Department of Physics, Washington University, St. Louis, Missouri 63130, USA.
Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany.
Phys Rev Lett. 2023 Mar 10;130(10):101602. doi: 10.1103/PhysRevLett.130.101602.
This Letter 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 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.
这封信研究了 Dyson-Schwinger（DS）方程作为量子场论计算工具的有效性。DS 方程是一个无限序列的耦合方程，由场论的关联格林函数 G_{n}精确满足。这些方程将低阶格林函数与高阶格林函数联系起来，如果对它们进行截断，那么得到的有限方程组是欠定的。解决欠定系统最简单的方法是将所有高阶格林函数置零，然后求解得到的几个格林函数的确定系统。所得到的 G_{1}或 G_{2}可以与可解模型中的精确结果进行比较，以查看高阶截断的准确性是否提高。研究了五个 D=0 模型：厄米特ϕ^{4}和ϕ^{6}以及非厄米特 iϕ^{3}、-ϕ^{4}和 iϕ^{5}理论。截断的 DS 方程给出了一个逼近序列，它缓慢地收敛到一个极限值，但这个极限值总是与精确值相差几个百分点。基于均值场类似近似的更复杂截断方案并不能解决这个棘手的计算问题。
