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A noncommutative Fourier transform establishes an invertible isometry between square-integrable functions on a Lie group and star-product functions on the dual of its Lie algebra, intertwining position and momentum representations.

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2026-08-03 14:34 UTC pith:VEEBLLKK

load-bearing objection Solid scaffolding for quantum mechanics on Lie groups with a genuinely nice SU(2) payoff; the formal projector/√δ normalization is the one load-bearing caveat, and the authors are honest about it. the 3 major comments →

arxiv 2512.19840 v2 pith:VEEBLLKK submitted 2025-12-22 quant-ph hep-thmath-phmath.CAmath.MP

Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain , Blagoje Oblak , Marios Petropoulos This is my paper
classification quant-ph hep-thmath-phmath.CAmath.MP MSC 43A3081S1022E7053D55 PACS 03.65.-w
keywords noncommutative Fourier transformLie groupsstar productmomentum representationPoisson summationquantizationWigner functionsgroup manifolds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a working Fourier duality for quantum mechanics on Lie groups: square-integrable wave functions on the group (position space) can be mapped invertibly to wave functions on the dual of the Lie algebra (momentum space), even though momenta do not commute. Because momentum operators fail to commute, multiplication of momentum-space wave functions is deformed into a star product, and the paper shows that the natural plane-wave transform weighted by the Haar-measure Jacobian is a Hilbert-space isometry. For groups with compact subgroups, one must quotient the operator algebra by the identity subset and project onto invariant states; the paper does this with an explicit projector and a careful renormalization by the rank-dependent volume factor. The construction yields a noncommutative Poisson summation formula valid for any compact Lie group, reducing to ordinary Fourier series for U(1) and to a new summation formula on R^3 for SU(2). If correct, this supplies the missing Fourier bridge for computing Wigner functions and path integrals on group manifolds.

Core claim

The central claim is that the map F[ψ](p)=∫_g J(X)d^nX e^{-i⟨p,X⟩}ψ(X), with J the Haar-measure Jacobian, is an invertible, isometric intertwiner between the large position representation on L^2(g) and the star-product momentum representation on L^2_⋆(g*). Its I-invariant projection F_I[ψ](p)=√|Z|^r ∫_G dg E^I(g,p)ψ(g) restricts to an isometry between L^2(G) and the invariant subspace P_I·L^2_⋆(g*). The paper proves Fourier inversion, the positive-definiteness of the star-product pairing on the image, transformation laws for left translations and momentum additions, and the convolution-to-star-product rules, and derives the noncommutative Poisson summation formula for any compact Lie group.

What carries the argument

The central object is the identity subset I = {e^{−i⟨p,X⟩} | exp X = e} of exponential operators—the logarithms of the identity. To reconcile compact subgroups with noncommuting momenta, the paper quotients the operator algebra by I via the averaging projector P_I = (1/|I|)Σ_{Y∈Logs(e)} e^{−i⟨p,Y⟩}, and uses the same projector to reduce both position and momentum Hilbert spaces. The argument is carried by the Baker-Campbell-Hausdorff group law B(X,Y) on the Lie algebra, which makes plane waves satisfy E(X,p)⋆E(Y,p)=E(B(X,Y),p) under the Gutt star product; this identity drives the proofs of covariance, isometry, and Poisson summation.

Load-bearing premise

The projector P_I = (1/|I|)Σ_{Y∈Logs(e)} e^{−i⟨p,Y⟩} is an infinite sum whose regularization by an infrared cutoff and renormalization by √|Z|^r is assumed to converge and to define a genuine Hilbert-space projector; if this formal limit does not exist, the isometry and the Poisson summation formula lose their justification.

What would settle it

Compute the left and right sides of the SU(2) Poisson summation formula (5.26) numerically for a smooth rapidly decaying test function such as ψ(X)=e^{-|X|^2} at a point X with, say, |X|=1 and a direction aligned with a coordinate axis; agreement of the two sides to high precision would support the formalism, while any discrepancy would localize the failure in the projector regularization. Alternatively, for any compact group, check the isometry identity ⟨F_I[ϕ]|F_I[ψ]⟩_{g*}=⟨ϕ|ψ⟩_G on a finite-dimensional truncation of the log-lattice to see whether the √|Z|^r normalization is correct.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Quantum mechanics on any Lie group (rigid bodies, spin chains, Lie-Poisson systems) now has a position–momentum duality with an explicit, invertible kernel, enabling Wigner functions and path integrals in the same way as on R^n.
  • The star-product scalar product on momentum space is positive-definite on the physical subspace, resolving an ambiguity in earlier definitions of noncommutative Fourier transforms.
  • A universal noncommutative Poisson summation formula follows for any compact Lie group; for SU(2) it yields an explicit summation over a one-dimensional lattice with integrals over planes perpendicular to the direction of X.
  • Fourier coefficients of characters of irreducible representations are localized on coadjoint orbits (for SU(2), spheres of radius 2λ+1), connecting the formalism to Kirillov’s character formula.
  • The construction reduces correctly to standard Fourier analysis for the Abelian case U(1), giving ordinary Fourier series and Poisson summation as a special case.

