Pith. sign in

REVIEW 3 major objections 4 minor 63 references

Seven Sphere Quantization

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The formal Fedosov quantization of the standard contact seven-sphere converges, at reciprocal-integer values of $\hbar$, to exact flat quantum connections on finite-dimensional subbundles.

desk verdict An explicit contact quantization of S7 that is probably right but currently rests on an 'easy to check' bracket verification that a referee should force into the open. read the letter →

arxiv 2507.14363 v1 pith:M7L4KI54 submitted 2025-07-18 math.SG hep-thmath-phmath.MPmath.QAmath.RT

Subhobrata Chatterjee , Can Görmez , Andrew Waldron This is my paper · ORCID
classification math.SGhep-thmath-phmath.MPmath.QAmath.RT MSC 53D5053D1053D55
keywords contactquantizationsevensphereFedosovconnectionHolstein-Primakoffformalembeddingquaternionicunitarygroupsymplecticspinorbundledeformation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper attempts to construct a quantization of the standard contact seven-sphere by solving a formal flat-connection problem on its symplectic spinor bundle, and then shows that the formal solution becomes a genuine flat connection precisely when $\hbar$ is the reciprocal of a positive integer. At those values the fibers truncate to finite-dimensional symmetric tensor representations of $U(2,\mathbb{H})$, so each subbundle carries a bona fide quantum dynamical system: parallel transport along paths gives Schrödinger equations and transition probabilities. A reader should care because a closed contact manifold, which packages position, momentum, and time together, is here shown to emerge from a formal deformation quantization as a family of finite-dimensional quantum theories whose dimensions grow in the classical limit.

What carries the argument

The load-bearing object is a reality-preserving formal embedding of $u(2,\mathbb{H})$ into the algebra of formal Laurent series in $\sqrt{\hbar}$ with coefficients in the Weyl algebra of the seven-dimensional Heisenberg algebra $\mathfrak{heis}_3$, given explicitly in Theorem 3.2. It generalizes the Holstein–Primakoff mechanism: the $u(2,\mathbb{H})$ generators are written as oscillator bilinears multiplied by polymeromorphic square-root factors $\sqrt{1/\hbar - N - n/2}$. The convergence of these square roots on the finite-dimensional spaces $\mathcal{H}^{1/m}$ truncates the Fock space and turns the formal embedding into honest unitary operators; the same mechanism lets the formal connection converge to the exact Cartan connection of the homogeneous model.

What would settle it

Compute the omitted brackets $[K,P]$ and $[K,K]$ for the generators displayed in Theorem 3.2 on the orthonormal basis of $\mathcal{H}^{1/m}$ for $m=2$ and $m=3$, using the polynomial-square-root operators of Section 4, and compare each result with the $u(2,\mathbb{H})$ structure constants; the paper's proof does not display these identities, so a mismatch at any basis vector would refute the central construction.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.4: for $\hbar \in I = \{1/m \mid m \in \mathbb{Z}_{>0}\}$, the symplectic spinor bundle $\mathcal{H}_\mathbb{Z}$ over $S^7$ admits a filtration $\mathcal{H}^1_\mathbb{Z} \subset \mathcal{H}^{1/2}_\mathbb{Z} \subset \mathcal{H}^{1/3}_\mathbb{Z} \subset \cdots \subset \mathcal{H}_\mathbb{Z}$, and the formal quantum connection of Theorem 1.3 induces on each subbundle $\mathcal{H}^{\hbar}_\mathbb{Z}$ an exact flat unitary connection. These subbundles are associated to the homogeneous model $U(2,\mathbb{H})/U(1,\mathbb{H})$ by the symmetric tensor representations of $U(2,\mathbb{H})$ of dimension $\binom{1/\hbar+2}{3}$, with fundamental weights $(1/\hbar - 1, 0)$. In short, requiring convergence of the formal Fedosov-type connection selects discrete values of $\hbar$ and truncates the infinite-dimensional Hilbert bundle to finite-dimensional unitary irreducible representations.

Load-bearing premise

The load-bearing premise is that the oscillator ansatz in Theorem 3.2 is complete and that every $u(2,\mathbb{H})$ bracket identity holds, even though the proof verifies only the $P$-$P$ brackets explicitly, sets $H_2$ to 1 by hand, and asserts the rest; if any unchecked identity fails, the flat connection and the convergence theorem collapse.

