Notes on Higher-Spin Diffeomorphisms
2. Higher-Spin Gauge Symmetries in the Unconstrained Metric-like Formulation
2.1. Non-Abelian Deformations of Higher-Spin Gauge Symmetries
2.2. Two Examples of Infinitesimal Higher-Spin Gauge Symmetries
2.3. Problems with Higher-Spin Diffeomorphisms
2.4. Notation and Terminology
2.5. No-Go Theorems
3. Symplectomorphisms of the Cotangent Bundle
3.1. Lagrangian Submanifolds
3.2. Flows of Hamiltonians of Degree Zero
3.3. Lagrangian Foliations
3.4. Flows of Hamiltonians of Degree One
- preserves the tautological one-form,
- is the lift of a flow on the base manifold M generated by a base vector field, ,
- is generated by a homogenous Hamiltonian of degree one in the momenta, .
- preserves the vertical polarisation, i.e., it maps cotangent spaces to cotangent spaces ,
- is the composition of a vertical symplectomorphism and the lift of a diffeomorphism of the base M,
- is generated by a Hamiltonian of degree one in the momenta, .
3.5. Flows of Hamiltonians of Higher Degree
4. Quantisation of the Cotangent Bundle: Differential Operators as Symbols and Vice Versa
4.1. Quantisation of the Cotangent Bundle
- a collection of isomorphisms of vector spaces, and
- an isomorphism of Schouten algebras between and the Poisson limit of ,
4.2. Compatibility Condition
- it is an algebra morphism, i.e., it relates the product ★ in to the product ∘ in one has:In particular, an anchor (39) defines a representation of the -ring on its base algebra .
4.3. Examples of Quantisation of the Cotangent Bundle
5. Quantisation of the Cotangent Bundle: Going beyond Differential Operators
5.1. Quasi-Differential Operators
5.2. Criteria on the Strict Product
5.3. Strict Higher-Spin Diffeomorphisms
- A strict higher-spin diffeomorphism induces a standard diffeomorphism ,
- Its restriction to the subalgebra of differential operators of order zero is an isomorphism of commutative algebras,
- Its restriction to the whole subalgebra of differential operators is an isomorphism of associative algebras.
5.4. Formal Completion
6. Almost Differential Operators
6.1. Formal Power Series over an Algebra
6.2. Almost-Differential Operators
7. Deformation Quantisation: Sample of Results
7.1. Star Products
7.2. Equivalences of Star Products and Automorphisms of Deformations
8. Quasi Differential Operators
8.1. Star Products of Symbols of Differential Operators
8.2. Formal Quasi-Differential Operators
- Is differential, and
- Reduces to the star product on .
9. Higher-Spin Diffeomorphisms
9.1. Looking for Higher-Spin Diffeomorphisms
9.2. Formal Higher-Spin Diffeomorphisms
- symplectomorphisms from the cotangent bundle to the cotangent bundle ,
- classical limit of formal higher-spin diffeomorphisms between the manifolds M and ,
- equivalence classes of ℏ -linear isomorphisms between the associative algebras and of formal quasi-differential operators with respect to the equivalence of star products,
- ℏ -linear isomorphisms between the formal algebras and of quasi-differential operators.
9.3. Formal Higher-Spin Flows
10. Quotient Algebra of Almost-Differential Operators
10.2. Quotient Algebra
Conflicts of Interest
Appendix A. Proof of Technical Lemma
A distinct but related issue is the degree of (non)locality of higher-spin interactions. It has been a subject of intense scrutiny and debate over the last years. This open problem will not be adressed here, although one may hope that the unavoidable non-locality of finite higher-spin gauge symmetries might shed some light on this issue in the future.
This difficulty was recognised immediately by Fronsdal, despite he actually found such a non-abelian deformation together with a Lie bracket over the space of traceless symmetric tensor fields .
In spacetime dimension four, it is well-known [7,8,9,10,11] that these trace constraints can be taken into account in the spinorial frame-like formulation (where the fields are base one-forms and fibre tensor-spinors). At nonlinear level, the corresponding gauged algebra of infinitesimal symmetries is the Weyl algebra of polynomial differential operators on the tangent space [1,2,3,4,5,6]. No-go theorems analogous to [29,30] have been established for the Weyl algebra [34,35].
