[Paper Review] Can the quark model be relativistic enough to include the parton model?

Y. S. Kim, Marilyn E. Noz|ArXiv.org|Mar 18, 2008

Quantum Mechanics and Applications7 references3 citations

TL;DR

This paper proposes a Lorentz-covariant quark model that unifies the nonrelativistic quark model and the relativistic parton model as limiting cases. By synthesizing four key papers by Paul A. M. Dirac, the authors construct a covariant oscillator formalism that preserves Lorentz invariance, successfully reproduces the dipole behavior of the proton form factor, and explains the parton picture in high-energy frames through relativistic wave function deformation, resolving long-standing inconsistencies between quantum mechanics and special relativity in hadronic physics.

ABSTRACT

Since quarks are regarded as the most fundamental particles which constitute hadrons that we observe in the real world, there are many theories about how many of them are needed and what quantum numbers they carry. Another important question is what keeps them inside the hadron, which is known to have space-time extension. Since they are relativistic objects, how would the hadron appear to observers in different Lorentz frames? The hadron moving with speed close to that of light appears as a collection of Feynman's partons. In other words, the same object looks differently to observers in two different frames, as Einstein's energy-momentum relation takes different forms for those observers. In order to explain this, it is necessary to construct a quantum bound-state picture valid in all Lorentz frames. It is noted that Paul A. M. Dirac studied this problem of constructing relativistic quantum mechanics beginning in 1927. It is noted further that he published major papers in this field in 1945, 1949, 1953, and in 1963. By combining these works by Dirac, it is possible to construct a Lorentz-covariant theory which can explain hadronic phenomena in the static and high-speed limits, as well as in between. It is shown also that this Lorentz-covariant bound-state picture can explain what we observe in high-energy laboratories, including the parton distribution function and the behavior of the proton form factor.

Motivation & Objective

• To resolve the long-standing inconsistency between the nonrelativistic quark model and the relativistic parton model in hadronic physics.
• To construct a Lorentz-covariant bound-state formalism that consistently describes hadrons in all inertial frames, including high-speed and rest frames.
• To address the failure of standard quantum mechanics to maintain Lorentz covariance, particularly regarding time-energy uncertainty and simultaneity in relativistic bound states.
• To demonstrate that Dirac’s four seminal papers (1945, 1949, 1953, 1963) provide a unified foundation for a relativistic quantum bound-state theory.
• To show that the proton form factor, including its dipole-like behavior, emerges naturally from this covariant framework, matching experimental data without ad hoc assumptions.

Proposed method

• Combines four foundational papers by Paul A. M. Dirac (1945, 1949, 1953, 1963) to construct a unified Lorentz-covariant formalism for relativistic bound states.
• Applies a covariant oscillator formalism with wave functions satisfying Lorentz-invariant boundary and orthogonality conditions.
• Uses the time-energy uncertainty relation carefully, distinguishing it from position-momentum uncertainty to avoid inconsistencies in excited states.
• Models the proton form factor using relativistic momentum-space wave functions that maintain overlapping regions even at high momentum transfer, preventing unphysical decay.
• Incorporates spin degrees of freedom via relativistic quark wave functions that couple to orbital motion, avoiding the use of free Dirac spinors which yield incorrect form factors.
• Analyzes the Lorentz-Dirac deformation of wave functions in form factor calculations, showing that relativistic overlap preserves physical behavior where nonrelativistic models fail.

Experimental results

Research questions

• RQ1 Can a single Lorentz-covariant quantum theory describe both the quark model (at rest) and the parton model (in high-speed frames) as limiting cases?
• RQ2 How can the time-energy uncertainty relation be consistently applied in relativistic bound states without violating Lorentz invariance or excluding excited states?
• RQ3 Why do nonrelativistic models fail to reproduce the dipole-like behavior of the proton form factor at high momentum transfer, and how does relativity restore this behavior?
• RQ4 What is the role of simultaneity and time separation in defining localized probability distributions in relativistic quantum mechanics?
• RQ5 Can Dirac’s formalism be synthesized into a unified framework that reconciles the Copenhagen interpretation with Einstein’s Lorentz covariance?

Key findings

• The Lorentz-covariant bound-state formalism successfully reproduces the dipole-like behavior of the proton form factor, matching experimental data.
• Relativistic wave function deformation ensures that momentum-space wave functions maintain an overlapping region at high momentum transfer, preventing unphysical suppression seen in nonrelativistic models.
• The model explains why high-energy protons appear as partons: the same hadron exhibits different internal structures in different Lorentz frames due to relativistic wave function deformation.
• The use of free Dirac spinors leads to incorrect form factor behavior, indicating that quark spin and orbital motion must be coupled relativistically in the bound state.
• Dirac’s four papers collectively form a consistent foundation for a relativistic quantum bound-state theory, resolving the conflict between quantum mechanics and special relativity.
• The absence of overlapping wave functions in nonrelativistic calculations leads to an unacceptable form factor behavior, highlighting the necessity of relativity for accurate hadronic structure description.