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Hypercomplex Representations of the Heisenberg Group and Mechanics

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Abstract

In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit ħ→0.

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References

  1. Agostini, F., Caprara, S., Ciccotti, G.: Do we have a consistent non-adiabatic quantum-classical mechanics? Europhys. Lett. 78(3), 6 (2007). doi:10.1209/0295-5075/78/30001. Art. 30001. MR2366698 (2008k:81004)

    Article  MathSciNet  Google Scholar 

  2. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1991). Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, corrected reprint of the second (1989) edition. MR96c:70001

    Google Scholar 

  3. Berezin, F.A.: Metod Vtorichnogo Kvantovaniya, 2nd edn. Nauka, Moscow (1986). Edited and with a preface by M.K. Polivanov. MR89c:81001

    MATH  Google Scholar 

  4. Boccaletti, D., Catoni, F., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P.: The Mathematics of Minkowski Space-time and an Introduction to Commutative Hypercomplex Numbers. Springer, Berlin (2007)

    Google Scholar 

  5. Brodlie, A., Kisil, V.V.: Observables and states in p-mechanics. Adv. Math. Res. 5, 101–136 (2003). arXiv:quant-ph/0304023. MR2117375

    MathSciNet  Google Scholar 

  6. Calzetta, E., Verdaguer, E.: Real-time approach to tunnelling in open quantum systems: decoherence and anomalous diffusion. J. Phys. A 39(30), 9503–9532 (2006). MR2246702 (2007f:82059)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Catoni, F., Cannata, R., Nichelatti, E.: The parabolic analytic functions and the derivative of real functions. Adv. Appl. Clifford Algebras 14(2), 185–190 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Davis, M.: Applied Nonstandard Analysis. Wiley-Interscience, New York (1977). Pure and Applied Mathematics. MR0505473 (58 #21590). 0-471-19897-8

    MATH  Google Scholar 

  9. De Bie, H., Eelbode, D., Sommen, F.: Spherical harmonics and integration in superspace: II. J. Phys. A, Math. Theor. 42(24), 245204 (2009) (English). Zbl1179.30053

    Article  ADS  Google Scholar 

  10. de Gosson, M.A.: Spectral properties of a class of generalized Landau operators. Commun. Partial Differ. Equ. 33(10–12), 2096–2104 (2008). MR2475331 (2010b:47128)

    Article  MATH  Google Scholar 

  11. de Gosson, M., Luef, F.: Symplectic capacities and the geometry of uncertainty: the irruption of symplectic topology in classical and quantum mechanics. Phys. Rep. 484(5), 131–179 (2009). MR2559681

    Article  ADS  MathSciNet  Google Scholar 

  12. Folland, G.B.: Harmonic analysis in phase space. In: Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989). MR92k:22017

    Google Scholar 

  13. Giachetta, G., Mangiarotti, L., Sardanashvily, G.: New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, River Edge (1997). MR2001723 (2004g:70049)

    MATH  Google Scholar 

  14. Gromov, N.A.: Контракции и Аналитические Продолжения Классических Групп. Единыи Подход. (in Russian) [Contractions and analytic extensions of classical groups. Unified approach]. Akad. Nauk SSSR Ural. Otdel. Komi Nauchn. Tsentr, Syktyvkar (1990). MR1092760 (91m:81078)

    Google Scholar 

  15. Gromov, N.A.: Transitions: contractions and analytical continuations of the Cayley-Klein groups. Int. J. Theor. Phys. 29, 607–620 (1990). doi:10.1007/BF00672035

    Article  MATH  MathSciNet  Google Scholar 

  16. Gromov, N.A., Kuratov, V.V.: All possible Cayley-Klein contractions of quantum orthogonal groups. Yad. Fiz. 68(10), 1752–1762 (2005). MR2189521 (2006g:81101)

    MathSciNet  Google Scholar 

  17. Günther, U., Kuzhel, S.: \(\mathcal{P}\mathcal{T}\)-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras. J. Phys. A, Math. Theor. 43(39), 392002 (2010)

