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Journal of Changsha University

Euler-Lagrange Equation of Dynamical Systems at the Quantum Level

Ziping Li

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

According to Faddeev-Popov, arule of path integral quantization, based on the generating Green function in Configuration-Space for a gauge-invariant system, Euler-Lagrange equations at the quantum level are derived. A Euler-Lagrange equation at the quantum level for electromagnetic field is reduced. It is pointed out that the results about the transverse shift are the classical results only, which must be corrected at the quantum level.

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According to Faddeev-Popov, arule of path integral quantization, based on the generating Green function in Configuration-Space for a gauge-invariant system, Euler-Lagrange equations at the quantum level are derived. A Euler-Lagrange equation at the quantum level for electromagnetic field is reduced. It is pointed out that the results about the transverse shift are the classical results only, which must be corrected at the quantum level.
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The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.
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It is demonstrated that the nature of optical parametric amplification is a quantum phenomenon. The system Lagrangian can be constructed by the path integral of coherent state. The equations of motion for photon operators are indeed the Euler-Lagrange equations of a Lagrangian. The quantum state evolution equation can also be obtained without resorting to quantum Hamiltonian or Lagrangian. Starting with classical Newton equation, quantum transition amplitude of the system can be educed by surface integral.