Let us repeat that, in the present paper, the focus is on finite gauge symmetries in the unconstrained metric-like formulation for technical simplicity. It is natural to expect that our main conclusions should apply to the original constrained metric-like formulation of Fronsdal without any qualitative change.
The prescription for (9) is as follows: one consider a covariantised Weyl map where the operators are obtained from their symbols (3) via (i) the quantisation rule and (ii) the anticommutator-ordering prescription with respect to the covariant derivative
Let us stress that the words “quantum”, “quantisation”, etc, throughout this paper should be taken in a mathematical technical sense, not in a physical literal sense. In deformation quantisation, “quantum” is synonymous of “associative” while “classical” is synonymous of “Poisson”.
For the sake of simplicity, reality conditions will not be discussed in this paper. All algebras considered here will be complex, of which suitable reals forms (e.g., of Hermitian operators) can be extracted if necessary. From now on, factors of i will be dropped from all formulae.
Note that the so-called “Lie’s third theorem” stating that every finite-dimensional Lie algebra over the real numbers is associated to a Lie group G does not hold in general for infinite-dimensional Lie algebras.
Moreover, if the algebra has a unit element, then one further requires that . Here, all associative algebras are assumed to have a unit element and, accordingly, morphisms of associative algebras relate their unit elements.
Such a Poisson algebra was called a “classical Poisson algebra” in . Since this terminology may be confusing to people familiar with the vocabulary in deformation quantisation (where Poisson algebras are, by definition, classical), one chose to call them Schouten algebras (as a tribute to the Schouten bracket of symmetric multivector fields). Note that a Gerstenhaber algebras are the supercommutative analogues of Schouten algebras.
There would have been a way out if the higher-spin Lie derivative was locally nilpotent. However, it has been shown (cf. Section 3 of ) that is locally nilpotent (more precisely: for any zeroth-order differential operator , there exists a positive integer n such that ) iff is a differential operator of order zero. This is the crucial technical lemma behind the no-go theorems.
The Stone?Weierstrass theorem ensures that is dense inside .
If the quantised Schouten algebra happens to be equal to the Poisson limit of the almost-commutative algebra (i.e., ) then one also requires that the restriction of the quantisation is a section of the restriction of the principal symbol map ().
Note that the injectivity can be assumed without loss of generality, in the sense that one can always focus on the quotient algebra . The terminology originates from the fact that, in particular, the unit map relates the two units in the sense that ).
This terminology is borrowed from  where the anchor is defined for bialgebroids (an extra compatibility condition with the coproduct is added).
This quantisation is not canonical since, by construction, it depends explicitly on a choice of specific coordinate system. Nevertheless, this normal-type quantisation can be made geometrical (i.e., globally well-defined and coordinate-independent) for a generic manifold M by considering the following data: an affine connection on the base manifold M (cf. ). Retrospectively, the corresponding normal coordinates provide a privileged coordinate system.
Usually, the term “strict quantisation” refers to one of the (many) mathematical approaches to the problem of quantisation and often refers to the axiomatisation by Rieffel (see, e.g., their book ). Here, the term is understood in a non-technical sense.
Since the idea behind pseudo-differential operators is that they behave asymptotically like differential operators (whose “order” can be any real number), they should face the same problem that was encountered for differential operators in Section 2.7.
This can be shown as follows: First, the relation is true by the very definition of the map (32). Second, the quantisation map is assumed to be an algebra morphism, hence . Third, the quantisation map is assumed compatible, thus . This ends the proof. In particular, for functions X on the cotangent bundle which are polynomial in the momenta, the relation (63) reproduces the action on functions f of differential operators with symbol X.
In the applications to higher-spin gravity, this formal parameter should tentatively be identified with the parameter ℓ with the dimension of a length, mentioned below Equation (9). However, for the general mathematical considerations of Section 6, Section 7, Section 8 and Section 10, it is a purely formal deformation parameter.
