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arxiv: 1907.00912 · v2 · pith:2K5S5CQCnew · submitted 2019-07-01 · 🌀 gr-qc

Problem of Time and Background Independence: classical version's higher Lie Theory

Edward Anderson This is my paper

Pith reviewed 2026-05-25 11:35 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Problem of TimeBackground IndependenceLie TheoryLie RigidityReallocation of Intermediary-Object InvarianceRefoliation InvarianceConstraint Closure
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The pith

Higher Lie Theory supplies selection principles for Background Independence via Lie Rigidity and Reallocation of Intermediary-Object Invariance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a local resolution of the Problem of Time, recast as a local theory of Background Independence, by incorporating higher aspects of Lie Theory. It generalizes Dirac's Algorithm to Lie's Algorithm for Generator Closure and treats the passage of theory families through closure procedures as the Dirac Rigidity subcase of a broader Lie Rigidity. The main advance is the introduction of Reallocation of Intermediary-Object Invariance as the theory-independent counterpart to general relativity's refoliation invariance, formulated as a commuting pentagon criterion, with both rigidity and the pentagon criterion proposed as selection principles within the Comparative Theory of Background Independence.

Core claim

The paper claims that Lie Rigidity provides a partial cohomological selection principle for comparing background-independent theories, while Reallocation of Intermediary-Object Invariance, defined as the requirement that different choices of intermediary objects in an evolution yield results differing by at most an automorphism of the final object, supplies a universal commuting-pentagon selection principle for the same comparative theory.

What carries the argument

Reallocation of Intermediary-Object Invariance, a commuting pentagon criterion that checks whether alternative intermediary paths in object evolution differ by at most an automorphism of the final object.

If this is right

  • Lie Rigidity supplies a cohomological criterion that selects among families of theories passed through closure algorithms.
  • Reallocation of Intermediary-Object Invariance extends refoliation invariance to a universal, theory-independent test.
  • Both criteria operate as selection principles inside the Comparative Theory of Background Independence.
  • Lattices of constraint substructures continue to induce dual lattices of observable subalgebras under the extended Lie structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These criteria could be used to filter candidate classical theories prior to attempting quantization.
  • The Foundations of Geometry example of Lie Rigidity indicates the criteria may apply to geometric structures outside gravitational theories.
  • Similar pentagon tests could be formulated for other evolution problems involving multiple paths between objects.

Load-bearing premise

That Lie Rigidity and the commuting pentagon criterion for Reallocation of Intermediary-Object Invariance function as selection principles for the Comparative Theory of Background Independence.

What would settle it

A calculation demonstrating that general relativity fails the commuting pentagon test for Reallocation of Intermediary-Object Invariance when applied to its own refoliation invariance.

Figures

Figures reproduced from arXiv: 1907.00912 by Edward Anderson.

Figure 2
Figure 2. Figure 2: c)]. This generalizes to the following universally poseable, if not necessarily realizable, structure. Definition 1 Reallocation of Intermediary-Object (RIO) is the commuting-pentagon property depicted in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Commuting pentagon from Oin to Ofin via two distinct allocations of intermediary objects, O1 and O2. Remark 1 One or both of O1 and O2 can be replaced with distinct arbitrary intermediate objects. Remark 2 Some theories will obey this property, and some will not (see Appendix A.2 and the Conclusion for examples). RIO invariance thus also has the status of a selection principle. 7 Conclusion The current Art… view at source ↗
Figure 2
Figure 2. Figure 2: a) Illustrating the nature of foliation f: a decorated (or, more precisely, rigged) version of the standard Differential-Geometric definition of chart on a manifold M. b) Posing Refoliation Invariance: is going from spatial hypersurfaces in to fin via the red (R) intermediary hypersurface being physically the same as going via the purple (P) intermediary surface? If so, the blue and black hypersurfaces wou… view at source ↗
read the original abstract

A local resolution of the Problem of Time has recently been given, alongside reformulation as a local theory of Background Independence. The classical part of this requires just Lie's Mathematics, much of which is basic: i) Lie derivatives to encode Relationalism. ii) Lie brackets for Closure giving Lie algebraic structures. iii) Observables defined by a Lie brackets relation, in the constrained canonical case as explicit PDEs to be solved using Lie's flow method, and themselves forming Lie algebras. iv) Lattices of constraint algebraic substructures induce dual lattices of observables subalgebras. The current Article focuses on two pieces of `higher Lie Theory' that are also required. Preliminarily, we extend Dirac's Algorithm for Constraint Closure to `Lie's Algorithm' for Generator Closure. 1) We then reinterpret `passing families of theories through the Dirac Algorithm' - a method used for Spacetime Construction (from space) and getting more structure from less structure assumed more generally - as the Dirac Rigidity subcase of Lie Rigidity. We also provide a Foundations of Geometry example of specifically Lie rather than Dirac Rigidity, to illustrate merit in extending from Dirac to Lie Algorithms. We point to such rigidity providing a partial cohomological (and thus global) selection principle for the Comparative Theory of Background Independence. 2) We finally pose the universal (theory-independent) analogue of GR's Refoliation Invariance for the general Lie Theory: Reallocation of Intermediary-Object Invariance. This is a commuting pentagon criterion: in evolving from an initial object to a final object, does switching which intermediary object one proceeds via amount to at most a difference by an automorphism of the final object? We argue for this to also be a selection principle in the Comparative Theory of Background Independence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to advance the classical Problem of Time and Background Independence via higher Lie Theory by extending Dirac's Algorithm to Lie's Algorithm for Generator Closure, reinterpreting passage of theory families through the Dirac Algorithm as the Dirac Rigidity subcase of a broader Lie Rigidity, supplying a Foundations of Geometry example of the latter, and introducing Reallocation of Intermediary-Object Invariance as a commuting pentagon criterion that serves as the universal analogue of GR Refoliation Invariance and, together with rigidity, as a selection principle within the Comparative Theory of Background Independence.

