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Deformation quantization as an appropriate guide to ontic structure

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Abstract

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation quantization procedure will be offered.

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Notes

  1. In the literature on OSR, ‘rather than’ might have different meanings. According to radical OSR (French and Ladyman 2003), for instance, there are no objects. According to its moderate version (Esfeld and Lam 2008), physical objects exist but characterized only by structures constituting the world. Our discussion is independent of these considerations.

  2. By ‘midway through a transition’ I do not mean during a historical period of theory change, but within a theoretical reconstruction of theory change.

  3. Different mathematical notions may fix the meaning of the compatibility, e.g. unitary transformation, isomorphism, diffeomorphism, etc. The context of use and the way in which these structures relate to the empirical content of theories determine the proper mathematical notion.

  4. A formulation underdetermintation does not necessarily lead to an ontological underdetermination. For a range of possibilities in which the former is tied to the latter, see Thébault (2014, p. 93). The underdetermination between Lagrangian and Hamiltonian formulations of CM is weak, meaning that the teleological picture of the world is a more natural interpretation of the latter, while the former naturally involves the instantaneous picture of the world (Thébault 2014, pp.100–103).

  5. A formulation of the theory T, \( T_F \), is generally covariant if and only if “the equations expressing its laws are written in a form that holds with respect to all members of a set of coordinate systems that are related by smooth but otherwise arbitrary transformations” (Pooley 2017, p. 115). To define diffeomorphism invariance, consider the theory T, its formulation \( T_F \) and its model \( <M, F, D> \), where M is a given manifold, F is a fixed field common to all kinematical models and D is a dynamical field which varies from model to model. \( T_F \) is diffeomorphically invariant if and only if, if \( <M, F, D> \) is a model of \( T_F \), then so is \( <M, F, d^*D> \), where \( d^* \) is the pull-back induced by the diffeomorphism d. The theory T might be called fully diffeomorphism invariant if it has no non-diffeomorphically invariant \( T_F \) (Pooley 2017, p. 117); the feature distinguishing the general theory of relativity.

  6. Provided that each formulation induces its own ontology.

  7. We shall see this sort of possibility in GQ.

  8. I owe the dichotomy between synchronic and diachronic SOU to an anonymous referee.

  9. In GQ, we will find out that the common structures are not equivalent. Therefore, GQ violates the consistency principle.

  10. The symplectic geometry is a theoretically useful framework for formulating CM, since it is a language in which Lagrangian and Hamiltonian mechanics are not only expressed, but are translated (not necessarily as equivalent formulations) into each other (Woodhouse 1997, Ch 2). Furthermore, as Belot (2007) has argued, the symplectic structure which Lagrangian and Hamiltonian mechanics both have is the prerequisite structure with which one can start the quantization process. Following mathematical-physicists (Abraham and Marsden 1978; Souriau 2012; Arnol’d 2013; Marsden and Ratiu 2013), philosophers of physics to understand the nature and the structure of CM in general (Butterfield 2007; Thébault 2012; Curiel 2013; Barrett 2014) and the transition from CM to QM in particular (Ruetsche 2011; Thébault 2014) are already familiar with the symplectic formulation of CM.

  11. For different methods of quantization, see Ali and Engliš (2005).

  12. For merits and demerits of different methods of quantization from a scientific point of view, see Ali and Engliš (2005).

  13. Introduced by Kostant (1970) and Souriau (1970), GQ has been extensively discussed in mathematical-physics textbooks (Guillemin and Sternberg 1990; Bates and Weinstein 1997; Woodhouse 1997; Puta 2012; Hall 2013) and briefly in philosophical literature (Landsman 2007; Thébault 2014).

  14. For the definitions, see “Appendix A”.

  15. A connection over L is \( \nabla :C^{\infty }_B(M) \rightarrow \varOmega ^1_B(M) \) such that \( \nabla (s_1 + s_2)=\nabla s_1 + \nabla s_2 \) and \( \nabla (\psi s)= (d\psi )s + \psi \nabla s \), where \( C^{\infty }_B(M) \) is the set of sections of B, \( \varOmega ^1_B(M) \) is the set of B valued 1-form on M and \( \psi \in C^{\infty }(M) \).

  16. For the notations, see “Appendix A”.

  17. The Lie derivative of \( \nu \) in the direction of \( X_f \), \( {\mathcal {L}}_{X_f}\nu \), may be expressed, in terms of the exterior derivative and interior multiplication, as \( {\mathcal {L}}_{X_f}\nu = i_X \circ d + d \circ i_X \).

