## Equivalence of Classical Klein-Gordon Field Theory to Correspondence-Principle First Quantization of the Spinless Relativistic Free Particle

**Authors:** Steven Kenneth Kauffmann

It has recently been shown that the classical electric and magnetic fields which satisfy the sourcefree
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 Schrödinger 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 complexvalued
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
Schrödinger 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 Schrödinger 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.

**Comments:** 8 pages, Also archived as arXiv:1012.5120 [physics.gen-ph].

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### Submission history

[v1] 24 Dec 2010

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