[Paper Review] The unified quantum wave equation

Arbab I. Arbab|arXiv (Cornell University)|May 14, 2011

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper proposes a unified quantum wave equation (UQWE) using quaternions to derive the Dirac, Klein-Gordon, and Schrödinger equations as special cases. The UQWE describes spin-0 and spin-1/2 particles as wavepackets undergoing periodic creation and annihilation, with the process governed by a time-translation symmetry and a characteristic timescale of ℏ/(m₀c²), reconciling quantum behavior with relativistic wave dynamics.

ABSTRACT

The quaterionic formulation of quantum mechanics yields the unified quantum wave equation (UQWEs). From these equations, Dirac, Klein - Gordon and Schrodinger equations can be derived. While the UQWEs represent a matter wave (de Broglie), the Maxwell equations represent a transverse wave (field). Owing to UQWEs, the spin-0 and spin-1/2 particle are described by a wavepacket consisting of waves traveling to the left and to the right with speed of light. UQWEs show that spin-0 and spin-1/2 are in continuous states of creation and annihilation that are compatible with Heisenberg uncertainty relation. The creation - annihilation process is a result of the time translation property of the particle wavefunction. These are $E'=E-im_0c^2$ and $E'=E\pm m_0c^2$, for Klein-Gordon' and Dirac' particles, respectively. It is found that $\frac{\hbar}{m_0c^2}$ is the period of the creation -annihilation process.

Motivation & Objective

• To develop a unified framework for quantum wave equations that generalizes and unifies the Dirac, Klein-Gordon, and Schrödinger equations.
• To resolve the issue of non-positive definite probability density in the Klein-Gordon equation by deriving it from a more fundamental wave equation.
• To explain the origin of particle spin and the creation-annihilation process in quantum particles using wavepacket dynamics and time-translation symmetry.
• To establish a formal analogy between quantum wave equations and Maxwell’s equations, distinguishing scalar (longitudinal) and vector (transverse) wave components.
• To show that the Dirac and Klein-Gordon equations emerge from a massless wave equation via energy translations (E′ = E ± m₀c² and E′ = E − im₀c²), linking them to gauge-like transformations.

Proposed method

• Formulate the unified quantum wave equation (UQWE) using quaternions, introducing scalar (ψ₀) and vector (ψ) fields as fundamental wave components.
• Derive the UQWE from a system of three coupled equations: ∇·ψ − (1/c²)(∂ψ₀/∂t) − (m₀/ℏ)ψ₀ = 0, ∇ψ₀ − (∂ψ/∂t) − (m₀c²/ℏ)ψ = 0, and ∇×ψ = 0.
• Eliminate auxiliary fields to obtain second-order wave equations for ψ₀ and ψ: (1/c²)(∂²ψ₀/∂t²) − ∇²ψ₀ + 2(m₀/ℏ)(∂ψ₀/∂t) + (m₀c/ℏ)²ψ₀ = 0 and similarly for ψ.
• Apply the transformation ψ₀ = exp(−m₀c²t/ℏ)φ to reduce the damped equation to the standard wave equation ∂²φ/∂t² − c²∇²φ = 0.
• Use the Arbab-Widatallah complex transformation m₀ → iβm₀ to derive the Dirac equation from the UQWE, showing equivalence to a massless wave equation under a modified gradient ∇′ = ∇ + i(m₀cβ/ℏ)α.
• Demonstrate that the creation-annihilation process arises from time-translation symmetry, with a period T = ℏ/(m₀c²), and that spin arises from the rotational motion of counter-propagating waves in the wavepacket.

Experimental results

Research questions

• RQ1 Can a single wave equation unify the Dirac, Klein-Gordon, and Schrödinger equations through a quaternionic formalism?
• RQ2 How does the UQWE explain the creation and annihilation of spin-0 and spin-1/2 particles in a way consistent with the Heisenberg uncertainty principle?
• RQ3 What is the physical origin of spin in the Dirac particle, and how does it emerge from wavepacket dynamics?
• RQ4 How do energy translations (E′ = E ± m₀c² and E′ = E − im₀c²) relate to the emergence of the Dirac and Klein-Gordon equations from a massless wave equation?
• RQ5 What is the role of the vector field ψ in the UQWE, and how does it differ from standard quantum mechanical formulations?

Key findings

• The UQWE yields the Dirac, Klein-Gordon, and Schrödinger equations as limiting cases through specific transformations, demonstrating a unified foundation for quantum wave mechanics.
• The creation-annihilation process for both spin-0 and spin-1/2 particles is periodic, with a characteristic timescale of ℏ/(m₀c²), derived from time-translation symmetry of the wavefunction.
• The wavefunction of a Dirac particle is a superposition of left- and right-moving waves at speed c, forming a wavepacket of length L = ℏ/(m₀c), consistent with the Compton wavelength.
• Spin-1/2 particles exhibit long-range interactions due to oscillatory wave nature, while spin-0 particles have short-range interactions due to exponential damping, consistent with Yukawa’s theory.
• The transformation ∇′ = ∇ + i(m₀cβ/ℏ)α maps the UQWE to a massless wave equation, showing that both Dirac and Klein-Gordon particles arise from a common massless origin via gauge-like shifts.
• The spin angular momentum of the electron is derived as S = (1/2)ℏ from the rotational motion of two counter-propagating waves, each with mass m₀/2, at radius r = L/2, confirming the quantum value.