  18. I thank two anonymous referees for pointing out these two worries to me.

  19. In the last section, I will discuss more about the theorem. For more details about the assumptions on which the theorem is based, see Ruetsche (2011, Ch. 2).

  20. GQ in linear spaces has its own difficulties (Ali and Engliš 2005, pp. 447–448).

  21. To characterize the problem of unitarily inequivalent representations in quantum field theory and quantum statistical mechanics - which I will discuss shortly - David Glick (2016, p. 5) has recognized two sorts of worries: “an underdetermination worry and a lack of common representation worry”. The first worry is that there are inequivalent structures to represent a single target system. The second is that while each inequivalent structure represents its own target system, different target systems are connected somehow. Hence one faces the question of which structure represents the connecting link. So construed, both worries are tightly tied to SOU.

  22. For some examples, see Kirwin et al. (2014) and the references in it.

  23. The origins of DQ go back to the works of the trailblazers of QM (Weyl 1927; Wigner 1932; von Neumann and Beyer 1955; Dirac 1966), but as an autonomous framework for quantization, it was proposed by Bayen et al. (1978a, b). For an accessible introduction to DQ, see Hirshfeld and Henselder (2002). For a detailed discussion, see Esposito (2014); Laurent-Gengoux et al. (2012). DQ approach pursued here is called formal deformation quantization. Strict deformation quantization suggested by Rieffel (1994) is formulated in terms of C*-algebras. For details about this latter flavor of DQ, see Landsman (2007).

  24. By ‘algebraic’ state I mean a functional which is defined, with appropriate qualities, on a \( C^* \)-algebra or \( ^* \)-algebra. By ‘physical’ state I mean an operator which is defined, with certain properties, on the representations of these algebras. For more on these, see footnote 26, Ruetsche (2011, Ch. 4) and Waldmann (2005).

  25. According to the algebraic form of canonical quantization, the Weyl relations are characterized by the map:

    $$\begin{aligned} \begin{aligned}&W: C^{\infty }(M) \rightarrow \mathcal {U(H)}\\&W(f)W(g)=exp\left( \frac{i}{2} \omega (X_f,X_g)\right) W(f+g) \end{aligned} \end{aligned}$$

    where \(\mathcal {U(H)}\) is the set of unitary operators acting on the Hilbert space \({\mathcal {H}}\). By setting \({\mathcal {U}}_W({\mathcal {H}})=W[C^{\infty }(M)]\), the structure \(({\mathcal {H}}, {\mathcal {U}}_W({\mathcal {H}}))\), called a representation of the Weyl relations, represents the corresponding quantum system. Two representations \(({\mathcal {H}}, {\mathcal {U}}_W({\mathcal {H}}))\) and \((\mathcal {H'}, \mathcal {U'}_{W'}(\mathcal {H'}))\) are unitarily equivalent if there exists a transformation \(T:{\mathcal {H}}\rightarrow \mathcal {H'}\) such that for every \(f\in C^{\infty }(M)\), \(T^{-1}W'(f)T=W(f)\). According to the Stone-von Neumann theorem, for any linear, finite dimensional symplectic manifold, all weakly continuous irreducible representations of Weyl relations are unitarily equivalent. For more details, see Ruetsche (Ruetsche (2011), Ch. 2).

  26. Let \( {\mathfrak {U}} \) be a \( C^* \)-algebra. A pair \( ({\mathcal {H}}, \pi ) \) is called a representation of \( {\mathfrak {U}} \) where \( {\mathcal {H}} \) is a Hilbert space and \( \pi \) is a \( * \)-homomorphism of \( {\mathfrak {U}} \) into the set of bounded operators acting on \( {\mathcal {H}} \), \( {\mathfrak {B}}({\mathcal {H}}) \). According to GNS theorem, for every \( {\mathfrak {U}} \) with algebraic state \( \omega \), there is a representation \( ({\mathcal {H}}_\omega , \pi _\omega ) \) and a cyclic vector \( |\xi _\omega \rangle \in {\mathcal {H}}_\omega \) such that for all \( A \in {\mathfrak {U}} \), \( \omega (A)= \langle {\xi _\omega }|{\pi _\omega (A)}|{\xi _\omega }\rangle \). \( ({\mathcal {H}}_\omega , \pi _\omega , |\xi _\omega \rangle ) \) is called a GNS construction or representation. For more on this, see Ruetsche (Ruetsche (2011), Ch. 4).

