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Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime

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James B. Hartle

General Relativity and Quantum Cosmology · Submitted 5 Apr 1993 · 151 references

Pith's verdict: The genuinely new work is the spacetime coarse-graining machinery in Sections V–VII; the gravitational section is a program rather than a construction, and the abstract's 'constructed' oversells it.

T0 · AI review

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arXiv IDgr-qc/9304006 (version 3)
Pith IDHXZCPNUE, Pith's permanent record of this paper and its review
CitationJames B. Hartle. "Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime." arXiv:gr-qc/9304006 (1993). https://pith.science/paper/HXZCPNUE
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The paper

The paper claims

Hartle argues that quantum mechanics can be written in fully four-dimensional form, with no privileged time and no measurement postulate, by assigning probabilities only to coarse-grained histories whose interference has vanished.

The reading

The paper's aim is to make quantum mechanics general enough to apply to the universe as a whole, where there is no external observer to measure anything and no fixed background geometry to define a moment of time. Hartle's proposal is that a quantum theory is fully specified by three things: the most refined set of alternative histories the system could have (fine-grained histories), the partitions of that set into coarser classes, and a complex-valued decoherence functional that measures interference between any two classes. Probabilities are then assigned only to those sets of histories whose off-diagonal interference terms are negligible as a consequence of the system's dynamics and initial condition; where that happens, the ordinary sum rules of probability follow automatically. The framework is built up through model cases — non-relativistic particles, the electromagnetic field, a single relativistic world line — and then applied to general relativity, where the fine-grained histories are four-dimensional metrics and matter fields and the coarse grainings are partitions invariant under diffeomorphisms. The payoff, if it works, is that the alternatives quantum theory talks about need no longer be located 'at a moment of time', which is what dissolves the conflict between general covariance and the Hamiltonian demand for a preferred family of spacelike surfaces.

What carries the argument

The decoherence functional is the load-bearing object: $D(α',α)=Tr[C_{α'}ρC_α^†]$ for a set of histories with class operators $C_α$, reducing in the sum-over-histories version to a double path integral with a factor $e^{i(S[q']-S[q])}$ and a final-condition kernel. A class operator $C_α$ is the amplitude for the system to follow any fine-grained history in the class $α$; when the classes are defined by ranges of configuration at fixed times these reduce to time-ordered chains of projection operators, and when they are defined by a spacetime region they become absorbing-potential evolution operators $U_R(T)$ that no longer factorize into unitary evolution plus reduction. The decoherence condition $D(α',α)≈0$ replaces the measurement postulate: it is what licenses probability assignment, and it is a property of the dynamics and initial condition rather than of any observer. The paper also carries a time-neutral variant with both initial and final density matrices, $D(α',α)=N Tr[ρ_f C_{α'}ρ_i C_α^†]$, which removes the built-in quantum arrow of time.

Extended reading notes

Core claim

The central claim is that the conflict between quantum mechanics and general covariance is an artifact of taking Hamiltonian quantum mechanics — states on spacelike surfaces, unitary evolution, wave-packet reduction — as the definition of quantum theory rather than as one special case of it. Hartle defines a generalized quantum mechanics by three elements: fine-grained histories, allowed coarse grainings (partitions of those histories into exhaustive exclusive classes), and a decoherence functional $D(α',α)$ satisfying Hermiticity, positivity, normalization, and a superposition principle across coarse grainings. Probabilities are the diagonal entries $p(α)=D(α,α)$, and they obey the probability sum rules whenever $D(α',α)≈0$ for $α'≠α$. For general relativity the fine-grained histories are four-metrics and matter fields, the coarse grainings are diffeomorphism-invariant partitions of these, and the class operators are sums over four-geometries restricted to a class, with the decoherence functional written as a double sum-over-histories weighted by initial and final conditions. Because the alternatives are four-dimensional regions of histories rather than configurations on a slice, no preferred time enters the formulation. Hamiltonian quantum mechanics on spacelike surfaces is then claimed to re-emerge as an approximation, valid for those initial conditions and coarse grainings for which the geometry behaves classically and supplies an effective notion of time.

Load-bearing premise

The argument assumes that the sum over four-geometries, together with a diffeomorphism-invariant class operator and a suitable measure, is a well-defined object that satisfies the four decoherence-functional axioms and admits at least some coarse grainings that actually decohere — a claim written down formally for gravity but never demonstrated, evaluated, or exemplified in these lectures.

