Differential geometry of canonical quantization

Hurt, Norman E.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 14 (1971) no. 2, pp. 153-170

[1] R. Abraham, Foundations of Mechanics, Benjamin, New York, 1967. | Zbl

[2] W.M. Boothby and H.C. Wang, On contact manifolds, Ann. of Math., t. 68, 1958, p. 721-734. | Zbl | MR

[3] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922. | MR | JFM

[4] H. Cartan, Notions d'algèbre différentielle, Colloque de Topologie, Bruxelles, 1950, p. 15-27. | Zbl | MR

[5] S.S. Chern, Pseudo-groupes continus infinis, Colloque International de Géométrie Différentielle, C. N. R. S., Paris, 1953, p. 119-136. | Zbl | MR

[6] C. Earle and J. Eells, Foliations and fibrations, J. Diff. Geom., t. 1, 1967, p. 33-41. | Zbl | MR

[7] C. Ehresmann, Les connexions infinitésimales dans un espace fibre différentiable, Colloque de Topologie, Bruxelles, 1950, p. 29-55. | Zbl | MR

[8] P. Garcia and A. Perez-Rendon, Symplectic approach to the theory of quantized fields. Comm. math. Phys., t. 13, 1969, p. 24-44. | Zbl | MR

[9] J.W. Gray, Some global properties of contact structures. Ann. of Math., t. 69, 1959, p. 421-450. | Zbl | MR

[10] Y. Hatakeyama, On the existence of Riemann metrics associated with a 2-form of rank 2. Tohoku Math. J., t. 14, 1962, p. 162-166. | Zbl | MR

[11] Y. Hatakeyama, Somes notes on differentiable manifolds with almost contact structures. Ibid., t. 15, 1963, p. 176-181. | Zbl | MR

[12] R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. A. M. S., t. 11, 1960, p. 236-242. | Zbl | MR

[13] R. Hermann, Remarks on the geometric nature of quantum phase space. J. Math. Phys., t. 6, 1965, p. 1768-1771. | MR

[14] R. Hermann, Differential Geometry and the Calculus of Variations. Academic Press, New York, 1968. | Zbl | MR

[15] N. Hurt, Remarks on canonical quantization. Il Nuoto Cimento, t. 55 A, 1968, p. 534-542. | Zbl

[16] N. Hurt, Remarks on Morse theory in canonical quantization. J. Math. Phys., t. 11, 1970, p. 539-551. | MR

[17] N. Hurt, On a classification of quantizable dynamical systems. Preprint, Univ. of Mass., 1969.

[18] J. Klein, Espaces variationnels et mécanique. Ann. Inst. Fourier, Grenoble, t. 12, 1962, p. 1-124. | Zbl | MR | Numdam

[19] B. Kostant, Orbits, representations of Lie groups and quantization. Lecture notes, unpublished.

[20] R. Lashof, Classification of fiber bundles by the loop space of the base. Ann. of Math., t. 64, 1956, p. 436-446. | Zbl | MR

[21] P. Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact. Colloque de Géométrie différentielle globale, Bruxelles, 1958, p. 37-59. | Zbl | MR

[22] A. Lichnerowicz, Les relations intégrales d'invariance. Bull. Soc. math., t. 70, 1946, p. 82-95. | Zbl | MR

[23] J.E. Marsden, Hamiltonian one parameter groups. Arch. Mech. Anal., t. 28, 1968, p. 362-396. | Zbl | MR

[24] J.E. Marsden and P. Chernoff, Hamiltonian Systems and Quantum Mechanics (to appear).

[25] A. Morimoto, On normal almost contact structures. J. Math. Soc. Japan, t. 15, 1963, p. 420-436. | Zbl | MR

[26] A. Morimoto, On normal almost contact structure with a regularity. Tohoku Math. J., t. 16, 1964, p. 90-104. | Zbl | MR

[27] Y. Ogawa, Some properties on manifolds with almost contact structures. Ibid., t. 15, 1963, p. 148-161. | Zbl | MR

[28] K. Ogiue, On fiberings of almost contact manifolds, Kodai Math. Sem. Reports, t. 17, 1965, p. 53-62. | Zbl | MR

[29] R.S. Palais, A global formulation of the Lie theory of transportation groups. Mem. A. M. S., t. 22, 1957, 123 p. | Zbl | MR

[30] G. Reeb, Quelques propriétés..., C. R. Acad. Sci. (Paris), t. 229, 1949, p. 969-971 ; Colloque de Topologie de Strasbourg, 1951, n° III ; Variétés symplectiques..., C. R. Acad. Sci. (Paris), t. 235, 1952, p. 776-778 ; Sur les éléments de contact..., J. f. reine ang. Math., t. 189, 1952, p. 186-189 ; Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques. Mem. Acad. royale Belgique, Cl. Sci., in-8° (2), 1952, n° 9, 64 p. | Zbl | MR

[31] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées. Hermann, Paris, 1952. | Zbl | MR

[32] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure. I. Tohoku Math. J., t. 12, 1960, p. 459-476. | Zbl | MR

[33] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structure. II. Ibid., t. 13, 1961, p. 281-294. | Zbl | MR

[34] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with contact metric structure. J. Math. Soc. Japan, t. 14, 1962, p. 249-271. | Zbl | MR

[35] I. Segal, Quantization of non linear systems. J. Math. Phys., t. 1, 1960, p. 468-488. | Zbl

[36] I. Segal, Mathematical problems of relativistic physics. Amer. Math. Soc., Providence, 1963. | Zbl | MR

[37] D. Shale and W.F. Stinespring, The quantum harmonic oscillator with hyperbolic phase space. J. Fnal. Anal., t. 1, 1967, p. 492-502. | MR

[38] J.-M. Souriau, Géométrie de l'espace de phases, calcul des variations et mécanique quantique (Marseille, 1965).

[39] J.-M. Souriau. Ouantification canonique (Marseille, 1962).

[40] J.-M. Souriau, Quantification canonique. Comm. math. Phys., t. 1, 1966, p. 374-398. | Zbl | MR

[41] J.-M. Souriau, Quantification canonique. II. Ann. Inst. H. Poincaré, t. 6, 4, 1967, p. 311-341. | Zbl | MR | Numdam

[42] S. Sternberg, Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs, 1964. | Zbl | MR

[43] S. Takizawa, On the contact structures of real and complex manifolds. Tohoku Math. J., t. 15, 1963, p. 227-252. | Zbl | MR

[44] S. Tanno, On fiberings of some non-compact contact manifolds. Ibid., t. 15, 1963, p. 289-297. | Zbl | MR

[45] S. Tanno, A theorem on regular vector fields and its applications to almost contact structures. Ibid., t. 17, 1965, p. 235-238. | Zbl | MR

[46] L. Van Hove, Sur certaines représentations unitaires d'un groupe infini de transformations. Mem. Acad. royale Belgique, Cl. Sc., in-8° (2), t. 26, 1952, n° 6, 102 p. | Zbl | MR

[47] J.A. Wolf, Differentiable fibre spaces and mappings compatible with Riemannian metrics. Mich. Math. J., t. 11, 1964, p. 65-70. | Zbl | MR