  27. Thanks to an anonymous referee for highlighting this issue and suggesting this reference to me.

  28. To do so, there are two different approaches. While the first approach uses non-regular, distributional potentials (Dias and Prata 2002, 2007; Dias et al. 2011), the second one utilizes regular potentials (Kryukov and Walton 2005, 2006; Belchev and Walton 2010). For their relation, see Dias and Prata (2006).

References

  • Abraham, R., & Marsden, J. (1978). Foundations of mechanics. Mathematical physics monograph series. San Francisco: Benjamin/Cummings Publishing Company.

    Google Scholar 

  • Ali, S. T., & Engliš, M. (2005). Quantization methods: a guide for physicists and analysts. Reviews in Mathematical Physics, 17(04), 391–490.

    Article  Google Scholar 

  • Arnol’d, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Berlin: Springer.

    Google Scholar 

  • Barrett, T. W. (2014). On the structure of classical mechanics. The British Journal for the Philosophy of Science, 66(4), 801–828.

    Article  Google Scholar 

  • Bates, S., & Weinstein, A. (1997). Lectures on the geometry of quantization (Vol. 8). Providence: American Mathematical Society.

    Google Scholar 

  • Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., & Sternheimer, D. (1978a). Deformation theory and quantization. I. deformations of symplectic structures. Annals of Physics, 111(1), 61–110.

    Article  Google Scholar 

  • Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., & Sternheimer, D. (1978b). Deformation theory and quantization. II. physical applications. Annals of Physics, 111(1), 111–151.

    Article  Google Scholar 

  • Belchev, B., & Walton, M. (2010). On robin boundary conditions and the morse potential in quantum mechanics. Journal of Physics A: Mathematical and Theoretical, 43(8), 085301.

    Article  Google Scholar 

  • Belot, G. (2007). The representation of time and change in mechanics. In Butterfield, J., & Earman, J. (Eds.), Philosophy of Physics, Handbook of the Philosophy of Science, pp. 133 – 227. North-Holland.

  • Bonneau, G., Faraut, J., & Valent, G. (2001). Self-adjoint extensions of operators and the teaching of quantum mechanics. American Journal of physics, 69(3), 322–331.

    Article  Google Scholar 

  • Bordemann, M., & Waldmann, S. (1998). Formal GNS construction and states in deformation quantization. Communications in Mathematical Physics, 195(3), 549–583.

    Article  Google Scholar 

  • Butterfield, J. (2007). On symplectic reduction in classical mechanics. In J. Butterfield & J. Earman (Eds.), Philosophy of physics, handbook of the philosophy of science (pp. 1–131). Amsterdam: North-Holland.

    Google Scholar 

  • Capri, A. Z. (1977). Self-adjointness and spontaneously broken symmetry. American Journal of Physics, 45(9), 823–825.

    Article  Google Scholar 

  • Cattaneo, A. S., & Felder, G. (2000). A path integral approach to the kontsevich quantization formula. Communications in Mathematical Physics, 212(3), 591–611.

    Article  Google Scholar 

  • Curiel, E. (2013). Classical mechanics is lagrangian; it is not hamiltonian. The British Journal for the Philosophy of Science, 65(2), 269–321.

    Article  Google Scholar 

  • Da Silva, A. C. (2001). Lectures on symplectic geometry (Vol. 3575). Berlin: Springer.

    Google Scholar 

  • Dias, N. C., Posilicano, A., & Prata, J. N. (2011). Self-adjoint, globally defined hamiltonian operators for systems with boundaries. Communications on Pure and Applied Analysis, 10(6), 1687–1706.

    Article  Google Scholar 

  • Dias, N. C., & Prata, J. N. (2002). Wigner functions with boundaries. Journal of Mathematical Physics, 43(10), 4602–4627.

    Article  Google Scholar 

  • Dias, N. C., & Prata, J. N. (2006). Comment on on infinite walls in deformation quantization. Annals of Physics, 321(2), 495–502.

    Article  Google Scholar 

  • Dias, N. C., & Prata, J. N. (2007). Deformation quantization of confined systems. International Journal of Quantum Information, 5(01n02), 257–263.

    Article  Google Scholar 

  • Dirac, P. (1966). Lectures on Quantum Mechanics. Monograph series. Belfer Graduate School of Science, Yeshiva University.

  • Dito, G. (2015). The necessity of wheels in universal quantization formulas. Communications in Mathematical Physics, 338(2), 523–532.

    Article  Google Scholar 

  • Esfeld, M., & Lam, V. (2008). Moderate structural realism about space-time. Synthese, 160(1), 27–46.

