe, suggested itself to me as the most interesting and significant topic that people could hear about. An earlier AAAS Symposium had led to a book, The Redshift Controversy, that was still in use, and we hope that this volume will prove itself of similarly lasting interest.

Two of the original speakers, Hyron Spinrad of the University of California at Berkeley, and Patrick Osmer, then of the National Optical Astronomy Observatories, revised their pieces to bring them up-to-date for inclusion in this book. Further, Ed Cheng of the COBE Science Team and NASA's Goddard Space Flight Center agreed to write a new piece for inclusion in the book. We appreciate his taking time during the period of his duties as Chief Scientist for the Hubble Space Telescope's repair mission to complete his piece. During the interval from the time of the symposium to the present, the Cosmic Background Explorer spacecraft was launched and has had its tremendous successes in showing that the Universe has a blackbody spectrum and in finding ripples in space that may be the seeds from which galaxy-formation began. Thus this book appears at an optimum time.

The technical ability of astronomers to obtain images and spectra of very faint galaxies has improved greatly over the last decade. Since galaxies are vast collections of gas and stars, they must physically evolve with time. We should be able to directly observe the time-evolution of galaxies by studying very distant systems; the look-back internal corresponding to the mostdistant galaxies known in 1992 now approaches 15 billion years (80% of the total expansion age of the Universe)!

The line spectra of these faint galaxies are invaluable for redshift determination and physical study. The realization that Ly α (121.6 nm), formed in neutral hydrogen gas, is a strong emission line in most active galaxies and perhaps normal star-forming galaxies also, has helped us measure much larger redshifts in 1987–92 than was previously possible. Recall that this wavelength is in the ultraviolet; it can be observed only by satellites. But when galaxies are very far away, their Doppler effect shifts this spectral line into the region of the spectrum that we can observe with large telescopes on Earth. The largest redshifts for radio galaxies now approach z=3.8. Differing selection effects control which galaxies can be seen/isolated that far away. At least some red galaxies must form at redshift zf>5 (where the subscript f stands for the epoch of star formation).

When we look out into space at night, we see the Moon, the planets, and the stars. The Moon is so close, only about 380000 kilometers (240000 miles) that we can send humans out to walk on it, as we did in the brief glorious period from 1969 to 1972. Even the planets are close enough that we can send spacecraft out to them, notably the Voyager spacecraft, one of which has passed Neptune. Whereas light and radio signals from spacecraft take only about a second to reach us from the Moon, the radio signals from Voyager 2 at Neptune took several hours to travel to waiting radio telescopes on Earth. We say that the distance to the Moon is 1 light-second and the distance to Neptune is several light-hours.

Aside from our Sun, the nearest star at 8 light-minutes away, the distances to the stars are measured in light-years. The nearest star system is Alpha Centauri, visible only in the southern sky, and the single nearest star is known as Proxima Centauri, about 4.2 light-years away. We know so little about the stars that new evidence in 1993 indicates that Proxima Centauri might not be a member of a triple-star system along with the other parts of alpha Centauri, as has long been thought. The speeds at which those stars are moving through space may be sufficiently different that Proxima is only temporarily near Alpha's components.

Looking up at the clear night sky, it is hard to avoid wondering about the many objects that we can see. It is simple to recognize with the naked eye that there are planets, countless stars, and the band of light from the disk of our own Galaxy, the Milky Way. With the help of binoculars or a small telescope, the complexity of the scene increases dramatically, and it becomes apparent that the glow of the Milky Way is the light from many faint stars. We also start to notice that there are numerous faint and fuzzy objects which are the nearby galaxies and the star-forming regions in our own Galaxy. Probing with more and more sophisticated instruments, the level of detail and structure that can be resolved using visible light increases until the light becomes so exceedingly faint that even the best detectors on the largest telescopes see only darkness. This is the regime of the farthest objects in the Universe.

Before discussing these objects in any detail, I would like to take a brief moment and address the question of how we can possibly know about things so remote in both distance and experience. After all, we invent and test the physical sciences here on Earth by making experiments, interacting with the world around us, and creating a system of beliefs (theories) that ties all these experiments together into a consistent and testable story.