Equivalence of classical Klein-Gordon field theory to correspondence-principle first quantization of the spinless relativistic free particle

TL;DR

This paper demonstrates that classical Klein-Gordon fields are equivalent to the first quantization of a relativistic free particle, linking classical field theory with quantum mechanics through canonical transformations.

Contribution

It establishes a direct canonical transformation mapping classical Klein-Gordon fields to a Schrödinger equation form, clarifying the connection between classical fields and quantum first quantization.

Findings

Classical Klein-Gordon fields correspond to the first quantized relativistic Schrödinger equation.

Canonical transformations relate classical fields to quantum wave functions with relativistic Hamiltonians.

Electromagnetic fields are treated similarly, connecting Maxwell's equations to quantum descriptions.

Abstract

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.