Editorial extensions

If this is right

  • For every positive integer $m$, there is a finite-dimensional quantum dynamical system $(\mathcal{H}^{1/m}_\mathbb{Z}, \nabla^{1/m})$ on $S^7$ whose flat connection is the $U(2,\mathbb{H})$-Cartan connection; parallel transport along Reeb orbits gives periodic Schrödinger evolutions.
  • The representation dimensions are $\binom{1/\hbar+2}{3}$, so as $\hbar \to 0$ the theories grow without bound, matching the classical limit of a closed contact manifold.
  • The probability rule computed from parallel transport of the flat connection is well-defined and independent of the chosen path because the connection is flat and the subbundles are finite-dimensional.
  • The formal asymptotic quantization 'shortens' to a bona fide quantization exactly at the discrete coupling values $\hbar = 1/m$, and this is the sense in which formality is resolved on $S^7$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The truncation mechanism should not be peculiar to $S^7$; any closed contact homogeneous space with a reductive model and a Heisenberg contraction may admit an analogous set of distinguished $\hbar$ values at which the formal series converges to finite-dimensional dynamics.
  • One can test the semiclassical limit directly: compute parallel-transport probabilities along the periodic Reeb orbits for small $\hbar$ and compare them with stationary-phase approximations to the contact action; the finite-dimensional theory should reproduce them.
  • The discrete set $\hbar = 1/m$ suggests an integrality condition in the contact analog of prequantization; a natural next step is to seek an index-theoretic or K-theoretic explanation for why exactly these values are selected.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a quantization scheme for the standard contact seven sphere S^7 = U(2,H)/U(1,H). It formulates a formal Fedosov-type connection problem (Problem 1.2) on the symplectic spinor bundle HZ and constructs a formal solution whose connection form is built from an embedding of the Lie algebra u(2,H) into a formal Laurent extension of the Weyl algebra of the Heisenberg algebra heis3. The embedding (Theorem 3.2) is a quaternionic generalization of the Holstein–Primakoff mechanism, with all generators given explicitly in terms of oscillators and square roots of number operators. The paper then shows that for ħ = 1/m, m ∈ Z_{>0}, the formal connection induces exact flat connections on finite-dimensional subbundles H^ħ_Z associated to symmetric tensor representations of U(2,H), yielding bona fide quantum dynamical systems (Theorem 1.4). The main supporting results are the partial verification of the Lie bracket identities in Theorem 3.2, the convergence of the square-root operators (Proposition 4.1), and the claim that the truncated operators form a representation (Proposition 4.2).

Significance. If the construction is fully correct, this is a substantial advance in contact quantization: it gives the first explicit convergence of a formal Fedosov-type contact connection to exact flat connections on finite-rank subbundles for a closed contact manifold, and it produces a family of unitary representations of U(2,H) whose dimensions grow as ħ → 0, in agreement with the classical limit. The paper's strengths are its concreteness and checkability: the generators in Theorem 3.2 are closed formulas, the classical Poisson analog is derived explicitly, and the convergence argument for the square-root operator is elementary and convincing. No data are fitted and no predictions are retrofitted; the embedding is solved from Lie bracket constraints. The main risk is that the central bracket verification is delegated to an unperformed check, and the truncated representation statement is asserted rather than proved in detail. These issues are local and fixable, but they are load-bearing for the main theorems.

