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Communications in Theoretical Physics

Phase Formulation and Exact Solutions for the Relativistic Klein–Gordon Field with a Time-Dependent Hamiltonian

Fu Jian, Gao Xiaochun, Xin‐Hua Li, Gao Jun

Published Jul 30, 1998• 0 citations
Publisher

Abstract

On the basis of the phase formulation, we find the quantum and classical exact solutions and corresponding total phases for the Klein–Gordon (KG) field with a time-dependent Hamiltonian. The total phase includes both the dynamical and geometric phases (Abaronov–Anandan phase). The connection between the quantum and classical solutions is then obtained. From this connection, we discuss the condition under which the geometric phase for the KG field can be defined.

DOI
10.1088/0253-6102/30/1/77

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It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.