    Article  Google Scholar 

  • Esposito, C. (2014). Formality theory: From poisson structures to deformation quantization (Vol. 2). Berlin: Springer.

    Google Scholar 

  • Fedosov, B. (2002). Deformation quantization: Pro and contra. Contemporary Mathematics, 315, 1–8.

    Article  Google Scholar 

  • French, S. (2011). Metaphysical underdetermination: Why worry? Synthese, 180(2), 205–221.

    Article  Google Scholar 

  • French, S. (2012). Unitary inequivalence as a problem for structural realism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 43(2), 121–136.

    Article  Google Scholar 

  • French, S. (2014). The structure of the world: Metaphysics and representation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • French, S., & Ladyman, J. (2003). Remodelling structural realism: Quantum physics and the metaphysics of structure. Synthese, 136(1), 31–56.

    Article  Google Scholar 

  • Ginzburg, V. L., & Montgomery, R. (2000). Geometric quantization and no-go theorems. Banach Center Publications, 51(1), 69–77.

    Google Scholar 

  • Glick, D. (2016). The ontology of quantum field theory: Structural realism vindicated? Studies in History and Philosophy of Science Part A, 59, 78–86.

    Article  Google Scholar 

  • Glymour, C. (2013). Theoretical equivalence and the semantic view of theories. Philosophy of Science, 80(2), 286–297.

    Article  Google Scholar 

  • Gotay, M. J. (1980). Functorial geometric quantization and van hove’s theorem. International Journal of Theoretical Physics, 19(2), 139–161.

    Article  Google Scholar 

  • Gotay, M. J. (1999). On the Groenewold-van Hove problem for \( R^{2n} \). Journal of Mathematical Physics, 40(4), 2107–2116.

    Article  Google Scholar 

  • Gotay, M. J., Grabowski, J., & Grundling, H. (2000). An obstruction to quantizing compact symplectic manifolds. Proceedings of the American Mathematical Society, 128(1), 237–243.

    Article  Google Scholar 

  • Gotay, M. J., Grundling, H., & Hurst, C. (1996). A Groenewold-van Hove theorem for \(S^2\). Transactions of the American Mathematical Society, 348(4), 1579–1597.

    Article  Google Scholar 

  • Gracia-Saz, A. (2003). On the kontsevich formula for deformation quantization.

  • Groenewold, H. J. (1946). On the principles of elementary quantum mechanics. In On the Principles of Elementary Quantum Mechanics, pp. 1–56. Springer.

  • Guillemin, V., & Sternberg, S. (1990). Symplectic techniques in physics. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hall, B. C. (2013). Quantum theory for mathematicians (Vol. 267). Berlin: Springer.

    Book  Google Scholar 

  • Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science, 79(2), 183–206.

    Article  Google Scholar 

  • Hirshfeld, A. C., & Henselder, P. (2002). Deformation quantization in the teaching of quantum mechanics. American Journal of Physics, 70(5), 537–547.

    Article  Google Scholar 

  • Isham, C. J. (1984). Topological and global aspects of quantum theory. In B. DeWitt & R. Stora (Eds.), Relativity, groups and topology. 2 (pp. 1059–1290). Amsterdam: North-Holland.

    Google Scholar 

  • Kirwin, W. D., Mourão, J. M., & Nunes, J. P. (2014). Coherent state transforms and the mackey-stone-von neumann theorem. Journal of Mathematical Physics, 55(10), 102101.

    Article  Google Scholar 

  • Kontsevich, M. (1997). Deformation quantization of poisson manifolds, i. arXiv:q-alg/9709040.

  • Kostant, B. (1970). Quantization and unitary representations. In C. T. Taam (Ed.), Lectures in modern analysis and applications III (pp. 87–208). Berlin: Springer.

    Chapter  Google Scholar 

  • Kryukov, S., & Walton, M. (2005). On infinite walls in deformation quantization. Annals of Physics, 317(2), 474–491.

    Article  Google Scholar 

  • Kryukov, S., & Walton, M. A. (2006). Star-quantization of an infinite wall. Canadian Journal of Physics, 84(6–7), 557–563.

    Article  Google Scholar 

  • Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science Part A, 29(3), 409–424.

    Article  Google Scholar 

  • Ladyman, J., Ross, D., Collier, J., & Spurrett, D. (2007). Every thing must go: Metaphysics naturalized. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Landsman, N. P. (2007). Between classical and quantum. In J. Butterfield & J. Earman (Eds.), Philosophy of Physics, Handbook of the Philosophy of Science (pp. 417–553). Amsterdam: North-Holland.

