Perturbation theory in large order
Perturbation theory in large order
复制标题
大阶微扰理论
DOI：
10.1016/0001-8708(78)90039-7
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发表时间：
1978
影响因子：
1.7
通讯作者：
C. Bender
中科院分区：
文献类型：
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作者：
C. Bender
对于许多量子力学模型，大阶微扰理论的行为是非常简单的。例如，在定义为-″+（x 24 4+ ϵx 4 4−E) y= 0, y（±∞）= 0的量子非谐振子中，基态能量E（基态）~∑n= 0∞A n的展开中的摄动系数A n显著地简化为n→∞:A n ~−(6 π 3) 1 2 Гn+ 1 2）。我们用应用数学的方法研究了量子力学中微扰理论的性质，并证明了它的大阶行为是由理论的半经典内容决定的。在量子场论中，微扰系数是通过对费曼图求和来计算的。我们在一个简单的λ φ 4模型中给出了一个统计过程，将所有图的集合求和为顶点数→∞。最后，我们讨论了量子电动力学中微扰理论的大阶行为与电子电荷α值之间的联系。
For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″+(x 2 4+ ϵx 4 4− E) y= 0, y (±∞)= 0, the perturbation coefficients A n in the expansion for the ground-state energy E (ground state)∼∑ n= 0∞ A n ϵ n simplify dramatically as n→∞: A n∼−(6 π 3) 1 2 Гn+ 1 2). We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λϕ 4 model for summing the set of all graphs as the number of vertices→∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.