The varational method with the Hamiltonian formalism of quantum field theory (QFT) is used to study the bound state for scalar particle and antiparticle (bosons with spin 0) in a reformulated model. We first recall some points about the two-body ground state energy solutions in 3 + 1 dimensions. In this case, some approximations are needed to obtain numerical solutions. Then, we study the upper and lower bound states in zero dimensional space for the two-body problem using simple trial states. It is shown that when the dimension of space is zero some analytical (exact) solutions can be obtained when the interactions are small.

中文翻译：

---

标量量子场论中零维空间的上下界态

使用量子场论 (QFT) 的哈密顿形式化的变分方法在重新制定的模型中研究标量粒子和反粒子（自旋为 0 的玻色子）的束缚态。我们首先回顾一下关于 3 + 1 维双体基态能量解的一些要点。在这种情况下，需要一些近似值来获得数值解。然后，我们使用简单的试验状态研究了二体问题在零维空间中的上下界状态。结果表明，当空间维数为零时，当相互作用很小时，可以获得一些解析（精确）解。