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Prog. Theor. Phys. Vol. 47 No. 3 (1972) pp. 1004-1025

On Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. II

Toshiei Kimura and Reiji Sugano^*

Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima-ken
^*Department of Physics, Osaka City University, Osaka

(Received October 9, 1971)

Abstract:

We examine whether a consistent quantization can be carried out for a non-linear Lagrangian when one cannot resort to the transformation from the given Lagrangian L = (1/8) g^-1/4 ×{g_ir, \dotq^r } g^1/2 g^ij {g_js, \dotq^s } g^-1/4 - v(q) to the standard from L = ½ \dotQ^a \dotQ^a - V(q). In order to obtain the variational equation the commutation relations for δqⁱ and δ\dotqⁱ are settled so as to be consistent with the Euler-Lagrange equation. Under the premise of δ\dotqⁱ = (d/dt) δqⁱ, it is shown that consistent Lagrangian and Hamiltonian formalisms exist when a space defined by metrics g_ij is of constant curvature. All consequences of a previous paper are derived and the dynamical model reduces to the case of flat space, if the commutability between the Euler equation and δqⁱ is supposed to hold. As a non-trivial example of the irreducible dynamical models, the case of the three-dimensional space with constant curvature is investigated. It is shown that for this model the explicit form of [\dotqⁱ, δq^j] and [\dotqⁱ, δ\dotq^j] are all determined and the consistent quantization is carried out.

URL : http://ptp.ipap.jp/link?PTP/47/1004/
DOI : 10.1143/PTP.47.1004

[ Full Text PDF : FREE ACCESS (1049K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

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References:

1. D. Kiang, K. Nakazawa and R. Sugano, Phys. Rev. 181 (1969), 1380[APS].
   H. E. Lin, W. C. Lin and R. Sugano, Nucl. Phys. B 16 (1970), 431[CrossRef].
2. R. Sugano, Prog. Theor. Phys. 46 (1971), 297[PTP].
3. T. Kimura, Prog. Theor. Phys. 46 (1971), 1261[PTP].
4. L. E. Eisenhart, Riemannian Geometry (Princeton University Press, 1949), p. 237.
5. Ref. 4), p. 84.
6. V. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, 1964), p. 175.
7. H. P. Robertson, Ann. of Math. 33 (1932), 496.
8. P. A. M. Dirac, Can. J. Math. 2 (1950), 129.

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 47 No. 5 (1972) pp. 1704-1713 :

   Variation Principle for Non-Linear Lagrangian in Quantum Mechanics

   Teruya Ohtani and Reiji Sugano

2. Progress of Theoretical Physics Vol. 47 No. 5 (1972) pp. 1722-1742 :

   Relativistically Covariant Formulation in Non-Linear Lagrangian Theories and Factor Ordering Problems

   Tsuneo Suzuki and Chuichiro Hattori

3. Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1395-1407 :

   On the Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. III

   Toshiei Kimura, Teruya Ohtani and Reiji Sugano

4. Progress of Theoretical Physics Vol. 48 No. 6 (1972) pp. 2082-2092 :

   Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. I

   Toshiharu Kawai

5. Progress of Theoretical Physics Vol. 49 No. 4 (1973) pp. 1352-1361 :

   Schwinger's Variation Principle by Means of Q-Number Variation for Non-Linear Lagrangian

   Reiji Sugano

6. Progress of Theoretical Physics Vol. 50 No. 1 (1973) pp. 277-289 :

   Quantization for Non-Linear Lagrangian

   Teruya Ohtani

7. Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 680-690 :

   Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. III

   Hideki Kamo and Toshiharu Kawai

8. Progress of Theoretical Physics Vol. 50 No. 5 (1973) pp. 1715-1728 :

   Q-Number Variational Method for Non-Linear Lagrangian in Quantum Mechanics

   Teruya Ohtani and Reiji Sugano

9. Progress of Theoretical Physics Vol. 52 No. 5 (1974) pp. 1687-1701 :

   A Quantum-Theoretical Lagrangian Formalism for Quasi-Linear Field Theories. I

   Takao Okabayashi and Hiroyuki Kikugawa

10. Progress of Theoretical Physics Vol. 83 No. 5 (1990) pp. 894-905 :

    Quantum Mechanics in Riemannian Manifold

    Naohisa Ogawa, Kanji Fujii and Alexander Kobushukin

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