    Article  Google Scholar 

  18. Herranz, F.J., Santander, M.: Conformal compactification of spacetimes. J. Phys. A 35(31), 6619–6629 (2002). arXiv:math-ph/0110019. MR1928852 (2004b:53123)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Herranz, F.J., Ortega, R., Santander, M.: Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry. J. Phys. A 33(24), 4525–4551 (2000). arXiv:math-ph/9910041. MR1768742 (2001k:53099)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. Soc. 3(2), 821–843 (1980). MR81h:22010

    Article  MATH  MathSciNet  Google Scholar 

  21. Howe, R.: Quantum mechanics and partial differential equations. J. Funct. Anal. 38(2), 188–254 (1980). MR83b:35166

    Article  MATH  MathSciNet  Google Scholar 

  22. Hudson, R.: Generalised translation-invariant mechanics. D.Phil. thesis, Bodleian Library, Oxford (1966)

  23. Hudson, R.: Translation invariant phase space mechanics. In: Proc. of the Conference Quantum Theory: Reconsideration of Foundations, vol. 2, 301–314. Vaxjo University Press, Vaxjo (2004). MR2111131 (2006e:81134)

    Google Scholar 

  24. Kanatchikov, I.V.: Precanonical quantum gravity: quantization without the space-time decomposition. Int. J. Theor. Phys. 40(6), 1121–1149 (2001). arXiv:gr-qc/0012074. MR2002m:83038

    Article  MATH  MathSciNet  Google Scholar 

  25. Khrennikov, A.: ‘Quantum probabilities’ as context depending probabilities (2001). arXiv:quant-ph/0106073

  26. Khrennikov, A.: Hyperbolic quantum mechanics. Adv. Appl. Clifford Algebras 13(1), 1–9 (2003). (in English). arXiv:quant-ph/0101002

    Article  MATH  MathSciNet  Google Scholar 

  27. Khrennikov, A.Yu.: Hyperbolic quantum mechanics. Dokl. Akad. Nauk, Ross. Akad. Nauk 402(2), 170–172 (2005). MR2162434 (2006d:81118)

    MathSciNet  Google Scholar 

  28. Khrennikov, A.: Hyperbolic quantization. Adv. Appl. Clifford Algebras 18(3–4), 843–852 (2008). MR2490591

    Article  MATH  MathSciNet  Google Scholar 

  29. Khrennikov, A., Segre, G.: Hyperbolic quantization. In: Quantum Probability and Infinite Dimensional Analysis, pp. 282–287 (2007). MR2359402

    Chapter  Google Scholar 

  30. Khrennikov, A.Y., Volovich, Y.I.: Numerical experiment on interference for macroscopic particles (2001). arXiv:quant-ph/0111159

  31. Kirillov, A.A.: Elements of the Theory of Representations. Springer, Berlin (1976). Translated from the Russian by E. Hewitt, Grundlehren der Mathematischen Wissenschaften, Band 220. MR54#447

    Book  MATH  Google Scholar 

  32. Kirillov, A.A.: Introduction to the Theory of Representations and Noncommutative Harmonic Analysis [MR90a:22005]. In: Representation Theory and Noncommutative Harmonic Analysis, i, pp. 1–156, 227–234 (1994). MR1311488. MR1 311 488

    Google Scholar 

  33. Kirillov, A.A.: Merits and demerits of the orbit method. Bull. Am. Math. Soc. 36(4), 433–488 (1999). MR2000h:22001

    Article  MATH  MathSciNet  Google Scholar 

  34. Kisil, V.V.: Clifford valued convolution operator algebras on the Heisenberg group. A quantum field theory model. In: Clifford Algebras and Their Applications in Mathematical Physics, Proceedings of the Third International Conference Held in Deinze, pp. 287–294 (1993). MR1266878

    Chapter  Google Scholar 

  35. Kisil, V.V.: Quantum probabilities and non-commutative Fourier transform on the Heisenberg group. In: Interaction Between Functional Analysis, Harmonic Analysis and Probability, Columbia, MO, 1994, pp. 255–266 (1995). MR97b:81060

