Affine quantization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>φ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mn>4</mml:mn></mml:msub></mml:math> succeeds while canonical quantization fails
Riccardo Fantoni, John R. Klauder
Abstract
Covariant scalar field quantization, nicknamed $({\ensuremath{\varphi}}^{r}{)}_{n}$, where $r$ denotes the power of the interaction term and $n=s+1$ where $s$ is the spatial dimension and 1 adds time. Models such that $r<2n/(n\ensuremath{-}2)$ can be treated by canonical quantization, while models such that $r>2n/(n\ensuremath{-}2)$ are nonrenormalizable, leading to perturbative infinities, or, if treated as a unit, emerge as `free theories'. Models such as $r=2n/(n\ensuremath{-}2)$, e.g., $r=n=4$, again using canonical quantization also become `free theories', which must be considered quantum failures. However, there exists a different approach called affine quantization that promotes a different set of classical variables to become the basic quantum operators and it offers different results, such as models for which $r>2n/(n\ensuremath{-}2)$, which has recently correctly quantized $({\ensuremath{\varphi}}^{12}{)}_{3}$. In the present paper we show, with the aid of a Monte Carlo analysis, that one of the special cases where $r=2n/(n\ensuremath{-}2)$, specifically the case $r=n=4$, can be acceptably quantized using affine quantization.