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A symmetrical interpretation of the Klein-Gordon equation

Michael B. Heaney, See Profile

Published Jan 1, 2013• 9 citations
OpenAlex

Abstract

Abstract. This paper presents a new Symmetrical Interpretation (SI) of relativistic quantum me-chanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This SI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein’s bubble experiment. This SI makes several experimentally testable predictions that differ from the CI, solves one part of the measurement problem, resolves some in-consistencies of the CI, and gives intuitive explanations of some previously mysterious quantum effects. 1.

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This paper presents a new Symmetrical Theory (ST) of nonrelativistic quantum mechanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This new ST is compared to the Conventional Theory (CT) of nonrelativistic quantum mechanics for the analysis of a beam-splitter experiment. The ST makes several experimentally testable predictions that differ from the CT, which can be checked using existing technology. The ST also solves one part of the CT measurement problem, and resolves some of the paradoxes of the CT. This nonrelativistic ST is the low energy limit of a relativistic ST presented in an earlier paper \cite{Heaney1}.

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From Primitives to the Klein–Gordon Equation

Allen Proxmire

The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² −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.

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Yulei Feng

In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrodinger field theory, obtaining the Schrodinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrodinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.

•Oct 24, 2012•0 citations
Abstract. This paper presents a new Symmetrical Interpretation (SI) of relativistic quantum me-chanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This SI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein’s bubble experiment. This SI makes several experimentally testable predictions that differ from the CI, solves one part of the measurement problem, resolves some in-consistencies of the CI, and gives intuitive explanations of some previously mysterious quantum effects. 1.
This paper presents a new Relativistic Symmetrical Interpretation (RSI) of the Dirac equation in (1+1)D which postulates: quantum mechanics is intrinsically time-symmetric, with no arrow of time; the fundamental objects of quantum mechanics are transitions; a transition is fully described by a complex transition amplitude density with specified initial and final boundary conditions; and transition amplitude densities never collapse. This RSI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein's bubble experiment with a spin-12\frac{1}{2}21​ particle. This RSI can predict the future and retrodict the past, has no zitterbewegung, resolves some inconsistencies of the CI, and eliminates some of the conceptual problems of the CI.
This paper presents a new Symmetrical Theory (ST) of nonrelativistic quantum mechanics which postulates: quantum mechanics is a theory about complete experiments, not particles; a complete experiment is maximally described by a complex transition amplitude density; and this transition amplitude density never collapses. This new ST is compared to the Conventional Theory (CT) of nonrelativistic quantum mechanics for the analysis of a beam-splitter experiment. The ST makes several experimentally testable predictions that differ from the CT, which can be checked using existing technology. The ST also solves one part of the CT measurement problem, and resolves some of the paradoxes of the CT. This nonrelativistic ST is the low energy limit of a relativistic ST presented in an earlier paper \cite{Heaney1}.
This paper presents a relativistic symmetrical interpretation of the Dirac equation in 1+1 dimensions which predicts no zitterbewegung for a free spin-1/2 particle. This could resolve the longstanding puzzle of zitterbewegung in relativistic quantum mechanics, and help clarify the physical meaning of the zitterbewegung currently under investigation in many-particle systems. Together with an earlier paper describing a relativistic symmetrical interpretation of the Klein-Gordon equation \cite{Heaney1}, this new interpretation resolves some inconsistencies in the conventional interpretations of both equations. This new interpretation also makes several experimental predictions that differ from the conventional interpretation of the Dirac equation.
Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions. Key words: relativistic quantum mechanics, Klein-Gordon equation, Bohmian interpretation. 1
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.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrodinger field theory, obtaining the Schrodinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrodinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.