四阶对称张量共正性的解析表达式及其应用
Analytical expressions of copositivity for 4th order symmetric tensors and applications
摘要：
在粒子物理学中，标量势必须从下到上有界，这样物理学才有意义。检验标量势的下界的精确表达式是必不可少的，它是检验由这种标量势给出的张量的正定性和正定性的解析表达式。由于一般标量势给出的张量是四阶对称的，所以本文的工作主要集中在寻找四阶张量的共正性和正定性的精确表达式.首先，给出了4阶二维对称张量正定性的一个解析充分必要条件。对于4阶3维对称张量，利用降阶或降维的证明技巧，给出了其（严格）正性的两个解析充分条件.进一步，给出了4阶二维对称张量的一个解析的共正性充分必要条件。我们也给出了4阶2维对称张量的（严格）共正性的几个明显的分析充分条件。最后，我们应用这些结果检验了标量势的下界，并给出了两个实标量场和Higgs玻色子势的解析真空稳定性条件。
英文摘要：
In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.
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