Pith. sign in

REVIEW 2 major objections 4 minor 55 references

This paper constructs a path-integral representation of quantum transition amplitudes on a general Lie group, handling compact directions through a sum over endpoint logarithms, and verifies it to two-loop order for Euler-Arnold systems.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 21:33 UTC pith:YYIMWUO2

load-bearing objection A genuinely novel path-integral framework for Lie groups, but the advertised generality fails for groups whose exponential map misses an open set (e.g., SL(2,R)). the 2 major comments →

arxiv 2607.16029 v1 pith:YYIMWUO2 submitted 2026-07-17 quant-ph hep-thmath-phmath.CAmath.MP

Quantum Mechanics on Lie Groups: II. Path Integrals

Mathieu Beauvillain , Blagoje Oblak , Marios Petropoulos This is my paper
classification quant-ph hep-thmath-phmath.CAmath.MP MSC 81S4022E70
keywords path integralsLie groupsEuler-Arnold systemsnoncommutative Fourier transformheat kernelwinding numbersFaddeev-Popov ghostssemiclassical expansion
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.

This paper derives a path-integral formula for the quantum propagator of a particle moving on a Lie group: the transition amplitude is written as a sum of ordinary path integrals over the Lie algebra, one for each logarithm of the final group element. That sum over logarithms is the higher-dimensional generalization of the winding-number sum familiar from the particle on a circle, and it is precisely what handles the compact directions of the group. The machinery is then applied to Euler-Arnold systems, whose quantized propagators are heat kernels, and the semiclassical propagator and partition function are computed to one loop, with the partition function carried to two loops where the standard R/12 heat-kernel coefficient is recovered. The construction makes quantum dynamics on group manifolds accessible to standard perturbative path-integral techniques.

Core claim

On its own terms, the paper's central claim is eq. (4.11): the propagator equals a sum over logarithms of the endpoint of path integrals over the Lie algebra, with the Hamiltonian action expressed through the left Maurer-Cartan form and a left-Haar measure at every time slice. This is obtained by 'decompactifying' the Hilbert space into periodic functions on the Lie algebra, then inserting resolutions of the identity coming from a noncommutative Fourier transform whose Gutt star product encodes the Baker-Campbell-Hausdorff formula. A change of variables from left to right Haar measures introduces a modular-function factor at the initial point, which matters for nonunimodular groups. Speciali

What carries the argument

The load-bearing object is the principal-branch logarithm map from a dense open subset of the group to its Lie algebra, together with the endpoint sets Logs(g_f), the set of Lie algebra elements whose exponential equals the final group element; the sum over these logarithms is the group-theoretic version of the winding-number sum on a circle, running over lattice translations along maximal tori. The path integral is assembled from two resolutions of the identity: position space uses the left Haar measure in exponential coordinates with its Jacobian, while momentum space uses a noncommutative Fourier transform whose star product is generated by the Baker-Campbell-Hausdorff formula. In the sem

Load-bearing premise

The whole construction rests on the assumption that the exponential map from the Lie algebra to the group covers the group up to a set of measure zero, with a principal-branch logarithm defined on a dense symmetric open set; endpoints with no real logarithm would get zero amplitude from the path-integral formula, which is wrong for open sets in groups like SL(2,R).

What would settle it

Take G = SL(2,R) with a standard Euler-Arnold Hamiltonian and compute the propagator from the identity to an element with trace < -2, which lies in the open set with no real logarithm. A spectral or heat-kernel computation gives a generically nonzero heat kernel, whereas the paper's formula would yield zero because the sum over logarithms is empty; a match would require modifying the endpoint sum.

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

If this is right

  • Euler-Arnold propagators are now computable to one loop via an explicit formula: sum over classical geodesics from the identity to the endpoint, weight each by the exponential of its length squared over 2T and the inverse square root of a determinant factor.
  • The two-loop partition function reproduces the R/12 term of the standard heat-kernel expansion, so the path-integral measure and ghost sector pass a nontrivial consistency check.
  • Measure-related effects (Jacobians and ghosts) enter only at two loops, so one-loop calculations can be done with flat measures and simple determinants.
  • The modular-function factors make the formalism applicable to nonunimodular groups, where left and right Haar measures differ.
  • The same path-integral representation gives a new derivation of characters of the regular representation, localizing the path integral on a single classical trajectory.

