Hamiltonian formalism for f(T) gravity
Hamiltonian formalism for f(T) gravity
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DOI：
10.1103/physrevd.97.104028
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发表时间：
2018-02
影响因子：
5
通讯作者：
R. Ferraro;M. J. Guzm'an
中科院分区：
文献类型：
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作者：
R. Ferraro;M. J. Guzm'an
我们给出了$f（T）$引力的哈密顿形式，并证明了该理论在$n$维中有$\frac{n（n-3）}{2}+1$$自由度.我们开始从标量张量的理论行动，这代表了一个标量场最低限度地与扭转标量T$，定义的TEGR拉格朗日耦合。$T$被写为非完整性的Vierbein系数的二次形式。通过对正则动量定义中涉及的多指标矩阵特征值结构的分析，得到了正则动量的主要约束条件。当与TEGR情况相比时，标量场产生一个额外的主要约束。次级约束是超哈密顿量和超动量约束，它们是从GR的ADM公式中保留下来的。有一组$\frac{n（n-1）}{2}$初级约束代表理论的局部洛伦兹变换，它们可以组合形成一组$\frac{n（n-1）}{2}-1 $第一类约束，而其中一个成为第二类。由于约束之间的括号矩阵的结构，该结果与维数无关。由于这种局部洛伦兹破坏，第一类正则哈密顿量被修改，并且唯一一个成为第二类的局部洛伦兹变换与第二类约束$\pi \approx 0$配对，以从$n^2+1$$对正则变量中删除一个自由度。剩下的$2n-1+\frac{n（n-1）}{2} - 1 $主约束删除了相同数量的自由度，使理论具有$\frac{n（n-3）}{2}+1 $自由度。这意味着$f（T）$引力只有一个额外的自由度，可以解释为标量自由度。
We present the Hamiltonian formalism for $f(T)$ gravity, and prove that the theory has $\frac{n(n-3)}{2}+1$ degrees of freedom in $n$ dimensions. We start from a scalar-tensor action for the theory, that represents a scalar field minimally coupled with the torsion scalar $T$ that defines the TEGR Lagrangian. $T$ is written as a quadratic form of the coefficients of anholonomy of the vierbein. We obtain the primary constraints through the analysis of the structure of the eigenvalues of the multi-index matrix involved in the definition of the canonical momenta. The auxiliar scalar field generates one extra primary constraint when compared with the TEGR case. The secondary constraints are the super-Hamiltonian and super-momenta constraints, that are preserved from the ADM formulation of GR. There is a set of $\frac{n(n-1)}{2}$ primary constraints that represent the local Lorentz transformations of the theory, which can be combined to form a set of $\frac{n(n-1)}{2}-1$ first class constraints, while one of them become second class. This result is irrespective of the dimension, due to the structure of the matrix of the brackets between the constraints. The first-class canonical Hamiltonian is modified due to this local Lorentz violation, and the only one local Lorentz transformation that becomes second class pairs up with the second class constraint $\pi \approx 0$ to remove one degree of freedom from the $n^2+1$ pairs of canonical variables. The remaining $2n-1+\frac{n(n-1)}{2} - 1 $ primary constraint remove the same number of degrees of freedom, leaving the theory with $\frac{n(n-3)}{2}+1$ degrees of freedom. This means that $f(T)$ gravity has only one extra degree of freedom, which could be interpreted as a scalar degree of freedom.