major comments (3)
  1. [Section 3.3.2, Theorem 3.2] The proof of the central embedding verifies only the [J,P] brackets and derives the coefficient relations from the six [P,P] brackets, then states 'It is easy to check that all remaining brackets are satisfied' without displaying them. The omitted identities include [K^{++}, K^{--}], the [K,P] brackets, and the K–K brackets. These are not cosmetic: the connection form in Eq. (5.3) is assembled by coupling the embedded generators to the exterior differential system (2.10), and its flatness to all orders is exactly the statement that these generators close under the u(2,H) commutation relations. A single failed bracket among the omitted identities would invalidate the curvature cancellation and hence Theorem 1.3. I request that the authors supply a complete verification, either by explicit computation in an appendix or by a structural argument using the stated |2|-grading together with Jacobi identities, rather than an assertion.
  2. [Section 4, Proposition 4.2] The proof that the truncated operators ρP^α_α̇ and ρK^α̇_β̇ form a representation of u(2,H) on H^ħ is not carried out. The text says 'One may use the formulas of the previous two displays to show that ρ^ħ([X,Y])|n1,n2,n3⟩ = [ρ^ħ(X),ρ^ħ(Y)]|n1,n2,n3⟩' without showing any of the mixed brackets. This step is the bridge from the formal embedding (Theorem 3.2) to the bona fide finite-dimensional representations, and it underlies both Theorem 4.4 (the commutativity of the 'embed' and 'shorten' maps) and Theorem 1.4. The boundary behavior is delicate because the operators do not preserve the total number operator; the stated mechanism is that square-root coefficients vanish exactly at the boundary of H^ħ. This must be checked for all bracket combinations, not only the ones that are diagonal in the number basis. Please provide a full verification or a reference to a detailed computation.
  3. [Section 3.3.1, Eq. (3.11)] The display of the |2|-grading of u(2,H) is corrupted in the text: it contains repeated non-mathematical strings such as '⟪rl⟫mo⟨⌟...' and the claimed direct sum decomposition cannot be read. The same corruption appears in the subsequent display of the contracted grading. Because the grading element −iK^{++} (and its contraction) is used to motivate the ansatz for the formal generators in Theorem 3.2, an unreadable grading display leaves the ansatz unjustified. The authors should restore the correct displays and verify that the eigenspace decomposition is correct as written.
minor comments (4)
  1. [Abstract and Introduction] Several words are missing spaces, e.g., 'tobona fidequantization', 'dimensionstendtoinfinity', and 'Itisnotsurprising'. These should be corrected in the final manuscript.
  2. [Section 4, after Eq. (4.1)] The switch from the formal parameter ħ to the numerical value ħ = 1/m is stated as 'It will be clear from context', but the convergence argument in Proposition 4.1 would be clearer if the text explicitly said that the partial sums S_ℓ(n,N,ħ) are evaluated at the positive real number 1/m while the connection forms in Section 5 continue to be formal Laurent series in √ħ.
  3. [Section 2.2] The Reeb vector field R = ∂/∂θ1 + ∂/∂θ2 + ∂/∂θ3 + ∂/∂θ4 is given in the torus coordinates of that subsection; a sentence clarifying that this expression is with respect to those coordinates would help avoid confusion with the earlier one-form notation.
  4. [Remark 4.3] The claim that the representations ρ^ħ on H^ħ are unitary irreducible with fundamental weights (1/ħ − 1, 0) is stated without proof or reference. Since this identification is used to describe the subbundles in Theorem 1.4, a short argument or citation to the representation theory of sp(4,C) would strengthen the exposition.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the u(2,H) embedding is solved from bracket constraints; the main weaknesses are an omitted bracket check and reliance on the authors' own contact-quantization framework, not a retrofitted prediction.

full rationale

The derivation chain does not contain a fitted parameter renamed as a prediction or an equation made true by definition. Theorem 3.2 constructs the formal embedding by ansatz with unknown polymeromorphic functions F, G, H, then solves the [J,P] and [P,P] bracket equations, obtaining H_1^2 = 1/hbar - n/2 - N and F = H_1 after fixing H_2 = 1; the remaining brackets are delegated to the sentence "It is easy to check that all remaining brackets are satisfied." That is an omitted verification, and it is load-bearing for the flatness of the connection form (5.3) and hence for Theorems 1.3 and 1.4, but it is a proof gap, not circularity. Likewise Proposition 4.2 asserts the truncated operators obey the u(2,H) brackets via "one may use the formulas ... to show", again an omitted check rather than a circular reduction. The paper also candidly states that the finite-dimensional quantum dynamical systems "could have been constructed directly from the data of the homogeneous model, its Cartan connection and the symmetric tensor representations of Remark 4.3"; this is a consistency observation, since Theorem 4.4 proves the formal connection limits agree with that direct construction, not an input used to define the formal connection. The main self-citation is the contact-quantization framework of [20,30]; it supplies the Fedosov-type formalism and the Holstein-Primakoff mechanism, but the novel u(2,H) embedding is derived from Lie-bracket constraints within this paper. No data are fitted, no prediction is retrofitted, and no uniqueness theorem is imported. The score reflects only minor, non-load-bearing reliance on the authors' own framework and the acknowledged direct-construction shortcut; the central claim still has independent content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

This construction has no empirical free parameters. Its inputs are the contact quantization framework developed in the authors' earlier papers [20,30], the standard homogeneous-space geometry of the seven-sphere, and an ansatz whose completeness is not proven. The formal parameter hbar is a genuine deformation parameter, and the choice hbar=1/m is dictated by convergence of the Holstein-Primakoff square root, not by data. The phase freedom in Remark 3.3 shows the quoted embedding is one member of a family.

