Two types of mimetic gravity models with higher derivatives of the mimetic field are analyzed in the Hamiltonian formalism. For the first type of mimetic gravity, the Ricci scalar only couples to the mimetic field and we demonstrate the number of degrees of freedom (DOFs) is three. Then in both Einstein frame and Jordan frame, we perform the Hamiltonian analysis for the extended mimetic gravity with higher derivatives directly coupled to the Ricci scalar. We show that different from previous studies working at the cosmological perturbation level, where only three propagating DOFs show up, this generalized mimetic model, in general, has four DOFs. To understand this discrepancy, we consider the unitary gauge and find out that the number of DOFs reduces to three. We conclude that the reason why this system looks peculiar is that the Dirac matrix of all secondary constraints becomes singular in the unitary gauge, resulting in extra secondary constraints and thus reducing the number of DOFs. Furthermore, we give a simple example of a dynamic system to illustrate how gauge choice can affect the number of secondary constraints as well as the DOFs when the rank of the Dirac matrix is gauge dependent.

中文翻译：

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具有更高导数的拟态引力的哈密顿分析

在哈密顿形式主义中分析了两种具有较高模拟场导数的模拟重力模型。对于第一种模拟重力，Ricci 标量仅与模拟场耦合，我们证明自由度 (DOF) 的数量为 3。然后在爱因斯坦坐标系和乔丹坐标系中，我们对扩展的模拟引力进行哈密顿分析，其中更高的导数直接耦合到 Ricci 标量。我们表明，与之前在宇宙学扰动水平上工作的研究不同，其中只有三个传播自由度出现，这个广义模拟模型通常有四个自由度。为了理解这种差异，我们考虑了酉规并发现 DOF 的数量减少到三个。我们得出结论，这个系统看起来很奇怪的原因是所有次要约束的狄拉克矩阵在酉规范中变得奇异，导致额外的次要约束，从而减少了自由度的数量。此外，我们给出了一个动态系统的简单示例，以说明当 Dirac 矩阵的秩与规范相关时规范选择如何影响次级约束的数量以及自由度。