[Paper Review] On the local limit of quantum field theories defined on the loop space

V. V. Belokurov, E. T. Shavgulidze|arXiv (Cornell University)|Sep 27, 2011

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper studies the local limit of a quantum field theory on loop space, showing that diffeomorphism invariance leads to convergence of Feynman diagrams in the limit $λ \to \infty$. By averaging over diffeomorphisms via a Wiener measure, the model recovers standard scalar field theory in the classical limit, while quantum amplitudes remain finite due to a built-in regularization mechanism from loop space structure.

ABSTRACT

The local limit of a quantum field theory on the loop space is studied. It is proved that the invariance of the theory with respect to the group of diffeomorphisms leads to Feynman diagrams convergence in the local limit.

Motivation & Objective

• To understand the behavior of quantum field theories defined on loop space in the local limit ($\lambda \to \infty$).
• To investigate how diffeomorphism invariance on the loop space affects the convergence of Feynman diagrams in the local limit.
• To establish a connection between loop space quantum field theory and standard scalar field theory in the classical limit.
• To demonstrate that quantum amplitudes remain finite despite the naive local limit suggesting divergence, due to memory of loop space structure.

Proposed method

• The model defines momentum space as $\mathcal{P} = C(S^1, \mathbb{R}^4)$, with fields parameterized by $p(\tau) = r + \frac{1}{\sqrt{\lambda}}\xi(\tau)$, where $\xi$ has zero average.
• The theory is invariant under $G = \mathrm{Diff}^2_+(S^1)$, acting via $gp(\tau) = p(g^{-1}(\tau)) / \sqrt{(g^{-1})'(\tau)}$.
• The Wiener measure $w_\lambda(dp) = \exp\left(-\frac{\lambda}{2}\int_{S^1}\|p'(\tau)\|^2 d\tau\right) dp$ is quasi-invariant under $G$, transforming with a Schwarz derivative factor.
• The interaction term is defined via a delta function constraint $\int_{S^1} \delta(g p_1(\tau) + \cdots + g p_4(\tau)) d\tau$, enforcing momentum conservation on the loop.
• Averaging over diffeomorphisms using a Wiener measure $w_\alpha(df)$ introduces a regularization that ensures convergence of quantum amplitudes.
• The local limit is taken by scaling $g'_{\lambda}(\tau) \to 1$, $g''_{\lambda}(\tau) \to 0$, with $g''/g' = f(\tau)/\sqrt{\lambda}$, leading to a controlled expansion in $\lambda^{-1}$.

Experimental results

Research questions

• RQ1 Does the local limit of a diffeomorphism-invariant quantum field theory on loop space yield a finite quantum field theory in the standard sense?
• RQ2 How does the memory of loop space structure affect the convergence of Feynman diagrams in the $\lambda \to \infty$ limit?
• RQ3 Can the averaging over diffeomorphisms via a Wiener measure provide a non-perturbative regularization mechanism for quantum amplitudes?
• RQ4 What is the role of the Schwarz derivative in mediating the transition from loop space to point-like field theory?
• RQ5 How does the arbitrariness in the parameter $\alpha$ of the Wiener measure relate to renormalization scale dependence in standard QFT?

Key findings

• The free action in the local limit reduces to the standard scalar field action $\int (\|r\|^2 + m^2) |\varphi(r)|^2 dr$, up to a $g$-dependent factor.
• The interaction term $\mathcal{A}_1^g$ in the local limit yields the standard $\varphi^4$ interaction, $\int \delta(r_1 + \cdots + r_4) \varphi(r_1)\cdots\varphi(r_4) dr_1\cdots dr_4$, up to a $g$-dependent factor.
• After averaging over diffeomorphisms via $w_\alpha(df)$, the fish diagram amplitude converges due to a damping factor $\exp\left(-\frac{1}{16}\|\rho\|^2 \int v^2(\tau) d\tau\right)$, ensuring integrability.
• The integral $I$ for the fish diagram is bounded by $C_2 \int \frac{1}{(\int v^2 d\tau)^2} \exp\left(-\frac{\alpha}{2} \int (v')^2 d\tau\right) dv < \infty$, proving convergence.
• The mechanism is generalizable: after integrating over loop fluctuations, each line acquires a factor $\exp\left(-\frac{1}{32}\|\rho\|^2 \int v_i^2 d\tau\right)$, ensuring convergence for all diagrams.
• The model exhibits a regularization mechanism analogous to renormalization, with the arbitrariness in $\alpha$ potentially corresponding to subtraction point dependence in standard QFT.