[Paper Review] Correlation functions of scalar field theories from homotopy algebras

Yuji Okawa|arXiv (Cornell University)|Mar 10, 2022

Advanced Topics in Algebra4 citations

TL;DR

This paper presents a novel algebraic framework for computing correlation functions in scalar field theories using quantum $A_{\infty}$ algebras, bypassing canonical quantization and path integrals. The method yields explicit, perturbative expressions that automatically satisfy the Schwinger-Dyson equations, as demonstrated for $\varphi^3$ theory in both Euclidean and Minkowski spacetime.

ABSTRACT

We present expressions for correlation functions of scalar field theories in perturbation theory using quantum $A_\infty$ algebras. Our expressions are highly explicit and can be used for theories both in Euclidean space and in Minkowski space including quantum mechanics. Correlation functions at a given order of perturbation theory can be calculated algebraically without using canonical quantization or the path integral, and we demonstrate it explicitly for $φ^3$ theory. We show that the Schwinger-Dyson equations are satisfied as an immediate consequence of the form of the expressions based on quantum $A_\infty$ algebras.

Motivation & Objective

• To develop a manifestly algebraic approach to perturbative correlation functions in scalar field theories, avoiding traditional quantization or path integral methods.
• To establish a direct link between quantum $A_{\infty}$ algebras and correlation functions, extending the known connection between homotopy algebras and on-shell amplitudes.
• To demonstrate that the Schwinger-Dyson equations emerge naturally from the algebraic structure of the proposed formalism.
• To provide explicit, computable expressions for correlation functions in $\varphi^3$ theory, including one-loop renormalization.
• To explore the role of the quasi-isomorphism component $\bm{f}$, particularly $\bm{f}\pi_0$, in generating correlation functions, highlighting the importance of the $\mathcal{H}^{\otimes 0}$ sector.

Proposed method

• Formalizes scalar field theories using a graded vector space $\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2$, with $\mathcal{H}_1$ for fields and $\mathcal{H}_2$ for sources, equipped with a symplectic form $\omega$.
• Represents the $A_{\infty}$ structure via a coderivation $\mathbf{Q} + \bm{m}$ on the tensor coalgebra $T\mathcal{H}$, encoding the free and interaction parts of the action.
• Introduces a quasi-isomorphism $\bm{f}$ from the cohomology of $\mathbf{Q}$ to the full $A_{\infty}$ algebra, with $\bm{f}\pi_0$ playing a key role in correlation functions.
• Derives a formula for correlation functions as a projection of the $A_{\infty}$ structure, expressed algebraically via $\bm{f}$ and the symplectic form.
• Applies the formalism to $\varphi^3$ theory, computing one-, two-, and three-point functions perturbatively using explicit algebraic contractions.
• Extends the framework to Minkowski space by defining time-ordered two-point functions via a retarded propagator with $i\epsilon$-prescription.

Experimental results

Research questions

• RQ1 Can correlation functions in scalar field theories be computed purely algebraically using quantum $A_{\infty}$ algebras without path integrals or canonical quantization?
• RQ2 How does the structure of the quasi-isomorphism $\bm{f}$, particularly $\bm{f}\pi_0$, relate to the computation of correlation functions?
• RQ3 Do the Schwinger-Dyson equations emerge as a direct consequence of the algebraic form of the correlation function formula?
• RQ4 Can the formalism be consistently extended to Minkowski spacetime and reproduce standard time-ordered correlation functions?
• RQ5 What is the role of the $\mathcal{H}^{\otimes 0}$ sector in the $A_{\infty}$ algebra framework for correlation functions?

Key findings

• The proposed formula for correlation functions in terms of quantum $A_{\infty}$ algebras reproduces the correct free theory two-point function, confirming agreement with Wick's theorem.
• For $\varphi^3$ theory, the one-, two-, and three-point functions computed algebraically match standard perturbative results, including one-loop renormalization effects.
• The Schwinger-Dyson equations are satisfied as an immediate consequence of the algebraic structure of the $A_{\infty}$ relations, specifically through the form of $\bm{f}$ and the coderivation $\mathbf{Q} + \bm{m}$.
• In Minkowski space, the two-point function is recovered as $\langle q(t_1)q(t_2)\rangle = \frac{\hbar}{2m\omega} e^{-i\omega|t_1 - t_2|}$ for $t_1 > t_2$, matching the vacuum expectation value of the time-ordered product.
• The component $\bm{f}\pi_0$ of the quasi-isomorphism is essential for generating correlation functions, indicating that the $\mathcal{H}^{\otimes 0}$ sector—often omitted in $A_{\infty}$ discussions—plays a crucial role.
• The framework provides a universal, algebraic foundation for correlation functions that could be extended to non-perturbative settings and potentially linked to Lefschetz thimbles or AdS/CFT via open string field theory.