The Klein–Gordon equation for spinless particles, as well as the Weyl–Pauli and Dirac equations for particles with spin, may be derived from one information-theoretic principle. Consider a gedanken experiment whereby the mean position of a particle in a central force field is to be estimated by means of one position measurement. An efficient (optimum) estimate obeys a condition of minimum Fisher information, or minimum precision, according to the second law of thermodynamics. When the Fisher information is minimized subject to a constraint on the mean-square kinetic energy for the particle, the solution is a probability law on position whose corresponding wave amplitude obeys the Klein-Gordon equation, the Weyl–Pauli equation, or the Dirac equation, depending on whether scalar (without spin) or vector (with spin) solutions are sought. The minimized Fisher information is proportional to the mean-square kinetic energy, minus the rest energy of the particle.
The Klein–Gordon equation for spinless particles, as well as the Weyl–Pauli and Dirac equations for particles with spin, may be derived from one information-theoretic principle. Consider a gedanken experiment whereby the mean position of a particle in a central force field is to be estimated by means of one position measurement. An efficient (optimum) estimate obeys a condition of minimum Fisher information, or minimum precision, according to the second law of thermodynamics. When the Fisher information is minimized subject to a constraint on the mean-square kinetic energy for the particle, the solution is a probability law on position whose corresponding wave amplitude obeys the Klein-Gordon equation, the Weyl–Pauli equation, or the Dirac equation, depending on whether scalar (without spin) or vector (with spin) solutions are sought. The minimized Fisher information is proportional to the mean-square kinetic energy, minus the rest energy of the particle.
The principle of minimum Fisher information (MFI) provides a basis for deriving most of the laws of physics.1 The Fisher I is defined as (mean-squares error)-1 in the efficient determination of the classical position θ of a particle. For a one-component pdf p(x) = q2(x), the Fisher information is a one-term functional I = 4 ʃdx[q’(x)]2. Thus, I decreases as q(x) becomes smoother, or as disorder increases, consistent with the second law of thermodynamics. In the above, p(x) was specifically assumed to have a single component q2(x). Then MFI gives a real-only wave equation. Here we consider multicomponent cases p(x) = q12(x) + … + qN2(x), where the qn(x) are independent degrees of freedom. We seek the Fisher I for the case N = 2 of nonrelativistic quantum mechanics or of scalar optics. A benchmark statistical situation arises when the N components of qn(x) do not overlap. Then their informations add, and the resulting mean-squares error value is efficient, i.e. an absolute minimum.
Consider an isolated statistical system specified by a coordinate x and its probability density p(x). A functional of p(x) called ``Fisher information'' can be used to measure the degree of disorder of the system due to the spread in p(x). Fisher information may be minimized, subject to a physical constraint, to attain a temporal equilibrium solution p(x). When the constraint is linear in the mean kinetic energy of the system, the equilibrium solution p(x) often obeys the correct differential equation for the system. In this way, the Schr\"odinger (energy) wave equation, Klein-Gordon equation, Helmholtz wave equation, diffusion equation, Boltzmann law, and Maxwell-Boltzmann law may be derived from one classical principle of disorder. The convergence rate for Fisher information is about that for alternative use of maximum entropy (in problems where both have the same equilibrium solution). This suggests that Fisher information defines an arrow of time. The arrow points in the direction of decreasing accuracy for the determination of the mean, or ideal, value of a parameter.
The principle of extreme physical information (EPI) places physics on an information-theoretic footing. It yields many of the fundamental wave equations of physics (of Klein-Gordon, Dirac, etc.). EPI is based upon the measurement of particle four-vectors ${\mathbf{\ensuremath{\theta}}}_{\mathit{n}}$ with random errors xn, n=1,...,N. This leads to an additive Fisher information form whose extremization provides the derivations. However, the model measurement procedure was, in the past, overly restrictive: the real and imaginary parts qn(x), n=1,...,N of the probability amplitude functions for the fluctuations were required to have nonoverlapping support regions. Here we rederive the requisite information form without this restriction. Our model measurement procedure is simply the efficient collection of four-data, that is, N independent measurements of four-coordinates of particles of the field. In effect, each measurement defines a different degree of freedom qn(x) of the scenario. \textcopyright{} 1996 The American Physical Society.
Derivation of Klein–Gordon equation Goals and prior knowledge The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl , and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions. In addition, the Schrödinger wave equation (SWE), which is non -relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G. Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law , and the zero-condition (3.18) gives the virial theorem , of classical mechanics. Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3). It is important to state what is assumed a priori . The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρx(x)ρy(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ]=h[ρx]+h[ρy]−h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ∥ρxρy) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨T⟩ψ−⟨T⟩Φ=8mℏ2J(ρ∥ρxρy),Φ=ρxρy. In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation.
This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² − ∇². The equation is second‑order in both time and space, treats them symmetrically, and reduces to the free‑particle Schrödinger equation in the non‑relativistic limit. Its plane‑wave solutions satisfy the relativistic mass‑shell condition E² = p²c² + m²c⁴. Why is a relativistic version needed at all? The Schrödinger equation describes how quantum systems evolve when they’re moving slowly compared to the speed of light — fine for atoms, molecules, and everyday chemistry, where electrons move at less than 1% of c. But the Schrödinger equation is first‑order in time and second‑order in space, treating the two on different footings. Special relativity says space and time are unified into a single four‑dimensional structure where Lorentz transformations mix them, and any equation that privileges time over space breaks Lorentz invariance. A truly fundamental quantum equation must respect this symmetry. Klein–Gordon is the minimal scalar‑field equation that does. The standard derivation runs by analogy: start with the relativistic energy–momentum relation E² = p²c² + m²c⁴, replace E by iℏ∂ₜ and p by −iℏ∇, apply both sides to a wavefunction. This produces the right equation but it’s not really a derivation — it’s a guess motivated by the Schrödinger operator‑substitution procedure. This walkthrough shows how the Event Density framework derives the Klein–Gordon equation from substrate primitives — the same primitives that gave the Born rule, the Schrödinger equation, and the Bell–Tsirelson bound — plus the commitment to Lorentz covariance. The participation measure becomes Lorentz‑covariant. The unique second‑order Lorentz‑scalar differential operator on a scalar field is the d’Alembertian; the minimum‑order Lorentz‑scalar linear equation is therefore (□ + M²) Ψ = 0. Dimensional analysis fixes M² = m²c²/ℏ². Local U(1) gauge invariance forces the covariant derivative and the conserved four‑current. The non‑relativistic limit, via rest‑energy factorization, recovers the free‑particle Schrödinger equation exactly. The structural payoff: the Klein–Gordon equation is what you get when the substrate ontology is pushed through a Lorentz‑covariant reformulation. No new primitives are required. Relativistic and non‑relativistic quantum mechanics are not two theories glued together by analogy; they are the same substrate ontology applied at different velocity regimes. The walkthrough is honest about the known caveats — negative‑energy solutions and the non‑positive‑definite j₀ current — both of which resolve at the Dirac level and at QFT.