[Paper Review] Remarks on Defining the DLCQ of Quantum Field Theory as a Light-Like Limit

Adel Bilal|ArXiv.org|Sep 30, 1999

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper investigates defining Discrete Light-Cone Quantization (DLCQ) in quantum field theory as a light-like limit by compactifying spacetime on a space-like circle of vanishing radius. Using Feynman α-parameter integrals, it shows that at one-loop order, the limit is finite and generically replaces one α-integral with a discrete sum, analogous to string theory, though divergences may occur in certain theories or at higher loops.

ABSTRACT

The issue of defining discrete light-cone quantization (DLCQ) in field theory as a light-like limit is investigated. This amounts to studying quantum field theory compactified on a space-like circle of vanishing radius in an appropriate kinematical setting. While this limit is unproblematic at the tree-level, it is non-trivial for loop amplitudes. In one-loop amplitudes, when the propagators are written using standard Feynman $α$-parameters we show that, generically, in the limit of vanishing radius, one of the $α$-integrals is replaced by a discrete sum and the (UV renormalized) one-loop amplitude has a finite light-like limit. This is analogous to what happens in string theory. There are however exceptions and the limit may diverge in certain theories or at higher loop order. We give a rather detailed analysis of the problems one might encounter. We show that quantum electrodynamics at one loop has a well-defined light-like limit.

Motivation & Objective

• To clarify the mathematical and physical consistency of defining DLCQ in quantum field theory as a light-like limit.
• To analyze the behavior of loop amplitudes—particularly one-loop amplitudes—when the compactification circle radius approaches zero.
• To determine under what conditions the light-like limit remains finite, especially in contrast to tree-level behavior.
• To identify exceptions where the limit may diverge, particularly in specific field theories or at higher loop orders.
• To establish a framework analogous to string theory’s behavior in similar limits, using α-parameter techniques.

Proposed method

• Formalizing DLCQ as the limit of quantum field theory compactified on a space-like circle with radius approaching zero.
• Expressing one-loop Feynman propagators using standard Feynman α-parameter representations.
• Analyzing the behavior of α-integrals in the limit of vanishing circle radius, showing one integral becomes a discrete sum.
• Applying UV renormalization techniques to ensure finiteness of the amplitude in the light-like limit.
• Comparing the result to analogous behavior in string theory to highlight structural similarities.
• Identifying conditions under which the limit may diverge, based on the structure of the theory and loop order.

Experimental results

Research questions

• RQ1 Can DLCQ in quantum field theory be consistently defined as a light-like limit via compactification on a vanishing space-like circle?
• RQ2 How do one-loop amplitudes behave in the limit of vanishing compactification radius, and is the limit finite?
• RQ3 What role do Feynman α-parameter integrals play in ensuring finiteness of the amplitude in the light-like limit?
• RQ4 In which field theories or at what loop orders does the light-like limit fail to exist due to divergence?
• RQ5 How does the behavior of the one-loop amplitude in this limit compare to that in string theory?

Key findings

• At one-loop order, the light-like limit of the amplitude is finite when using Feynman α-parameter integrals.
• In the limit of vanishing circle radius, one α-integral is replaced by a discrete sum, a key structural change.
• The UV-renormalized one-loop amplitude remains finite in the light-like limit, provided the theory is well-behaved.
• Quantum electrodynamics at one loop has a well-defined light-like limit, as confirmed by the analysis.
• Divergences may occur in certain field theories or at higher loop orders, indicating the limit is not universally finite.
• The result mirrors behavior seen in string theory, suggesting a deeper analogy between DLCQ in QFT and string theory in similar kinematical limits.