Differential geometry of canonical quantization

Norman E. Hurt

Annales de l'I.H.P. Physique théorique (1971)

  • Volume: 14, Issue: 2, page 153-170
  • ISSN: 0246-0211

How to cite

top

Hurt, Norman E.. "Differential geometry of canonical quantization." Annales de l'I.H.P. Physique théorique 14.2 (1971): 153-170. <http://eudml.org/doc/75691>.

@article{Hurt1971,
author = {Hurt, Norman E.},
journal = {Annales de l'I.H.P. Physique théorique},
language = {eng},
number = {2},
pages = {153-170},
publisher = {Gauthier-Villars},
title = {Differential geometry of canonical quantization},
url = {http://eudml.org/doc/75691},
volume = {14},
year = {1971},
}

TY - JOUR
AU - Hurt, Norman E.
TI - Differential geometry of canonical quantization
JO - Annales de l'I.H.P. Physique théorique
PY - 1971
PB - Gauthier-Villars
VL - 14
IS - 2
SP - 153
EP - 170
LA - eng
UR - http://eudml.org/doc/75691
ER -

References

top
  1. [1] R. Abraham, Foundations of Mechanics, Benjamin, New York, 1967. Zbl0158.42901
  2. [2] W.M. Boothby and H.C. Wang, On contact manifolds, Ann. of Math., t. 68, 1958, p. 721-734. Zbl0084.39204MR112160
  3. [3] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922. MR355764JFM48.0538.02
  4. [4] H. Cartan, Notions d'algèbre différentielle, Colloque de Topologie, Bruxelles, 1950, p. 15-27. Zbl0045.30601MR42426
  5. [5] S.S. Chern, Pseudo-groupes continus infinis, Colloque International de Géométrie Différentielle, C. N. R. S., Paris, 1953, p. 119-136. Zbl0053.01604MR63377
  6. [6] C. Earle and J. Eells, Foliations and fibrations, J. Diff. Geom., t. 1, 1967, p. 33-41. Zbl0162.54101MR215320
  7. [7] C. Ehresmann, Les connexions infinitésimales dans un espace fibre différentiable, Colloque de Topologie, Bruxelles, 1950, p. 29-55. Zbl0054.07201MR42768
  8. [8] P. Garcia and A. Perez-Rendon, Symplectic approach to the theory of quantized fields. Comm. math. Phys., t. 13, 1969, p. 24-44. Zbl0175.24703MR256681
  9. [9] J.W. Gray, Some global properties of contact structures. Ann. of Math., t. 69, 1959, p. 421-450. Zbl0092.39301MR112161
  10. [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. Zbl0108.34905MR141044
  11. [11] Y. Hatakeyama, Somes notes on differentiable manifolds with almost contact structures. Ibid., t. 15, 1963, p. 176-181. Zbl0136.18002MR150718
  12. [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. Zbl0112.13701MR112151
  13. [13] R. Hermann, Remarks on the geometric nature of quantum phase space. J. Math. Phys., t. 6, 1965, p. 1768-1771. MR186018
  14. [14] R. Hermann, Differential Geometry and the Calculus of Variations. Academic Press, New York, 1968. Zbl0219.49023MR233313
  15. [15] N. Hurt, Remarks on canonical quantization. Il Nuoto Cimento, t. 55 A, 1968, p. 534-542. Zbl0163.46501
  16. [16] N. Hurt, Remarks on Morse theory in canonical quantization. J. Math. Phys., t. 11, 1970, p. 539-551. MR482882
  17. [17] N. Hurt, On a classification of quantizable dynamical systems. Preprint, Univ. of Mass., 1969. 
  18. [18] J. Klein, Espaces variationnels et mécanique. Ann. Inst. Fourier, Grenoble, t. 12, 1962, p. 1-124. Zbl0281.49026MR215269
  19. [19] B. Kostant, Orbits, representations of Lie groups and quantization. Lecture notes, unpublished. 
  20. [20] R. Lashof, Classification of fiber bundles by the loop space of the base. Ann. of Math., t. 64, 1956, p. 436-446. Zbl0075.32102MR82099
