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THE QUANTUM MECHANICS of COSMOLOGY James B. Hartle

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THE QUANTUM MECHANICS of COSMOLOGY James B. Hartle THE QUANTUM MECHANICS OF COSMOLOGY James B. Hartle Department of Physics University of California Santa Barbara, CA 93106 USA TABLE OF CONTENTS I. INTRODUCTION II. POST-EVERETT QUANTUM MECHANICS II.1. Probability II.1.1. Probabilities in General II.1.2. Probabilities in Quantum Mechanics II.2. Decoherent Histories II.2.1. Fine and Coarse-grained Histories II.2.2. Decohering Sets of Coarse-grained Histories II.2.3. No Moment by Moment Definition of Decoherence II.3. Prediction, Retrodiction, and History II.3.1. Prediction and Retrodiction II.3.2. The Reconstruction of History II.4. Branches (Illustrated by a Pure ρ) II.5. Sets of Histories with the Same Probabilities II.6. The Origins of Decoherence in Our Universe II.6.1. On What Does Decoherence Depend? II.6.2. A Two Slit Model II.6.3. The Caldeira-Leggett Oscillator Model II.6.4. The Evolution of Reduced Density Matrices II.7. Towards a Classical Domain II.8. The Branch Dependence of Decoherence II.9. Measurement II.10. The Ideal Measurement Model and The Copenhagen Approximation to Quantum Mechanics II.11. Approximate Probabilities Again II.12. Complex Adaptive Systems II.13. Open Questions 1 III. GENERALIZED QUANTUM MECHANICS III.1. General Features III.2. Hamiltonian Quantum Mechanics III.3. Sum-Over-Histories Quantum Mechanics for Theories with a Time III.4. Differences and Equivalences between Hamiltonian and Sum-Over-Histories Quantum Mechanics for Theories with a Time. III.5. Classical Physics and the Classical Limit of Quantum Mechanics III.6. Generalizations of Hamiltonian Quantum Mechanics IV. TIME IN QUANTUM MECHANICS IV.1. Observables on Spacetime Regions IV.2. The Arrow of Time in Quantum Mechanics IV.3. Topology in Time IV.4. The Generality of Sum-Over-History Quantum Mechanics V. THE QUANTUM MECHANICS OF SPACETIME V.1. The Problem of Time V.1.1. General Covariance and Time in Hamiltonian Quantum Mechanics V.1.2. The \Marvelous Moment" V.2. A Quantum Mechanics for Spacetime V.2.1. What We Need V.2.2. Sum-Over-Histories Quantum Mechanics for Theories without a Time V.2.3. Sum-Over-Spacetime-Histories Quantum Mechanics V.2.4. Extensions and Contractions V.3. The Construction of Sums Over Spacetime Histories V.4. Some Open Questions VI. PRACTICAL QUANTUM COSMOLOGY VI.1. The Semiclassical Regime VI.2. The Semiclassical Approximation to the Quantum Mechanics of a Non-Relativistic Particle VI.3. Semiclassical Prediction in Quantum Mechanics ACKNOWLEDGEMENTS REFERENCES APPENDIX: BUZZWORDS 2 I. INTRODUCTION It is an inescapable inference from the physics of the last sixty years that we live in a quantum mechanical universe | a world in which the basic laws of physics conform to that framework for prediction we call quantum mechanics. If this inference is correct, then there must be a description of the universe as a whole and everything in it in quantum mechanical terms. The nature of this description and its observable consequences are the subject of quantum cosmology. Our observations of the present universe on the largest scales are crude and a classical description of them is entirely adequate. Providing a quantum mechanical description of these observations alone might be an interesting intellectual chal- lenge, but it would be unlikely to yield testable predictions differing from those of classical physics. Today, however, we have a more ambitious aim. We aim, in quantum cosmology, to provide a theory of the initial condition of the universe which will predict testable correlations among observations today. There are no realistic predictions of any kind that do not depend on this initial condition, if only very weakly. Predictions of certain observations may be testably sensitive to its details. These include the large scale homogeneity and isotropy of the universe, its approximate spatial flatness, the spectrum of density fluctuations that produced the galaxies, the homogeneity of the thermodynamic arrow of time, and the existence of classical spacetime. Further, one of the main topics of this school is the question of whether the coupling constants of the effective interactions of the elementary particles at accessible energy scales may depend, in part, on the initial condition of the universe. It is for such reasons that the search for a theory of the initial condition of the universe is just as necessary and just as fundamental as the search for a theory of the dynamics of the elementary particles.