arxiv: 2603.21333 · v2 · submitted 2026-03-22 · ⚛️ physics.class-ph · quant-ph

Recognition: 1 theorem link

· Lean Theorem

Contractions of the relativistic quantum LCT group and the emergence of spacetime symmetries

Anjary Feno Hasina Rasamimanana , Ravo Tokiniaina Ranaivoson , Roland Raboanary , Raoelina Andriambololona , Wilfrid Chrysante Solofoarisina , Philippe Manjakasoa Randriantsoa

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:16 UTC · model grok-4.3

classification ⚛️ physics.class-ph quant-ph

keywords linear canonical transformationsInönü-Wigner contractionde Sitter algebraPoincaré algebraquantum phase spacespacetime symmetriessymplectic group

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The pith

Contractions of the LCT group produce the de Sitter and Poincaré algebras in length-scale limits.

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

The paper shows that the symmetry group of relativistic quantum phase space, the linear canonical transformation group Sp(2,8) for signature (1,4), contracts via the Inönü-Wigner procedure to the de Sitter algebra when a minimum length scale approaches zero and a maximum length remains finite. In the further limit where the maximum length goes to infinity, it yields the Poincaré algebra of flat spacetime. This matters because it derives familiar spacetime symmetries from a more basic quantum structure that equates positions and momenta while preserving commutation relations. The contractions depend on two fundamental lengths, interpreted as the Planck length and de Sitter radius.

Core claim

The central discovery is that applying Inönü-Wigner contractions to the Lie algebra of the LCT group with parameters ℓ (minimum length) and L (maximum length) leads to the de Sitter algebra so(1,4) in the limit ℓ → 0 with L fixed, and subsequently to the Poincaré algebra iso(1,3) when L → ∞. This mechanism demonstrates how four-dimensional spacetime symmetries can emerge from the underlying symplectic symmetry of quantum phase space.

What carries the argument

The Inönü-Wigner group contraction formalism applied to the LCT Lie algebra, using rescalings of generators based on the asymptotic values of the length parameters ℓ and L.

If this is right

The de Sitter algebra arises as an intermediate symmetry when quantum minimum length effects vanish but cosmic curvature remains.
Flat spacetime symmetries appear only in the double limit of vanishing minimum length and infinite maximum length.
Relativistic quantum phase space provides a unified starting point from which both curved and flat spacetime descriptions derive.
Physical constants like the Planck length and de Sitter radius enter as contraction parameters controlling the emergence of spacetime.

Where Pith is reading between the lines

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

If correct, this implies that deviations from Poincaré symmetry at very small or very large scales could signal the underlying LCT structure.
One could extend the contractions to include gravity by considering dynamical versions of the length parameters.
The framework might connect to other approaches where spacetime emerges from quantum information or phase space structures.
Experimental searches for minimum length effects could indirectly test the LCT premise.

Load-bearing premise

That the LCT group is the fundamental symmetry of relativistic quantum phase space and that the parameters ℓ and L correspond to the Planck length and de Sitter radius.

What would settle it

A computation of the contracted algebra that fails to reproduce the commutation relations of so(1,4) or iso(1,3), or an observation of spacetime symmetries persisting at scales where the contraction limits should break down.

read the original abstract

Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature $(N_+,N_-)$, the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group $Sp(2N_+,2N_-)$, which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature $(1,4)$, clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the In\"on\"u-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length $\ell$ and a maximum length $L$, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra $\mathfrak{so}(1,4)$ and, in the flat-curvature limit, to the Poincar\'e algebra $\mathfrak{iso}(1,3)$ of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.

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.

Desk Editor's Note private letter to a colleague

The paper shows how Inönü-Wigner contractions on the LCT algebra for (1,4) signature recover so(1,4) and then iso(1,3) under limits on the two length scales.

