关于从Maxwell-Dirac电动力学中消除旋量场的可能性
On the possibility of differential-algebraic elimination of the spinor field from the Maxwell--Dirac electrodynamics
AI总结：
研究在特定规范下Maxwell-Dirac方程中旋量场能否通过微分-代数方法消除，通过构造截断幂级数解并线性化延长系统，分析系数矩阵的秩，发现电磁场及其导数可唯一确定旋量场，电磁四势的四阶时间导数也由低阶导数决定，表明旋量场可被消除，电磁场可独立演化。
摘要：
我们研究是否可以在特定规范下从Maxwell-Dirac方程中微分-代数地消除旋量场。为此，我们构造了一个通用的截断幂级数解，并围绕该解线性化延长的Maxwell-Dirac方程组。然后分析线性化系统的系数矩阵的秩。我们的结果表明，一般来说，旋量分量由电磁场及其导数唯一确定。此外，电磁四势分量的四阶时间导数由时间的低阶导数决定。这些发现强烈表明，旋量场可以微分-代数地被消除，所得到的方程描述电磁场的独立演化，即可以仅用电磁变量来制定初值问题。
英文摘要：
We investigate whether the spinor field can be differential-algebraically eliminated from the Maxwell--Dirac equations in a particular gauge. To this end, we construct a generic truncated power-series solution and linearize the prolonged system of the Maxwell--Dirac equations about this solution. We then analyze the ranks of the coefficient matrices associated with the linearized system. Our results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. Furthermore, the fourth-order time derivatives of the components of the electromagnetic four-potential are uniquely determined by derivatives of the lower order with respect to time. These findings strongly suggest that the spinor field can be differential-algebraically eliminated, and the resulting equations describe independent evolution of the electromagnetic field, i.e., a Cauchy problem can be formulated in terms of the electromagnetic variables alone.
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