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arxiv: 2302.10293 · v3 · submitted 2023-02-02 · ⚛️ physics.gen-ph

Time-like definition of quaternions in exterior algebra

Ivano Colombaro This is my paper

Pith reviewed 2026-05-24 09:59 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords quaternionsexterior algebraexterior calculusrotationstime-like coordinatesspace-time signaturethree-dimensional space-time
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The pith

Quaternions arise from exterior algebra in a three-dimensional space-time with three time-like coordinates.

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

The paper shows how exterior algebra and calculus on a three-dimensional space-time with all time-like coordinates produce the standard quaternion multiplication rules. It then uses this construction to express rotations in the same framework. A sympathetic reader would care because the approach ties quaternions directly to the wedge product and exterior derivative rather than to complex numbers or vector cross products, and it does so in a spacetime signature that differs from the usual one time-like direction.

Core claim

Considering a three-dimensional space-time characterized by three time-like coordinates, a suitable formulation of quaternions is recovered by means of the properties arising from exterior algebra and calculus. Rotations may be written in terms of these quaternions in accordance with the definition provided in exterior algebra.

What carries the argument

Exterior algebra and calculus on a three-dimensional manifold with all time-like coordinates, whose wedge products and derivative rules generate the quaternion multiplication table.

If this is right

  • Quaternion multiplication is realized through the wedge product and other operations of exterior calculus in the chosen signature.
  • Rotations in three dimensions are represented by the same quaternion objects constructed from exterior algebra.
  • The entire quaternion structure remains consistent with the graded-commutative and nilpotent properties of the exterior algebra.
  • The formulation supplies a direct geometric origin for quaternions inside differential forms rather than an external imposition.

Where Pith is reading between the lines

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

  • If the construction holds, similar exterior-algebra definitions of quaternions might be possible in other non-Lorentzian signatures.
  • The approach could be tested by deriving the same quaternion rules from the exterior derivative on explicit coordinate charts with three time-like directions.
  • It raises the question whether the usual one-time signature admits an equally direct exterior-algebra route to quaternions.

Load-bearing premise

That the algebraic properties of exterior calculus remain consistent and sufficient to define quaternion multiplication and rotation representations when the underlying space-time has three time-like rather than one time-like coordinate.

What would settle it

An explicit computation showing that the quaternion multiplication table or the rotation representation fails to emerge from the exterior products and derivatives in this three-time-like coordinate system would disprove the recovery.

read the original abstract

A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions, in accordance with definition provided in exterior algebra.

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

0 major / 3 minor

Summary. The manuscript claims that quaternions can be recovered from the algebraic properties of exterior calculus when the underlying three-dimensional manifold has three time-like coordinates rather than the conventional (1,2) signature. It asserts that this yields a consistent formulation of quaternion multiplication and demonstrates an application to rotations expressed via the exterior-algebra definition.

Significance. If the derivation holds, the result is a valid but unsurprising observation: the bivector algebra generated by the exterior product yields the quaternion relations whenever all three basis vectors share the same metric sign, since (e_i ∧ e_j)^2 = −e_i² e_j² = −1. This is algebraically insensitive to overall signature and reproduces the standard quaternion group. The paper therefore supplies an explicit exterior-calculus presentation rather than a new algebraic structure. Credit is due for making the construction explicit in a uniform-signature setting, though the result does not alter known quaternion identities or rotation representations.

minor comments (3)
  1. [main text] The abstract states consistency but the main text should include an explicit multiplication table or step-by-step verification that the recovered bivector products satisfy the quaternion relations i² = j² = k² = ijk = −1 (or equivalent).
  2. [introduction / §2] Notation for the three time-like basis vectors and the induced metric should be introduced once at the beginning and used consistently; the current presentation leaves the metric signature implicit in places.
  3. [application to rotations] The rotation section would benefit from a short comparison with the conventional (1,2) case to clarify what, if anything, changes when all coordinates are time-like.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for acknowledging the explicit construction in the uniform-signature setting. Below we address the points raised in the report.

read point-by-point responses

  1. Referee: The manuscript claims that quaternions can be recovered from the algebraic properties of exterior calculus when the underlying three-dimensional manifold has three time-like coordinates rather than the conventional (1,2) signature. It asserts that this yields a consistent formulation of quaternion multiplication and demonstrates an application to rotations expressed via the exterior-algebra definition.

    Authors: The manuscript presents precisely this recovery: starting from the exterior product and the metric in the (3,0) signature, the bivector basis elements satisfy the quaternion multiplication rules, and the rotation application is derived directly from the same exterior-algebra operations. revision: no

  2. Referee: If the derivation holds, the result is a valid but unsurprising observation: the bivector algebra generated by the exterior product yields the quaternion relations whenever all three basis vectors share the same metric sign, since (e_i ∧ e_j)^2 = −e_i² e_j² = −1. This is algebraically insensitive to overall signature and reproduces the standard quaternion group. The paper therefore supplies an explicit exterior-calculus presentation rather than a new algebraic structure.

    Authors: We agree that the underlying relation (e_i ∧ e_j)^2 = −1 follows immediately once all three vectors have the same metric sign. The manuscript does not claim a new algebraic structure; its contribution is the explicit, step-by-step derivation of the full quaternion multiplication table from the exterior product and wedge-product rules in the three-time-like setting, together with the corresponding rotation formula expressed in the same language. revision: no

  3. Referee: Credit is due for making the construction explicit in a uniform-signature setting, though the result does not alter known quaternion identities or rotation representations.

    Authors: We appreciate the referee’s recognition of the explicit construction. The manuscript makes no claim to modify the classical quaternion identities or rotation representations; it simply exhibits them as consequences of exterior algebra when the metric signature is (3,0). revision: no

Circularity Check

0 steps flagged

No significant circularity; standard bivector-to-quaternion isomorphism.

full rationale

The derivation identifies quaternion units with bivectors in 3D exterior algebra and verifies the multiplication table from the wedge product and metric. The key identity (e_i ∧ e_j)^2 = −e_i² e_j² equals −1 for any uniform signature, which is an algebraic fact independent of the paper's inputs. Rotation representations follow from the same bivector exponential without additional fitting or self-citation. The construction is self-contained against the exterior algebra axioms and does not reduce any claimed result to a renamed input or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard axioms of exterior algebra applied inside a non-standard space-time signature; no free parameters or new entities with independent evidence are introduced in the abstract.

axioms (2)
  • domain assumption The algebraic identities of exterior algebra and calculus continue to hold in a three-dimensional manifold with three time-like coordinates.
    Invoked to recover quaternion multiplication and rotation representations.
  • domain assumption Exterior algebra supplies a complete and consistent foundation for quaternion structure without additional postulates.
    Required for the recovery claim to succeed.
invented entities (1)
  • Three-time-like space-time manifold no independent evidence
    purpose: To serve as the ambient space in which exterior algebra yields quaternions.
    Postulated to enable the claimed recovery; no independent evidence supplied.

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Reference graph

Works this paper leans on

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