平凡性、可重整性与禁闭
Triviality, Renormalizability and Confinement
摘要：
根据最近的结果，四维Gell-Mann - Low函数β（g）是非交错的，并且在无穷远处具有线性渐近性。根据Bogoliubov和Shirkov分类，这意味着在大距离上构造有限相互作用的连续理论的可能性。这一结论与晶格结果明显矛盾，晶格结果表明了量子场论的平凡性。这一矛盾通过Rw^4理论中可重整化的一个特殊性质得以解决：为了得到连续的重整化理论，不需要从裸理论中消去一个格。事实上，这种重正化不是偶然的，而是可以在Wilson的多参数重整化群的框架下理解的。这些思想在QCD中的应用表明，威尔逊的禁闭理论不是纯粹的说明性的，而是与实际情况有直接的关系。因此，禁闭和质量间隙的分析证明问题可以被认为是解决了，至少在物理严格水平上。
英文摘要：
According to recent results, the Gell-Mann - Low function β(g) of four-dimensional ϕ^4 theory is non-alternating and has a linear asymptotics at infinity. According to the Bogoliubov and Shirkov classification, it means possibility to construct the continuous theory with finite interaction at large distances. This conclusion is in visible contradiction with the lattice results indicating triviality of ϕ^4 theory. This contradiction is resolved by a special character of renormalizability in ϕ^4 theory: to obtain the continuous renormalized theory, there is no need to eliminate a lattice from the bare theory. In fact, such kind of renormalizability is not accidental and can be understood in the framework of Wilson's many-parameter renormalization group. Application of these ideas to QCD shows that Wilson's theory of confinement is not purely illustrative, but has a direct relation to a real situation. As a result, the problem of analytical proof of confinement and a mass gap can be considered as solved, at least on the physical level of rigor.
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