free parameters (2)
  • hbar specialization hbar=1/m = 1/m (m a positive integer)
    The square-root series only converges to a finite-dimensional truncation at these discrete values; the paper selects them via the convergence requirement, not from data.
  • Phase factors psi, phi1, phi2 in the embedding ansatz = 0 (phases set to unity)
    Remark 3.3 allows polynomial phase functions in (n,N) that preserve the reality conditions; setting them to unity is a choice among a family of embeddings.
assumptions (4)
  • domain assumption The contact Fedosov quantization framework of [20,30] supplies the formal connection scheme for Problems 1.1 and 1.2.
    Invoked in Sections 3 and 5; the paper assumes this quantization program rather than deriving it from first principles.
  • standard math The seven-sphere is the reductive homogeneous space U(2,H)/U(1,H) with the local exterior differential system (2.10).
    Used in Theorem 1.3 and Theorem 1.4; standard Cartan geometry, but the explicit local one-forms are the input for the connection construction.
  • standard math The metaplectic-c structure over S7 exists and is unique, giving HZ isomorphic to U(2,H) times_{U(1,H)} H.
    Section 5.1.1 uses H^2(S7,Z)=0 and Mpc structure theory from [31]; no independent verification is provided in this paper.
  • ad hoc to paper The embedding ansatz of Theorem 3.2 is complete and all unshown bracket relations hold.
    The proof explicitly delegates the remaining bracket checks; the flatness of the formal connection and the convergence theorem depend on this untested completeness.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Seven Sphere Quantization." pith.science (2026). https://pith.science/paper/M7L4KI54

@misc{pith2026250714363,
  author       = {Pith},
  title        = {Pith review of: Seven Sphere Quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7L4KI54}},
  note         = {Machine review of arXiv:2507.14363}
}
read the original abstract

Co-oriented contact manifolds quite generally describe classical dynamical systems. Quantization is achieved by suitably associating a Schr\"odinger equation to every path in the contact manifold. We quantize the standard contact seven sphere by treating it as a homogeneous space of the quaternionic unitary group in order to construct a contact analog of Fedosov's formal connection on symplectic spinor bundles. We show that requiring convergence of the formal connection naturally filters the symplectic spinor bundle and yields an exact flat connection on each corresponding subbundle. A key ingredient is a generalization of the Holstein--Primakoff mechanism to the quaternionic unitary group. The passage from formal to bona fide quantization determines unitary irreducible representations of the quaternionic unitary group, whose dimensions tend to infinity as the formal deformation parameter approaches its classical limit. This appearance of finite-dimensional representations is not surprising since the contact seven sphere is closed and physically describes generalized positions, momenta and time variables.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 46 canonical work pages

  1. [1]

    Quantification géométrique,

    J.-M. Souriau, “Quantification géométrique,”Comm. Math. Phys.1 (1966) 374–398. http://projecteuclid.org/euclid.cmp/1103758996. [Page 1.]

    work page arXiv 1966

  2. [2]

    Quantization and unitary representations. I. Prequantization,

    B. Kostant, “Quantization and unitary representations. I. Prequantization,” inLectures in Modern Analysis and Applications III, vol. 170 ofLecture Notes in Mathematics, pp. 87–208. Springer, Berlin, Heidelberg, 1970.https://doi.org/10.1007/BFb0079068. [Page 1.]

    doi:10.1007/bfb0079068 1970

  3. [3]

    Souriau,Structure of dynamical systems: A symplectic view of physics, vol

    J.-M. Souriau,Structure of dynamical systems: A symplectic view of physics, vol. 149 ofProgress in Mathematics. Birkhäuser, 1997. Translated from the French by C. H. Cushman-de Vries, Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. [Page 1.]

    work page 1997

  4. [4]

    N. M. J. Woodhouse,Geometric quantization. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second ed., 1992. [Page 1.]

    work page 1992

  5. [5]

    Geometric quantization,

    A. A. Kirillov, “Geometric quantization,” inDynamical systems, IV, vol. 4 ofEncyclopaedia of Mathematical Sciences, pp. 139–176. Springer, Berlin, 2001. https://doi.org/10.1007/978-3-662-06791-8_2 . [Page 1.]

    doi:10.1007/978-3-662-06791-8_2 2001

  6. [6]

