[Paper Review] Gauge Invariance in Nonlocal Regularized QED

M. A. Clayton|ArXiv.org|Jul 13, 1993

Black Holes and Theoretical Physics3 citations

TL;DR

This paper establishes that the invariant measure in nonlocal regularized quantum electrodynamics (QED) exists to all orders, ensuring gauge invariance at one loop and perturbative unitarity. The method, based on a nonlocal regularization scheme from Moffat and Wood, preserves gauge symmetry and avoids anomalies in non-anomalous theories, with implications for extending the formalism to other field theories.

ABSTRACT

The existence of the invariant measure in nonlocal regularized actions is discussed. It is shown that the measure for nonlocally regularized QED, as presented in\cite{Moff-Wood}, exists to all orders, and is precisely what is required to maintain gauge invariance at one loop and guarantees perturbative unitarity. We also demonstrate how the given procedure breaks down in anomalous theories, and discuss its generalization to other actions.

Motivation & Objective

• To establish the existence of the invariant measure in nonlocal regularized QED actions.
• To demonstrate that this measure preserves gauge invariance at one-loop order.
• To ensure perturbative unitarity through the structure of the regularized action.
• To analyze the breakdown of the procedure in anomalous field theories.
• To generalize the nonlocal regularization framework to other quantum field theories.

Proposed method

• The paper employs a nonlocal regularization scheme previously proposed by Moffat and Wood, which modifies the action with nonlocal operators to suppress ultraviolet divergences.
• It analyzes the functional measure in the path integral formulation, proving its existence to all orders in perturbation theory.
• The invariance of the measure under local U(1) gauge transformations is rigorously examined in the context of one-loop diagrams.
• The authors use Feynman diagrams (with five figures generated) to analyze the structure of the regularized vertices and propagators.
• The method involves checking Ward identities and the consistency of the regularization with gauge symmetry.
• The formalism is extended to discuss its applicability in general gauge theories, particularly in non-anomalous cases.

Experimental results

Research questions

• RQ1 Does the invariant measure in nonlocal regularized QED exist to all orders in perturbation theory?
• RQ2 Can the nonlocal regularization procedure preserve gauge invariance at one-loop order?
• RQ3 Does the existence of the invariant measure guarantee perturbative unitarity in the theory?
• RQ4 How does the regularization procedure fail in anomalous quantum field theories?
• RQ5 To what extent can this nonlocal regularization scheme be generalized to other field theories?

Key findings

• The invariant measure in nonlocal regularized QED exists to all orders in perturbation theory, confirming the consistency of the regularization scheme.
• The measure preserves gauge invariance at one-loop order, ensuring that Ward identities are satisfied.
• Perturbative unitarity is guaranteed by the structure of the regularized action and the existence of the invariant measure.
• The method breaks down in anomalous theories, as expected, due to the absence of a consistent invariant measure.
• The formalism provides a viable framework for nonlocal regularization in non-anomalous gauge theories, with potential for broader application.
• The results support the viability of nonlocal regularization as a method to maintain fundamental symmetries in quantum field theories.