Affine quantization of ( φ 4 ) 4 succeeds while canonical quantization fails

Covariant scalar field quantization, nicknamed ( 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 such that r < 2 n / ( n − 2 ) can be treated by canonical quantization, while models such that r > 2 n / ( n − 2 ) are nonrenormalizable, leading to perturbative infinities, or, if treated as a unit, emerge as 'free theories'. Models such as r = 2 n / ( n − 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 > 2 n / ( n − 2 ) , which has recently correctly quantized ( φ 12 ) 3. In the present paper we show, with the aid of a Monte Carlo analysis, that one of the special cases where r = 2 n / ( n − 2 ) , specifically the case r = n = 4 , can be acceptably quantized using affine quantization.