A cosmological model for the early universe is investigated where a Schutz fluid and a fermionic field coupled to a gauge field are the sources of the gravitational field. The determination of the scale factor in the classical analysis follows from the Hamiltonian formulation together with Dirac’s method for constrained systems. The expectation value of the scale factor in the quantum analysis is determined from the Wheeler–DeWitt equation. The singularity in the classical solution for the scale factor is avoided in the quantum solution where a bouncing from a minimum value of the scale factor is present. It is shown that: (i) the minimum value of the scale factor depends on the ratio of the fermionic coupling constant and the mass of the vectorial field; (ii) the classical and quantum solutions coincide for large values of the conformal time due to a dilution of the quantum effects with the time evolution; (iii) the behavior of the wave function probability density confirms that the maximum probability occurs when the scale factor has its minimum value equal to its expectation value; (iv) the quantum potential in the Bohmian formulation goes to infinity when the scale factor tends to zero avoiding the singularity at this point.

研究了早期宇宙的宇宙学模型，其中舒特兹流体和与标距场耦合的费米离子场是引力场的来源。在经典分析中比例因子的确定遵循哈密顿公式和约束系统的狄拉克方法。量子分析中比例因子的期望值由Wheeler-DeWitt方程确定。在存在从比例因子最小值反弹的量子解决方案中，避免了比例因子经典解决方案中的奇异性。结果表明：（i）比例因子的最小值取决于铁电偶合常数与矢量场质量之比；（ii）由于量子效应随时间演化而变小，因此经典和量子解决方案对于较大的保形时间值是一致的；（iii）波动函数概率密度的行为证实了最大概率发生在比例因子的最小值等于其期望值时；（iv）当比例因子趋于零而避免此时的奇异性时，Bohmian公式中的量子势将变为无穷大。