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Parameterization of all path integral trajectories

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  • Yang, Ciann-Dong
  • Wei, Chia-Hung

Abstract

It is well known that the differentiation of the propagator obtained by path integral formalism leads to the Schrödinger equation. In this paper, we will prove the complementary result that the integration of the Schrödinger equation will lead to the path integral trajectories forming the propagator. The proposed Schrödinger’s approach to path integral is helpful in explaining the origin of the multiple quantum paths connecting two fixed points and in providing a means to find all these multiple paths. We point out that path integral trajectories are governed by quantum Hamilton equations derived from the Schrödinger equation and can be continuously parameterized in terms of a free parameter so that an infinite dimensional path integral can be transformed into a one-dimensional normal integral over this free parameter.

Suggested Citation

  • Yang, Ciann-Dong & Wei, Chia-Hung, 2007. "Parameterization of all path integral trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 118-134.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:1:p:118-134
    DOI: 10.1016/j.chaos.2006.10.008
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    References listed on IDEAS

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    1. Yang, Ciann-Dong, 2007. "The origin and proof of quantization axiom p→pˆ=-iℏ∇ in complex spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 274-283.
    2. El Naschie, M.S., 2005. "A new solution for the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 935-939.
    3. Yang, Ciann-Dong, 2006. "Modeling quantum harmonic oscillator in complex domain," Chaos, Solitons & Fractals, Elsevier, vol. 30(2), pages 342-362.
    4. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    5. El Naschie, M.S., 2006. "Hilbert space, the number of Higgs particles and the quantum two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 9-13.
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    Cited by:

    1. Mukhamedov, A.M., 2009. "Meso-structures of dynamical chaos and E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1930-1938.
    2. Yang, Ciann-Dong & Wei, Chia-Hung, 2008. "Strong chaos in one-dimensional quantum system," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 988-1001.
    3. Yang, Ciann-Dong, 2009. "Stability and quantization of complex-valued nonlinear quantum systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 711-723.
    4. Yang, Ciann-Dong, 2009. "A new hydrodynamic formulation of complex-valued quantum mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 453-468.
    5. Yang, Ciann-Dong, 2008. "On the existence of complex spacetime in relativistic quantum mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 316-331.

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