Where Pith is reading between the lines

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

  • The decompactified form suggests a direct numerical strategy: discretize the path integral on the Lie algebra and sample the endpoint-log sum, which may be cheaper than simulating paths constrained to the group manifold.
  • If the surjectivity obstruction is set aside, the framework naturally extends to any Lie group whose exponential map is onto (compact groups, connected nilpotent groups); for groups like SL(2,R), the formula would need an analytic continuation or a more refined treatment of endpoints without real logarithms.
  • The explicit cancellation of delta-function divergences at two loops hints at an all-orders structure in which ghosts can be integrated out once and for all, yielding a local effective counterterm action for the curved measure.
  • The winding-sum structure parallels the one used for compactified scalars in thermal field theory, so the methods may transfer to finite-temperature path integrals on group manifolds.

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

2 major / 4 minor

Summary. The paper constructs a path-integral representation for transition amplitudes in L²(G) for a Lie group G, using the noncommutative Fourier transforms developed in the companion paper [1]. The key idea is to decompactify G to its Lie algebra by summing over logarithms of the endpoint, Eq. (4.2), and then to derive a phase-space path integral, Eq. (4.11), with a careful treatment of left/right Haar measures and the modular function, Eq. (4.16). For Euler–Arnold systems the authors perform a semiclassical expansion to two loops, obtaining a one-loop propagator (5.33) and a two-loop partition-function density (6.9) whose β/12 term is identified with the standard heat-kernel coefficient R/12. The U(1) case is treated in detail in Appendix A and is used as a consistency check.

Significance. If the results are valid for the claimed class of groups, this is a useful and nontrivial framework: it extends earlier noncommutative-Fourier path integrals by handling compact directions and non-unimodular groups, and it provides explicit two-loop formulas with ghost-mediated cancellation of δ(0) divergences. The derivation is largely self-contained, and the paper is explicit about the star-product symbol (4.15), the modular-function factors, and the matching of the heat-kernel coefficient (6.9) against known results. The main weakness is that the central claim is stated for an arbitrary Lie group G, whereas a load-bearing assumption—surjectivity of the exponential map up to measure zero—fails for standard noncompact groups such as SL(2,R). This limits the actual scope of the construction and needs to be addressed before the paper can be accepted as a general treatment.

major comments (2)
  1. [§3.1 and Eq. (4.11)] The derivation of the path integral relies on the assumption, stated in §3.1, that exp: g → G is surjective up to a set of measure zero and that a principal-branch logarithm exists on a dense symmetric set. This assumption is false for many noncompact semisimple groups. For SL(2,R), the set U = {g : Tr(g) < −2} is open and nonempty, hence of positive Haar measure, and no element of U has a real logarithm. For such endpoints the sum over Y_f ∈ Logs(g_f) in Eq. (4.11) is empty, so the formula gives a vanishing propagator, whereas the heat kernel on SL(2,R) is nonzero on U. Thus Eq. (4.11) is not valid for arbitrary Lie groups, and the abstract's claim for 'a Lie group G' is too broad. The same obstruction affects the 'global BCH formula' defined by path lifting in §3.1: for g(t)=exp(tX)exp(tY) that passes through U, no continuous lift into g exists. The paper should either restrict its sta
  2. [§6.1, Eq. (6.3)] The partition-function density (6.3) restricts the sum over logarithms of the identity to a fixed Cartan subalgebra h. This restriction is not generally justified. For a group such as SL(2,R), the logarithms of the identity include elements in compact Cartan subalgebras (e.g. rotations by 2πn), while a fixed split Cartan subalgebra contains only the trivial logarithm of the identity; different Cartan conjugacy classes contribute different winding sectors. The argument that the identity is a measure-zero set of exceptional points does not determine the value of the density (e|e^{-βH}|e), which is computed exactly at the identity. The high-temperature two-loop result (6.9) is local and is not affected, but the global partition-function formula (6.3) is. The authors should state the additional assumptions needed for this restriction (e.g. a single conjugacy class of Cartan subalgebras, or a
minor comments (4)
  1. [Eq. (3.6)] Typo: 'r is the the dimension' should read 'r is the dimension'.
  2. [Eq. (4.8)] The dummy index in the discretized measure is inconsistent: the product runs over j but the integrand uses X_k. This should be cleaned up.
  3. [§5.4 and Appendix C] The diagrammatic notation is hard to follow in plain text, and some diagrams are not rendered. The textual explanation would benefit from either actual figures or an explicit table of the diagrams together with their analytic expressions.
  4. [Appendix C, Eq. (C.2)] In the '4Y1D term', the displayed correlator is ⟨Y^a Y^b Y^c Y^d⟩, but from Eq. (5.43) the 4Y1D vertex is contracted with Ẋ Y^b Y^c Y^d. The missing derivative should be restored for consistency.

