*2.1. Poincaré Gauge Gravity Theory: The Basics*

Recalling that the Standard Model in the fundamental particle physics is formulated as a gauge theory for the internal unitary symmetry groups, one may say that, apart from the gravitational interaction, the gauge-theoretic approach underlies the modern physics. There exist, however, a natural extension of Einstein's GR that is based on the Poincaré symmetry group *G* = *T*<sup>4</sup> ⋊ *SO*(1, 3), the semi-direct product of the four-parameter translation group *T*<sup>4</sup> and the six-parameter Lorentz group *SO*(1, 3), with the energy-momentum current and the spin angular momentum current as the sources of the gravitational field [9,10,16–18].

The gauge fields act as mediators of physical interactions for the fermion matter source. Specializing to the electromagnetic and gravitational interactions, in the framework of the standard Yang–Mills–Sciama–Kibble approach [10], one then describes electromagnetism by the *U*(1) gauge field potential *A<sup>i</sup>* , and in similar way, one describes gravity by the Poincaré gauge potentials *e α i* and Γ*<sup>i</sup> αβ*. Geometrically, the 4 potentials *e α i* of the translation subgroup *T*<sup>4</sup> are naturally interpreted as the coframe (or the tetrad) field of a physical observer on the spacetime manifold *M*4, whereas the 6 potentials Γ*<sup>i</sup> αβ* <sup>=</sup> <sup>−</sup> <sup>Γ</sup>*<sup>i</sup> βα* for the Lorentz subgroup *SO*(1, 3) are identified with the local connection that introduces the parallel transport on the spacetime *M*4.

The multiplet of gauge potentials,

$$\left\{ \begin{array}{cccc} A\_{i\prime} & e^{\alpha}\_{i\prime} & \Gamma\_{i}^{a\beta} \end{array} \right\} \,\,\, \tag{1}$$

determines the corresponding multiplet of the "Yang–Mills" gauge field strengths:

$$F\_{\text{ij}} \quad = \ \ \partial\_{\text{i}} A\_{\text{j}} - \partial\_{\text{j}} A\_{\text{i} \nu} \tag{2}$$

$$\left(T\_{i\dot{j}}{}^{a}{}\_{~} = \ \ \ \ \ \partial\_{i}e\_{\dot{j}}^{a} - \ \partial\_{\dot{j}}e\_{i}^{a} + \Gamma\_{i\beta}{}^{a}e\_{\dot{j}}^{\beta} - \Gamma\_{j\beta}{}^{a}e\_{\dot{i}}^{\beta}{}\_{~} \right. \tag{3}$$

$$R\_{i\dot{j}}{}^{a\beta}{}^{b\beta}{}^{c} = \,^{\Im}\partial\_{\dot{i}}\Gamma\_{\dot{j}}{}^{a\beta} - \partial\_{\dot{j}}\Gamma\_{\dot{i}}{}^{a\beta} + \Gamma\_{i\gamma}{}^{\beta}\Gamma\_{\dot{j}}{}^{a\gamma} - \Gamma\_{\dot{j}\gamma}{}^{\beta}\Gamma\_{\dot{i}}{}^{a\gamma}.\tag{4}$$

Thereby, we derive the Maxwell tensor *Fij* as the *U*(1) gauge field strength for the electromagnetic field, and the spacetime torsion tensor *Tij <sup>α</sup>* and the curvature tensor *Rij αβ* = − *Rij βα* as the two Poincaré (*T*<sup>4</sup> "translational" and *SO*(1, 3) "rotational", respectively) gauge field strengths for the gravitational field.

The nontrivial "mixed" form of the torsion (3) is explained by the semi-direct structure of the Poincaré symmetry group. The resulting Riemann–Cartan geometry on the spacetime *M*<sup>4</sup> is characterized by the nonvanishing torsion and curvature, whereas in the special case *Tij <sup>α</sup>* = 0 we recover the Riemannian geometry, and for *Rij αβ* = 0 one finds the Weitzenböck space of distant parallelism.

