We derive conditions which are sufficient for theories consisting of multiple vector fields, which could also couple to non-dynamical external fields, to have the required structure of constraints of multi-field generalised Proca theories, so that the number of degrees of freedom is correct. The Faddeev–Jackiw constraint analysis is used and is cross-checked by Lagrangian constraint analysis. To ensure the theory is constraint, we impose a standard special form of Hessian matrix. The derivation benefits from the realisation that the theories are diffeomorphism invariance. The sufficient conditions obtained include a refinement of secondary-constraint enforcing relations derived previously in literature, as well as a condition which ensures that the iteration process of constraint analysis terminates. Some examples of theories are analysed to show whether they satisfy the sufficient conditions. Most notably, due to the obtained refinement on some of the conditions, some theories which are previously interpreted as being undesirable are in fact legitimate, and vice versa. This in turn affects the previous interpretations of cosmological implications which should later be reinvestigated.

我们导出了由多个向量场组成的理论（也可以耦合到非动态外部场）的充分条件，以具有多场广义普罗卡理论所需的约束结构，从而使自由度数正确。使用 Faddeev-Jackiw 约束分析并通过拉格朗日约束分析进行交叉检查。为了确保理论是约束性的，我们施加了 Hessian 矩阵的标准特殊形式。推导受益于理论是微分同胚不变性的认识。获得的充分条件包括对先前文献中导出的次要约束执行关系的细化，以及确保约束分析的迭代过程终止的条件。分析一些理论的例子来表明它们是否满足充分条件。最值得注意的是，由于对某些条件的改进，一些以前被解释为不受欢迎的理论实际上是合法的，反之亦然。这反过来又影响了之前对宇宙学含义的解释，以后应该重新研究这些解释。