    Google Scholar 

  • Laurent-Gengoux, C., Pichereau, A., & Vanhaecke, P. (2012). Poisson structures (Vol. 347). Berlin: Springer.

    Google Scholar 

  • Lee, J. (2013). Introduction to smooth manifolds, graduate texts in mathematics. New York: Springer.

    Google Scholar 

  • Lutz, S. (2017). What was the syntax-semantics debate in the philosophy of science about? Philosophy and Phenomenological Research, 95(2), 319–352.

    Article  Google Scholar 

  • Marsden, J. E., & Ratiu, T. S. (2013). Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems (Vol. 17). Berlin: Springer.

    Google Scholar 

  • Nguyen, J. (2017). Scientific representation and theoretical equivalence. Philosophy of Science, 84(5), 982–995.

    Article  Google Scholar 

  • North, J. (2009). The “structure” of physics: A case study. The Journal of Philosophy, 106(2), 57–88.

    Article  Google Scholar 

  • Onofri, E. (1976). Dynamical quantization of the kepler manifold. Journal of Mathematical Physics, 17(3), 401–408.

    Article  Google Scholar 

  • Pooley, O. (2006). Points, particles, and structural realism. In D. Rickles, S. French, & J. T. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 83–120). Oxford: Oxford University Press.

    Chapter  Google Scholar 

  • Pooley, O. (2017). Background independence, diffeomorphism invariance and the meaning of coordinates. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 105–143). New York: Springer.

    Chapter  Google Scholar 

  • Puta, M. (2012). Hamiltonian mechanical systems and geometric quantization (Vol. 260). Berlin: Springer.

    Google Scholar 

  • Rawnsley, J. (1979). A nonunitary pairing of polarizations for the kepler problem. Transactions of the American Mathematical Society, 250, 167–180.

    Article  Google Scholar 

  • Rieffel, M. A. (1994). Quantization and C*-algebras. Contemporary Mathematics, 167, 67–67.

    Google Scholar 

  • Ruetsche, L. (2011). Interpreting quantum theories. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Souriau, J.-M. (1970). Structure des Systèmes Dynamiques. Paris: Dunod.

    Google Scholar 

  • Souriau, J.-M. (2012). Structure of dynamical systems: A symplectic view of physics (Vol. 149). Berlin: Springer.

    Google Scholar 

  • Thébault, K. P. (2012). Symplectic reduction and the problem of time in nonrelativistic mechanics. The British Journal for the Philosophy of Science, 63(4), 789–824.

    Article  Google Scholar 

  • Thébault, K. P. (2014). Quantization as a guide to ontic structure. The British Journal for the Philosophy of Science, 67(1), 89–114.

    Article  Google Scholar 

  • Tu, L. (2010). An introduction to manifolds. Universitext. New York: Springer.

    Google Scholar 

  • Tuynman, G. (1988). Studies in Geometric Quantization. Amsterdam: Universiteit van Amsterdam.

    Google Scholar 

  • Van Fraassen, B. C. (2014). One or two gentle remarks about hans halvorson’s critique of the semantic view. Philosophy of Science, 81(2), 276–283.

    Article  Google Scholar 

  • von Neumann, J., & Beyer, R. T. (1955). Mathematische Grundlagen der Quantenmechanik. Mathematical Foundations of Quantum Mechanics... Translated... by Robert T. Beyer. Princeton: Princeton University Press.

    Google Scholar 

  • Waldmann, S. (2001). A remark on the deformation of GNS representations of \( ^* \)-algebras. Reports on Mathematical Physics, 48(3), 389–396.

    Article  Google Scholar 

  • Waldmann, S. (2005). States and representations in deformation quantization. Reviews in Mathematical Physics, 17(01), 15–75.

    Article  Google Scholar 

  • Waldmann, S. (2016). Recent developments in deformation quantization. In F. Finster, J. Kleiner, C. Röken, & J. Tolksdorf (Eds.), Quantum mathematical physics: A bridge between mathematics and physics (pp. 421–439). Cham: Springer.

    Chapter  Google Scholar 

  • Weyl, H. (1927). Quantenmechanik und gruppentheorie. Zeitschrift für Physik, 46(1–2), 1–46.

    Article  Google Scholar 

  • Wigner, E. P. (1932). On the quantum correction for thermodynamic equilibrium. Physical Review, 40, 749–759.

    Article  Google Scholar 

  • Willwacher, T. (2014). The obstruction to the existence of a loopless star product. Comptes Rendus Mathematique, 352(11), 881–883.

