We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}$ , and define $m=E/c^2$ , which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.

arXiv:1008.4224Aug 25, 20103 citations

The Phase-Space Structure of the Klein-Gordon Field

C. Best, P. Górnicki, Walter Greiner

arXiv:hep-ph/9301275Annals of PhysicsAug 1, 199321 citations

From the main equation to the Klein-Gordon equation

Austin Blaquière

Journal of Optimization Theory and ApplicationsJan 1, 197910 citations

Time-Scalar Field Theory and Quantum Mechanics: Schrodinger, Dirac, and Klein-Gordon Equations from Scalar-Time Continuity

Jordan Gabriel Farrell

We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dOdΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standardquantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale coherence and experiment-first falsifiability.We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dOdΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standardquantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale coherence and experiment-first falsifiability.

Dec 26, 20250 citations

Another form of the Klein-Gordon equation

Morton D. Kostin

Physics Letters ANov 1, 19877 citations

1 Comment on ’Another form of the Klein-Gordon equation’

Andreas Aste

It is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. PACS code: 11.10.-z Keywords: Klein-Gordon equation, relativistic field theory. 2 The alternative formulation of the Klein-Gordon equation [1, 2] proposed by M.D. Kostin reads [3] i¯h ∂φ ∂t = +mc2φ + c ( ˆ⃗p ⃗ ψ) (1) i¯h ∂ ⃗ ψ ∂t = −mc2 ⃗ ψ + c ˆ ⃗pφ, (2) where φ(⃗r,t) and ⃗ ψ(⃗r,t) are ’scalar ’ and ’vector ’ probability amplitudes, respectively, and ˆ ⃗p = −i¯h ⃗ ∇. Defining the probability density P = φ ∗ φ + ( ⃗ ψ ∗ ⃗ ψ) (3) and the probability current density one readily derives the probability conservation equation ⃗S = c(φ ∗ ⃗ ψ + φ ⃗ ψIt is shown that the alternative Klein-Gordon equation with positive definite probability density proposed in a letter by M.D. Kostin does not meet the requirements of relativistic (quantum) field theory and therefore does not allow for a meaningful physical interpretation. PACS code: 11.10.-z Keywords: Klein-Gordon equation, relativistic field theory. 2 The alternative formulation of the Klein-Gordon equation [1, 2] proposed by M.D. Kostin reads [3] i¯h ∂φ ∂t = +mc2φ + c ( ˆ⃗p ⃗ ψ) (1) i¯h ∂ ⃗ ψ ∂t = −mc2 ⃗ ψ + c ˆ ⃗pφ, (2) where φ(⃗r,t) and ⃗ ψ(⃗r,t) are ’scalar ’ and ’vector ’ probability amplitudes, respectively, and ˆ ⃗p = −i¯h ⃗ ∇. Defining the probability density P = φ ∗ φ + ( ⃗ ψ ∗ ⃗ ψ) (3) and the probability current density one readily derives the probability conservation equation ⃗S = c(φ ∗ ⃗ ψ + φ ⃗ ψ

Jan 1, 20011 citation

Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}

, and define

m=E/c^2

, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.