    Quantum mechanics as a deformation of classical mechanics,

    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Quantum mechanics as a deformation of classical mechanics,”Lett. Math. Phys.1 no. 6, (1975/77) 521–530. https://doi.org/10.1007/BF00399745. [Page 1.]

    work page doi:10.1007/bf00399745 1975

  7. [7]

    Deformation theory and quantization. I. Deformations of symplectic structures,

    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Deformation theory and quantization. I. Deformations of symplectic structures,”Ann. Physics 111 no. 1, (1978) 61–110. https://doi.org/10.1016/0003-4916(78)90224-5. [Page 1.]

    doi:10.1016/0003-4916(78)90224-5 1978

  8. [8]

    Deformation theory and quantization. II. Physical applications,

    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Deformation theory and quantization. II. Physical applications,”Ann. Physics 111 no. 1, (1978) 111–151. https://doi.org/10.1016/0003-4916(78)90225-7. [Page 1.]

    doi:10.1016/0003-4916(78)90225-7 1978

Show all 63 references
  1. [9]

    Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds,

    M. De Wilde and P. B. A. Lecomte, “Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds,”Lett. Math. Phys.7 no. 6, (1983) 487–496. https://doi.org/10.1007/BF00402248. [Page 1.]

    1983 doi

  2. [10]

    A simple geometrical construction of deformation quantization,

    B. V. Fedosov, “A simple geometrical construction of deformation quantization,”J. Differential Geom. 40 no. 2, (1994) 213–238.http://projecteuclid.org/euclid.jdg/1214455536. [Pages 1, 1, and 1.]

    1994

  3. [11]

    Deformation quantization of Poisson manifolds,

    M. Kontsevich, “Deformation quantization of Poisson manifolds,”Lett. Math. Phys.66 no. 3, (2003) 157–216. https://doi.org/10.1023/B:MATH.0000027508.00421.bf. [Page 1.]

    2003

  4. [12]

    General concept of quantization,

    F. A. Berezin, “General concept of quantization,”Comm. Math. Phys.40 (1975) 153–174. http://projecteuclid.org/euclid.cmp/1103860463. [Page 1.]

    1975

  5. [13]

    Quantum Darboux theorem,

    O. Corradini, E. Latini, and A. Waldron, “Quantum Darboux theorem,”Phys. Rev. D103 no. 10, (2021) Paper No. 105021, 19.https://doi.org/10.1103/physrevd.103.105021. [Pages 1 and 1.]

    2021 doi

  6. [14]

    He,Odd dimensional symplectic manifolds

    Z. He,Odd dimensional symplectic manifolds. ProQuest LLC, Ann Arbor, MI, 2010. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt: kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:0822892. Thesis (Ph.D.)–Massachusetts Institute of Techno...

    2010

  7. [15]

    Lefschetz contact manifolds and odd dimensional symplectic geometry,

    Y. Lin, “Lefschetz contact manifolds and odd dimensional symplectic geometry,”Optics Communications 296 (June, 2013) 17–24. arXiv:1311.1431 [math]. [Page 1.]

    2013 arXiv

  8. [16]

    Geometric quantization on presymplectic manifolds,

    I. Vaisman, “Geometric quantization on presymplectic manifolds,”Monatsh. Math. 96 no. 4, (1983) 293–310. https://doi.org/10.1007/BF01471212. [Page 1.] 26 Chatterjee, Görmez, & Waldron

    1983 doi

  9. [17]

    On the quantization of presymplectic dynamical systems via coisotropic imbeddings,

    M. J. Gotay and J. e. Śniatycki, “On the quantization of presymplectic dynamical systems via coisotropic imbeddings,” Comm. Math. Phys.82 no. 3, (1981/82) 377–389. http://projecteuclid.org/euclid.cmp/1103920596. [Page 1.]

    1981

  10. [18]

    Presymplectic manifolds and the quantization of relativistic particle systems,

    C. Günther, “Presymplectic manifolds and the quantization of relativistic particle systems,” in Differential geometrical methods in mathematical physics, vol. 836 ofLecture Notes in Mathematics, pp. 383–400. Springer, Berlin, 1980. [Page 1.]

    1980

  11. [19]

    Contact Geometry and Quantum Mechanics,

    G. Herczeg and A. Waldron, “Contact Geometry and Quantum Mechanics,”Phys. Lett. B781 (2018) 312–315, arXiv:1709.04557 [hep-th] . [Pages 1 and 1.]