Circularity Check

0 steps flagged

No significant circularity: the path integral is constructed from the operator algebra and checked against independent heat-kernel results.

full rationale

The central formula (4.11) is derived, not assumed: the propagator on L^2(G) is decompactified via the logarithm sum (4.2), then obtained by inserting the position/momentum resolutions of identity (3.8)/(3.18) from the noncommutative Fourier toolkit, producing the phase-space action (4.9) and eventually the Lagrangian form (4.19). No parameter is fitted to the final propagator or partition function; the one-loop expressions (5.33)/(6.4) and two-loop coefficient (6.9) are compared with the standard heat-kernel/De Witt expansions and with the U(1) propagator (A.18), which are used as checks rather than inputs. The paper does rely on the companion paper [1] for Fourier transforms and the periodic realization of L^2(G), but this is a normal import of prior published work with stated assumptions, not a self-citation that smuggles in the target result or forbids alternatives. The stated surjectivity assumption on exp in §3.1 is a substantive correctness limitation (e.g. it fails for SL(2,R) elements with trace < -2), but that is a mathematical validity concern, not a circularity: it does not make the derivation equivalent to its inputs. No circular step could be identified with a quotable reduction of a prediction to a fit or to a definition.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claim rests on the exponential-map surjectivity assumption (fails for e.g. SL(2,R)), the log-lattice characterization (3.10), the Cartan-subalgebra restriction for the identity (6.3), and the imported noncommutative Fourier transform from [1]. No free parameters are fitted: I, T, β are physical inputs; structure constants come from the group. No new entities are invented: Faddeev-Popov ghosts are standard tools. The heaviest burden is the surjectivity axiom, which limits the class of Lie groups to which the decompactified path integral applies.

axioms (5)
  • domain assumption The exponential map exp: g → G is surjective up to a set of measure zero, and a principal branch of log exists (open, symmetric, dense image).
    Stated in §3.1 (p.8). Fails for SL(2,R), where trace<−2 elements have no real logarithm and form an open positive-measure set. Without it, the sum over Logs(g_f) in (4.2) is empty for such endpoints and the path-integral formula cannot give the true propagator.
  • domain assumption For generic g, Logs(g) = {X + 2π n^i a_i(X) | n^i ∈ Z^r} along a maximal torus (eq. 3.10).
    Used to justify the Dirac-comb inclusion (3.9) and the winding-number interpretation. Describes compact (elliptic) directions; hyperbolic and parabolic elements of noncompact semisimple groups have different (often unique) logarithms, and elements outside exp(g) have none.
  • ad hoc to paper The identity's logarithms may be restricted to a fixed Cartan subalgebra h when computing the partition-function density (eq. 6.3).
    Footnote 4 notes Logs(e) is much larger than (3.10). §6.1 argues the exceptional measure-zero set does not affect the group integral (6.1), so the density is evaluated as if e were generic. This is a heuristic localization, not derived from the trace over L²(G).
  • standard math The Gutt star product (3.14) defines a bijective isometry between position and momentum Hilbert spaces (noncommutative Fourier transform of [1]).
    Borrowed from the authors' companion paper [1] and deformation quantization [47-49]. Standard within the LQG group-Fourier literature; the paper cites it as established but does not re-derive it.
  • domain assumption The measure Jacobian can be exponentiated with Grassmann-odd ghosts (5.9), and the semiclassical propagator obeys the De Witt ansatz (5.34) with Wick rotation β = -iT.
    Standard Faddeev-Popov/heat-kernel technology [43,52]; the ghosts are responsible for the claimed δ(0) cancellations at two loops (§5.4).

pith-pipeline@v1.3.0-alltime-deepseek · 34180 in / 21408 out tokens · 197483 ms · 2026-08-01T21:33:56.490946+00:00 · methodology

0 comments
read the original abstract

We continue our study of quantum dynamics on a Lie group $G$, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space $L^2(G)$. This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in $G$. We show that compactness can be handled through a sum over winding numbers in maximal tori of $G$, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 29 linked inside Pith

