[Paper Review] Effective Hamiltonians with Relativistic Corrections II: Application to Compton Scattering by a Proton

S. Scherer, G. I. Poulis+1 more|arXiv (Cornell University)|Feb 25, 1993

Quantum and Classical Electrodynamics4 references4 citations

TL;DR

This paper compares two methods for deriving effective 2×2 Hamiltonians with relativistic corrections: the Foldy-Wouthuysen transformation and direct reduction of matrix elements. Applied to proton Compton scattering, the Foldy-Wouthuysen approach reproduces the covariant result and satisfies the low-energy theorem, while the direct reduction method fails even after gauge restoration, incorrectly identifying Z-diagrams with contact terms beyond leading order.

ABSTRACT

We discuss two different methods of obtaining ``effective $2 imes 2$ Hamiltonians'' of the electromagnetic interaction which include relativistic corrections. One is the standard Foldy--Wouthuysen transformation which we compare with the Hamiltonian obtained from a direct reduction of the matrix element of the interaction Hamiltonian between positive--energy solutions of the free Dirac equation. The two approaches are applied to Compton scattering by a proton for which a low--energy theorem exists. It is found that the Foldy--Wouthuysen Hamiltonian yields the same result as a covariant calculation. This is not true for the direct reduction method which will in general lead to incorrect results even after restoring the gauge invariance property of the Hamiltonian. Furthermore, it is shown that an identification of the Z--diagrams of the usual Dirac representation with the contact graphs of the Foldy--Wouthuysen representation is incorrect beyond the order of the low--energy theorem.

Motivation & Objective

• To evaluate the consistency of effective Hamiltonians with relativistic corrections in the context of proton Compton scattering.
• To compare the Foldy-Wouthuysen transformation with direct matrix element reduction for constructing 2×2 Hamiltonians.
• To test whether both methods reproduce the established low-energy theorem for Compton scattering by a proton.
• To examine the validity of identifying Z-diagrams in the Dirac representation with contact interactions in the Foldy-Wouthuysen representation.
• To determine which method preserves gauge invariance and yields correct physical results beyond leading order.

Proposed method

• Employing the Foldy-Wouthuysen transformation to decouple positive- and negative-energy states in the Dirac equation, yielding a 2×2 effective Hamiltonian.
• Applying direct reduction of the interaction matrix element between positive-energy solutions of the free Dirac equation to obtain an alternative effective Hamiltonian.
• Comparing both effective Hamiltonians against the result from a fully covariant field-theoretic calculation.
• Testing gauge invariance restoration in the direct reduction method and assessing its physical consistency.
• Analyzing the correspondence between Z-diagrams in the Dirac representation and contact terms in the Foldy-Wouthuysen representation at higher orders.
• Using the low-energy theorem as a benchmark to validate the correctness of each method.

Experimental results

Research questions

• RQ1 Does the Foldy-Wouthuysen-transformed effective Hamiltonian reproduce the covariant result for proton Compton scattering?
• RQ2 Can the direct reduction method, even after gauge invariance restoration, yield the correct low-energy limit?
• RQ3 Is the identification of Z-diagrams in the Dirac representation with contact interactions in the Foldy-Wouthuysen representation valid beyond the leading-order low-energy theorem?
• RQ4 What are the differences in physical predictions between the two effective Hamiltonian approaches at higher orders?
• RQ5 Which method correctly describes the structure of the electromagnetic interaction in the non-relativistic limit with relativistic corrections?

Key findings

• The Foldy-Wouthuysen-transformed effective Hamiltonian correctly reproduces the result of the covariant calculation for proton Compton scattering.
• The direct reduction method fails to produce the correct result even after restoring gauge invariance, indicating fundamental flaws in the approach.
• The identification of Z-diagrams in the Dirac representation with contact terms in the Foldy-Wouthuysen representation is invalid beyond the order of the low-energy theorem.
• The low-energy theorem serves as a reliable consistency check, and only the Foldy-Wouthuysen method satisfies it.
• The failure of the direct reduction method suggests it cannot be used reliably for higher-order relativistic corrections.
• The study confirms that the Foldy-Wouthuysen transformation provides a physically consistent framework for effective Hamiltonians with relativistic corrections in proton Compton scattering.