    2018 arXiv

  12. [20]

    Contact quantization: quantum mechanics = parallel transport,

    G. Herczeg, E. Latini, and A. Waldron, “Contact quantization: quantum mechanics = parallel transport,” Arch. Math. (Brno)54 no. 5, (2018) 281–298. https://doi.org/10.5817/am2018-5-281. [Pages 1, 1, and 1.]

    2018 doi

  13. [21]

    Contact homogeneous spaces,

    D. V. Alekseevskii, “Contact homogeneous spaces,”Functional Analysis and Its Applications24 no. 4, (1990) 324–325.https://doi.org/10.1007/BF01077337. [Page 1.]

    1990 doi

  14. [22]

    Semisimple symmetric contact spaces,

    D. V. Alekseevskii and C. Gorodski, “Semisimple symmetric contact spaces,”Indagationes Mathematicae 31 no. 6, (2020) 1110–1133.https://doi.org/10.1016/j.indag.2020.09.008. [Page 1.]

    2020 doi

  15. [23]

    Geiges,An introduction to contact topology, vol

    H. Geiges,An introduction to contact topology, vol. 109 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008. https://doi.org/10.1017/CBO9780511611438. [Page 1.]

    2008 doi

  16. [24]

    Libermann and C.-M

    P. Libermann and C.-M. Marle,Symplectic geometry and analytical mechanics, vol. 35 of Mathematics and Its Applications. D. Reidel Publishing Company, Dordrecht, 1987. https://doi.org/10.1007/978-94-009-3807-6 . Translated from the French by Bertram Eugene Schwarzbach. [Page 1.]

    1987 doi

  17. [25]

    Contact manifolds and dissipation, classical and quantum,

    F. M. Ciaglia, H. Cruz, and G. Marmo, “Contact manifolds and dissipation, classical and quantum,” Ann. Physics 398 (2018) 159–179. https://doi.org/10.1016/j.aop.2018.09.012. [Page 1.]

    2018 doi

  18. [26]

    Contact Hamiltonian mechanics,

    A. Bravetti, H. Cruz, and D. Tapias, “Contact Hamiltonian mechanics,”Ann. Physics 376 (2017) 17–39. https://doi.org/10.1016/j.aop.2016.11.003. [Page 1.]

    2017 doi

  19. [27]

    Contact Hamiltonian dynamics: the concept and its use,

    A. Bravetti, “Contact Hamiltonian dynamics: the concept and its use,”Entropy 19 no. 10, (2017) Paper No. 535, 12.https://doi.org/10.3390/e19100535. [Page 1.]

    2017 doi

  20. [28]

    Contact Hamiltonian systems,

    M. de León and M. Lainz Valcázar, “Contact Hamiltonian systems,”J. Math. Phys.60 no. 10, (2019) 102902, 18.https://doi.org/10.1063/1.5096475. [Page 1.]

    2019 doi

  21. [29]

    A review on contact hamiltonian and lagrangian systems,

    M. d. León and M. Lainz, “A review on contact hamiltonian and lagrangian systems,”. http://arxiv.org/abs/2011.05579. arXiv:2011.05579 [math-ph]. [Page 1.]

    2011 arXiv

  22. [30]

    Dynamical quantization of contact structures,

    R. Casals, G. Herczeg, and A. Waldron, “Dynamical quantization of contact structures,”Adv. Theor. Math. Phys.27 no. 3, (2023) 881–959.https://doi.org/10.4310/atmp.2023.v27.n3.a7. [Pages 1, 1, and 3.1.]

    2023 doi

  23. [31]

    The metaplectic representation,Mpc structures and geometric quantization,

    P. L. Robinson and J. H. Rawnsley, “The metaplectic representation,Mpc structures and geometric quantization,”Mem. Amer. Math. Soc.81 no. 410, (1989) iv+92. https://doi.org/10.1090/memo/0410. [Pages 1, 5.1, 5.1, and 5.1.1.]

    1989 doi

  24. [32]

    Habermann and L

    K. Habermann and L. Habermann,Introduction to symplectic Dirac operators, vol. 1887 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2006.https://doi.org/10.1007/b138212. [Pages 1 and 5.1.1.]

    2006 doi

  25. [33]

    Symplectic Dirac operators andM pc-structures,

    M. Cahen, S. Gutt, and J. Rawnsley, “Symplectic Dirac operators andM pc-structures,” Gen. Relativity Gravitation43 no. 12, (2011) 3593–3617. https://doi.org/10.1007/s10714-011-1239-x. [Pages 1 and 5.1.]