    Google Scholar 

  36. Kisil, V.V.: Plain mechanics: classical and quantum. J. Nat. Geom. 9(1), 1–14 (1996). arXiv:funct-an/9405002. MR1374912 (96m:81112)

    MATH  MathSciNet  Google Scholar 

  37. Kisil, V.V.: Wavelets in Banach spaces. Acta Appl. Math. 59(1), 79–109 (1999). arXiv:math/9807141. MR1740458 (2001c:43013)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kisil, V.V.: Nilpotent Lie groups in Clifford analysis and mathematical physics. In: Clifford Analysis and Its Applications, Prague, 2000, pp. 135–141 (2001). arXiv:math-ph/0009013. MR2003b:30059

    Chapter  Google Scholar 

  39. Kisil, V.V.: Quantum and classical brackets. Int. J. Theor. Phys. 41(1), 63–77 (2002). arXiv:math-ph/0007030. MR2003b:81105

    Article  MATH  MathSciNet  Google Scholar 

  40. Kisil, V.V.: Two slits interference is compatible with particles’ trajectories. In: Quantum Theory: Reconsideration of Foundations, pp. 215–226 (2002). arXiv:quant-ph/0111094

    Google Scholar 

  41. Kisil, V.V.: p-Mechanics as a physical theory: an introduction. J. Phys. A 37(1), 183–204 (2004). arXiv:quant-ph/0212101, On-line. Zbl1045.81032. MR2044764 (2005c:81078)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. Kisil, V.V.: Spectrum as the support of functional calculus. In: Functional Analysis and Its Applications, pp. 133–141 (2004). arXiv:math.FA/0208249. MR2098877

    Chapter  Google Scholar 

  43. Kisil, V.V.: p-Mechanics and field theory. Rep. Math. Phys. 56(2), 161–174 (2005). arXiv:quant-ph/0402035, On-line. MR2176789 (2006h:53104)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Kisil, V.V.: A quantum-classical bracket from p-mechanics. Europhys. Lett. 72(6), 873–879 (2005). arXiv:quant-ph/0506122, On-line. MR2213328 (2006k:81134)

    Article  ADS  MathSciNet  Google Scholar 

  45. Kisil, V.V.: Erlangen program at large–0: starting with the group SL2(R). Not. Am. Math. Soc. 54(11), 1458–1465 (2007). arXiv:math/0607387, On-line. MR2361159

    MATH  MathSciNet  Google Scholar 

  46. Kisil, V.V.: Two-dimensional conformal models of space-time and their compactification. J. Math. Phys. 48(7), 073506 (2007). arXiv:math-ph/0611053. MR2337687

    Article  ADS  MathSciNet  Google Scholar 

  47. Kisil, V.V.: Erlangen program at large—2 1/2: Induced representations and hypercomplex numbers. Известия Коми Научного Центра УрО РАН 5(1), 4–10 (2009). arXiv:0909.4464

    MathSciNet  Google Scholar 

  48. Kisil, V.V.: Erlangen program at large–1: geometry of invariants. Symmetry Integr. Geom. Methods Appl. 6(076), 45 (2010). arXiv:math.CV/0512416

    Google Scholar 

  49. Kisil, V.V.: Computation and dynamics: classical and quantum. AIP Conf. Proc. 1232(1), 306–312 (2010). arXiv:0909.1594

    Article  ADS  Google Scholar 

  50. Kisil, V.V.: Erlangen program at large—2: inventing a wheel. The parabolic one. Trans. Inst. Math. of the NAS of Ukraine, 89–98 (2010). arXiv:0707.4024

  51. Kisil, V.V.: Erlangen Programme at Large 3.2: Ladder operators in hypercomplex mechanics. Acta Polytech. 51(4), 44–53 (2011). arXiv:1103.1120

    Google Scholar 

  52. Kisil, V.V.: Erlangen programme at large: an overview. In: Advances in Applied Analysis, pp. 1–65 (2012). arXiv:1106.1686 (submitted)

