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Physical Review

Quantum Mechanics and the Relativistic Hamilton-Jacobi Equation

Lloyd Motz, Adolph Selzer

Published Mar 23, 1964• 13 citations
Publisher

Abstract

In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where $S$ is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.

DOI
10.1103/physrev.133.b1622

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In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function eiS\ensuremathℏ{e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}e\ensuremathℏiS​, where SSS is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.
It is shown that for a relativistic particle moving in an electromagnetic field its equations of motion written in a form of the second law of Newton can be reduced with the help of elementary operations to the Hamilton-Jacobi equation. The derivation is based on a possibility of transforming the equation of motion to a completely antisymmetric form. Moreover, by perturbing the Hamilton-Jacobi equation we obtain the principle of least action.\ The analogous procedure is easily extended to a general relativistic motion of a charged relativistic particle in an electromagnetic field. It sis also shown that the special-relativistic Hamilton-Jacobi equation for a free particle allows one to easily demonstrate the wave-particle duality inherent to this equation and, in addition, to obtain the operators of the four-momentum whose eigenvalues are the classical four-momentum
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
An investigation of origins of the quantum mechanical momentum operator has shown that it corresponds to the nonrelativistic momentum of classical special relativity theory rather than the relativistic one, as has been unconditionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy operators are defined. Schrödinger equations with relativistic kinematics are introduced and investigated for a free particle and a particle trapped in the deep potential well.
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.