Title: Quantum Cosmology of $f(R,T)$ gravity Show Chinese title

Title: Quantum Cosmology of $f(R,T)$ gravity

Abstract: Modified gravity theories have the potential of explaining the recent acceleration of the Universe without resorting to the mysterious concept of dark energy. In particular, it has been pointed out that matter-geometry coupling may be responsible for the recent cosmological dynamics of the Universe, and matter itself may play a more fundamental role in the description of the gravitational processes that usually assumed. We study the quantum cosmology of the $f(R,T)$ gravity theory, in which the effective Lagrangian of the gravitational field is given by an arbitrary function of the Ricci scalar, and the trace of the matter energy-momentum tensor, respectively. For the background geometry we adopt the Friedmann--Robertson--Walker metric, and we assume that matter content of the Universe consists of a perfect fluid. We obtain the general form of the gravitational Hamiltonian, of the quantum potential, and of the canonical momenta, respectively. This allows us to formulate the full Wheeler-de Witt equation describing the quantum properties of this modified gravity model. As a specific application we consider in detail the quantum cosmology of the $f(R,T)=F^0(R)+\theta RT$ model, in which $F^0(R)$ is an arbitrary function of the Ricci scalar, and $\theta $ is a function of the scale factor only. The Hamiltonian form of the equations of motion, and the Wheeler-de Witt equations are obtained, and a time parameter for the corresponding dynamical system is identified, which allows to formulate the Schr\"{o}dinger--Wheeler--de Witt equation for the quantum-mechanical description of the model under consideration. A perturbative approach for the study of this equation is developed, and the energy levels of the Universe are obtained by using a twofold degenerate perturbation approach. A second quantization approach for the description of quantum time is also proposed, and briefly discussed. Show Chinese abstract

Abstract: 修改引力理论有可能在不借助神秘的暗能量概念的情况下解释宇宙的近期加速膨胀。 特别是，已经指出物质-几何耦合可能是最近宇宙宇宙学动力学的原因，物质本身可能在描述通常所假设的引力过程中起着更基本的作用。 我们研究了$f(R,T)$引力理论的量子宇宙学，其中引力场的有效拉格朗日量分别由里奇标量和物质能动张量的迹的任意函数给出。 对于背景几何，我们采用弗里德曼-罗伯逊-沃尔克度规，并假设宇宙的物质内容由完美流体组成。 我们分别得到了引力哈密顿量、量子势和规范动量的一般形式。 这使我们能够制定完整的惠勒-德维特方程，以描述这种修改引力模型的量子特性。 作为具体应用，我们详细考虑了$f(R,T)=F^0(R)+\theta RT$模型的量子宇宙学，其中$F^0(R)$是里奇标量的任意函数，$\theta $仅是尺度因子的函数。 得到了运动方程的哈密顿形式和惠勒-德维特方程，并识别了对应动力系统的时态参数，这使得可以为该模型的量子力学描述制定薛定谔-惠勒-德维特方程。 发展了一种研究该方程的微扰方法，并通过双重简并微扰方法获得了宇宙的能量级。 还提出了描述量子时间的第二种量子化方法，并进行了简要讨论。