[Paper Review] The wedge form of relativistic dynamics

A. Makhlin|ArXiv.org|Aug 7, 1996

High-Energy Particle Collisions Research1 references3 citations

TL;DR

This paper proposes a novel formulation of relativistic dynamics in high-energy collisions by confining quantum field theory interactions within the light-cone wedge, where Lorentz contraction localizes the interaction. It introduces a boundary condition on the light cone, derives one-particle states and propagators for scalar and fermion fields, and establishes a connection between inclusive processes and temporal evolution order, offering a new framework for ultra-relativistic scattering theory.

ABSTRACT

It is commonly accepted that in hadronic or nuclear collisions at extremely high energies the shortest scales are explored. At the classical level, this property of the interaction is closely related to the Lorentz contraction of the fields of colliding particles which provides instantaneous switching the interaction on. I argue that the underlying quantum dynamics should be confined to within the light wedge of the two-dimensional plane where the first interaction takes place and suggest to include this property as the boundary condition for the quantum field theory which describes the collision process. Connection between the type of inclusive process and the temporal order of its dynamical evolution is discussed. The one-particle states and propagators of the perturbation theory for the scalar and fermion fields are found.

Motivation & Objective

• To address the challenge of describing ultra-relativistic particle collisions where Lorentz contraction localizes interactions to short spacetime scales.
• To propose a boundary condition based on the light-cone wedge as a fundamental constraint on quantum field dynamics in high-energy processes.
• To clarify the relationship between the temporal order of dynamical evolution and the type of inclusive process observed.
• To derive one-particle states and perturbative propagators for scalar and fermion fields under the wedge constraint.
• To provide a new theoretical framework for relativistic scattering that respects the causal structure of high-energy collisions.

Proposed method

• Formulate quantum field theory with a boundary condition restricting dynamics to the light-cone wedge in a 2D spacetime plane.
• Apply Lorentz invariance and relativistic causality to define the wedge as the region where the first interaction occurs.
• Use canonical quantization to derive one-particle states and time-ordered propagators for scalar and spinor fields.
• Analyze the temporal evolution of matrix elements to relate ordering to inclusive process types.
• Employ perturbation theory within the wedge-restricted framework to compute propagators and transition amplitudes.
• Utilize the structure of the light cone to simplify the dynamics and enforce instantaneous switching of interaction.

Experimental results

Research questions

• RQ1 How can quantum field theory be constrained to the light-cone wedge to model ultra-relativistic collisions?
• RQ2 What is the role of the temporal order of interaction in determining the nature of inclusive processes?
• RQ3 How do one-particle states and propagators transform under the wedge boundary condition?
• RQ4 Can the wedge formulation consistently describe both scalar and fermionic fields in high-energy dynamics?
• RQ5 What is the causal and dynamical significance of restricting dynamics to the light-cone region?

Key findings

• The paper derives explicit expressions for one-particle states and time-ordered propagators of scalar and fermion fields under the wedge boundary condition.
• The temporal evolution of matrix elements is shown to depend on the order of interactions, linking this order to the type of inclusive process.
• The wedge constraint naturally enforces instantaneous switching of interaction due to Lorentz contraction, aligning with classical expectations.
• The framework provides a consistent quantum field theory formulation confined to the causal region of the first interaction.
• The method yields a well-defined perturbative structure for scattering amplitudes within the wedge, preserving relativistic invariance.
• The results suggest a new way to model high-energy collisions by focusing dynamics on the light-cone geometry.