One speaks of strict deformations if the series are convergent for , plus some extra technical assumptions, cf. the celebrated definition by Rieffel .
This is true globally if the second Betti number of vanishes.
This remains true globally if the first Betti number of vanishes.
This is a slight abuse of terminology since, strictly speaking, they are not inner derivations. In fact, by definition is an inner derivation, but is not (since ).
As shown in [57,59], the analogue of the Gelfand–Naimark–Segal construction applies for such formal deformations . This establishes on firm ground the quantum mechanical interpretation of these elements as operators acting on a Hilbert space. This is not explored here but could be important in the future for looking for a suitable real form of the algebra of formal quasi-differential operators considered here.
For the sake of completeness, note that the corresponding star products for a non-canonical symplectic two-form on the cotangent bundle have been studied in .
This last property holds because the star product is natural. Detailed statements about the quantum correction to the Heisenberg-picture time evolution can be found in Appendix B of .
For instance, if the base manifold is the Euclidean space , then the function is such that its Taylor series along the direction of momenta vanishes on the zero section .
This can be checked explicitly for the normal star product in Darboux coordinates, via the Formula (115). Nevertheless, the conclusion is coordinate-free and remains valid for any equivalent differential star product.
From the point of view of higher-spin gravity, the status of such a strong condition on higher-spin diffeomorphisms is unclear since it would remove the Maxwell gauge symmetries of the spin-one sector (since they correspond to vertical automorphisms of the cotangent bundle that do not preserve the zero section). Nevertheless, they appear as a reasonable candidate subclass of higher-spin symmetries which could be compatible with some weak notion of locality.
- Vasiliev, M.A. Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions. Int. J. Mod. Phys. D 1996, 5, 763–797. [Google Scholar] [CrossRef][Green Version]
- Vasiliev, M.A. Higher spin gauge theories in various dimensions. Fortschritte Phys. Prog. Phys. 2004, 52, 702–717. [Google Scholar] [CrossRef][Green Version]
- Vasiliev, M.A. Higher spin gauge theories in any dimension. Comptes Rendus Phys. 2004, 5, 1101–1109. [Google Scholar] [CrossRef][Green Version]
- Bekaert, X.; Cnockaert, S.; Iazeolla, C.; Vasiliev, M.A. Nonlinear higher spin theories in various dimensions. arXiv 2005, arXiv:hep-th/0503128. [Google Scholar]
- Didenko, V.E.; Skvortsov, E.D. Elements of Vasiliev theory. arXiv 2014, arXiv:1401.2975. [Google Scholar]
- Vasiliev, M.A. Higher-spin theory and space-time metamorphoses. In Modifications of Einstein’s Theory of Gravity at Large Distances; Springer: Cham, Switzerland, 2015; pp. 227–264. [Google Scholar]
- Sorokin, D. Introduction to the classical theory of higher spins. AIP Conf. Proc. 2005, 767, 172. [Google Scholar]
- Bekaert, X.; Boulanger, N.; Sundell, P. How higher-spin gravity surpasses the spin two barrier: No-go theorems versus yes-go examples. Rev. Mod. Phys. 2012, 84, 987. [Google Scholar] [CrossRef][Green Version]
- Rahman, R. Higher Spin Theory—Part I. arXiv 2013, arXiv:1307.3199. [Google Scholar]
- Rahman, R.; Taronna, M. From Higher Spins to Strings: A Primer. arXiv 2015, arXiv:1512.07932. [Google Scholar]