    Google Scholar 

  53. Kisil, V.V.: Comment on “Do we have a consistent non-adiabatic quantum-classical mechanics?” by Agostini F. et al. Europhys. Lett. 89, 50005 (2010). arXiv:0907.0855

    Article  ADS  Google Scholar 

  54. Lang, S.: Sl2(R). Graduate Texts in Mathematics, vol. 105. Springer, New York (1985). Reprint of the 1975 edition. MR803508 (86j:22018)

    MATH  Google Scholar 

  55. Lévy-Leblond, J.-M.: Une nouvelle limite non-relativiste du groupe de Poincaré. Ann. Inst. H. Poincaré Sect. A 3, 1–12 (1965). MR0192900 (33 #1125)

    MATH  Google Scholar 

  56. Low, S.G.: Noninertial symmetry group of Hamilton’s mechanics. ArXiv e-prints (March 2009), available at 0903.4397

  57. Percival, I., Richards, D.: Introduction to Dynamics, vol. VIII. Cambridge University Press, Cambridge (1982). 228 p. (English)

    MATH  Google Scholar 

  58. Pimenov, R.I.: Unified axiomatics of spaces with maximal movement group. Litov. Mat. Sb. 5, 457–486 (1965). (in Russian). Zbl0139.37806

    MATH  MathSciNet  Google Scholar 

  59. Plaksa, S.: Commutative algebras of hypercomplex monogenic functions and solutions of elliptic type equations degenerating on an axis. In: Further Progress in Analysis. Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, August 13–18, 2007, pp. 977–986 (2009)

    Chapter  Google Scholar 

  60. Taylor, M.E.: Noncommutative Harmonic Analysis. Mathematical Surveys and Monographs, vol. 22. American Mathematical Society, Providence (1986). MR88a:22021

    MATH  Google Scholar 

  61. Torre, A.: Linear and quadratic exponential modulation of the solutions of the paraxial wave equation. J. Opt. A, Pure Appl. Opt. 12(3), 035701 (2010) (11pp)

    Google Scholar 

  62. Ulrych, S.: Relativistic quantum physics with hyperbolic numbers. Phys. Lett. B 625(3–4), 313–323 (2005). MR2170329 (2006e:81103a)

    ADS  MathSciNet  Google Scholar 

  63. Ulrych, S.: Representations of Clifford algebras with hyperbolic numbers. Adv. Appl. Clifford Algebras 18(1), 93–114 (2008). MR2377525 (2009d:81139)

    Article  MATH  MathSciNet  Google Scholar 

  64. Ulrych, S.: Considerations on the hyperbolic complex Klein–Gordon equation. J. Math. Phys. 51(6), 063510 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  65. Vourdas, A.: Analytic representations in quantum mechanics. J. Phys. A 39(7), R65–R141 (2006). MR2210163 (2007g:81069)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  66. Yaglom, I.M.: A simple non-Euclidean geometry and its physical basis. In: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity, Heidelberg Science Library. Springer, New York (1979). Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon. MR520230 (80c:51007)

    Google Scholar 

  67. Zachos, C.: Deformation quantization: quantum mechanics lives and works in phase-space. Int. J. Mod. Phys. A 17(3), 297–316 (2002). arXiv:hep-th/0110114. MR1888 937

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Acknowledgements

I am grateful to A.Yu. Khrennikov and S. Ulrych for useful discussion on relation between double numbers and physics. S. Plaksa advised me on various aspect of commutative hypercomplex algebras. U. Güenther draw my attention to the connection between \(\mathcal{P}\mathcal{T}\)-symmetric Hamiltonians and Krein spaces. Prof. N.A. Gromov made several useful suggestions of methodological nature. Constructive comments of anonymous referees provided further ground for paper’s improvement.

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  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Vladimir V. Kisil

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Correspondence to Vladimir V. Kisil.

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Dedicated to the memory of V.I. Arnold

On leave from the Odessa University.

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Kisil, V.V. Hypercomplex Representations of the Heisenberg Group and Mechanics. Int J Theor Phys 51, 964–984 (2012). https://doi.org/10.1007/s10773-011-0970-0

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