Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime

摘要

These are the author's lectures at the 1992 Les Houches Summer School,"Gravitation and Quantizations". They develop a generalized sum-over-historiesquantum mechanics for quantum cosmology that does not require either apreferred notion of time or a definition of measurement. The "post-Everett"quantum mechanics of closed systems is reviewed. Generalized quantum theoriesare defined by three elements (1) the set of fine-grained histories of theclosed system which are its most refined possible description, (2) the allowedcoarse grainings which are partitions of the fine-grained histories intoclasses, and (3) a decoherence functional which measures interference betweencoarse grained histories. Probabilities are assigned to sets of alternativecoarse-grained histories that decohere as a consequence of the closed system'sdynamics and initial condition. Generalized sum-over histories quantum theoriesare constructed for non-relativistic quantum mechanics, abelian gauge theories,a single relativistic world line, and for general relativity. For relativitythe fine-grained histories are four-metrics and matter fields. Coarse grainingsare four-dimensional diffeomorphism invariant partitions of these. Thedecoherence function is expressed in sum-over-histories form. The quantummechanics of spacetime is thus expressed in fully spacetime form. Thecoarse-grainings are most general notion of alternative for quantum theoryexpressible in spacetime terms. Hamiltonian quantum mechanics of matter fieldswith its notion of unitarily evolving state on a spacelike surface is recoveredas an approximation to this generalized quantum mechanics appropriate for thoseinitial conditions and coarse-grainings such that spacetime geometry