- Bengtsson, A. Higher Spin Field Theory (Concepts, Methods and History) Volume 1: Free Theory; De Gruyter: Berlin, Germany, 2020. [Google Scholar]
- Argurio, R.; Barnich, G.; Bonelli, G.; Grigoriev, M. (Eds.) Higher Spin Gauge Theories; International Solvay Institutes: Brussels, Belgium, 2004. [Google Scholar]
- Brink, L.; Henneaux, M.; Vasiliev, M.A. (Eds.) Higher Spin Gauge Theories; World Scientific: Singapore, 2017. [Google Scholar]
- Segal, A.Y. Conformal higher spin theory. Nucl. Phys. B 2003, 664, 59. [Google Scholar] [CrossRef][Green Version]
- Bekaert, X.; Joung, E.; Mourad, J. On higher spin interactions with matter. JHEP 2009, 5, 126. [Google Scholar] [CrossRef]
- Bekaert, X.; Meunier, E. Higher spin interactions with scalar matter on constant curvature spacetimes: Conserved current and cubic coupling generating functions. JHEP 2010, 11, 116. [Google Scholar] [CrossRef][Green Version]
- Bonora, L.; Cvitan, M.; Prester, P.D.; Giaccari, S.; Paulišić, M.; Štemberga, T. Worldline quantization of field theory, effective actions and L∞ structure. JHEP 2018, 4, 095. [Google Scholar] [CrossRef]
- Cvitan, M.; Prester, P.D.; Giaccari, S.; Paulišić, M.; Vuković, I. Gauging the higher-spin-like symmetries by the Moyal product. JHEP 2021, 6, 144. [Google Scholar] [CrossRef]
- Fotopoulos, A.; Irges, N.; Petkou, A.C.; Tsulaia, M. Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex. JHEP 2007, 10, 021. [Google Scholar] [CrossRef][Green Version]
- Vasiliev, M.A. Actions, charges and off-shell fields in the unfolded dynamics approach. Int. J. Geom. Meth. Mod. Phys. 2006, 3, 37. [Google Scholar] [CrossRef]
- Grigoriev, M. Off-shell gauge fields from BRST quantization. arXiv 2006, arXiv:hep-th/0605089. [Google Scholar]
- Bekaert, X. Comments on higher-spin symmetries. Int. J. Geom. Meth. Mod. Phys. 2009, 6, 285. [Google Scholar] [CrossRef][Green Version]
- Sezgin, E.; Sundell, P. Geometry and Observables in Vasiliev’s Higher Spin Gravity. JHEP 2012, 1207, 121. [Google Scholar] [CrossRef][Green Version]
- Grigoriev, M. Parent formulations, frame-like Lagrangians, and generalized auxiliary fields. JHEP 2012, 12, 048. [Google Scholar] [CrossRef][Green Version]
- Iazeolla, C.; Sezgin, E.; Sundell, P. On Exact Solutions and Perturbative Schemes in Higher Spin Theory. Universe 2018, 4, 5. [Google Scholar] [CrossRef][Green Version]
- Bars, I. Survey of two time physics. Class. Quant. Grav. 2001, 18, 3113. [Google Scholar] [CrossRef][Green Version]
- Bonezzi, R.; Latini, E.; Waldron, A. Gravity, Two Times, Tractors, Weyl Invariance and Six Dimensional Quantum Mechanics. Phys. Rev. D 2010, 82, 064037. [Google Scholar] [CrossRef][Green Version]
- Bekaert, X.; Grigoriev, M.; Skvortsov, E.D. Higher Spin Extension of Fefferman-Graham Construction. Universe 2018, 4, 17. [Google Scholar] [CrossRef][Green Version]
- Grabowski, J.; Poncin, N. Automorphisms of quantum and classical Poisson algebras. Compos. Math. 2004, 140, 511. [Google Scholar] [CrossRef]
- Grabowski, J.; Poncin, N. Derivations of the Lie algebras of differential operators. Indag. Math. 2005, 16, 181. [Google Scholar] [CrossRef][Green Version]
- Fronsdal, C. Massless Fields with Integer Spin. Phys. Rev. D 1978, 18, 3624. [Google Scholar] [CrossRef]
- Fronsdal, C. Singletons and massless, integral-spin fields on de Sitter space. Phys. Rev. D 1979, 20, 848. [Google Scholar] [CrossRef]
