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.
Highlights
The family of covariant scalar field theories under consideration have classical Hamiltonians given by Z Hðπ; φÞ 1⁄41⁄2πðxÞ2 þ ð∇⃗ φÞðxÞ2 þ m2φðxÞ2þ gφðxÞr dsx; ð1Þ where the mass m > 0, the coupling constant g ≥ 0, r is the power of the interaction term, and s is the number of spatial dimensions
The focus hereafter, in this paper, is on the case r 1⁄4 2n=ðn − 2Þ, and more explicitly, we choose r 1⁄4 n 1⁄4 4.1 This model has been studied by applying canonical quantization, and the universally accepted result [3,4,5] is that this model becomes a “free model” despite the presence of the interaction and g > 0
The model on which this paper has focused is ðφ4Þ4 and normally uses canonical quantization that was the only procedure, or other procedures designed to get equivalent results
Summary
Þ gφðxÞr dsx; ð1Þ where the mass m > 0, the coupling constant g ≥ 0, r is the power of the interaction term, and s is the number of spatial dimensions As classical elements they lead to suitable equations of motion and these solutions automatically guaraRnteRe that, for T > 0, all such solutions obey the rule that. The domain of the example in (1) includes the complete set of arbitrary continuous paths, πðx; tÞ and φðx; tÞ, that determine the domain This expression for the domain is unchanged if the interaction term is excluded, while, on the other hand, the given domain will be dramatically reduced from the true free theory domain if the interaction term is sufficiently strong and has been introduced. These questions can be answered if we show the domains may be studied
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