  1. [1]

    Quantum mechanics on Lie groups: I. Noncommutative Fourier transforms,

    M. Beauvillain, B. Oblak, and M. Petropoulos, “Quantum mechanics on Lie groups: I. Noncommutative Fourier transforms,”J. Phys. A59(2026), no. 25, 255303, 2512.19840

    Pith/arXiv arXiv 2026

  2. [2]

    How much does the rigid body rotate? A Berry’s phase from the 18th century,

    R. Montgomery, “How much does the rigid body rotate? A Berry’s phase from the 18th century,”Amer. J. Phys.59(1991), no. 5, 394–398

    1991

  3. [3]

    J. E. Marsden and T. S. Ratiu,Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, vol. 17. Springer Science & Business Media, 2013

    2013

  4. [4]

    31 Lectures on Geometric Mechanics,

    D. D. Holm, “31 Lectures on Geometric Mechanics,”2408.09564

    Pith/arXiv arXiv

  5. [5]

    V. I. Arnold and B. A. Khesin,Topological methods in hydrodynamics, vol. 125. Springer Science & Business Media, 1999

    1999

  6. [6]

    Khesin and R

    B. Khesin and R. Wendt,The geometry of infinite-dimensional groups, vol. 51. Springer Science & Business Media, 2008

    2008

  7. [7]

    Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,

    V. I. Arnold, “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits,”Ann. Inst. Fourier16(1966), no. 1, 319–361

    1966

  8. [8]

    The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid,

    S. Vasylkevych and J. E. Marsden, “The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid,”Dyn. PDE2(2005), no. 4, 281–300,0711.4875

    Pith/arXiv arXiv 2005

  9. [9]

    Geometric Hydrodynamics: from Euler, to Poincaré, to Arnold,

    K. Modin, “Geometric Hydrodynamics: from Euler, to Poincaré, to Arnold,” 1910.03301

    Pith/arXiv arXiv 1910

  10. [10]

    The Euler–Poincaré equations and semidirect products with applications to continuum theories,

    D. D. Holm, J. E. Marsden, and T. S. Ratiu, “The Euler–Poincaré equations and semidirect products with applications to continuum theories,”Adv. Math.137(1998), no. 1, 1–81,chao-dyn/9801015. 38

    Pith/arXiv arXiv 1998

  11. [11]

    The Euler-Poincaré equations in geo- physical fluid dynamics,

    D. D. Holm, J. E. Marsden, and T. S. Ratiu, “The Euler-Poincaré equations in geo- physical fluid dynamics,”chao-dyn/9903035

    Pith/arXiv arXiv

  12. [12]

    Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,

    F. Gay-Balmaz and T. S. Ratiu, “Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,”0903.4287

    Pith/arXiv arXiv

  13. [13]

    Berry Phases in the Reconstructed KdV Equation,

    B. Oblak and G. Kozyreff, “Berry Phases in the Reconstructed KdV Equation,”Chaos 30(2020) 113114,2002.01780

    Pith/arXiv arXiv 2020

  14. [14]

    Quantum Field Theory of Fluids,

    B. Gripaios and D. Sutherland, “Quantum Field Theory of Fluids,”Phys. Rev. Lett. 114(2015), no. 7, 071601,1406.4422

    Pith/arXiv arXiv 2015

  15. [15]

    The Quantum mechanics of perfect fluids,

    S. Endlich, A. Nicolis, R. Rattazzi, and J. Wang, “The Quantum mechanics of perfect fluids,”JHEP04(2011) 102,1011.6396

    Pith/arXiv arXiv 2011

  16. [16]

    The quantum perfect fluid in 2D,

    A. Dersy, A. Khmelnitsky, and R. Rattazzi, “The quantum perfect fluid in 2D,”SciPost Phys.17(2024), no. 1, 019,2211.09820

    Pith/arXiv arXiv 2024

  17. [17]

    On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,

    L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,”Phys. Zeitsch. der Sow.8(1935) 153–169

    1935

  18. [18]

    The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview,

    M. Lakshmanan, “The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview,”Philos. Trans. Roy. Soc. A369(2011), no. 1939, 1280–1300,1101.1005

    Pith/arXiv arXiv 2011

  19. [19]

