Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unitary perturbation theory on the light cone using adiabatic switching

Stéphane Munier

TL;DR

This work compares two perturbation-theory formalisms for light-cone quantization: the standard time-ordered approach and a unitary adiabatic-switching scheme. The authors develop a systematic, diagrammatic perturbation theory with adiabatic switching, enabling normalized wave functions and a clear Gell-Mann–Low framework, and illustrate its machinery in a simple quantum-mechanical model and in a massive scalar φ^3 field theory at one loop. They demonstrate that, after proper renormalization (e.g., MS-bar in d = 4 and d = 6), the adiabatic prescription yields well-defined wave functions whose phases encode energy shifts, while preserving unitarity at each order and matching standard perturbation theory where applicable. The results highlight the benefits of a fully diagrammatic, unitary LCPT approach and outline extensions toward infrared structure and gauge theories like QED/QCD, potentially improving perturbative organization and unitarity in high-energy scattering on the light cone.

Abstract

Light-cone perturbation theory is a powerful tool for calculating high-energy scattering amplitudes, particularly for quantum particles such as electrons, photons, or protons scattering off heavy nuclei, a process analogous to potential scattering. Central to these computations are the light-cone wave functions of incoming and outgoing particles, representing the projection of dressed initial and final states onto partonic Fock states. The dressed states are obtained by applying an evolution operator in the Dirac picture to bare partonic states, which may be interpreted physically as a time evolution from preparation to interaction. In standard approaches, a non-unitary operator is used, and proper normalization is imposed a posteriori. Here, we systematically develop perturbation theory from a perturbatively unitary evolution operator, using adiabatic switching to regularize the infinite-time limits. This provides a theoretically coherent framework for organizing calculations, reproducing known results entirely diagrammatically without enforcing unitarity by hand. We illustrate the method with a simple quantum mechanical model, enabling calculations to arbitrary perturbative orders, and then evaluate wave functions in field theories quantized on the light cone, focusing on a massive scalar theory with cubic interaction at one-loop accuracy.

Unitary perturbation theory on the light cone using adiabatic switching

TL;DR

This work compares two perturbation-theory formalisms for light-cone quantization: the standard time-ordered approach and a unitary adiabatic-switching scheme. The authors develop a systematic, diagrammatic perturbation theory with adiabatic switching, enabling normalized wave functions and a clear Gell-Mann–Low framework, and illustrate its machinery in a simple quantum-mechanical model and in a massive scalar φ^3 field theory at one loop. They demonstrate that, after proper renormalization (e.g., MS-bar in d = 4 and d = 6), the adiabatic prescription yields well-defined wave functions whose phases encode energy shifts, while preserving unitarity at each order and matching standard perturbation theory where applicable. The results highlight the benefits of a fully diagrammatic, unitary LCPT approach and outline extensions toward infrared structure and gauge theories like QED/QCD, potentially improving perturbative organization and unitarity in high-energy scattering on the light cone.

Abstract

Light-cone perturbation theory is a powerful tool for calculating high-energy scattering amplitudes, particularly for quantum particles such as electrons, photons, or protons scattering off heavy nuclei, a process analogous to potential scattering. Central to these computations are the light-cone wave functions of incoming and outgoing particles, representing the projection of dressed initial and final states onto partonic Fock states. The dressed states are obtained by applying an evolution operator in the Dirac picture to bare partonic states, which may be interpreted physically as a time evolution from preparation to interaction. In standard approaches, a non-unitary operator is used, and proper normalization is imposed a posteriori. Here, we systematically develop perturbation theory from a perturbatively unitary evolution operator, using adiabatic switching to regularize the infinite-time limits. This provides a theoretically coherent framework for organizing calculations, reproducing known results entirely diagrammatically without enforcing unitarity by hand. We illustrate the method with a simple quantum mechanical model, enabling calculations to arbitrary perturbative orders, and then evaluate wave functions in field theories quantized on the light cone, focusing on a massive scalar theory with cubic interaction at one-loop accuracy.

Paper Structure

This paper contains 38 sections, 151 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Perturbation theory
  3. Theoretical setup
  4. Standard prescription
  5. Formulation
  6. Constructing eigenstates of $H$ from the perturbative series
  7. Fock state expansion for normalized states
  8. Alternative calculation
  9. Adiabatic switching
  10. Formulation
  11. Perturbative series
  12. The Gell-Mann and Low theorem
  13. Energy shift from the phase of the dressed state
  14. Fock state expansion
  15. State vector of a particle in models with no space dimension
  16. ...and 23 more sections

Figures (3)

  • Figure 1: Order $\bar{\lambda}_R^2$ contributions to the $1\to 1$ wave function. The number "2" above the counterterm vertex represents its order in powers of $\bar{\lambda}_R$.
  • Figure 2: Order $\bar{\lambda}_R^3$ contributions to the $\phi\to\varphi\varphi$ wave function.
  • Figure 3: One-loop vertex correction diagrams.