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arxiv: 1906.10083 · v1 · pith:XTITIZWAnew · submitted 2019-06-24 · 🧮 math-ph · hep-th· math.MP· nucl-th

Relativistic invariance in Euclidean formulations of quantum mechanics

Gohin Shaikh Samad , Wayne Polyzou This is my paper

Pith reviewed 2026-05-25 16:47 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MPnucl-th
keywords Poincaré groupEuclidean quantum mechanicsrelativistic invariancereflection positivityirreducible representationsgeneratorsself-adjointness
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The pith

Poincaré generators have explicit representations in Euclidean space-time variables for all positive-mass positive-energy irreducible representations.

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

The paper derives explicit representations of the Poincaré generators in Euclidean space-time variables. This is done for every positive-mass positive-energy irreducible representation of the Poincaré group. The construction preserves the commutation relations, establishes hermiticity and self-adjointness of the generators, and verifies reflection positivity of the kernels. A sympathetic reader would care because Euclidean formulations offer computational advantages for hadronic systems while the physical inner product remains expressible directly in Euclidean variables with no analytic continuation required.

Core claim

The identification of the complex Euclidean group with the complex Poincaré group relates the infinitesimal generators of both groups. In this work explicit representations of the Poincaré generators in Euclidean space-time variables for all positive-mass positive-energy irreducible representations of the Poincaré group are derived. The commutation relations are checked, both hermiticity and self-adjointness are established, and reflection positivity of the kernels is verified.

What carries the argument

The identification of the complex Euclidean group with the complex Poincaré group, which maps the infinitesimal generators into Euclidean variables while preserving the algebra.

If this is right

  • The derived generators satisfy the Poincaré algebra commutation relations.
  • Hermiticity and self-adjointness hold for the generators in Euclidean variables.
  • The kernels satisfy reflection positivity, so the Euclidean inner product defines a valid Hilbert space.
  • The representations apply to every positive-mass positive-energy irreducible representation.

Where Pith is reading between the lines

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

  • Numerical work on relativistic bound states could proceed entirely in Euclidean variables without returning to Minkowski space.
  • Similar group identifications might simplify Euclidean treatments of other space-time symmetries.
  • The separation between representation theory and metric signature could be tested in model calculations with finite degrees of freedom.

Load-bearing premise

The complex Euclidean group can be identified with the complex Poincaré group so that their infinitesimal generators are directly related.

What would settle it

An explicit check that the proposed Euclidean expressions for the generators violate the Poincaré commutation relations or fail to produce reflection-positive kernels would falsify the construction.

read the original abstract

Relativistic invariance in Euclidean formulations of quantum mechanics is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. Euclidean formulations of relativistic quantum mechanics have some computational advantages. In the Euclidean representation the physical Hilbert space inner product is expressed in terms of Euclidean space-time variables with no need for any analytic continuation. The identification of the complex Euclidean group with the complex Poincar\'e group relates the infinitesimal generators of both groups. In this work explicit representations of the Poincar\'e generators in Euclidean space-time variables for all positive-mass positive-energy irreducible representations of the Poincar\'e group are derived. The commutation relations are checked, both hermiticity and self-adjointness are established, and reflection positivity of the kernels is verified.

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 / 2 minor

Summary. The paper claims that identifying the complex Euclidean group with the complex Poincaré group allows explicit representations of the Poincaré generators to be constructed in Euclidean space-time variables for all positive-mass positive-energy irreducible representations. These representations are shown to satisfy the Poincaré commutation relations, to be Hermitian and self-adjoint, and to yield kernels that satisfy reflection positivity, thereby providing a Euclidean formulation of relativistic quantum mechanics in which the physical inner product is expressed directly in Euclidean variables without analytic continuation.

Significance. If the construction and verifications hold, the work supplies concrete, explicit Poincaré generators in Euclidean variables together with direct algebraic and positivity checks for the full class of positive-mass positive-energy irreps. This removes the need for Wick rotation in the inner product and supplies a practical bridge between Euclidean computational methods and relativistic invariance, which is relevant for hadronic systems at sub-hadronic scales. The explicit character of the representations and the independent verification of the algebra, hermiticity/self-adjointness, and reflection positivity are clear strengths.

minor comments (2)
  1. [Abstract] The abstract states that the commutation relations, hermiticity, self-adjointness, and reflection positivity are verified for the stated class; a brief sentence in the introduction or §2 indicating the precise range of representations covered (e.g., any restrictions on spin or mass) would help the reader locate the corresponding theorems.
  2. Notation for the Euclidean four-vector and the complexified generators is introduced gradually; a short table or paragraph collecting the mapping between Minkowski and Euclidean generators at the beginning of the main construction section would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct construction with independent verification

full rationale

The paper performs an explicit construction of Poincaré generators in Euclidean variables using the stated group identification as the enabling premise, then verifies the commutation relations, hermiticity, self-adjointness, and reflection positivity by direct computation for the positive-mass positive-energy irreps. No equations reduce to fitted inputs renamed as predictions, no self-definitional loops appear, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation chains. The central results are self-contained algebraic and positivity checks that stand apart from the initial identification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions from relativistic quantum mechanics and representation theory; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Positive mass and positive energy irreducible representations of the Poincaré group.
    The paper restricts attention to these representations as stated in the abstract.
  • domain assumption Identification of the complex Euclidean group with the complex Poincaré group.
    This identification is invoked to relate the infinitesimal generators.

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Reference graph

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