[Paper Review] Lorentz Group in Feynman's World

Y. S. Kim, Marilyn E. Noz|ArXiv.org|May 2, 2001

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper unifies Feynman's parton model with Wigner's little group formalism of the Lorentz group, showing that the parton picture emerges as a Lorentz-boosted quark model through harmonic oscillator representations. The key result is that the parton model's decoherence and entropy increase arise from the $E(2)$-like little group symmetry of massless particles, with the interaction time ratio $e^{-2\eta} \sim 10^{-6}$ at Fermilab energies, explaining the parton picture as a consequence of Lorentz covariance and internal space-time symmetry.

ABSTRACT

R. P. Feynman was quite fond of inventing new physics. It is shown that some of his physical ideas can be supported by the mathematical instruments available from the Lorentz group. As a consequence, it is possible to construct a Lorentz-covariant picture of the parton model. It is shown first how the Lorentz group can be used for studying the internal space-time symmetries of relativistic particles. These symmetries are dictated by Wigner's little groups, whose transformations leave the energy-momentum four-vector of a given particle invariant. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted that the E(2)-like symmetry of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like little group for massive particles. It is shown that the formalism can be extended to cover relativistic particles with space-time extensions, such as heavy ions and hadrons in the quark model. It is possible to construct representations of the little group using harmonic oscillators, which Feynman et al. used for studying relativistic extended hadrons. This oscillator formalism allows us to show that Feynman's parton model is a Lorentz-boosted quark model. The formalism also allows us to explain in detail Feynman's rest of the universe which is contained in his parton picture.

Motivation & Objective

• To unify Feynman's phenomenological parton model with Wigner's mathematical framework of the Lorentz group's little groups.
• To demonstrate that the parton model is a Lorentz-boosted version of the quark model using harmonic oscillator wave functions.
• To explain the physical origin of Feynman's 'rest of the universe' and decoherence in the parton picture through internal space-time symmetries.
• To show that the $E(2)$-like little group for massless particles arises as a limit of the $O(3)$-like little group for massive particles via group contraction.

Proposed method

• Use of Wigner's little groups to describe internal space-time symmetries of relativistic particles, with $O(3)$ for massive particles and $E(2)$ for massless ones.
• Application of Inonu-Wigner group contraction to derive the $E(2)$-like symmetry from the $O(3)$-like little group in the infinite-momentum or zero-mass limit.
• Construction of Lorentz-covariant harmonic oscillator wave functions that transform under the little group, enabling Lorentz boosts of internal structure.
• Use of light-cone variables and normal coordinates in coupled-oscillator systems to model the time separation between quark interactions and external probes.
• Analysis of the ratio of interaction time to oscillator period as $e^{-2\eta}$, showing it becomes $10^{-6}$ at 900 GeV, leading to decoherence.
• Interpretation of the unobserved time-separation variable as contributing to entropy increase in the parton model.

Experimental results

Research questions

• RQ1 How can Feynman's parton model be derived from the Lorentz group and Wigner's little groups?
• RQ2 What is the role of the $E(2)$-like little group in describing massless particles and its relation to the $O(3)$-like little group for massive particles?
• RQ3 How do harmonic oscillator representations of the little group provide a covariant description of relativistic hadrons?
• RQ4 What is the physical origin of the decoherence observed in the parton model, and how is it linked to Lorentz covariance?
• RQ5 How does the 'rest of the universe' in Feynman's parton picture relate to unobserved internal space-time variables?

Key findings

• The parton model is shown to be a Lorentz-boosted version of the quark model, with the internal structure of hadrons described by harmonic oscillator wave functions transforming under the little group.
• The $E(2)$-like little group for massless particles emerges as a group contraction of the $O(3)$-like little group for massive particles, consistent with the infinite-momentum or zero-mass limit.
• The ratio of interaction time to oscillator period in the parton model is $e^{-2\eta}$, which reaches $10^{-6}$ at 900 GeV proton energy, explaining why quark interactions are not resolved.
• The unobserved time-separation variable in the parton model contributes to entropy increase, linking decoherence to thermodynamic behavior in high-energy physics.
• Feynman's 'rest of the universe' corresponds to the unmeasured internal space-time degrees of freedom, which become inaccessible under Lorentz boosts.
• The formalism unifies Einstein's $E=\sqrt{m^2 + p^2}$ with Wigner's internal symmetries and Feynman's parton dynamics into a single covariant framework.