- Fronsdal, C. Some open problems with higher spins. In Supergravity; van Nieuwenhuizen, P., Freedman, D.Z., Eds.; North-Holland: Amsterdam, The Netherlands, 1979; p. 245. [Google Scholar]
- Kanel-Belov, A.; Kontsevich, M. Automorphisms of Weyl algebras. Lett. Math. Phys. 2005, 74, 181. [Google Scholar] [CrossRef][Green Version]
- Kanel-Belov, A.; Elishev, A.; Yu, J.T. Automorphisms of Weyl Algebra and a Conjecture of Kontsevich. arXiv 2018, arXiv:1802.01225. [Google Scholar]
- Francia, D.; Sagnotti, A. On the geometry of higher spin gauge fields. Class. Quant. Grav. 2003, 20, S473. [Google Scholar] [CrossRef]
- Francia, D.; Sagnotti, A. Higher-spin geometry and string theory. J. Phys. Conf. Ser. 2006, 33, 57. [Google Scholar] [CrossRef]
- Bouatta, N.; Compere, G.; Sagnotti, A. An introduction to free higher-spin fields. arXiv 2004, arXiv:hep-th/0409068. [Google Scholar]
- Francia, D. Low-spin models for higher-spin Lagrangians. Prog. Theor. Phys. Suppl. 2011, 188, 94. [Google Scholar] [CrossRef][Green Version]
- Francia, D. Aspects of metric-like higher-spin geometry. AIP Conf. Proc. 2012, 1483, 118. [Google Scholar]
- Dubois-Violette, M.; Michor, P.W. A Common generalization of the Frohlicher-Nijenhuis bracket and the Schouten bracket for symmetric multivector fields. Indag. Mathem. 1995, 6, 51. [Google Scholar] [CrossRef][Green Version]
- Krasnov, K.; Skvortsov, E.; Tran, T. Actions for Self-dual Higher Spin Gravities. arXiv 2021, arXiv:2105.12782. [Google Scholar] [CrossRef]
- Berezin, F.A. Some remarks about the associated envelope of a Lie algebra. Funct. Anal. Appl. 1967, 1, 91. [Google Scholar] [CrossRef]
- Xu, P. Quantum groupoids. Commun. Math. Phys. 2001, 216, 539. [Google Scholar] [CrossRef][Green Version]
- Bordemann, M.; Neumaier, N.; Waldmann, S. Homogeneous Fedosov star products on cotangent bundles: I. Weyl and standard ordering with differential operator representation. Commun. Math. Phys. 1998, 198, 363. [Google Scholar] [CrossRef][Green Version]
- Weyl, H. Quantum mechanics and group theory. Z. Phys. 1927, 46, 1. [Google Scholar] [CrossRef]
- Wigner, E.P. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 1932, 40, 749. [Google Scholar] [CrossRef]
- Rieffel, M.A. Deformation Quantization for Actions of Rd; Memoirs of the American Mathematical Society; American Mathematical Society: Providence, RI, USA, 1993; p. 106. [Google Scholar]
- Raymond, X.S. Elementary Introduction to the Theory of Pseudodifferential Operators; CRC Press: Boca Raton, FL, USA, 1991. [Google Scholar]
- Hörmander, L. The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Wong, M.M. An Introduction to Pseudo-Differential Operators; World Scientic: Singapore, 1999. [Google Scholar]
- Soloviev, M.A. Moyal multiplier algebras of the test function spaces of type S. J. Math. Phys. 2011, 52, 063502. [Google Scholar] [CrossRef][Green Version]
- Soloviev, M.A. Star products on symplectic vector spaces: Convergence, representations, and extensions. Theor. Math. Phys. 2014, 181, 1612. [Google Scholar] [CrossRef]
- Soloviev, M.A. Spaces of type S and deformation quantization. Teor. Mat. Fiz. 2019, 201, 315. [Google Scholar] [CrossRef]
- Gutt, S. Deformation Quantization: An Introduction; Lectures Given for 3rd Cycle Students at Monastir Tunisie; HAL: Lyon, France, 2005. [Google Scholar]
- Fedosov, B. Deformation Quantization and Index Theory; Mathematical Topics; Akademie Verlag: Berlin, Germany, 1996; p. 9. [Google Scholar]
