[Paper Review] Hydrodynamics of Spinning Particles

Giovanni Salesi, Luis C. Kretly+2 more|ArXiv.org|Feb 15, 1998

Rheology and Fluid Dynamics Studies3 references21 citations

TL;DR

This paper proposes that the quantum potential in non-relativistic quantum mechanics arises from the internal Zitterbewegung (Zitter) motion of spinning particles, decomposing the total velocity into center-of-mass (classical) and internal (quantum) components. By expressing the kinetic energy in terms of this decomposition and identifying the non-classical term with the internal motion, the authors derive the quantum potential without stochastic assumptions, showing that spin and Zitterbewegung are fundamental to quantum behavior, with ℏ naturally emerging as twice the spin magnitude.

ABSTRACT

In this note, we first obtain the decomposition of the non-relativistic field velocity into the classical part (i.e., the velocity w=p/m OF the center-of-mass (CM), and the so-called quantum part (i.e., the velocity V of the motion IN the CM frame (namely, the internal spin-motion or Zitterbewegung), these two parts being orthogonal. Our starting point is the Pauli current. Then, by inserting such a composite expression of the velocity into the kinetic energy term of the non-relativistic newtonian lagrangian, we get the appearance of the so-called "quantum potential" (which makes the difference between classical and quantum behaviour) as a pure consequence of the internal motion. Such a result carries further evidence about the possibility that the quantum behaviour of micro-systems be a direct consequence of the fundamental existence of spin.

Motivation & Objective

• To clarify the physical origin of the quantum potential in the Madelung fluid formulation of quantum mechanics.
• To demonstrate that the non-classical term in the Schrödinger equation arises from internal motion (Zitterbewegung) in the center-of-mass frame.
• To show that spin is the fundamental source of quantum behavior, with ℏ emerging from the spin vector magnitude.
• To provide a non-stochastic, classical-geometric derivation of the quantum potential using velocity decomposition and tensor algebra.

Proposed method

• Decompose the field velocity into center-of-mass velocity w and internal Zitterbewegung velocity V, using a Gordon-like decomposition in tensor algebra.
• Express the kinetic energy term in the non-relativistic Lagrangian as the sum of center-of-mass and internal motion contributions.
• Identify the non-classical term (7) in the Lagrangian as the kinetic energy of the internal Zitter motion, leading to the quantum potential (6).
• Use the König theorem to justify the additive nature of internal and external kinetic energies in the Lagrangian.
• Derive the relation |s| = ℏ/2 from the condition that the internal velocity V² matches the non-classical energy term, linking spin to ℏ.
• Assume the smallness of the small component in the Dirac bispinor to justify ∇ρ·s ≈ 0, simplifying the internal velocity expression to V² = s²(∇ρ/mρ)².

Experimental results

Research questions

• RQ1 Can the quantum potential in the Madelung formulation be derived from a classical internal motion rather than postulated?
• RQ2 Is the origin of the non-classical term in the Schrödinger equation traceable to spin-induced Zitterbewegung?
• RQ3 Does the value of Planck’s constant ℏ emerge naturally from the magnitude of the spin vector s?
• RQ4 What is the physical interpretation of the velocity decomposition v = w + V in the context of spinning particles?
• RQ5 How does the internal motion in the CM frame relate to the quantum potential and the total energy in the Lagrangian?

Key findings

• The non-classical term in the Lagrangian, responsible for the quantum potential, is identified as the kinetic energy of the internal Zitterbewegung motion in the center-of-mass frame.
• The internal velocity V is shown to satisfy V² = s²(∇ρ/mρ)² under the condition ∇ρ·s ≈ 0, which holds in the non-relativistic limit.
• The relation |s| = ℏ/2 is derived from the requirement that the internal kinetic energy matches the non-classical term, linking spin magnitude directly to Planck’s constant.
• When spin s = 0, the quantum potential vanishes, reducing the Hamilton–Jacobi equation to a classical Newtonian form.
• The quantum potential is not postulated but emerges naturally from the internal motion, providing a physical basis for its origin without stochastic assumptions.
• The value ℏ is not fundamental but emerges as ℏ = 2|s|, suggesting that spin is the true origin of quantum behavior.