    2011 doi

  26. [34]

    Deformation quantization of contact manifolds,

    B. M. Elfimov and A. A. Sharapov, “Deformation quantization of contact manifolds,”Lett. Math. Phys. 112 no. 6, (2022) Paper No. 124, 21.https://doi.org/10.1007/s11005-022-01621-3. [Page 1.]

    2022 doi

  27. [35]

    Deformation quantization of Heisenberg manifolds,

    M. A. Rieffel, “Deformation quantization of Heisenberg manifolds,”Comm. Math. Phys.122 no. 4, (1989) 531–562. http://projecteuclid.org/euclid.cmp/1104178588. [Page 1.]

    1989

  28. [36]

    Deformation quantization and operator algebras,

    M. A. Rieffel, “Deformation quantization and operator algebras,” inOperator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), vol. 51, Part 1 ofProc. Sympos. Pure Math., pp. 411–423. Amer. Math. Soc., Providence, RI, 1990. https://doi.org/10.1090/pspum/051...

    1988 doi

  29. [37]

    Questions on quantization,

    M. A. Rieffel, “Questions on quantization,” inOperator algebras and operator theory (Shanghai, 1997), vol. 228 ofContemp. Math., pp. 315–326. Amer. Math. Soc., Providence, RI, 1998. https://doi.org/10.1090/conm/228/03294. [Page 1.] Seven Sphere Quantization 27

    1997 doi

  30. [38]

    Lie group convolution algebras as deformation quantizations of linear Poisson structures,

    M. A. Rieffel, “Lie group convolution algebras as deformation quantizations of linear Poisson structures,” Amer. J. Math.112 no. 4, (1990) 657–685.https://doi.org/10.2307/2374874. [Page 1.]

    1990 doi

  31. [39]

    Convergence of star products: from examples to a general framework,

    S. Waldmann, “Convergence of star products: from examples to a general framework,”EMS Surv. Math. Sci. 6 no. 1-2, (2019) 1–31.https://doi.org/10.4171/emss/31. [Page 1.]

    2019 doi

  32. [40]

    Strict quantization of coadjoint orbits,

    N. P. Landsman, “Strict quantization of coadjoint orbits,”J. Math. Phys.39 no. 12, (1998) 6372–6383. https://doi.org/10.1063/1.532644. [Pages 1, 1, and 4.5.]

    1998 doi

  33. [41]

    An obstruction to quantization of the sphere,

    E. Hawkins, “An obstruction to quantization of the sphere,”Comm. Math. Phys.283 no. 3, (2008) 675–699. https://doi.org/10.1007/s00220-008-0517-2. [Page 1.]

    2008 doi

  34. [42]

    B. V. Fedosov,Deformation quantization and index theory, vol. 9 ofMathematical Topics. Akademie Verlag, Berlin, 1996. [Page 1.]

    1996

  35. [43]

    Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet,

    T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet,” Physical Review58 no. 12, (Dec., 1940) 1098–1113. [Pages 1, 3.1, and 4.]

    1940

  36. [44]

    Algebraic identities amongU (n) infinitesimal generators,

    S. Okubo, “Algebraic identities amongU (n) infinitesimal generators,” J. Mathematical Phys.16 (1975) 528–535. https://doi.org/10.1063/1.522550. [Page 1.]

    1975 doi

  37. [45]

    Boson realizations of Lie algebras with applications to nuclear physics,

    A. Klein and E. R. Marshalek, “Boson realizations of Lie algebras with applications to nuclear physics,” Rev. Modern Phys.63 no. 2, (1991) 375–558. https://doi.org/10.1103/RevModPhys.63.375. [Page 1.]

    1991 doi

  38. [46]

    The Holstein-Primakoff and Dyson realizations for the Lie superalgebra gl(m/n+ 1),

    T. D. Palev, “The Holstein-Primakoff and Dyson realizations for the Lie superalgebra gl(m/n+ 1),” J. Phys. A30 no. 23, (1997) 8273–8278. https://doi.org/10.1088/0305-4470/30/23/023. [Page 1.]

    1997 doi

  39. [47]

    Holstein-Primakoff realizations on coadjoint orbits,

    P. Oh and C. Rim, “Holstein-Primakoff realizations on coadjoint orbits,”Modern Phys. Lett. A12 no. 3, (1997) 163–172.https://doi.org/10.1142/S0217732397000169. [Page 1.]