    Article  Google Scholar 

  • Woodhouse, N. M. J. (1997). Geometric quantization. Oxford: Oxford University Press.

    Google Scholar 

Download references

Acknowledgements

The paper has benefited greatly from valuable suggestions and constructive criticisms of two anonymous referees of this journal; I sincerely thank them. Funding for this research was provided by Shahid Beheshti University (grant number dal/600/4749).

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Appendices

Appendix A: Symplectic Geometry

In the following, I will use standard notations (Tu 2010; Lee 2013) to denote differential geometric notions. In particular, I will denote the set of smooth functions on Manifold M by \( C^{\infty }(M) \), the tangent space to M at p by \( T_pM \), the space of smooth k-vector fields on M by \( {\mathfrak {X}}^k(M) \) and the space of smooth k-forms on M by \(\varOmega ^k(M) \). My exposition of symplectic geometry formulation of CM is essentially extracted from Da Silva (2001), Lee (2013, Ch. 22) and Esposito (2014, Ch. 2).

Symplectic manifold Let M be a smooth manifold. Structure \((M,\omega )\) is symplectic manifold if:

  1. (a)

    M is a smooth manifold;

  2. (b)

    \( \omega :{\mathfrak {X}}^1(M)\times {\mathfrak {X}}^1(M)\rightarrow C^{\infty }(M) \) is a 2-form (\( \omega \in \varOmega ^2(M) \)) such that:

    1. (b.1)

      \( d\omega =0 \), where d is the exterior derivative (\( \omega \) is closed);

    2. (b.2)

      \( \forall p \in M, V_p \in T_p M: \omega _p(W_p,V_p)=0\rightarrow W_P=0 \) (\( \omega \) is non-degenrate).

    \( \omega \) is called the symplectic 2-form.

Hamiltonian vector field Let \( (M,\omega ) \) be a symplectic manifold and H, the Hamiltonian function, a smooth function on M (\( H\in C^{\infty }(M) \)). \( X_H \in {\mathfrak {X}}^1(M) \) is called the Hamiltonian vector field associated with H if \( \omega (X_H,.)=dH \).

Poisson structure (first characterization) Let \( (M,\omega ) \) be a symplectic manifold. \( (C^{\infty }(M),\circ , \left\{ ,\right\} ) \) is called the Poisson Structure if:

  1. (a)

    \( \circ : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M) \) is a communicative multiplication;

  2. (b)

    \( \left\{ ,\right\} : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M) \) is defined as \( (f,g)\mapsto \left\{ f,g\right\} :=\omega (X_f,X_g) \), where \( X_f \) and \( X_g \) are the Hamiltonian vector fields associated with the smooth functions f and g.

Poisson structure (second characterization) Let \( (M,\omega ) \) be a symplectic manifold. \( (C^{\infty }(M),\circ , \alpha ) \) is called the Poisson Structure if:

  1. (a)

    \( \circ : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M) \) is a communicative multiplication;

  2. (b)

    \( \alpha : \varOmega ^1(M)\times \varOmega ^1(M) \rightarrow C^{\infty }(M) \) is a bi-vector field (\( \alpha \in {\mathfrak {X}}^2(M)) \) such that \( \left[ \alpha ,\alpha \right] _S=0 \), which \( \left[ ,\right] _S: {\mathfrak {X}}^k(M)\times {\mathfrak {X}}^l(M)\rightarrow {\mathfrak {X}}^{k+l-1}(M) \) is the Schouten-Nijenhuis bracket defined as:

    $$\begin{aligned} \begin{aligned} \left[ \alpha ,\alpha \right] _S= \alpha ^{ij}(\partial _i\alpha ^{kl})\partial _j\wedge \partial _k\wedge \partial _l \end{aligned} \end{aligned}$$
    (A.1)

where \( \partial ^{ij}\)s are bi-vector coefficients in local coordinates. \( \alpha \) is called the Poisson bi-vector field and relates to the symplectic 2-form \( \omega \) by:

$$\begin{aligned} \begin{aligned} \alpha (df,dg)=\omega (X_f,X_g) \end{aligned} \end{aligned}$$
(A.2)

Hamiltonian system Strcuture \( (M_H, \omega _H, H) \) is called the Hamiltonian system if:

  1. (a)

    \( (M_H,\omega _H) \) is a symplectic manifold such that:

    1. (a.1)

      \( M_H={\mathbb {R}}^{2n} \) with coordinate maps \( q_1,\ldots ,q_n,p_1,\ldots ,p_n \);

    2. (a.2)

      \( \omega _H= \sum \limits _{j=1}^{n} dq^{i}\wedge dp^{j} \);

  2. (b)

    \( H=C^{\infty }(M) \) is the Hamiltonian function.