Here we do not discuss the construction of the complete dynamical scheme of the Poincaré gauge theory that requires the introduction of the corresponding gravitational field Lagrangian, and consider the electromagnetic and the gravitational fields as a nondynamical background. It is important to recall, though, that the variation of the Lagrange density of matter with respect to the gauge field potentials (1) gives rise to the corresponding dynamical currents: the electric current, the canonical energy-momentum tensor, and the spin angular momentum tensor, respectively. Further details can be found in [9,10,16–18], and we conclude this section with the following technical points which are needed for the subsequent discussion.

One can decompose the local Lorentz connection into the sum:

$$
\Gamma\_i{}^{a\beta} = \bar{\Gamma}\_i{}^{a\beta} - \mathcal{K}\_i{}^{a\beta} \tag{5}
$$

of the Riemannian connection (denoted by the tilde), which is torsionless *∂<sup>i</sup> e α <sup>j</sup>* − *∂<sup>j</sup> e α <sup>i</sup>* + Γe *iβ α e β <sup>j</sup>* − <sup>Γ</sup><sup>e</sup> *jβ α e β <sup>i</sup>* = 0 and metric-compatible, plus the post-Riemannian contortion tensor,

$$K\_{ia\beta} = \frac{1}{2} \left( T\_{a\beta i} - T\_{ia\beta} + T\_{i\beta a} \right). \tag{6}$$

On the other hand, the torsion tensor *Tµν <sup>α</sup>* = *e i µ e j <sup>ν</sup>Tij α* can be decomposed into the three irreducible parts:

$$T\_{\mu\nu}{}^{a} = \frac{1}{3} (\delta^{a}\_{\mu} T\_{\nu} - \delta^{a}\_{\nu} T\_{\mu}) + \frac{1}{3} \eta\_{\mu\nu\lambda}{}^{a} \dot{T}^{\lambda} + \mathcal{F}^{a}\_{\mu\nu}{}^{a} \tag{7}$$

where ր*Tµν α* is the trace-free and axial trace-free tensor, the torsion trace vector *T<sup>µ</sup>* = *Tαµ α* , and the axial trace vector:

$$\dot{T}^{\alpha} = -\frac{1}{2} \eta^{\alpha \mu \nu \lambda} T\_{\mu \nu \lambda \nu} \tag{8}$$

with the totally antisymmetric Levi-Civita tensor *η αµνλ* .

### *2.2. Hamiltonian for the Dirac Fermion*

The Pauli-like equation for a fermion particle, moving under the action of the torsion field had been derived in [15] for the flat Minkowski spacetime, and in [19] for an arbitrary curved space background. The relativistic dynamics of the Dirac particle with spin 1/2, electric charge *q*, and mass *m* minimally coupled to the gravitational and electromagnetic fields is described by the invariant action:

$$S = \int d^4 \mathbf{x} \sqrt{-\mathbf{g}} \,\mathrm{L},\tag{9}$$

where the Lagrangian of the spinor wave function *ψ* and *ψ* = *ψ* †*γ* 0ˆ has the form:

$$L = \frac{i\hbar}{2} \left( \overline{\psi}\gamma^a D\_a \psi - D\_a \overline{\psi}\gamma^a \psi \right) - mc \,\overline{\psi}\psi \,. \tag{10}$$

The spinor covariant derivative describes the minimal coupling of the charged Dirac particle with the external electromagnetic and gravitational gauge fields (1):

$$D\_{\mathfrak{a}} = e\_{\mathfrak{a}}^i \left( \partial\_i - \frac{i\mathfrak{q}}{\hbar} A\_i + \frac{i}{4} \Gamma\_i^{\beta\gamma} \sigma\_{\beta\gamma} \right). \tag{11}$$

Here, *c* and *h*¯ are the speed of light and Planck's constant, respectively, the 4-potential of the electromagnetic field *A<sup>i</sup>* = (−*φ*, *A*) encompasses the scalar *φ* and vector *A* potentials, and *σαβ* = *<sup>i</sup>* 2 *γαγ<sup>β</sup>* − *γβγ<sup>α</sup>* are the Lorentz algebra generators, where the flat Dirac matrices *γ <sup>α</sup>* are defined in local Lorentz frames.