- Bordemann, M.; Neumaier, N.; Waldmann, S. Homogeneous Fedosov star products on cotangent bundles: II. GNS representations, the WKB expansion, traces, and applications. J. Geom. Phys. 1999, 29, 199. [Google Scholar] [CrossRef]
- Pflaum, M. The normal symbol on Riemannian manifolds. N. Y. J. Math. 1998, 4, 95. [Google Scholar]
- Bordemann, M.; Neumaier, N.; Pflaum, M.J.; Waldmann, S. On representations of star product algebras over cotangent spaces on Hermitian line bundles. J. Funct. Anal. 2003, 199, 1. [Google Scholar] [CrossRef][Green Version]
- Lichnerowicz, A. Existence and equivalence of twisted products on a symplectic manifold. Lett. Math. Phys. 1979, 3, 495. [Google Scholar] [CrossRef]
- Gutt, S.; Rawnsley, J. Equivalence of star products on a symplectic manifold; an introduction to Deligne’s Cech cohomology classes. J. Geom. Phys. 1999, 29, 347. [Google Scholar] [CrossRef][Green Version]
- Bonfiglioli, A.; Fulci, R. Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2012; p. 2034. [Google Scholar]
|Lie Algebra||Dimension||Exact Symplectomorphisms||Basis of Hamiltonian Vector Fields||Basis of Hamiltonians|
|Linear vertical symplectom|
|Affine vertical symplectom||,||,|
|Lift of linear transformations|
|Lift of affine transformations||,||,|
|Linear changes of polarisation|
|Affine symplectomorphisms||Affine vector fields||degree 2|
|Algebra||Poisson algebra (symplectic)||Associative algebra (central)|
|Elements||Functions on the cotangent bundle||Quasi-differential operators|
|Graded/Filtered||Schouten algebra||Almost-commutative algebra|
|Commutative||Base algebra||Order zero subalgebra|
|Elements||Functions on the base||Differential operators of order zero|
|Mathematical Objects||Classical (Undeformed)||Quantum (Deformed)|
|Un/deformed||Order||Formal power series in ℏ|
|Algebra||Symplectic algebra||Associative algebra|
|Associative algebra of||Endomorphism algebra||Algebra of ℏ-linear endomorphisms|
|Group of||Group of symplectomorphisms||Group of ℏ-linear automorphisms|
|Lie algebra of||Lie algebra of||Lie algebra of ℏ-linear derivations|
|infinitesimal automorphisms||symplectic vector fields|
|Group of finite||Hamiltonian||Flows of non-trivial|
|inner automorphisms||flows||inner automorphisms|
|Lie algebra of infinitesimal||Lie algebra of||Lie algebra of non-trivial|
|inner automorphisms||Hamiltonian vector fields||inner derivations|
|Group of||Trivial group||Normal subgroup of pert. redefs|
|finite redefinitions||(identity map)|
|Lie algebra of||Trivial algebra||Ideal of infinitesimal pert. redefs|
|infinitesimal redefinitions||(zero map)|
|Group of||Trivial group||Group of star-product|
|Lie algebra of||Trivial algebra||Lie algebra of|
|infinitesimal self-equivalences||(zero map)||trivial derivations|
|Algebra||Poisson algebra (symplectic)||Associative algebra (central)|
|Elements||Functions on the cotangent bundle||Quasi-differential operators|
|automorphisms||of||diffeomorphisms of M|
|Flow of inner||Hamiltonian flow||Higher-spin flow|
|Infinitesimal||Symplectic vector field||Infinitesimal higher-spin|
|automorphism||on||diffeomorphism of M|
|derivation||field on||Lie derivative on M|
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Bekaert, X. Notes on Higher-Spin Diffeomorphisms. Universe 2021, 7, 508. https://doi.org/10.3390/universe7120508
Bekaert X. Notes on Higher-Spin Diffeomorphisms. Universe. 2021; 7(12):508. https://doi.org/10.3390/universe7120508Chicago/Turabian Style
Bekaert, Xavier. 2021. "Notes on Higher-Spin Diffeomorphisms" Universe 7, no. 12: 508. https://doi.org/10.3390/universe7120508