Editorial extensions

If this is right

  • Probabilities can be assigned to alternatives that are not 'at a moment of time' — for example, whether a particle's path ever enters a given spacetime region — so quantum theory acquires genuine four-dimensional alternatives rather than slice-by-slice ones.
  • In quantum cosmology, the initial condition of the universe plus the dynamics determine which sets of histories decohere, so the classical domain of everyday experience becomes a derived prediction rather than a postulated background.
  • The observed arrows of time (thermodynamic, psychological, radiative, cosmological) can all be traced to asymmetric boundary conditions rather than to a fundamental asymmetry in the laws, since the time-neutral form of the decoherence functional is symmetric between past and future.
  • Where a coarse graining produces a decohering set of quasiclassical variables obeying a Langevin-like equation, classical deterministic behavior is recovered with calculable noise and dissipation, giving a quantitative account of the classical limit.
  • Hamiltonian quantum mechanics with its unitarily evolving state is demoted to an approximation, valid only in epochs and at scales where spacetime geometry is itself effectively classical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the program holds, the 'problem of time' in quantum gravity is not solved by finding the right internal clock but by dropping the requirement that alternatives be located on a clock surface at all — a reframing that would make most proposed clock-based resolutions look like approximations to a more general scheme.
  • The framework suggests a concrete calculational target for quantum cosmology: compute the decoherence functional on coarse grainings of four-geometries for a proposed initial condition and check whether any of them decohere at all. Absent such a demonstration, the claim that our classical spacetime is emergent remains a promissory note.
  • The time-neutral version with a final condition raises the possibility that residual quantum correlations between early and late universe could be observable in principle as violations of the usual retarded/advanced asymmetry, which would be a rare cosmological handle on the boundary conditions of the universe.
  • The emphasis on partitions of histories rather than on algebras of observables puts this framework in direct dialogue with consistent-histories approaches; the interesting open question is whether the two notions of 'set of alternatives' coincide in cases with gauge symmetry, where only invariant partitions are admissible.
Original 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

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,860 input tokens, $0.0004 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.37 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

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

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 These are the author's lectures at the 1992 Les Houches Summer School, "Gravitation and Quantizations". background 1.00 0.30 0.01 asserted 0.98
s1 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. contribution 1.00 0.85 0.46 asserted 1.00
s2 The "post-Everett" quantum mechanics of closed systems is reviewed. background 0.96 0.13 0.04 asserted 0.89
s3 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. method 0.65 0.33 0.10 asserted 1.00
s4 Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. method 0.63 0.48 0.14 asserted 1.00
s5 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. contribution 0.94 0.80 0.11 asserted 1.00
s6 For relativity the fine-grained histories are four-metrics and matter fields. method 0.73 0.52 0.05 asserted 1.00
s7 Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. method 0.80 0.63 0.06 asserted 1.00
s8 The decoherence function is expressed in sum-over-histories form. method 0.71 0.77 0.07 asserted 1.00
s9 The quantum mechanics of spacetime is thus expressed in fully spacetime form. interpretation 0.54 0.71 0.27 asserted 1.00
s10 The coarse-grainings are most general notion of alternative for quantum theory expressible in spacetime terms. interpretation 0.95 0.55 0.26 asserted 1.00
s11 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 interpretation 0.54 0.70 0.24 asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.51 · evidence from proof or derivation 0.68.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: The two-slit experiment. An electron gun at right emits an electron traveling towards
Figure 2. FIG. 2: A model closed quantum system. At one fundamental level of description this system con
Figure 3. FIG. 3: Branching structure of a set of alternative histories. This figure illustrates the set of
Figure 4. FIG. 4: The two-slit experiment with an interacting gas. Near the slits light particles of a gas
Figure 5. FIG. 5: Factoring a sum over paths single-valued in time across a surface of constant time. Shown
Figure 6. FIG. 6: The decoherence of histories coarse-grained by intervals of a distinguished set of config
Figure 7. FIG. 7: The schematic structure of the space of
Figure 8. FIG. 8: The schematic structure of the space of sets of histories in sum-over-histories quantum
Figure 12. FIG. 12: Two-dimensional simplicial geometries. Two-dimensional surfaces can be made up by
Figure 13. FIG. 13: Superspace. A cosmological history is a four-dimensional cosmological spacetime with
Figure 14. FIG. 14: The semiclassical approximation to the quantum mechanics of a non-relativistic particle.

Showing 11 of the paper's 14 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9304006 by the authors.

References
How later papers used it

58 later papers cite it

Evidence: substantial. 45 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 111 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 2 of 5 papers with this label
How the literature uses it

59 arXiv papers cite this in 112 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 61 sentences without a confident label

Uses its method or tool 5 papers
  • In this paper, I focus on Hartle's definition of the class operator via path integrals [ref] .

    Decoherent Histories of Spin Networks · Introduction · 0.97

  • We assume that the matter content of the universe can be described by a massless scalar field and we consider the consistent histories approach of [ref] to explore the existence of a quantum bounce.

    The Wheeler-DeWitt Quantization Can Solve the Singularity Problem · The Craig and Singh result · 0.85

  • We use sum-over-histories quantum mechanics [ref] to construct a toy model of the quantum dynamics of gravitational collapse in a scenario like what is envisioned in the fuzzball proposal[See e.g. [ref] for recent studies of possible quantum gravity signatures in alternative models of gravitational collapse.].

    Observational Implications of Fuzzball Formation · Introduction · 0.98

Compares with it 3 papers
Cites it as background 36 papers
  • The violation of Bell inequality in the entangled Einstein-Podolsky-Rosen (EPR) pair [ref] indicates that two particles have an “instant interaction”, in contrast to theories of hidden variables that preserve strict locality [ref] .

    Bell Inequality in the Holographic EPR Pair · Introduction · 0.94

  • One attempt to do this was made by J. Hartle who set out new, histories-based axioms for Generalised Quantum Mechanics (GQM) which do not require the existence of a Hilbert space of states [ref] .

    Hilbert Spaces from Path Integrals · Introduction · 0.90

  • The idea that quantum mechanics can be well defined even if the notion of time is absent has been proposed [ref] and developed in a number of different strategies [ref] .

    Timeless path integral for relativistic quantum mechanics · Introduction · 0.99

Discussion

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