Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1802.04266 · Physical Review A

Probability density of relativistic spinless particles

MohammadJavad Kazemi, Hadi Hashamipour, Mohammad Hosein Barati

Published Jul 19, 2018• 9 citations
arXivHTMLPDF

Abstract

In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for ($1+1$) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than ${m}_{0}c$.

DOI
10.1103/physreva.98.012125

Related papers

On The Probability Density of Spinless Particles

MohammadJavad Kazemi, Hadi Hashamipour, Mohammad Hosein Barati

In this paper, we shall derive a new conserved current for Klein-Gordon equation. The first component of this current is non-negative and reduces to $|\phi|^2$ in the non-relativistic limit. Therefore, it can be interpreted as a suitable probability density for spinless particles. In addition, this current is time-like and so prevents faster than light particle propagation. We will see the probability density has a considerable deviation from $|\phi|^2$ providing the uncertainty in momentum is much greater than $m_0c$.

•Feb 12, 2018•0 citations

Conserved Current Densities, Localization Probabilities, and a New Global Gauge Symmetry of Klein-Gordon Fields

Mostafazadeh, A., Farhad Zamani

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$, with $a\in(-1,1)$, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.

arXiv:quant-ph/0312078•Dec 9, 2003•4 citations

The Klein-Gordon equation with relativistic mass: a relativistic Schrödinger equation

P.-A. Gourdain

The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.

arXiv:2501.07798•Jan 14, 2025•0 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 ⃗ ψ

•Jan 1, 2001•1 citation

PROBABILITY CURRENT IN ZERO-SPIN RELATIVISTIC QUANTUM MECHANICS

T. FÜLÖP, T. Matolcsi

We show that the antisymmetric spinor tensor representation of spin-0 relativistic quantum mechanics provides a conserved current with positive definite timelike component, interpretable as probability density. The construction runs in complete analogy to the spin-1/2 case, and provides an analogously natural one-particle Hilbert space description for spin 0. Except for the free particle, the obtained formulation proves to be inequivalent to the one based on the Klein--Gordon equation. The second quantized version may lead to new field theoretical interaction terms for zero-spin particles.

arXiv:1712.08220•WORLD SCIENTIFIC eBooks•Apr 1, 2021•0 citations

Relativistic Quantum Mechanics

Arjun Sharma

The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.

CRC Press eBooks•Jun 18, 2021•1 citation
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ∣\ensuremathϕ∣2{|\ensuremath{\phi}|}^{2}∣\ensuremathϕ∣2 in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ∣\ensuremathϕ∣2{|\ensuremath{\phi}|}^{2}∣\ensuremathϕ∣2 only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.
In this paper, we shall derive a new conserved current for Klein-Gordon equation. The first component of this current is non-negative and reduces to ∣ϕ∣2|\phi|^2∣ϕ∣2 in the non-relativistic limit. Therefore, it can be interpreted as a suitable probability density for spinless particles. In addition, this current is time-like and so prevents faster than light particle propagation. We will see the probability density has a considerable deviation from ∣ϕ∣2|\phi|^2∣ϕ∣2 providing the uncertainty in momentum is much greater than m0cm_0cm0​c.
For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities JaμJ_a^μJaμ​, with a∈(−1,1)a\in(-1,1)a∈(−1,1), and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density Jaμ{\cal J}_a^μJaμ​ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both JaμJ_a^μJaμ​ and Jaμ{\cal J}_a^μJaμ​ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike JaμJ_a^μJaμ​ the current density Jaμ{\cal J}_a^μJaμ​ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of JaμJ_a^μJaμ​. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter aaa is rational. It is the multiplicative group of positive real numbers if aaa is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.
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 ⃗ ψ
We show that the antisymmetric spinor tensor representation of spin-0 relativistic quantum mechanics provides a conserved current with positive definite timelike component, interpretable as probability density. The construction runs in complete analogy to the spin-1/2 case, and provides an analogously natural one-particle Hilbert space description for spin 0. Except for the free particle, the obtained formulation proves to be inequivalent to the one based on the Klein--Gordon equation. The second quantized version may lead to new field theoretical interaction terms for zero-spin particles.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.