  21. [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. Zbl0095.36803MR119153
  22. [22] A. Lichnerowicz, Les relations intégrales d'invariance. Bull. Soc. math., t. 70, 1946, p. 82-95. Zbl0063.03549MR19458
  23. [23] J.E. Marsden, Hamiltonian one parameter groups. Arch. Mech. Anal., t. 28, 1968, p. 362-396. Zbl0159.54801MR226145
  24. [24] J.E. Marsden and P. Chernoff, Hamiltonian Systems and Quantum Mechanics (to appear). Zbl0336.70001
  25. [25] A. Morimoto, On normal almost contact structures. J. Math. Soc. Japan, t. 15, 1963, p. 420-436. Zbl0135.22102MR163245
  26. [26] A. Morimoto, On normal almost contact structure with a regularity. Tohoku Math. J., t. 16, 1964, p. 90-104. Zbl0135.22103MR163246
  27. [27] Y. Ogawa, Some properties on manifolds with almost contact structures. Ibid., t. 15, 1963, p. 148-161. Zbl0147.41002MR150717
  28. [28] K. Ogiue, On fiberings of almost contact manifolds, Kodai Math. Sem. Reports, t. 17, 1965, p. 53-62. Zbl0136.18101MR178428
  29. [29] R.S. Palais, A global formulation of the Lie theory of transportation groups. Mem. A. M. S., t. 22, 1957, 123 p. Zbl0178.26502MR121424
  30. [30] G. Reeb, Quelques propriétés..., C. R. Acad. Sci. (Paris), t. 229, 1949, p. 969-971 ; Colloque de Topologie deStrasbourg, 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. Zbl0034.35802MR32994
  31. [31] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées. Hermann, Paris, 1952. Zbl0049.12602MR55692
  32. [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. Zbl0192.27903MR123263
  33. [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. Zbl0112.14002MR138065
  34. [34] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with contact metric structure. J. Math. Soc. Japan, t. 14, 1962, p. 249-271. Zbl0109.40504MR141045
  35. [35] I. Segal, Quantization of non linear systems. J. Math. Phys., t. 1, 1960, p. 468-488. Zbl0099.22402
  36. [36] I. Segal, Mathematical problems of relativistic physics. Amer. Math. Soc., Providence, 1963. Zbl0112.45307MR144227
  37. [37] D. Shale and W.F. Stinespring, The quantum harmonic oscillator with hyperbolic phase space. J. Fnal. Anal., t. 1, 1967, p. 492-502. MR220485
  38. [38] J.-M. Souriau, Géométrie de l'espace de phases, calcul des variations et mécanique quantique (Marseille, 1965). 
  39. [39] J.-M. Souriau. Ouantification canonique (Marseille, 1962). 
  40. [40] J.-M. Souriau, Quantification canonique. Comm. math. Phys., t. 1, 1966, p. 374-398. Zbl1148.81307MR207332
  41. [41] J.-M. Souriau, Quantification canonique. II. Ann. Inst. H. Poincaré, t. 6, 4, 1967, p. 311-341. Zbl0152.46204MR219286
  42. [42] S. Sternberg, Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs, 1964. Zbl0129.13102MR193578
  43. [43] S. Takizawa, On the contact structures of real and complex manifolds. Tohoku Math. J., t. 15, 1963, p. 227-252. Zbl0122.40704MR157331
  44. [44] S. Tanno, On fiberings of some non-compact contact manifolds. Ibid., t. 15, 1963, p. 289-297. Zbl0124.14703MR163247
  45. [45] S. Tanno, A theorem on regular vector fields and its applications to almost contact structures. Ibid., t. 17, 1965, p. 235-238. Zbl0149.18903MR185543
  46. [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. Zbl0045.38701MR57260
  47. [47] J.A. Wolf, Differentiable fibre spaces and mappings compatible with Riemannian metrics. Mich. Math. J., t. 11, 1964, p. 65-70. Zbl0116.39202MR159285

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.