[Paper Review] Relativistic spin operator and Dirac equation

Paweł Caban, Jakub Rembieliński+1 more|arXiv (Cornell University)|Jun 14, 2012

Quantum Mechanics and Applications7 references3 citations

TL;DR

This paper establishes a direct link between the unitary representation theory of the Poincaré group and the Dirac equation for spin-1/2 particles, showing that the Foldy-Woutheysen mean-spin operator emerges naturally from this framework. It demonstrates that the Dirac equation arises from manifest Lorentz covariance in the bispinor representation, and confirms that the standard quantum field theory spin operator for fermions corresponds exactly to the Foldy-Woutheysen mean-spin operator.

ABSTRACT

We give a direct link between description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss in detail the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin $s=1/2$ corresponds to the Foldy-Woutheysen mean-spin operator.

Motivation & Objective

• To establish a rigorous connection between the abstract unitary representation of the Poincaré group and the Dirac equation for massive spin-1/2 particles.
• To clarify the nature of the relativistic spin observable in the context of quantum field theory and quantum information.
• To show that the Foldy-Woutheysen mean-spin operator is the physically consistent spin operator for Dirac particles.
• To demonstrate that the Dirac equation is a consequence of manifest Lorentz covariance in the bispinor representation.

Proposed method

• Formulate the Dirac formalism in an abstract Hilbert space that is the direct sum of positive and negative energy unitary representations of the Poincaré group.
• Introduce a covariant basis transforming under the bispinor representation of the Lorentz group, which naturally satisfies the Dirac equation.
• Define the spin basis using the standard Wigner construction for unitary representations, ensuring proper transformation under Lorentz boosts.
• Use the Foldy-Woutheysen transformation to relate the covariant basis to the spin basis, thereby diagonalizing the Dirac Hamiltonian.
• Derive the transformation properties of the spin operator under Lorentz transformations, showing it transforms via Wigner rotations.
• Explicitly compute the matrix elements of the spin operator and verify its equivalence to the Foldy-Woutheysen mean-spin operator using Dirac matrix identities.

Experimental results

Research questions

• RQ1 How can the Dirac equation be derived from the principles of manifest Lorentz covariance and unitary representation theory?
• RQ2 What is the precise relationship between the Foldy-Woutheysen mean-spin operator and the standard quantum field theory spin observable for spin-1/2 particles?
• RQ3 How does the spin operator transform under Lorentz transformations, and does it transform as a three-vector?
• RQ4 Why is the standard choice of momentum basis vectors in the Dirac formalism inconsistent with Lorentz covariance, and what is the correct alternative?
• RQ5 What is the physical meaning of the spin operator defined via the Foldy-Woutheysen transformation in the context of relativistic quantum information?

Key findings

• The Dirac equation emerges as a consequence of demanding manifest Lorentz covariance in the bispinor representation, with the covariant basis vectors satisfying the equation by construction.
• The Foldy-Woutheysen mean-spin operator is rigorously shown to be equivalent to the standard quantum field theory spin operator for s=1/2 particles.
• The spin operator transforms under Lorentz transformations according to a Wigner rotation, confirming its correct transformation behavior as a three-vector under spatial rotations.
• The choice of momentum basis vectors as $|\mathsf{\epsilon}p,\sigma\rangle$ ensures compatibility with Lorentz covariance, while the naive choice $p^{\pi}$ fails to preserve the transformation law.
• The matrix elements $\overline{v}^{\mathsf{\epsilon}}(p)\gamma^{\mu}v^{\mathsf{\epsilon}}(p) = \frac{p^{\mu}}{m}I_2$ confirm the correct normalization and current conservation.
• The spin operator's matrix elements satisfy $\overline{v}^{\mathsf{\epsilon}}(p)\gamma^5 v^{\mathsf{\epsilon}}(p) = 0$, confirming the absence of scalar components and consistency with the mean-spin definition.