    Semiclassical Spectrum of the Continuous Heisenberg Spin Chain,

    A. Jevicki and N. Papanicolaou, “Semiclassical Spectrum of the Continuous Heisenberg Spin Chain,”Annals Phys.120(1979) 107

    1979

  20. [20]

    Gaudin,The Bethe wavefunction

    M. Gaudin,The Bethe wavefunction. Cambridge University Press, 2014

    2014

  21. [21]

    A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics,

    J. Lamers, “A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics,”PoSModave2014(2015) 001,1501.06805

    Pith/arXiv arXiv 2015

  22. [22]

    Path-integral spin dynamics with ex- change and external field,

    T. Nussle, S. Nicolis, I. Sofos, and J. Barker, “Path-integral spin dynamics with ex- change and external field,”Phys. Rev. B112(2025), no. 5, 054404,2502.19113

    Pith/arXiv arXiv 2025

  23. [23]

    Nonlinear bosonization of Fermi surfaces: The method of coadjoint orbits,

    L. V. Delacretaz, Y.-H. Du, U. Mehta, and D. T. Son, “Nonlinear bosonization of Fermi surfaces: The method of coadjoint orbits,”Phys. Rev. Res.4(2022), no. 3, 033131, 2203.05004

    Pith/arXiv arXiv 2022

  24. [24]

    EffectivefieldtheoryofBerryFermiliquidfromthecoadjointorbitmethod,

    X.Huang, “EffectivefieldtheoryofBerryFermiliquidfromthecoadjointorbitmethod,” Phys. Rev. B109(2024), no. 23, 235146,2312.00877

    Pith/arXiv arXiv 2024

  25. [25]

    Berry phases in the bosonization of nonlinear edge modes,

    M. Beauvillain, B. Oblak, and M. Petropoulos, “Berry phases in the bosonization of nonlinear edge modes,”Phys. Rev. B112(2025), no. 12, 125136,2408.03991

    arXiv 2025

  26. [26]

    H. C. Allen and P. C. Cross,Molecular Vib-rotors: The theory and interpretation of high resolution infrared spectra. Wiley, 1963

    1963

  27. [27]

    Bauder,Fundamentals of rotational spectroscopy

    A. Bauder,Fundamentals of rotational spectroscopy. Wiley Online Library, 2011

    2011

  28. [28]

    L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Physics: The- ory and Application. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1984

    1984

  29. [29]

    Quantum mechanics of a generalised rigid body,

    B. Gripaios and D. Sutherland, “Quantum mechanics of a generalised rigid body,”J. Phys. A49(2016), no. 19, 195201,1504.01406. 39

    Pith/arXiv arXiv 2016

  30. [30]

    Ponzano-Regge model revisited III: Feynman diagrams and effective field theory,

    L. Freidel and E. R. Livine, “Ponzano-Regge model revisited III: Feynman diagrams and effective field theory,”Class. Quant. Grav.23(2006) 2021–2062,hep-th/0502106

    Pith/arXiv arXiv 2006

  31. [31]

    3D Quantum Gravity and Effective Noncommutative Quantum Field Theory,

    L. Freidel and E. R. Livine, “3D Quantum Gravity and Effective Noncommutative Quantum Field Theory,”Phys. Rev. Lett.96(2006) 221301,hep-th/0512113

    Pith/arXiv arXiv 2006

  32. [32]

    Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity,

    L. Freidel and S. Majid, “Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity,”Class. Quant. Grav.25(2008) 045006, hep-th/0601004

    Pith/arXiv arXiv 2008

  33. [33]

    Three Dimensional Quantum Geometry and De- formed Poincare Symmetry,

    E. Joung, J. Mourad, and K. Noui, “Three Dimensional Quantum Geometry and De- formed Poincare Symmetry,”J. Math. Phys.50(2009) 052503,0806.4121

    Pith/arXiv arXiv 2009

  34. [34]

    Spinors and Voros star-product for Group Field Theory: First Contact,

    M. Dupuis, F. Girelli, and E. R. Livine, “Spinors and Voros star-product for Group Field Theory: First Contact,”Phys. Rev. D86(2012) 105034,1107.5693

    Pith/arXiv arXiv 2012

  35. [35]

    Quantum Mechanics on SO(3) via Non-commutative Dual Variables,

    D. Oriti and M. Raasakka, “Quantum Mechanics on SO(3) via Non-commutative Dual Variables,”Phys. Rev. D84(2011) 025003,1103.2098