We denote the local spatial and time coordinates by *x <sup>i</sup>* = (*t*, *x a* ), *a*, *b*, *c* = 1, 2, 3. An orthonormal coframe (tetrad) is needed to attach spinor spaces at every point of the space–time manifold. Then the dynamics of the Dirac particle can be investigated in an arbitrary Poincaré gauge field (*e α i* , Γ*i αβ*), where the components of tetrads in the Schwinger gauge [19] read:

$$e\_i^{\hat{0}} = V \,\delta\_i^0, \qquad e\_i^{\hat{a}} = \mathcal{W}^{\hat{a}}{}\_b \left(\delta\_i^b - cK^b \,\delta\_i^0\right), \qquad a, b = 1, 2, 3. \tag{12}$$

As was shown in reference [19], the Hermitian Hamiltonian of the fermion particle has the form:

$$\begin{split} \mathcal{H} &= -\beta m c^2 V + q \Phi + \frac{c}{2} \left( \pi\_b \, \_b \mathcal{F}^b{}\_a a^a + a^a \mathcal{F}^b{}\_a \pi\_b \right) \\ &+ \frac{c}{2} (\mathbf{K} \cdot \boldsymbol{\pi} + \boldsymbol{\pi} \cdot \mathbf{K}) + \frac{\hbar c}{4} (\boldsymbol{\Xi} \cdot \boldsymbol{\Sigma} - \mathbf{Y} \gamma\_5), \end{split} \tag{13}$$

where the kinetic 3-momentum operator *π<sup>a</sup>* = −*ih*¯ *∂<sup>a</sup>* − *qA<sup>a</sup>* = *p<sup>a</sup>* − *qA<sup>a</sup>* accounts of the interaction with the electromagnetic field, and we denoted:

$$\mathcal{F}^{b}{}\_{a} = V W^{b}{}\_{\widehat{a}} \qquad \text{Y} = V \varepsilon^{\widehat{a}\widehat{b}\widehat{c}} \Gamma\_{\widehat{a}\widehat{b}\widehat{c}\prime} \qquad \Sigma^{a} = \frac{V}{c} \varepsilon^{\widehat{a}\widehat{b}\widehat{c}} (\Gamma\_{\widehat{0}\widehat{b}\widehat{c}} + \Gamma\_{\widehat{b}\widehat{c}\widehat{0}} + \Gamma\_{\widehat{c}\widehat{0}\widehat{b}}).\tag{14}$$

As usual, *α <sup>a</sup>* = *βγ<sup>a</sup>* (*a*, *<sup>b</sup>*, *<sup>c</sup>*, · · · = 1, 2, 3) and the spin matrices <sup>Σ</sup> <sup>1</sup> = *iγ* 2ˆ *γ* 3ˆ , Σ <sup>2</sup> = *iγ* 3ˆ *γ* 1ˆ , Σ <sup>3</sup> = *iγ* 1ˆ *γ* 2ˆ and *γ*<sup>5</sup> = *iα* 1ˆ *α* 2ˆ *α* 3ˆ . Boldface notation is used for 3-vectors *K* = {*K <sup>a</sup>*}, *<sup>π</sup>* <sup>=</sup> {*πa*}, *<sup>α</sup>* <sup>=</sup> {*α <sup>a</sup>*}, **<sup>Σ</sup>** <sup>=</sup> {<sup>Σ</sup> *a*}.

Taking into account the decomposition of the connection (5) into the Riemannian and post-Riemannian parts, we find that the Pauli-like equation with the Hermitian Hamiltonian (13) encompasses the spin-torsion coupling:

$$
\Upsilon = \widetilde{\Upsilon} + Vc\widetilde{\Upsilon}^0, \qquad \Xi^{\widehat{a}} = \widetilde{\Xi}^{\widehat{a}} - V\widetilde{\Upsilon}^{\widehat{a}}.\tag{15}
$$

The tilde denotes the Riemannian quantities. The post-Riemannian contributions come from the components *T*ˇ *<sup>α</sup>* = (*T*ˇ b0 , *T*ˇ <sup>b</sup>*<sup>a</sup>* ) of the axial torsion vector (8). Accordingly, the spintorsion coupling terms read explicitly

$$-\frac{\hbar cV}{4}\left(\boldsymbol{\Sigma}\cdot\boldsymbol{\mathcal{T}} + c\gamma\_5\boldsymbol{\mathcal{T}}^0\right).\tag{16}$$

The above general formalism can be applied to the study of fermion's dynamics in arbitrary external electromagnetic and gravitational (including the post-Riemannian one) fields.

