We have studied the vacuum and thermal fluctuation energies of a soliton at finite temperatures. First the vacuum energy coming from the Dirac sea is calculated by a summation of the discrete and continuum energy spectrum of the Dirac equation in the background field of a soliton. And all the divergences are removed by the same renormalization scheme at zero temperature. Then the vacuum energy and thermal fluctuation energy at finite temperatures in a temperature dependent soliton background are calculated. And the numerical results are analyzed and discussed.

中文翻译：

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孤子在有限温度下的真空和热波动能量

我们研究了孤子在有限温度下的真空和热波动能量。首先，来自狄拉克海的真空能量是通过对孤子背景场中狄拉克方程的离散能谱和连续能谱求和来计算的。并且在零温度下通过相同的重整化方案消除所有发散。然后计算了温度相关孤子背景中有限温度下的真空能量和热波动能量。并对数值结果进行了分析和讨论。