Significance. If the central claims hold, the work would supply a Lie-theoretic language for comparing background-independent theories and a pair of criteria (Lie Rigidity and the pentagon) that could function as partial selection principles. The generalization from GR refoliation invariance to a theory-independent pentagon is conceptually coherent with the paper's prior Lie-algebraic treatment of observables and constraint substructures, but the absence of explicit derivations or reductions limits the immediate impact.

major comments (2)
  1. [Reallocation of Intermediary-Object Invariance] In the section posing Reallocation of Intermediary-Object Invariance (final paragraph of the abstract and the corresponding development): the commuting pentagon criterion is defined in terms of intermediary objects differing by at most an automorphism, yet no derivation from the Lie bracket closure or the dual lattices of observables subalgebras is supplied, nor is it shown by explicit specialization that the pentagon recovers GR refoliation invariance when the underlying Lie structures are taken to be the Lie derivative and the GR constraint algebra. This renders the claim that the criterion functions as a selection principle for the Comparative Theory an assertion rather than a demonstrated result.
  2. [Lie Rigidity] In the subsection on Lie Rigidity and its Dirac subcase: the reinterpretation of passing families of theories through the Dirac Algorithm as Dirac Rigidity is presented, and a partial cohomological selection principle is asserted, but no concrete computation is given showing how the rigidity criterion ranks or excludes specific theories, nor is the Foundations of Geometry example worked out in sufficient detail to illustrate a non-Dirac instance of Lie Rigidity that actually selects among candidate structures.
minor comments (2)
  1. The manuscript would benefit from an explicit statement, early in the text, of which results are new versus which are carried over from the author's earlier papers on the same topic.
  2. [Reallocation of Intermediary-Object Invariance] The pentagon criterion would be clearer if accompanied by a commutative diagram or a short algebraic formulation rather than a purely verbal description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the report. We address each major comment below and agree to strengthen the manuscript with explicit derivations and expanded examples.

read point-by-point responses

  1. Referee: [Reallocation of Intermediary-Object Invariance] In the section posing Reallocation of Intermediary-Object Invariance (final paragraph of the abstract and the corresponding development): the commuting pentagon criterion is defined in terms of intermediary objects differing by at most an automorphism, yet no derivation from the Lie bracket closure or the dual lattices of observables subalgebras is supplied, nor is it shown by explicit specialization that the pentagon recovers GR refoliation invariance when the underlying Lie structures are taken to be the Lie derivative and the GR constraint algebra. This renders the claim that the criterion functions as a selection principle for the Comparative Theory an assertion rather than a demonstrated result.

    Authors: The pentagon is motivated directly by the dual lattices of observables subalgebras and the automorphism actions arising from Lie bracket closure in the preceding sections on observables and constraints. However, we agree that a step-by-step derivation linking the definition to bracket closure and an explicit reduction recovering GR refoliation invariance is not supplied. We will add a dedicated subsection providing these derivations and the specialization, thereby substantiating the selection-principle claim. revision: yes

  2. Referee: [Lie Rigidity] In the subsection on Lie Rigidity and its Dirac subcase: the reinterpretation of passing families of theories through the Dirac Algorithm as Dirac Rigidity is presented, and a partial cohomological selection principle is asserted, but no concrete computation is given showing how the rigidity criterion ranks or excludes specific theories, nor is the Foundations of Geometry example worked out in sufficient detail to illustrate a non-Dirac instance of Lie Rigidity that actually selects among candidate structures.

    Authors: The Foundations of Geometry example is presented to exhibit a non-Dirac instance of Lie Rigidity. We acknowledge that it remains schematic and that no explicit ranking computation for concrete theories is performed. We will expand the example with the missing calculations to demonstrate selection among candidate structures. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper references a recently given local resolution of the Problem of Time as background and then develops two new higher-Lie elements: reinterpretation of the Dirac Algorithm as the Dirac Rigidity subcase of a broader Lie Rigidity, plus the definition of Reallocation of Intermediary-Object Invariance via a commuting pentagon criterion. These are introduced by explicit definition and analogy to GR refoliation, with the selection-principle status argued rather than derived from prior equations. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claims to unverified prior assertions appear in the supplied text; the contributions remain independent extensions of Lie structures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on standard Lie-algebra axioms plus two newly introduced constructs whose independent support is not shown in the abstract.

axioms (2)
  • domain assumption Lie derivatives encode Relationalism and Lie brackets give Closure yielding Lie algebraic structures
    Stated explicitly in the abstract as the basic Lie mathematics required for the local resolution.
  • standard math Observables defined by Lie-bracket relations form Lie algebras and lattices of constraint substructures induce dual observable subalgebras
    Invoked as part of the classical Lie framework in the abstract.
invented entities (2)
  • Lie's Algorithm for Generator Closure no independent evidence
    purpose: Generalize Dirac's Algorithm beyond constraints to arbitrary generators
    Newly proposed extension; no independent evidence supplied in abstract.
  • Reallocation of Intermediary-Object Invariance (commuting pentagon criterion) no independent evidence
    purpose: Serve as universal, theory-independent analogue of GR refoliation invariance and selection principle
    Posed in the abstract; no independent evidence or derivation supplied.

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Reference graph

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