We formulate the gauge invariant Lorentz covariant Ginzburg–Landau theory which describes nonstationary regimes: relaxation of a superconducting system accompanied by eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) and also forced oscillations under the action of an external gauge field. The theory describes Lorentz covariant electrodynamics of superconductors where Anderson–Higgs mechanism occurs, at the same time the dynamics of conduction electrons remains non-relativistic. It is demonstrated that Goldstone oscillations cannot be accompanied by oscillations of charge density and they generate the transverse field only. In addition, we consider Goldstone modes and features of Anderson–Higgs mechanism in two-band superconductors. We study dissipative processes, which are caused by movement of the normal component of electron liquid and violate the Lorentz covariance, on the examples of the damped oscillations of the order parameter and the skin-effect for electromagnetic waves. An experimental consequence of the extended time-dependent Ginzburg–Landau theory regarding the penetration of the electromagnetic field into a superconductor is proposed.

我们制定了轨距不变的Lorentz协变量Ginzburg-Landau理论，该理论描述了非平稳状态：超导系统的松弛伴随着内部自由度的本征振荡（希格斯模式和戈德斯通模态），以及在外部轨距场的作用下的强迫振荡。该理论描述了发生安德森-希格斯机制的超导体的洛伦兹协变电动力学，同时传导电子的动力学仍然是非相对论的。结果表明，戈德斯通振荡不能伴随电荷密度的振荡，它们仅产生横向场。另外，我们考虑了两频带超导体中的Goldstone模式和Anderson–Higgs机制的特征。我们研究耗散过程，它们是由电子液体的正常成分的运动引起的，并违反了洛伦兹协方差，例如在阶数参数的阻尼振荡和电磁波集肤效应的示例中。提出了扩展时间相关的Ginzburg-Landau理论关于电磁场渗透到超导体中的实验结果。