Where Pith is reading between the lines

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

  • If the isometry extends to infinite-dimensional Lie-Fréchet groups as conjectured in the paper, the same Fourier bridge could provide a phase-space formulation for perfect fluids and other Lie-Poisson systems with infinitely many degrees of freedom.
  • The dependence of the SU(2) Poisson formula on derivatives of the Fourier transform suggests a link to the Weyl quantization on the Heisenberg group; one could test whether the formula is equivalent to a non-Abelian version of the Poisson summation for the radial Fourier transform.
  • A direct numerical check of the SU(2) summation formula with, say, a Gaussian ψ would provide a quick falsifier without needing full rigor of the projector.
  • The authors’ handling of infinite factors by infrared regularization is analogous to thermodynamic-limit procedures; one may expect a fully rigorous version to require treating P_I as a principal value or using a lattice of logarithms, which could change the normalization by |Z|^r.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper develops a noncommutative Fourier transform on Lie groups, mapping square-integrable wave functions on the group (or on the Lie algebra, in the 'large' picture) to star-product functions on the dual Lie algebra. The main construction is the intertwiner F[ψ](p)=∫_g J(X)d^nX e^{-i⟨p,X⟩}ψ(X), together with its I-invariant projection for compact subgroups: F^I[ψ](p)=√|Z|^r ∫_G dg E^I(g,p)ψ(g). The authors claim this provides an invertible isometry between L^2(G) and a renormalized I-invariant star-product momentum space, and they derive a noncommutative Poisson summation formula for any compact Lie group, with explicit U(1) and SU(2) examples. The paper is explicitly written at physicist level of rigor and repeatedly cautions that infinite factors and distributional identities are treated formally.

Significance. If the advertised Hilbert-space statements could be made rigorous, this would be a useful unified framework for quantum mechanics on group manifolds, with direct applications to Wigner functions and path integrals. The explicit construction of the intertwiner, the U(1) and SU(2) worked examples, and the SU(2) character computation (5.21)–(5.23) are valuable and checkable. The paper is also unusually transparent about its formal manipulations. However, the central 'isometry of Hilbert spaces' claim is not literally a theorem about L^2_⋆(g*) as it stands, because the projector onto I-invariant functions and the renormalization by √|Z|^r involve objects such as √δ(p−n) that do not define a Hilbert-space structure. This is the main correctness risk and must be addressed before the results can be accepted as stated.

major comments (3)
  1. [§3.3, (3.13); §4.3, (4.26), (4.30)–(4.32); §5.1, (5.6)–(5.7)] The load-bearing 'isometry' claim is only formal. For a compact G with nontrivial identity subset I, no nonzero I-invariant function on g* is square-integrable; the paper itself notes this for U(1) in (3.18)–(3.20), where a projected periodic function has vanishing ordinary L^2 norm. The subsequent renormalization by √|Z|^r, and in particular the use of √δ(p−n) in (5.6)–(5.7), does not define a separable Hilbert space: √δ is not a tempered distribution, and products such as √δ(p−m)√δ(p−n) are not defined. Since the scalar-product identity (4.26) and the inversion relations (4.30)–(4.32) all pass through this projector and renormalization, they are not literal statements about L^2_⋆(g*) or its I-invariant subspace. The §1 'word of caution' acknowledges formal methods, but the abstract and the 'isometry' statements go beyond that. Please either supply a rigorous renormalized/rigged-Hilbert
  2. [§4.2, Eq. (4.17)] For a group of rank r>1, the normalization in (4.17) is inconsistent. The first equality has a factor 1/|Z|^r in front of the sum over Z^r; applying the Poisson summation formula separately to each of the r sums gives 1/|Z|^r ∑_{k∈Z^r} ∏_i δ(⟨p,a_i(X)⟩−k_i), not 1/|Z| as written. As it stands, (4.17) is off by |Z|^{r-1} for r>1. This propagates into the definitions of I-invariant plane waves and Fourier coefficients in (4.19)–(4.20), and into the general Poisson formula (5.34). Since the paper claims results for any compact Lie group, this must be corrected and the higher-rank normalization rechecked.
  3. [§5.3, Eqs. (5.29)–(5.34)] The general Poisson summation formula for arbitrary compact G is only sketched and rests on two unproven assumptions. First, the Jacobian is assumed to decompose as J(X)=N(X)/D(X) with N periodic under translations by 2πa_i(X) and D polynomial; this is verified for SU(2) but not established for a general compact Lie group. Second, in passing from (5.31) to (5.33), derivatives in the root-space directions are discarded as total derivatives; this requires a boundary/decay justification that is not given. As a result, the statement 'Poisson summation for any group' is not yet supported by the derivation. Either prove these structural claims or restrict the theorem to the cases where they are shown to hold.
minor comments (3)
  1. [§5.1, around (5.6)] There is a duplicated word: 'enforces enforces square-integrability' should read 'enforces square-integrability'.
  2. [§3.3, (3.13) and §2.2, (2.15)–(2.17)] The symbol I is used both for the set of exponential operators and for a subset of the Lie algebra, and |I| is treated sometimes as a cardinality and sometimes as a volume. For SU(2), where Logs(e) is a union of spheres, the 'sum' in (3.13) is formally a continuous integral over those spheres; the notation and the regularization prescription should be made explicit.
  3. [§4.3, (4.40)–(4.41)] The lone star lemma is stated for functions decaying sufficiently fast at infinity, but the passage from (4.40) to the inverse-Fourier form (4.41) is not fully spelled out. The factor √|Z|^r in (4.41) also deserves a remark: for the unprojected case I={0} one has r=0, so the factor is absent; this could be stated explicitly to avoid confusion.