It can be shown that (Da Silva 2001, p. 107):

$$\begin{aligned} \begin{aligned} X_H= \sum _{j=1}^{n} \left( \frac{\partial H}{\partial p_j}\frac{\partial }{\partial q_j}-\frac{\partial H}{\partial p_j}\frac{\partial }{\partial q_j}\right) \end{aligned} \end{aligned}$$
(A.3)

which satisfies the Hamiltonian equations. The Darboux theorem (Lee 2013, p. 571) guarantees that for every 2n-dimensional symplectic manifold \( (M,\omega ) \) and for every \( p\in M \), we can find a smooth chart \( (U, q_1,\ldots ,q_n,p_1,\ldots ,p_n) \) centered at p in which \( \omega = \sum \nolimits _{j=1}^{n} dq^{i}\wedge dp^{j} \). The collection of these smooth charts forms an atlas for \( (M,\omega ) \).

Structure of classical mechanics (SCM) Let \( (M,\omega , H) \) be a symplectic manifold equipped with the Hamiltonian function. The classical mechanics is presented by the associated Poisson structure, i.e. \( (C^{\infty }(M),\circ , \alpha ) \).

Appendix B: Formal Concepts

Deformation quantization proceeds on the basis of the framework of formal power series. In the following, its key notions and structures will be defined. For more details, see Gracia-Saz (2003), Laurent-Gengoux et al. (2012) and Esposito (2014).

Formal power series with coefficients in an algebra Let A be an algebra over the commutative ring K. Formal Power series with coefficient in A is defined as:

$$\begin{aligned} \begin{aligned} a=\sum _{n=0}^{\infty } t^na_n; \quad a_n\in A \end{aligned} \end{aligned}$$
(B.1)

where t is a formal indeterminate. We denote the set of formal power series by A[t].

Formal power series with coefficients in a ring Let K be a commutative ring. Formal Power series with coefficient in K is defined as:

$$\begin{aligned} \begin{aligned} b=\sum _{n=0}^{\infty } t^n b_n; \quad b_n\in K \end{aligned} \end{aligned}$$
(B.2)

We denote the set of formal power series by K[t].

Formal ring Let K[t] be a commutative ring. \(K[\![ t ]\!]=(K[t], +, \circ )\), the formal ring, has the ring structure in which addition is defined by:

$$\begin{aligned} \begin{aligned}&+: K[t], \times K[t] \rightarrow K[t]\\&(a,b)\mapsto a+b=\sum _{n=0}^{\infty } t^n (a_n+b_n); a_n,b_n \in K \end{aligned} \end{aligned}$$
(B.3)

and multiplication by:

$$\begin{aligned} \begin{aligned}&\circ : K[t] \times K[t] \rightarrow K[t]\\&(a,b)\mapsto a\circ b=\sum _{n=0}^{\infty } t^n (c_n); \; c_n=\sum _{k=0}^{\infty } a_n \circ b_{n-k}; \; a_n,b_n \in K \end{aligned} \end{aligned}$$
(B.4)

Formal algebra Let A[t] be an algebra. \(A[\![ t ]\!]=(A[t], \circ )\), the formal algebra, is an algebra in which multiplication is defined by:

$$\begin{aligned} \begin{aligned}&\circ : A[t] \times A[t] \rightarrow A[t]\\&(a,b)\mapsto a\circ b=\sum _{n=0}^{\infty } t^n (c_n); \; c_n=\sum _{k=0}^{\infty } a_n \circ b_{n-k}; \; a_n,b_n \in A \end{aligned} \end{aligned}$$
(B.5)

Regarding these definitions, the algebra of smooth functions on manifold M is denoted by \(C^\infty (M)[\![ t ]\!]=(C^\infty (M)[t], \circ )\). To define the formal structure of classical mechanics as a formal Poisson structure, we need first to define the formal k-vector field on M.

Formal K-vector field Let \(X_n\)’s be k-vector fields on M (\(X_n \in {\mathfrak {X}}^k)\). The formal k-vector field X as a formal power series is defined as:

$$\begin{aligned} \begin{aligned} X=\sum _{n=0}^{\infty } t^n X_n \end{aligned} \end{aligned}$$
(B.6)

The space of formal k-vector fields is denoted by \({\mathfrak {X}}^k[\![ t ]\!]\). Given an SCM \( (C^{\infty }(M),\circ , \alpha ) \), the formal Poisson bi-vector field is:

$$\begin{aligned} \begin{aligned} \alpha _t=\left( \sum _{n=0}^{\infty } t^n \alpha _n\right) \in {\mathfrak {X}}^2[\![ t ]\!] \end{aligned} \end{aligned}$$
(B.7)

where \(\alpha _i\)’s are skew-symmetric vector fields such that \([\alpha _t,\alpha _t]_S=0\).

