[Paper Review] The Necessary and Sufficient Conditions for Transformation from Dirac Representation to Foldy-Wouthuysen Representation

В. П. Незнамов|ArXiv.org|Apr 2, 2008

Matrix Theory and Algorithms3 citations

TL;DR

This paper establishes the necessary and sufficient conditions for transforming the Dirac representation to the Foldy-Wouthuysen (FW) representation in relativistic quantum mechanics. It proves that a block-diagonal Hamiltonian transformation is necessary, while the wave function transformation law given by equation (6) is sufficient. The study shows that previous unitary transformations proposed in references [14]–[17] fail the sufficiency condition and thus are not valid FW transformations, providing a rigorous criterion for identifying correct FW-like transformations.

ABSTRACT

The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.

Motivation & Objective

• To define the precise mathematical conditions under which a transformation from the Dirac representation to the Foldy-Wouthuysen representation is valid.
• To resolve ambiguity in the literature regarding which unitary transformations qualify as true FW transformations.
• To provide a criterion that distinguishes valid FW transformations from those that do not satisfy the required wave function transformation law.
• To enable accurate calculation of matrix elements in the FW representation using normalized two-component wave functions from the Dirac representation.

Proposed method

• Derives the necessary condition: the Hamiltonian must transform into a block-diagonal form with respect to the upper and lower components of the Dirac spinor.
• Identifies the sufficient condition through the wave function transformation law expressed in equation (6), which governs how the two-component spinors transform.
• Applies the derived conditions to evaluate previously proposed unitary transformations from references [14]–[17].
• Uses the normalized two-component wave functions from the Dirac representation to compute matrix elements in the FW representation via the transformation law (6).
• Analyzes the structure of the transformation operators to verify compliance with the sufficiency condition.
• Demonstrates that transformations not satisfying equation (6) cannot be considered proper FW transformations, even if unitary.

Experimental results

Research questions

• RQ1 What conditions must be satisfied for a unitary transformation to qualify as a valid Foldy-Wouthuysen transformation from the Dirac representation?
• RQ2 Why do certain unitary transformations previously labeled as FW transformations fail to meet the criteria for being true FW transformations?
• RQ3 How can matrix elements in the Foldy-Wouthuysen representation be consistently computed using wave functions from the Dirac representation?
• RQ4 What is the precise role of the wave function transformation law in defining the FW representation?
• RQ5 Can a systematic criterion be established to distinguish correct FW transformations from incorrect ones in the literature?

Key findings

• The block-diagonal structure of the Hamiltonian in the transformed basis is a necessary condition for a valid Foldy-Wouthuysen transformation.
• The wave function transformation law described by equation (6) is the sufficient condition for a transformation to be classified as a proper FW transformation.
• Unitary transformations proposed in references [14]–[17] do not satisfy the sufficiency condition (6), and therefore are not valid Foldy-Wouthuysen transformations.
• Matrix elements in the FW representation can be reliably computed using normalized two-component wave functions from the Dirac representation, provided the transformation law (6) is applied.
• The study establishes a clear, mathematically rigorous criterion to distinguish correct FW transformations from incorrect ones, resolving longstanding ambiguity in the literature.
• The results clarify that unitarity alone is insufficient to qualify as a FW transformation—adherence to the specific wave function transformation law (6) is essential.