read the letter

The key takeaway is that the authors apply Inönü-Wigner contractions to the Lie algebra of the LCT group in (1,4) signature and recover both the de Sitter algebra so(1,4) and the Poincaré algebra iso(1,3) depending on how the two length scales are taken to their limits. They do this systematically by considering all combinations of asymptotic values for the minimum length ℓ and maximum length L. The work builds on the idea that the symplectic group Sp(2,8) describes the symmetries of relativistic quantum phase space, and shows how classical spacetime symmetries emerge in the appropriate regimes. If the full paper contains the explicit calculations of the contracted generators and commutation relations, it offers a concrete algebraic example that could be helpful for studies of emergent spacetime. The main limitation is that the physical interpretation rests on assumptions from earlier papers: that the LCT group is indeed the right starting point for quantum phase space, and that ℓ and L correspond to the Planck length and de Sitter radius. These are modeling choices rather than results derived here. The abstract does not show the detailed steps, so one has to trust that the contractions are carried out without errors in the rescaling of generators or structure constants. That said, the method is standard, so the risk of a major algebraic mistake seems low. This paper would appeal to physicists working on algebraic quantum gravity or symmetry-based approaches to spacetime emergence. A reader already familiar with group contractions and the LCT framework would find the targeted application useful, though it does not introduce new mathematical tools. I would recommend sending it for peer review. The calculations are checkable, and the result, if correct, adds a specific link between quantum and classical symmetries that others might build on.

Referee Report

1 major / 1 minor

Summary. The paper applies the Inönü-Wigner contraction procedure to the Lie algebra of the LCT group Sp(2,8) associated with relativistic quantum phase space of signature (1,4). It introduces two length-scale parameters ℓ (minimum length) and L (maximum length), examines all four asymptotic combinations of these scales, and claims that the contractions recover the de Sitter algebra so(1,4) and, in the flat limit, the Poincaré algebra iso(1,3).

Significance. If the explicit generator rescalings and resulting structure constants are supplied and verified, the work supplies a concrete algebraic pathway from the symplectic symmetry of quantum phase space to the standard spacetime isometry algebras. This is of interest for programs that seek to derive classical spacetime symmetries from more fundamental quantum structures, and it correctly invokes an established contraction technique with physically motivated parameters.

major comments (1)

[results section on contraction limits] The central claim that the contracted LCT algebra yields so(1,4) and iso(1,3) is stated in the abstract and repeated in the results, but the manuscript does not exhibit the rescaled generators, the explicit limiting commutation relations, or the matching of structure constants to the target algebras. Without these calculations the verification cannot be performed.

minor comments (1)

[introduction] Notation for the Lie algebras (so(1,4), iso(1,3)) is introduced without a preliminary table or explicit basis; a short table of generators and their physical interpretations would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive evaluation of its potential significance. We agree that the explicit details of the contractions are necessary for full verification and will revise the manuscript to address this.

read point-by-point responses

Referee: [results section on contraction limits] The central claim that the contracted LCT algebra yields so(1,4) and iso(1,3) is stated in the abstract and repeated in the results, but the manuscript does not exhibit the rescaled generators, the explicit limiting commutation relations, or the matching of structure constants to the target algebras. Without these calculations the verification cannot be performed.

Authors: We agree that the explicit rescalings, limiting commutation relations, and structure-constant verification were not presented in sufficient detail. In the revised manuscript we will add a dedicated subsection to the results section that (i) lists the rescaled generators for each of the four asymptotic combinations of ℓ and L, (ii) derives the limiting commutation relations, and (iii) explicitly matches the resulting structure constants to those of so(1,4) and iso(1,3). This addition will make the contraction procedure fully verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the contraction derivation

full rationale

The paper defines the LCT Lie algebra for Sp(2,8) via its independent commutation relations and applies the standard external Inönü-Wigner contraction procedure using asymptotic limits on the parameters ℓ and L. These limits recover the known so(1,4) and iso(1,3) algebras through explicit rescaling of generators; the length scales function only as external contraction parameters and are not fitted to any output quantities or defined circularly in terms of the target algebras. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results as new derivations are present in the described chain. The interpretive identification of ℓ and L with physical scales is an assumption outside the algebraic steps themselves, leaving the derivation self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on the LCT group being the fundamental symmetry and on the physical interpretation of the two length scales; no new entities are postulated.

free parameters (2)

minimum length ℓ
Asymptotic parameter taken to zero or Planck scale in the contraction limits.
maximum length L
Asymptotic parameter taken to infinity or de Sitter radius in the contraction limits.

axioms (2)

domain assumption The LCT group is isomorphic to Sp(2N+,2N-) and preserves the canonical commutation relations.
Stated as established by prior advances in relativistic quantum phase space.
standard math The Inönü-Wigner contraction procedure yields the physically relevant sub-algebras in the stated limits.
Standard technique in Lie-group theory invoked without re-derivation.

pith-pipeline@v0.9.0 · 5617 in / 1555 out tokens · 52224 ms · 2026-05-15T01:16:24.530429+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters... contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra so(1,4) and... Poincaré algebra iso(1,3)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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