Circularity Check

0 steps flagged

No circularity: the Fourier transform and isometry are derived from the chosen star product and standard lemmas, with self-citations only as announcements.

full rationale

The paper's central derivation is self-contained. The noncommutative Fourier transform (4.9) is not assumed but derived from the intertwining conditions (4.5)–(4.8), and the star product is fixed by the symmetric-ordering quantization map (2.9). The isometry (4.26) is proven directly from the plane-wave product rule (4.11) and the delta-function identity (4.24), which in turn follows from the definition of the I-invariant plane waves and the standard representation of the delta function (4.12). The normalizations involving |Z|^r are dictated by the group-theoretic counting of logarithm branches, not fitted to make the target result true. The Poisson summation formulas (5.9), (5.26), and (5.34) are derived using the standard Poisson summation lemma, which is an external proven result; the non-Abelian variants are genuinely new identities obtained by combining this lemma with the I-invariant plane-wave structure. The self-citations to [26] and [27] are only forward-looking announcements and play no role in the derivations. The paper's caveat about formal treatment of the infinite projector P_I and singular distributions is a mathematical-rigor limitation, not a circularity: the authors explicitly state they are not claiming rigorous Hilbert-space status for all intermediate objects.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The construction relies on standard Lie theory (BCH formula, Haar measure, structure of logarithms on compact groups) plus the specific choices of symmetric ordering and the formal projector regularization. No free parameters are fitted; the |Z|^r and δ(0) factors are formal regularizations, not adjustable parameters.

axioms (4)
  • domain assumption G is finite-dimensional and weakly exponential (exp: g->G has dense image)
    Stated in §1 ('A word of caution') and used to justify principal-branch coordinates on a dense subset; all later constructions rely on these coordinates.
  • domain assumption Quantization uses symmetric ordering (2.9) for momentum monomials, giving the Gutt star product
    The star product (3.6) and plane-wave composition (3.8) depend on this choice; the alternative Duflo map is treated in Appendix A.
  • domain assumption The pairing (3.10) defines a positive-definite scalar product on L^2_⋆(g*)
    Stated as not manifestly positive-definite in §3.2; proved later in §4.3 via the Fourier isometry. It is assumed when calling L^2_⋆(g*) a Hilbert space.
  • domain assumption For compact G, the Haar Jacobian factorizes as J(X)=N(X)/D(X) with N periodic under translations by 2π a_i(X) and D polynomial
    Explicitly assumed below (5.29) for the general Poisson summation; true for compact groups via the root product formula, but not proved in this paper.

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0 comments
read the original abstract

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

Figures

Figures reproduced from arXiv: 2512.19840 by Blagoje Oblak, Marios Petropoulos, Mathieu Beauvillain.

Figure 1
Figure 1. Figure 1: A cartoon of the principal branch (blue disk) of the logarithm from a Lie algebra [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logarithms of the identity in the u(1) Lie algebra (left) and the su(2) Lie algebra (right). In the U(1) case, logarithms of the identity form a one-dimensional lattice of 2π￾separated points in the Lie algebra u(1) ∼= R. This is because the exponential is exp : x 7→ e ix. In the SU(2) case, we take the exponential map to be exp : X⃗ 7→ e iX⃗ ·⃗σ in terms of Pauli matrices (see section 5). With this conven… view at source ↗
Figure 3
Figure 3. Figure 3: Localization of I-invariant plane waves (5.17) in X⃗ at fixed p (left), and in p at fixed X⃗ (right). On the left panel, we chose ∥p∥ = 2.2 so that only five values of m are allowed in (5.17). On the right panel, the plane waves are localized on planes perpendicular to ⃗uX that are evenly spaced by integer multiples of ⃗uX. Fourier modes. Let us now turn to the I-invariant plane waves (4.16). Since the set… view at source ↗

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Forward citations

Cited by 1 Pith paper

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    quant-ph 2026-07 conditional novelty 6.0

    A path integral on the Hilbert space of a Lie group is built by decompactifying to the Lie algebra and summing over winding sectors in maximal tori, yielding two-loop heat-kernel coefficients for Euler-Arnold systems.

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