Using the Cornwall-Jackiw-Tomboulis effective action $Γ(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the massless (2 + 1)-dimensional Thirring model with four-component spinors is investigated in the Hartree-Fock (HF) approximation. In this case both $Γ(S)$ and its stationary (or HF) equation for the full fermion propagator $S$ are calculated in the first order of the bare coupling constant $G$. We have shown that there exist a well-defined dependence of $G\equiv G(Λ)$ on the cutoff parameter $Λ$ under which the HF equation is renormalized. In general, it has two sets, (i) and (ii), of solutions for fermion propagator corresponding to dynamical appearance of different mass terms in the model. In the case of set (i) the mass terms are Hermitian, but the solutions from the set (ii) correspond to a dynamical generation of the non-Hermitiam mass terms, i.e. to a spontaneous non-Hermiticity of the Thirring model. Despite this, the mass spectrum of the quasiparticle excitations of all non-Hermitian ground states is real. In addition, among these non-Hermitian phases there are both $\cP\cT$ symmetrical and non-symmetrical phases. Moreover, in contrast with previous investigations of this effect in other models, we have observed the spontaneous non-Hermiticity phenomenon also in the {\it massive} (2+1)-dimensional Thirring model.
Quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-time as a test for the massless Thirring model
我们分析了不同的方法来描述的量子场论的自由无质量（伪）标量场定义在1+1维时空中描述玻色子化版本的无质量Thirring模型。这些是（i）公理化的量子场论，（ii）电流代数和（iii）路径积分。我们证明了定义在Schwartz检验函数类S_0（R^2）上的自由无质量（伪）标量场的量子场论将所有这些方法联系起来。这个量子场论在怀特曼公理和怀特曼正定性条件的框架内得到了很好的定义。根据Wightman公理的要求，在S_0（R^2）而不是S（R^2）的测试函数类上定义Wightman观测量的物理意义是，与自由无质量（伪）标量场的"质心“集体运动有关的集体零模无关，它可以从关联函数的中间态中删除（Eur. J. C 24，653（2002））。在这样的理论中，由无质量（伪）标量场的位移引起的连续对称性被自发破缺，并且存在不消失的自发磁化。所得到的结果进行了讨论与科尔曼定理断言戈德斯通玻色子和自发破缺连续对称性的情况下，在量子场论定义在1+1维时空。