    Pith/arXiv arXiv 2011

  36. [36]

    Group Fourier transform and the phase space path integral for finite dimensional Lie groups,

    M. Raasakka, “Group Fourier transform and the phase space path integral for finite dimensional Lie groups,”1111.6481

    Pith/arXiv arXiv

  37. [37]

    Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups,

    C. Guedes, D. Oriti, and M. Raasakka, “Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups,”J. Math. Phys.54(2013) 083508, 1301.7750

    Pith/arXiv arXiv 2013

  38. [38]

    Dover Publications, 2012

    Schulman,Techniques and Applications of Path Integration. Dover Publications, 2012

    2012

  39. [39]

    The heat kernel onsl(2,R),

    S. Mori, “The heat kernel onsl(2,R),” 2019

    2019

  40. [40]

    Fundamental solutions for the two dimensional affine group and hn+1,

    M. Craddock, “Fundamental solutions for the two dimensional affine group and hn+1,” Journal of Mathematical Analysis and Applications445(2017), no. 1, 953–970

    2017

  41. [41]

    The heat kernel on ads 3 and its applications,

    J. R. David, M. R. Gaberdiel, and R. Gopakumar, “The heat kernel on ads 3 and its applications,”Journal of High Energy Physics2010(Apr., 2010)

    2010

  42. [42]

    Heat kernel asymptotics on symmetric spaces,

    I. G. Avramidi, “Heat kernel asymptotics on symmetric spaces,” 2006

    2006

  43. [43]

    Heat kernel expansion: User’s manual,

    D. V. Vassilevich, “Heat kernel expansion: User’s manual,”Phys. Rept.388(2003) 279–360,hep-th/0306138

    Pith/arXiv arXiv 2003

  44. [44]

    Noncommutative geometry and path integrals,

    M. Kapranov, “Noncommutative geometry and path integrals,”math/0612411

    Pith/arXiv arXiv

  45. [45]

    Woodhouse,Geometric Quantization

    N. Woodhouse,Geometric Quantization. Oxford mathematical monographs. Clarendon Press, 1997

    1997

  46. [46]

    Nonabelian Bosonization in Two-Dimensions,

    E. Witten, “Nonabelian Bosonization in Two-Dimensions,”Commun. Math. Phys.92 (1984) 455–472

    1984

  47. [47]

    An explicit *-product on the cotangent bundle of a lie group,

    S. Gutt, “An explicit *-product on the cotangent bundle of a lie group,”Letters in Mathematical Physics7(1983) 249–258

    1983

  48. [48]

    Kontsevich star-product on the dual of a Lie algebra,

    G. Dito, “Kontsevich star-product on the dual of a Lie algebra,”Lett. Math. Phys.48 (1999) 307–322,math/9905080

    Pith/arXiv arXiv 1999

  49. [49]

    Kontsevich’s universal formula for deformation quantization and the Campbell–Baker–Hausdorff formula,

    V. Kathotia, “Kontsevich’s universal formula for deformation quantization and the Campbell–Baker–Hausdorff formula,”International Journal of Mathematics11(2000), no. 04, 523–551,math/9811174. 40

    Pith/arXiv arXiv 2000

  50. [50]

    Integration in functional spaces and it applications in quantum physics,

    I. M. Gelfand and A. M. Yaglom, “Integration in functional spaces and it applications in quantum physics,”J. Math. Phys.1(1960) 48

    1960

  51. [51]

    Functional determinants and geometry,

    R. Forman, “Functional determinants and geometry,”Invent. Math.88(1987), no. 3, 447–493

    1987

  52. [52]

    The semiclassical expansion,

    C. DeWitt-Morette, “The semiclassical expansion,”Annals of Physics97(1976), no. 2, 367–399

    1976

  53. [53]

    Curvatures of left invariant metrics on lie groups,

    J. Milnor, “Curvatures of left invariant metrics on lie groups,”Advances in Mathematics 21(1976), no. 3, 293–329

    1976

  54. [54]

    The Peter-Weyl Theorem for Compact Groups,

    D. P. Williams, “The Peter-Weyl Theorem for Compact Groups,”Lecture notes at Dartmouth College(1991)

    1991

  55. [55]

    Oblak,BMS Particles in Three Dimensions

    B. Oblak,BMS Particles in Three Dimensions. Springer Theses, 2017. 41

    2017