    1997 doi

  40. [48]

    A. W. Knapp,Lie groups beyond an introduction, vol. 140 ofProgress in Mathematics. Birkhäuser, Boston, second ed., 2002. [Page 2.]

    2002

  41. [49]

    Transformation groups of spheres,

    D. Montgomery and H. Samelson, “Transformation groups of spheres,”Ann. of Math. (2)44 (1943) 454–470. https://doi.org/10.2307/1968975. [Page 2.]

    1943 doi

  42. [50]

    Some remarks about Lie groups transitive on spheres and tori,

    A. Borel, “Some remarks about Lie groups transitive on spheres and tori,”Bull. Amer. Math. Soc. 55 (1949) 580–587. https://doi.org/10.1090/S0002-9904-1949-09251-0 . [Page 2.]

    1949 doi

  43. [51]

    A. L. Besse,Einstein manifolds, vol. 10 ofClassics in Mathematics. Springer, Berlin, Heidelberg,

  44. [52]

    R. W. Sharpe,Differential geometry: Cartan’s generalization of Klein’s Erlangen program, vol. 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. With a foreword by S. S. Chern. [Page 2.1.]

    1997

  45. [53]

    Kobayashi and K

    S. Kobayashi and K. Nomizu,Foundations of differential geometry, Volume II. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original, A Wiley-Interscience Publication. [Page 2.1.]

    1996

  46. [54]

    Čap and J

    A. Čap and J. Slovák,Parabolic geometries I: Background and general theory, vol. 154 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009. https://doi.org/10.1090/surv/154. [Pages 2.1, 2.1, and 5.2.]

    2009 doi

  47. [55]

    Kobayashi and K

    S. Kobayashi and K. Nomizu,Foundations of differential geometry, Volume I. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication. [Page 2.1.]

    1996

  48. [56]

    On the contraction of groups and their representations,

    E. Inonu and E. P. Wigner, “On the contraction of groups and their representations,”Proc. Nat. Acad. Sci. U.S.A.39 (1953) 510–524. https://doi.org/10.1073/pnas.39.6.510. [Page 3.1.]

    1953 doi

  49. [57]

    On the principles of elementary quantum mechanics,

    H. J. Groenewold, “On the principles of elementary quantum mechanics,”Physica 12 (1946) 405–460. [Page 3.2.]

    1946

  50. [58]

    A Groenewold-van Hove theorem forS2,

    M. J. Gotay, H. Grundling, and C. A. Hurst, “A Groenewold-van Hove theorem forS2,” Trans. Amer. Math. Soc.348 no. 4, (1996) 1579–1597. https://doi.org/10.1090/S0002-9947-96-01559-0 . [Page 3.2.]

    1996 doi

  51. [59]

    Schmüdgen,Unbounded operator algebras and representation theory, vol

    K. Schmüdgen,Unbounded operator algebras and representation theory, vol. 37 ofOperator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990. https://doi.org/10.1007/978-3-0348-7469-4 . [Page 4.]

    1990 doi

  52. [60]

    Schmüdgen,An invitation to unbounded representations of∗-algebras on Hilbert space, vol

    K. Schmüdgen,An invitation to unbounded representations of∗-algebras on Hilbert space, vol. 285 of Graduate Texts in Mathematics. Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-46366-3 . [Page 4.]

    2020 doi

  53. [61]

    Universal metaplectic structures and geometric quantization,

    M. Forger and H. Hess, “Universal metaplectic structures and geometric quantization,”Comm. Math. Phys. 64 no. 3, (1979) 269–278.http://projecteuclid.org/euclid.cmp/1103904723. [Pages 5.1 and 5.1.1.] 28 Chatterjee, Görmez, & Waldron

    1979

  54. [62]

    On a geometric quantization scheme generalizing those of Kostant-Souriau and Czyż,

    H. Hess, “On a geometric quantization scheme generalizing those of Kostant-Souriau and Czyż,” in Differential geometric methods in mathematical physics, vol. 139 ofLecture Notes in Physics, pp. 1–35. Springer, Berlin, Heidelberg, 1981.https://doi.org/10.1007/3-540-10578-6_19. ...

    1981 doi

  55. [1987]

    [Page 2.]

    https://doi.org/10.1007/978-3-540-74311-8 . [Page 2.]

    doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.