Let us now specialize to the analysis of the possible physical effects of the spacetime torsion and the inertial forces on the non-relativistic particle in the rotating reference frame (such as the Earth), [46]:

$$W = 1, \quad W^{\hat{a}}{}\_{b} = \delta^{a}\_{b'} \quad K^{a} = -\frac{(\omega \times r)^{a}}{c}, \quad \quad \Gamma\_{\mathbb{G}}{}^{\underline{a}\hat{b}} = -\frac{\varepsilon^{abc}\omega\_{c}}{c}, \quad \Gamma\_{\mathbb{G}}{}^{\underline{a}\hat{b}} = 0. \tag{17}$$

Substituting this configuration into the Hamiltonian (13) we derive:

$$\mathcal{H} = \beta m c^2 + c\boldsymbol{\omega} \cdot \boldsymbol{\pi} - \boldsymbol{\omega} \cdot (\boldsymbol{r} \times \boldsymbol{\pi}) - \frac{\hbar}{2} \boldsymbol{\omega} \cdot \boldsymbol{\Sigma} - \frac{\hbar c}{4} \left( \dot{\boldsymbol{\Gamma}}^0 c \gamma\_5 + \boldsymbol{\mathcal{T}} \cdot \boldsymbol{\Sigma} \right). \tag{18}$$

In order to reveal the physical contents of the Schrödinger equation, we need to go to the Foldy–Wouthuysen (FW) representation. Applying the methods developed in [19], we find the FW Hamiltonian:

$$\begin{split} H &= \quad \beta \epsilon + \eta \phi - \omega \cdot (\mathbf{r} \times \boldsymbol{\pi}) - \frac{\hbar}{2} \boldsymbol{\omega} \cdot \boldsymbol{\Sigma} - \frac{\eta \hbar c^{2}}{4} \left\{ \frac{1}{\epsilon}, \mathbf{B} \cdot \boldsymbol{\Pi} \right\} \\ &+ \frac{\hbar c^{3}}{8} \left\{ \frac{\boldsymbol{\pi} \cdot \boldsymbol{\Pi}}{\epsilon}, \check{\boldsymbol{\Gamma}} \boldsymbol{\delta} \right\} - \frac{\hbar c}{8} \left\{ \frac{mc^{2}}{\epsilon}, \check{\boldsymbol{\Gamma}} \cdot \boldsymbol{\Sigma} \right\} \\ &- \frac{\hbar c^{3}}{8} \left[ \frac{\boldsymbol{\Sigma} \cdot \boldsymbol{\pi}}{\epsilon (\epsilon + mc^{2})} \, \boldsymbol{\pi} \cdot \check{\boldsymbol{\Gamma}} + \check{\boldsymbol{\Gamma}} \cdot \boldsymbol{\pi} \frac{\boldsymbol{\Sigma} \cdot \boldsymbol{\pi}}{\epsilon (\epsilon + mc^{2})} \right] \\ &- \frac{\eta \hbar c}{8} \left\{ \frac{1}{\epsilon (\epsilon + mc^{2})}, \boldsymbol{\Sigma} \cdot (\boldsymbol{\mathfrak{C}} \times \boldsymbol{\pi} - \boldsymbol{\pi} \times \boldsymbol{\mathfrak{C}}) \right\}. \tag{19}$$

Here, **<sup>Π</sup>** <sup>=</sup> *<sup>β</sup>***Σ**, { , } denotes anticommutators, *<sup>ǫ</sup>* <sup>=</sup> √ *m*2*c* <sup>4</sup> + *c* <sup>2</sup>*π*<sup>2</sup> , and E = *E* + *B* × (*ω* × *r*) is the physical electric field as seen in the noninertial rotating reference frame.