∗ The physics of the very early universe is likely to be quantum mechanical in an essential way. The singularity theorems of classical general relativityy suggest that an early era preceded ours in which even the geometry of spacetime exhibited significant quantum fluctuations. It is for a theory of the initial condition that de- scribes this era, and all later ones, that we need a quantum mechanics of cosmology. That quantum mechanics is the subject of these lectures. The \Copenhagen" frameworks for quantum mechanics, as they were for- mulated in the 1930's and '40's and as they exist in most textbooks today,z are inadequate for quantum cosmology on at least two counts. First, these formu- lations characteristically assumed a possible division of the world into \obsever" and \observed", assumed that \measurements" are the primary focus of scientific statements and, in effect, posited the existence of an external \classical domain". However, in a theory of the whole thing there can be no fundamental division into observer and observed. Measurements and observers cannot be fundamental no- tions in a theory that seeks to describe the early universe when neither existed. In ∗ For reviews of quantum cosmology see lectures by Halliwell in this volume and Hartle (1988c, 1990a). For a bibliography of papers on the subject through 1989 see Halliwell (1990). y For a review of the singularity theorems of classical general relativity see Geroch and Horowitz (1979). For the specific application to cosmology see Hawking and Ellis (1968). z There are various \Copenhagen" formulations. For a classic exposition of one of them see London and Bauer (1939). 3 a basic formulation of quantum mechanics there is no reason in general for there to be any variables that exhibit classical behavior in all circumstances. Copenhagen quantum mechanics thus needs to be generalized to provide a quantum framework for cosmology. The second count on which the familiar formulations of quantum mechanics are inadequate for cosmology concerns their central use of a preferred time. Time is a special observable in Hamiltonian quantum mechanics. Probabilities are pre- dicted for observations \at one moment of time". Time is the only observable for which there are no interfering alternatives as a measurement of momentum is an interfering alternative to a measurement of position. Time is the sole observable not represented by an operator in the familiar quantum mechanical formalism, but rather enters the theory as a parameter describing evolution. Were there a fixed geometry for spacetime, that background geometry would give physical meaning to the preferred time of Hamiltonian quantum mechanics. Thus, the Newtonian time of non-relativistic spacetime is unambiguously taken over as the preferred time of non-relativistic quantum mechanics. The spacetime of special relativity has many timelike directions. Any one of them could supply the preferred time of a quantum theory of relativistic fields. It makes no difference which is used for all such quantum theories are unitarily equivalent because of relativistic causality. However, spacetime is not fixed fundamentally. In a quantum theory of gravity, in the domain of the very early universe, it will fluctuate and be without definite value. If spacetime is quantum mechanically variable, there is no one fixed spacetime to supply a unique notion of \timelike, \spacelike" or \spacelike surface". In a quantum theory of spacetime there is no one background geometry to give meaning to the preferred time of Hamiltonian quantum mechanics. There is thus a conflict between the general covariance of the physics of gravitation and the preferred time of Hamiltonian quantum mechanics. This is the \problem of time" in quantum gravity.∗ Again the familiar framework of quantum mechanics needs to be generalized. These lectures discuss the generalizations of Copenhagen quantum mechanics necessary to deal with cosmology and with quantum cosmological spacetimes. Their purpose is to sketch a coherent framework for the process of quantum mechanical prediction in general, and for extracting the predictions of theories of the initial condition in particular. Throughout I shall assume quantum mechanics and I shall assume spacetime. Some hold the view that \quantum mechanics is obviously absurd but not obviously wrong" in its application to the macroscopic domain. I hope to show that it is not obviously absurd. Some believe that spacetime is not fundamental. If so, then a quantum mechanical framework at least as general as that sketched here will be needed to discuss the effective physics of spacetime on all scales above the Planck length and revisions of the familiar framework at least as radical as those suggested here will be needed below it. The strategy of these lectures is to discuss the generalizations needed for a quantum mechanics of cosmology in two steps. I shall begin by assuming in Section II a fixed background spacetime that supplies a preferred family of timelike directions for quantum mechanics. This of course, is an excellent approximation on accessible scales for times later than 10−43 sec
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