[Paper Review] Scaled Affine Quantization of Ultralocal $φ^4_2$ a comparative Path Integral Monte Carlo study with Canonical Quantization

Riccardo Fantoni, John R. Klauder|arXiv (Cornell University)|Sep 28, 2021

Matrix Theory and Algorithms4 citations

TL;DR

This paper presents a path integral Monte Carlo study comparing scaled canonical quantization (CQ) and scaled affine quantization (AQ) for the ultralocal $\varphi^4_2$ scalar field theory. While both approaches use scaling to control divergences, AQ yields a non-trivial, non-free continuum limit with a finite renormalized coupling, unlike CQ, which becomes free. This demonstrates AQ's superiority in handling non-renormalizable interactions in low dimensions.

ABSTRACT

After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, $φ^r_n$, where $r$ denotes the power of the interaction term and $n = s + 1$ with $s$ the spatial dimension and $1$ adds imaginary time, such that $r \geq 2n/(n-2)$ can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for $r>2$ and any $n$ even for the ultralocal field theory. In particular we consider the ultralocal $φ^4_2$ model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.

Motivation & Objective

• To investigate whether affine quantization can produce a non-trivial, renormalizable quantum field theory for $\varphi^4_2$ in 2D spacetime, where canonical quantization fails.
• To compare the continuum limits of scaled canonical and scaled affine quantization in the ultralocal $\varphi^4_2$ model.
• To test the hypothesis that affine quantization avoids triviality for $r>2$ interactions, even in the absence of spacetime derivatives.
• To analyze the behavior of renormalized mass and coupling constants under scaling and lattice refinement.
• To assess the role of symmetry and field tunneling in the continuum limit, particularly regarding $\varphi \to -\varphi$ symmetry preservation.

Proposed method

• Employed path integral Monte Carlo (PIMC) simulations with $10^8$ Monte Carlo steps to sample field configurations for both scaled CQ and AQ versions of the $\varphi^4_2$ model.
• Used Metropolis algorithm with adaptive field displacement to maintain acceptance ratio near 0.5, ensuring efficient sampling.
• Applied block averaging and jackknife error estimation to account for autocorrelation and estimate statistical uncertainties.
• Implemented scaling $g \to a^{s(r-2)/2}g$ with $s=1$, $r=4$, to control divergences and probe the continuum limit as $N \to \infty$ ($a=1/N$).
• Tracked renormalized mass $m_R$, coupling $g_R$, and $g_R m_R^n$ across lattice sizes to assess convergence and non-triviality.
• Monitored vacuum expectation value $\langle \varphi \rangle$ to assess symmetry behavior and tunneling through $\varphi=0$ barrier due to affine potential $\frac{3}{8}(\hbar/\varphi)^2$.

Experimental results

Research questions

• RQ1 Does scaled affine quantization produce a non-trivial, non-free continuum limit for the ultralocal $\varphi^4_2$ theory, unlike scaled canonical quantization?
• RQ2 How do the renormalized coupling constants $g_R$ behave as the lattice spacing $a \to 0$ in both CQ and AQ frameworks?
• RQ3 Can the affine quantization framework maintain a finite, non-zero $g_R$ in the continuum limit despite scaling, indicating renormalizability?
• RQ4 To what extent does the $\varphi \to -\varphi$ symmetry remain unbroken in the continuum, and how does the affine potential influence field tunneling?
• RQ5 What is the role of the Fubini-Study metric and constant negative curvature in enabling non-trivial dynamics in AQ compared to the flat space of CQ?

Key findings

• In the scaled affine quantization approach, the renormalized coupling $g_R$ does not vanish in the continuum limit ($N \to \infty$), indicating a non-trivial, non-free quantum field theory.
• In contrast, scaled canonical quantization leads to $g_R \to 0$ in the continuum limit, confirming a trivial, free theory despite $g > 0$ in the bare action.
• The renormalized mass $m_R$ approaches the physical mass $m$ as $g \to 0$ in both CQ and AQ, satisfying expected sum rules.
• The vacuum expectation value $\langle \varphi \rangle = 0$ is preserved in all simulations, consistent with $\varphi \to -\varphi$ symmetry, though spontaneous symmetry breaking is expected in the continuum limit due to exclusion of $\varphi = 0$.
• The affine effective potential term $\frac{3}{8}(\hbar/\varphi)^2$ enables tunneling through the $\varphi = 0$ barrier at finite $N$, supporting symmetry realization in the simulation.
• Despite the mathematical nature of the scaling, the results strongly suggest that un-scaled affine quantization would yield a non-trivial, renormalizable theory for $r>2$, unlike canonical quantization.