[Paper Review] Fractional spin - a property of particles described with a fractional Schroedinger equation

Richard Herrmann|ArXiv.org|May 22, 2008

Experimental and Theoretical Physics Studies2 references7 citations

TL;DR

This paper proposes that particles governed by a fractional Schrödinger equation exhibit an intrinsic fractional spin, arising from invariance under spatial rotations when using fractional derivatives. The key result is that for $α \neq 1$, the Hamiltonian commutes with a fractional angular momentum operator $J^{2\alpha-1}_z$, which decomposes into standard orbital and a new fractional spin component $S^{2\alpha-1}_z = \vec{r} \times \delta\vec{p}$, where $\delta\vec{p}$ is the momentum difference between fractional and ordinary derivatives.

ABSTRACT

It is shown, that the requirement of invariance under spatial rotations reveales an intrinsic fractional extended translation-rotation-like property for particles described with the fractional Schroedinger equation, which we call fractional spin.

Motivation & Objective

• To investigate the internal symmetry structure of particles described by the fractional Schrödinger equation.
• To determine whether such particles possess intrinsic angular momentum properties beyond standard spin.
• To identify a new type of spin-like quantum number arising from fractional derivatives in the Hamiltonian.
• To establish a mathematical framework for fractional angular momentum operators that commute with the fractional Hamiltonian.
• To clarify the physical interpretation of fractional derivatives in quantum mechanics through rotational symmetry and commutation relations.

Proposed method

• The paper employs fractional calculus, using the Riemann-Liouville or Liouville-type fractional derivative $\partial^\alpha_x$ as a general operator.
• It defines generalized momentum and position operators $\hat{P}_\mu$ and $\hat{X}_\mu$ using fractional powers of $\hbar/mc$, introducing non-integer scaling in coordinate and momentum representations.
• A fractional angular momentum operator $K^\beta_z$ is introduced, with $\beta = 2\alpha - 1$ chosen so that $J^{2\alpha-1}_z$ commutes with the Hamiltonian $H^\alpha$, ensuring conservation.
• The fractional spin is defined as $S^{2\alpha-1}_z = x\delta p_y - y\delta p_x$, where $\delta p_i = i\left(\left(\hbar/mc\right)^{2\alpha-1}mc\partial^{2\alpha-1}_i - \hbar\partial_i\right)$, representing a momentum shift due to fractional derivatives.
• Commutation relations for the fractional total angular momentum are derived, showing non-standard algebraic structure dependent on $\alpha$, $\hbar$, and $mc$.
• The analysis uses the Leibniz product rule for fractional derivatives to compute commutators between $J^{2\alpha-1}_z$ and the Hamiltonian, confirming its conservation for $\alpha \neq 1$.

Experimental results

Research questions

• RQ1 Does the fractional Schrödinger equation give rise to a new type of intrinsic angular momentum not present in the standard case ($\alpha = 1$)?
• RQ2 Can a fractional spin quantum number be consistently defined via rotational symmetry and commutation relations in fractional quantum mechanics?
• RQ3 What is the algebraic structure of the fractional angular momentum operators, and how does it differ from the standard $su(2)$ algebra?
• RQ4 How does the momentum operator in the fractional framework differ from the standard one, and what is its physical interpretation?
• RQ5 What is the role of the parameter $\alpha$ in determining the nature of the internal degrees of freedom in fractional particles?

Key findings

• For $\alpha \neq 1$, the fractional Schrödinger equation describes particles with an intrinsic internal structure, revealed through non-vanishing commutators with standard angular momentum.
• The operator $J^{2\alpha-1}_z = K^{2\alpha-1}_z$ commutes with the Hamiltonian $H^\alpha$, establishing it as a conserved fractional total angular momentum.
• The fractional spin component $S^{2\alpha-1}_z$ is defined as $x\delta p_y - y\delta p_x$, where $\delta p_i$ is the difference between fractional and standard momentum operators.
• The fractional spin arises from a non-trivial transformation property under rotations, linked to the fractional derivative's non-locality and non-integer scaling.
• The commutation relations for $J^{2\alpha-1}_{x,y,z}$ exhibit a modified algebra: $[J^{2\alpha-1}_x, J^{2\alpha-1}_y] = (2\alpha-1)\frac{\hbar}{mc} J^{2\alpha-1}_z p_z^{2(\alpha-1)}$, indicating a non-standard Lie algebra structure.
• For $\alpha = 1$, the fractional spin operator vanishes, recovering the standard quantum mechanical case, confirming that fractional spin is a distinct feature for $\alpha \neq 1$.