Abstract & Overview
Covariant scalar field quantizations will be called (φr)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 where r < 2n/(n − 2) can be treated by canonical quantization, while models where r > 2n/(n−2) are trivial or, if treated as a unit, emerge as ‘free theories’. Moreover, according to canonical quantization, models where r = 2n/(n−2), e.g., r = n = 4, also become ‘free theories’, which must be considered quantum failures. However, there exists a different approach called affine quantization. This approach promotes a different set of classical variables to become the basic quantum operators and offers different results. It is well known that the canonical quantization of φ44 fails. This article addresses this failure alongside solutions to other problems.