Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime

TL;DR

The work develops a covariant, time-neutral framework for quantum mechanics suitable for quantum cosmology and quantum gravity, built on a generalized sum-over-histories with fine-grained histories, coarse grainings, and a decoherence functional $D(\alpha',\alpha)$. It shows Hamiltonian quantum mechanics emerges as an approximation in regimes with a fixed background time, while the full formalism accommodates spacetime, reparametrization, and diffeomorphism invariance across nonrelativistic, gauge, relativistic, and gravitational theories. Through solvable models (e.g., Emsch, linear-oscillator baths), environment-induced decoherence and the emergence of a quasiclassical domain are demonstrated, providing a mechanism for the classical world and spacetime to arise from quantum dynamics and initial conditions. The framework offers a principled route to quantum cosmology and quantum gravity, clarifying how predictions can be made without external observers and how classical spacetime and causality can emerge from fundamentally quantum, covariant descriptions.

Abstract

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

Figures (14)

• Figure 1: The two-slit experiment. An electron gun at right emits an electron traveling towards a screen with two slits, its progress in space recapitulating its evolution in time. When precise detections are made of an ensemble of such electrons at the screen it is not possible, because of interference, to assign a probability to the alternatives of whether an individual electron went through the upper slit or the lower slit. However, if the electron interacts with apparatus that measures which slit it passed through, then these alternatives decohere and probabilities can be assigned.
• Figure 2: A model closed quantum system. At one fundamental level of description this system consists of a large number of electrons, nucleons, and excitations of the electromagnetic field. However, the initial state of the system is such that at a coarser level description it contains an observer together with the necessary apparatus for carrying out a two-slit experiment. Alternatives for the system include whether the "system" contains a two-slit experiment or not, whether it contains an observer or not, whether the observer measured which slit the electron passed through or did not, whether the electron passed through the upper or lower slit, the alternative positions of arrival of the electron at the screen, the alternative arrival positions registered by the apparatus, the registration of these in the brain of the observer, etc., etc., etc. Each exhaustive set of exclusive alternatives is represented by an exhaustive set of orthogonal projection operators on the Hilbert space of the closed system. Time sequences of such sets of alternatives describe sets of alternative coarse-grained histories of the closed system. Quantum theory assigns probabilities to the individual alternative histories in such a set when there is negligible quantum mechanical interference between them, that is, when the set of histories decoheres.
• Figure 3: Branching structure of a set of alternative histories. This figure illustrates the set of alternative histories for the model closed system of Figure 2 defined by the alternatives of whether the observer decided to measure or did not decide to measure which slit the electron went through at time $t_1$, whether the electron went through the upper slit or through the lower slit at time $t_2$, and the alternative positions of arrival at the screen at time $t_3$. A single branch corresponding to the alternatives that the measurement was carried out, the electron went through the upper slit, and arrived at point $y_9$ on the screen is illustrated by the heavy line. The illustrated set of histories does not decohere because there is significant quantum mechanical interference between the branch where no measurement was carried out and the electron went through the upper slit and the similar branch where it went through the lower slit. A related set of histories that does decohere can be obtained by replacing the alternatives at time $t_2$ by the following set of three alternatives: (a record of the decision shows a measurement was initiated and the electron went through the upper slit); (a record of the decision shows a measurement was initiated and the electron went through the lower slit); (a record of the decision shows that the measurement was not initiated). The vanishing of the interference between the alternative values of the record and the alternative configurations of apparatus ensures the decoherence of this set of alternative histories.
• Figure 4: The two-slit experiment with an interacting gas. Near the slits light particles of a gas collide with the electrons. Even if the collisions do not affect the trajectories of the electrons very much they can still carry away the phase correlations between the histories in which the electron arrived at point $y_k$ on the screen by passing through the upper slit and that in which it arrived at the same point by passing through the lower slit. A coarse graining that consisted only of these two alternative histories of the electron would approximately decohere as a consequence of the interactions with the gas given adequate density, cross-section, etc. Interference is destroyed and probabilities can be assigned to these alternative histories of the electron in a way that they could not be if the gas were not present ( cf. Fig. 1). The lost phase information is still available in correlations between states of the gas and states of the electron. The alternative histories of the electron would not decohere in a coarse graining that included both the histories of the electron and operators that were sensitive to the correlations between the electrons and the gas. This model illustrates a widely occurring mechanism by which certain types of coarse-grained sets of alternative histories decohere in the universe.
• Figure 5: Factoring a sum over paths single-valued in time across a surface of constant time. Shown at left is the sum over paths defining the amplitude to start from $q_0$ at time $t=0$, proceed through interval $\Delta_k$ at time $t_k$, and wind up at $q_f$ at time $T$. If the histories are such that each path intersects each surface of constant time once and only once, then the sum on the left can be factored as indicated at right. The factored sum consists of a sum over paths before time $t_k$, a sum over paths after time $t_k$, followed by a sum over the values of $q_k$ at time $t_k$ inside the interval $\Delta_k$. The possibility of this factorization is what allows the Hamiltonian form of quantum mechanics to be derived from a sum-over-histories formulation. The sums over paths before and after $t_k$ define wave functions on that time-slice and the integration over $q_k$ defines their inner product. The notion of state at a moment of time and the Hilbert space of such states is thus recovered. If the sum on the left were over paths that were multiple valued in time, the factorization on the right would not be possible.