**5. Experimental Manifestations of Spin-Torsion Coupling**

Experimental search for the nontrivial torsion effects is naturally embedded into the broader framework of the studies of the spin-dependent long-range forces [21–40]. By making use of the corresponding experimental techniques, it is possible to find strong limits on the values of the gauge gravity spin-torsion coupling constants and on the torsion field itself. A good example of an efficient approach in this respect gives an observation of the nuclear spin precession in gaseous spin polarized <sup>3</sup>He or <sup>129</sup>Xe samples with the help of a highly sensitive low-field magnetometer [61–63] detecting a sidereal variation of the relative spin precession frequency in a new type of <sup>3</sup>He/129Xe clock comparison test. In a similar experiment [64], the ratio of nuclear spin-precession frequencies of <sup>199</sup>Hg and <sup>201</sup>Hg atoms was measured in the magnetic field and the Earth's gravitational field. Based on the corresponding experimental data from [64], the analysis of dynamics of the minimally coupled Dirac fermion [19] in external electromagnetic and gravitational fields revealed the strong bounds on the possible background space-time torsion:

$$\frac{c}{2} \left| \tilde{T} \right| \cdot \left| \cos \theta \right| < 6.45 \times 10^{-6} \text{s}^{-1} \text{ } \tag{55}$$

where *θ* is the angle between the magnetic *B* and torsion *T***ˇ** fields. On the other hand, by making use of the experimental data from [61], one finds the restriction:

$$\frac{c}{2} \left| \check{T} \right| \cdot \left| \cos \theta \right| < 3.59 \times 10^{-7} s^{-1}. \tag{56}$$

These results are consistent with the alternative empirical estimates for the torsion limits [30–40]. As another powerful tool one can mention the use of the quantum interferometry to probe the spacetime structure, including the search for possible post-Riemannian deviations, focusing on the detection of the phase shift and polarization rotation effects for the neutron and atom beams [65–69].

As an application of the quantum hydrodynamics formalism, let us investigate a simple model of a continuous medium of particles with spin and consider the dynamics of the spin waves in such a particle system. Neglecting the spin-thermal coupling in the spin dynamical Equation (46) and assuming the small perturbations of the spin *s* = *s*<sup>0</sup> + *δs* around an undisturbed value |*s*0| = *h*¯ /2, we find, in the first order,

$$
\delta\partial\_t\delta\mathbf{s} = \mathbf{\hat{D}}^{(0)} \times \delta\mathbf{s} + \mathbf{\hat{D}}^{(1)} \times \mathbf{s}\_0. \tag{57}
$$

Here, the equilibrium values of the external background fields are encoded in **Ωˆ** (0) = − *ω*<sup>0</sup> − *q m <sup>B</sup>*<sup>0</sup> <sup>−</sup> *<sup>c</sup>* 2 *T***ˇ**, where *B*0, *ω*<sup>0</sup> and *T***ˇ** are the external uniform magnetic field, the Earth's angular velocity and the background torsion, respectively, and the small disturbance reads:

$$
\hat{\mathbf{M}}^{(1)} = -\frac{\Delta\delta\mathbf{s}}{m},
\tag{58}
$$

Assuming that the perturbations of the spin vary as *δs* ∼ exp(−*iωst* + *ik* · *r*), we then derive the dispersion law relating the wave frequency *ω<sup>s</sup>* and the wave vector *k* for the spin waves excited in the external magnetic and torsion fields:

$$
\omega\_s^2 = \Omega\_c^2 + \frac{c^2}{4}\vec{T}^2 + \omega\_0^2 + 2\Omega\_c\omega\_0\cos\theta\_1 + c\,\Omega\_c\vec{T}\cos\theta\_2 + c\,\omega\_0\vec{T}\cos(\theta\_2 - \theta\_1). \tag{59}
$$

Here, Ω*<sup>c</sup>* = *qB*0 *<sup>m</sup>* <sup>+</sup> *hk*¯ 2 2*m* , whereas *θ*<sup>1</sup> and *θ*<sup>2</sup> are, respectively, the angle between the external magnetic field and Earth's angular velocity, and the angle between *B*<sup>0</sup> and the background torsion *T***ˇ**. Equation (59) is a generalization for the dispersion relation of spin waves found in Reference [70] and it takes into account the contribution of the spin part of the quantum Bohm potential as an additional spin torque due to the self-action inside the system of particles, which leads to the propagation of spin waves. The square of the frequency *ω*<sup>2</sup> *s* encompasses a contribution proportional to the square of the modulus of the wave vector ∼ *k* 2 . As we can see from the dispersion relation (59) the torsion effect is maximal when the pseudovector field *T***ˇ** is aligned along the external magnetic field.

### **6. Discussion and Conclusions**

In this paper, we for the first time developed the quantum hydrodynamics for the many-particle system of massive Dirac fermion spin-1/2 particles interacting with external electromagnetic, metric gravitational/inertial and torsion fields. This essentially extends the single-particle quantum hydrodynamical approach which was developed for the flat spacetime, see [71–76] and the references therein. Taking as the basis of the earlier general formalism [19], the consistent hydrodynamical formulation was constructed for the manyparticle quantum system of fermions, and the explicit relations between the microscopic and macroscopic fluid variables were derived with help of the Madelung decomposition approach. In the present study, we have focused on the physically important situation with *T*ˇ 0ˆ = 0, a more exotic case with *T*ˇ 0ˆ 6= 0 (which may be realized in cosmology, for example, see the earlier work [15], or in the more general models) will be considered elsewhere.

The resulting system of hydrodynamical equations consists of the continuity Equation (44), the momentum balance Equation (45) and the spin dynamics Equation (46). The momentum balance equation includes the contributions in the form of the quantum Bohm potential and a new spin part of the Bohm quantum potential which are proportional to the square of Planck's constant. In addition, the dynamical equations take into account the thermal effects resulting from the fluctuations of the spin and velocity near their average values. As an application of the formalism, we evaluated the possible effects of the spacetime torsion and the spin part of quantum Bohm potential on the dispersion characteristics of the spin waves (59) excited in the many-particle fermion system. The

developed hydrodynamical model can be used in the future studies of various types of transport phenomena in spinning matter with an account of external electromagnetic and gravitational fields.

**Author Contributions:** All authors discussed the results and contributed to the final manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of MIT was supported by the Russian Science Foundation under the grant 19-72-00017, and the research of YNO was supported in part by the Russian Foundation for Basic Research (Grant No. 18-02-40056-mega).

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

