论文标题
不确定的Dyson-Schwinger方程
Underdetermined Dyson-Schwinger equations
论文作者
论文摘要
本文研究了Dyson-Schwinger（DS）方程作为量子场理论中的计算工具的有效性。 DS方程是耦合方程的无限序列，它完全由连接的绿色功能$ g_n $满足。这些方程式链接到较高的绿色功能，如果它们被截断，则无法确定的有限方程系统。解决不确定的系统的最简单方法是将所有较高绿色的功能设置为零，然后求解最初几个绿色功能的确定系统。可以将$ g_1 $或$ g_2 $与可解决的模型中的确切结果进行比较，以查看高阶截断的精度是否提高。研究了五个$ d = 0 $型号：Hermitian $ ϕ^4 $和$ ϕ^6 $和非Hermitian $ iDcartian $ ix^3 $，$ - ϕ^4 $，以及$ ix^5 $理论。截断的ds方程给出了一系列近似值，这些近似值会缓慢收敛到限制值，但是该限制值始终与确切值的差异始终不同。基于平均场状近似值的更复杂的截断方案不会解决此强大的计算问题。
This paper examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green's functions $G_n$ of the field theory. These equations link lower to higher Green's functions and, if they are truncated, the resulting finite system of equations is underdetermined. The simplest way to solve the underdetermined system is to set all higher Green's function(s) to zero and then to solve the resulting determined system for the first few Green's functions. The $G_1$ or $G_2$ so obtained can be compared with exact results in solvable models to see if the accuracy improves for high-order truncations. Five $D=0$ models are studied: Hermitian $ϕ^4$ and $ϕ^6$ and non-Hermitian $iϕ^3$, $-ϕ^4$, and $iϕ^5$ theories. The truncated DS equations give a sequence of approximants that converge slowly to a limiting value but this limiting value always {\it differs} from the exact value by a few percent. More sophisticated truncation schemes based on mean-field-like approximations do not fix this formidable calculational problem.
