In this work, with the help of the quantum hydrodynamic formalism, the gravitational equation associated with the classical Dirac field is derived. The hydrodynamic representation of the Dirac equation described by the evolution of four mass densities, subject to the theory-defined quantum potential, has been generalized to the curved space-time in the covariant form. Thence, the metric of space-time has been defined by imposing the minimum action principle. The derived gravity shows the spontaneous emergence of the “cosmological” gravity tensor (CGT), a generalization of the classical cosmological constant (CC), as a part of the energy-impulse tensor density (EITD). Even if the classical cosmological constant is set to zero, the CGT is non-zero, allowing a stable quantum vacuum (out of the collapsed branched polymer phase). The theory shows that in the classical macroscopic limit, the general relativity equation is recovered. In the perturbative approach, the CGT leads to a second-order correction to Newtonian gravity that takes contribution from the space where the mass is localized (and the space-time is curvilinear), while it tends to zero as the space-time approaches the flat vacuum, leading, as a means, to an overall cosmological constant that may possibly be compatible with the astronomical observations. The Dirac field gravity shows analogies with the modified Brans–Dicke gravity, where each spinor term brings an effective gravity constant G divided by its field squared. The work shows that in order to obtain the classical minimum action principle and the general relativity limit of the macroscopic classical scale, quantum decoherence is necessary.

中文翻译：

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经典狄拉克场的自旋张量引力

在这项工作中，借助量子流体动力学形式主义，导出了与经典狄拉克场相关的引力方程。狄拉克方程的流体动力学表示由四个质量密度的演化描述，受理论定义的量子势的约束，已被推广到协变形式的弯曲时空。因此，时空的度量是通过施加最小作用原理来定义的。推导出的引力显示了“宇宙学”引力张量 (CGT) 的自发出现，它是经典宇宙学常数 (CC) 的概括，作为能量脉冲张量密度 (EITD) 的一部分。即使经典宇宙学常数设置为零，CGT 也不为零，从而允许稳定的量子真空（从坍塌的支链聚合物相中分离出来）。该理论表明，在经典宏观极限下，广义相对论方程得到恢复。在微扰方法中，CGT 导致对牛顿引力的二阶校正，该校正来自质量局部化的空间（并且时空是曲线的），而随着时空接近，它趋于零平坦的真空，作为一种手段，导致可能与天文观测兼容的整体宇宙学常数。Dirac 场引力显示出与修正的 Brans-Dicke 引力的类比，其中每个自旋项带来有效引力常数 G 除以其场的平方。工作表明，为了获得经典最小作用力原理和宏观经典尺度的广义相对论极限，量子退相干是必要的。