[Paper Review] Worldline quantization of field theory, effective actions and $L_\infty$ structure

L. Bonora, M. Cvitan+4 more|arXiv (Cornell University)|Feb 8, 2018

Black Holes and Theoretical Physics23 references3 citations

TL;DR

This paper introduces a worldline quantization approach to derive the effective action of massive fermion and scalar fields coupled to external higher-spin sources, demonstrating that the resulting regularized effective action possesses an $L_\infty$ symmetry. The method systematically encodes gauge symmetries through deformation quantization, revealing that the $L_\infty$ structure arises naturally from Ward identities of current correlators, providing a non-perturbative framework for constructing consistent higher-spin field theories.

ABSTRACT

We formulate the worldline quantization of a massive fermion model coupled to external higher spin sources. We use the relations obtained in this way to show that its regularized effective action is endowed with an $L_\infty$ symmetry. The same result holds also for a massive scalar model.

Motivation & Objective

• To establish a systematic method for deriving effective actions of matter fields coupled to higher-spin sources using worldline quantization.
• To demonstrate that the resulting effective action, even in its non-local form, carries an $L_\infty$ symmetry.
• To show that this $L_\infty$ symmetry originates from Ward identities of current correlators, linking it to the dynamics of the higher-spin fields.
• To provide a robust alternative to Feynman diagram methods, which lack prior knowledge of gauge symmetry forms at high spin.
• To lay a foundation for generating consistent higher-spin field theories via integration of matter fields, with $L_\infty$ symmetry as a key organizing principle.

Proposed method

• Worldline quantization is applied to massive Dirac and scalar fields coupled to external higher-spin sources via Weyl quantization of phase space operators.
• The full action is expressed as an expectation value of operators, with momentum and field operators replacing classical variables.
• The effective action is derived by integrating out the matter fields, using the worldline path integral formalism to compute amplitudes via heuristic rules analogous to Feynman diagrams.
• The $L_\infty$ structure is uncovered by analyzing the Ward identities of current correlators, which encode the symmetry of the effective action.
• The method ensures that the full (non-local) effective action inherits the $L_\infty$ symmetry directly from the quantum consistency conditions.
• The formalism is extended to show that the same $L_\infty$ symmetry holds for both fermionic and scalar models, with symmetric $\mathcal{W}^{(n)}$ tensors in the scalar case.

Experimental results

Research questions

• RQ1 Does worldline quantization of a massive fermion model coupled to higher-spin sources yield an effective action with an $L_\infty$ symmetry?
• RQ2 Can the $L_\infty$ symmetry of the effective action be derived directly from Ward identities of current correlators without prior knowledge of the gauge transformations?
• RQ3 How does the worldline quantization method compare to standard Feynman diagram techniques in preserving and revealing gauge symmetries at high spin?
• RQ4 Is the $L_\infty$ symmetry structure robust across different matter fields, such as scalars and fermions, when coupled to the same external sources?
• RQ5 Can the $L_\infty$ symmetry in the effective action be interpreted as the dynamical origin of higher-spin gauge symmetry in a non-local, one-loop framework?

Key findings

• The regularized effective action obtained via worldline quantization of a massive fermion coupled to external higher-spin sources exhibits an $L_\infty$ symmetry.
• The $L_\infty$ symmetry is derived from the Ward identities of current correlators, which are preserved under the worldline quantization procedure.
• The same $L_\infty$ symmetry is shown to hold for a massive scalar model coupled to the same external sources, with the symmetry arising automatically due to symmetric $\mathcal{W}^{(n)}$ tensors.
• The $L_\infty$ symmetry is not just a local property but characterizes the full non-local effective action, including its non-local structure.
• The method provides a systematic way to derive the correct gauge symmetry of the effective action without guessing the form of higher-spin transformations.
• Anomalies in the Ward identities—potential obstructions to consistent higher-spin theory construction—would satisfy a consistency condition analogous to the Wess-Zumino condition, suggesting a deeper algebraic structure.