Formal structure of classical mechanics (FSCM) Let \( (C^{\infty }(M),\circ , \alpha ) \) be an SCM. \((C^\infty (M)[t], \circ , \alpha _t )\) is an FSCM, if:

  1. a)

    \(C^\infty (M)[\![ t ]\!]=(C^\infty (M)[t], \circ )\) is the formal algebra associated with the algebra \((C^\infty (M), \circ )\);

  2. b)

    \(\alpha _t\) is the formal Poisson bi-vector field by setting \(\alpha _0=\alpha \).

Equivalent FSCMs \((C^\infty (M)[t], \circ , \alpha _t )\) and \((C^\infty (M)[t], \circ , {\tilde{\alpha }}_t )\) are equivalent if there exists a formal vector field \(X=\sum _{n=0}^{\infty } t^n X_n \) such that:

$$\begin{aligned} \begin{aligned} {\tilde{\alpha }}_t=exp(t{\mathcal {L}}_X)\alpha _t \end{aligned} \end{aligned}$$
(B.8)

where \({\mathcal {L}}_X\) is the Lie derivative. The associated equivalence class is denoted by \([A_t]\).

Formal star product Let \(A[\![ t ]\!]\) be a formal algebra over the formal ring \(K[\![ t ]\!]\). The formal star product is a \(K[\![ t ]\!]\)-bilinear map:

$$\begin{aligned} \begin{aligned}&\star : A[t] \times A[t] \rightarrow A[t]\\&(f,g)\mapsto f \star g = \sum _{n=0} ^ {\infty } B_n(f,g)t^n; \; f,g\in A \end{aligned} \end{aligned}$$
(B.9)

where \(B_n\)’s\(:A \times A \rightarrow A\) are bi-differential operators, and \(f\star g = f \circ g\) for \(t=0\). The star product extends to A[t] by:

$$\begin{aligned} \begin{aligned} \left( \sum _{k=0} ^ {\infty } f_k t^k\right) \star \left( \sum _{l=0} ^ {\infty } g_l t^l\right) = \sum _{n=0} ^ {\infty } \left( \sum _{k+l+m=n} B_m(f_k,g_l)t^n\right) ; \; f_k,g_l\in A \end{aligned} \end{aligned}$$
(B.10)

Now we can deform FSCM by the formal star product.

Deformed formal structure of classical mechanics (DFSCM) Suppose that a classical system is re-presented by an FSCM denoted by \((C^\infty (M)[t], \circ , \alpha _t )\). The associated quantized system is re-presented by the deformed structure denoted by \((C^\infty (M)[t], \circ , \star _{\alpha _t})\).

Equivalent DFSCMs \((C^\infty (M)[t], \circ , \star _{\alpha _t} )\) and \((C^\infty (M)[t], \circ , \star _{{\tilde{\alpha }}_t})\) are equivalent if there exists a gauge action:

$$\begin{aligned} \begin{aligned}&D: A[t]\rightarrow A[t]\\&f\mapsto D(f) =D_0(f)+\sum _{n=1} ^ {\infty } D_n(f)t^n \end{aligned} \end{aligned}$$
(B.11)

(where \(D_i\)s are differential operators and \(D_0(f)=f\)) such that:

$$\begin{aligned} \begin{aligned} f \star _{{\tilde{\alpha }}_t} g= D(E(f)\star _{\alpha _t} E(g)) \end{aligned} \end{aligned}$$
(B.12)

where E is the inverse of D in the group of gauge actions. The associated equivalence class is denoted by \([\star _{A_t}]\).

Kontsevich’s Theorem (existence and classification of quantization) Let \( (C^{\infty }(M),\circ , \alpha ) \) be the SCM. There exists a one-to-one correspondence between the equivalence classes of FSCMs on it, \(\left\{ [A_t]\right\} \), and the equivalence classes of DFSCMs, \( \left\{ [\star _{A_t}]\right\} \).

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Yaghmaie, A. Deformation quantization as an appropriate guide to ontic structure. Synthese 198, 10793–10815 (2021). https://doi.org/10.1007/s11229-020-02752-7

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