arXiv 1402.39222014-02-17hep-th

一维标量势Dirac场的真空极化和Casimir能量

Vacuum Polarization and Casimir Energy of a Dirac Field Induced by a Scalar Potential in One Spatial Dimension

Siamak S. Gousheh, Soheila S. Mousavi, Leila Shahkarami

备注：: 16 pages, 7 figures
DOI：: 10.1103/PhysRevD.90.045027

摘要：

研究了一维Dirac场与标量场耦合的真空极化和Casimir能量。这两种效应都有一个共同的原因，即由于与背景场的耦合而导致的光谱畸变。选择势为对称方势阱，问题变得精确可解，我们可以解析地找到系统的整个谱。我们表明，总的状态数和总密度保持不变，与自由的情况相比，正如人们所期望的。此外，由于费米子的正能量和负能量本征态是彼此的费米子数共轭，并且不存在零能量束缚态，因此总密度和负能量态和正能量态的总数分别保持不变。因此，在这个模型中的真空极化是零的任何选择的参数的潜力。值得注意的是，虽然由于模型的对称性，真空极化为零，但系统的Casimir能量一般不为零。在Casimir能量作为阱深度的函数的曲线图中，当束缚能级改变方向并移回它们的起源连续区时，近似存在最大值。对于固定的深度值，Casimir能量是宽度的线性函数，并且总是正值。此外，卡西米尔能量密度（所有负能态的能量密度）和所有正能态的能量密度恰好是彼此的镜像。最后，考虑Casimir能量，计算了价费米子最低束缚态的总能量，发现对于标量势，最低束缚态几乎总是不稳定的。

英文摘要：

We investigate the vacuum polarization and the Casimir energy of a Dirac field coupled to a scalar potential in one spatial dimension. Both of these effects have a common cause which is the distortion of the spectrum due to the coupling with the background field. Choosing the potential to be a symmetrical square-well, the problem becomes exactly solvable and we can find the whole spectrum of the system, analytically. We show that the total number of states and the total density remain unchanged as compared with the free case, as one expects. Furthermore, since the positive- and negative-energy eigenstates of the fermion are fermion-number conjugates of each other and there is no zero-energy bound state, the total density and the total number of negative and positive states remain unchanged, separately. Therefore, the vacuum polarization in this model is zero for any choice of the parameters of the potential. It is important to note that although the vacuum polarization is zero due to the symmetries of the model, the Casimir energy of the system is not zero in general. In the graph of the Casimir energy as a function of the depth of the well there is a maximum approximately when the bound energy levels change direction and move back towards their continuum of origin. The Casimir energy for a fixed value of the depth is a linear function of the width and is always positive. Moreover, the Casimir energy density (the energy density of all the negative-energy states) and the energy density of all the positive-energy states are exactly the mirror images of each other. Finally, computing the total energy of a valence fermion present in the lowest fermionic bound state, taking into account the Casimir energy, we find that the lowest bound state is almost always unstable for the scalar potential.

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