We consider a whole class of theories, each of which describes dynamics of a vector field coupled to any external background field. Furthermore, each of these theories is diffeomorphism invariance, has time–time and time-space components of Hessian of the vector sector equal zero but with space–space part being non-degenerate, and the vector sector is free from Ostrogradski instability. By using the Faddeev–Jackiw constrained analysis, we derive the conditions that theories in this class have to satisfy in order for the vector sector to have $$d-1$$ d - 1 propagating degrees of freedom in $$d$$ d -dimensional spacetime. Most of these conditions are trivialised due to diffeomorphism invariance requirements. This leaves only a condition that a complicated combination of terms should not be trivially zero. Being an inequality, this condition is naturally easy to be fulfilled. We verify our result by using the Dirac constrained analysis to obtain the same result. Furthermore, we have also made a small remark on how the absence of diffeomorphism invariance affects the form of Faddeev–Jackiw brackets.

中文翻译：

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广义Proca理论的约束分析与微分同胚不变性

我们考虑一整类理论，每个理论都描述了与任何外部背景场耦合的矢量场的动力学。此外，这些理论中的每一个都是微分同胚不变性，向量扇区的 Hessian 的时-时和时-空分量为零，但空间-空间部分是非退化的，并且向量扇区没有 Ostrogradski 不稳定性。通过使用 Faddeev-Jackiw 约束分析，我们推导出此类理论必须满足的条件，以便向量扇区在 $$d$$ d 中具有 $$d-1$$ d - 1 传播自由度维时空。由于微分同胚不变性要求，这些条件中的大多数是微不足道的。这仅留下一个条件，即复杂的项组合不应微不足道地为零。作为一个不平等，这个条件自然容易满足。我们通过使用狄拉克约束分析来验证我们的结果以获得相同的结果。此外，我们还对微分同胚不变性的缺失如何影响 Faddeev-Jackiw 括号的形式做了一点评论。