Under ordinary conditions we assume <sup>|</sup>*ehB*¯ | ≪ *<sup>m</sup>*<sup>2</sup> *c* 2 , that is the magnetic field is much smaller than the critical field <sup>|</sup>*B*| ≪ *<sup>B</sup><sup>c</sup>* <sup>=</sup> *<sup>m</sup>*<sup>2</sup> *c* <sup>2</sup>/*eh*¯, and particle's velocity is much smaller than the speed of light, <sup>|</sup>*π*|/*<sup>m</sup>* <sup>≪</sup> *<sup>c</sup>*. Then *<sup>ǫ</sup>* <sup>=</sup> *mc*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*2/2*m*, and in the semiclassical limit of (19) we finally obtain the Pauli-type equation *ih*¯ *∂ψ <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>H</sup>*nr*<sup>ψ</sup>* with the non-relativistic Hamiltonian:

$$H^{\text{nr}} = \frac{\pi^2}{2m} + q\phi - \omega \cdot (\mathbf{r} \times \boldsymbol{\pi}) - \frac{\hbar}{2}\omega \cdot \boldsymbol{\sigma} - \frac{q\hbar}{2m}\mathbf{B} \cdot \boldsymbol{\sigma} + \frac{\hbar c}{8m} \left\{ \boldsymbol{\pi} \cdot \boldsymbol{\sigma} \, \vec{\mathcal{I}}^0 \right\} - \frac{\hbar c}{4} \, \check{\mathbf{T}} \cdot \boldsymbol{\sigma} . \tag{20}$$

This result is consistent with an alternative analysis based on the method of exact FW transformations [20], see the relevant discussion in [47,48].

In the physically important situations, the torsion pseudovector is spacelike, and <sup>|</sup>*T*ˇ| ≫ *cT*<sup>ˇ</sup> 0ˆ . Taking this into account, we now switch to the physically interesting case *T*ˇ 0ˆ = 0. This is the true for the fermions (10) and (11) minimally coupled to gravity.

### **3. Quantum Hydrodynamics for Spin-Torsion Coupling**

In this section, we derive the many-particle quantum hydrodynamics (MPQHD) equations from the many-particle Pauli-like equation for the system of charged particles with spin-1/2. The method of MPQHD allows to present the dynamics of a system of interacting quantum particles in terms of the functions defined in the three-dimensional physical space. This is important for the study of wave process, which take place in a three-dimensional physical space [49,50]. In flat spacetime, the MPQHD formalism for many-particles fermion systems was previously developed in [51–53], whereas the case of the noninertial reference frames was considered in [54]. The methods of MPQHD can be used for the analysis of a wide variety of systems of many interacting particles. In particular, the finite temperature hydrodynamic model has been derived recently in [55] for the spin-1 ultracold bosons. In Reference [56] the method was applied to the study of the polarization dynamics in a system of quantum particles with nontrivial electric dipole moments.

After applying the Foldy–Wouthuysen transformation for Dirac particle in combination with the method of many-particle quantum hydrodynamics, we arrive at the manyparticle Pauli-like equation:

$$i\hbar\frac{\partial\psi\_s}{\partial t} = \hat{H}\psi\_{s\prime} \tag{21}$$

where the many-particle wave function of the system of *N* spinning particles:

$$
\psi\_s(\mathbf{R}, t) = \psi\_s(\mathbf{r}\_1, \mathbf{r}\_2, \dots, \mathbf{r}\_N, t) \tag{22}
$$

is a spinor function in the 3*N*-dimensional configuration space (*s* is the spin index), and the many-particle Hamiltonian reads:

$$
\hat{H} = \sum\_{p=1}^{N} \left( \frac{\hbar\_p^2}{2m\_p} + \frac{\hbar}{2} \boldsymbol{\sigma} \cdot \boldsymbol{\Omega}\_p + \phi\_p \right). \tag{23}
$$

Here, we introduced:

$$\mathbf{\dot{\Omega}}\_{p} = \begin{aligned} -\boldsymbol{\omega} - \frac{q\_{p}}{m\_{p}} \mathbf{B}\_{p} - \frac{c}{2} \mathbf{\dot{\mathcal{T}}}\_{p\prime} \end{aligned} \tag{24}$$

$$\phi\_p \quad = \quad q\_p \phi(r\_p) - \frac{m\_p}{2} [\omega \times r\_p]^2 \,\text{.}\tag{25}$$

$$\hat{\pi}\_p \quad = \ -i\hbar \nabla\_p - q\_p A\_p - m\_p \,\omega \times \mathbf{r}\_p \,. \tag{26}$$

and *m<sup>p</sup>* and *q<sup>p</sup>* denote the mass and the charge of *p*-th particle, respectively. In particular, *q<sup>p</sup>* stands for the charge of electrons *q<sup>e</sup>* = − *e*, or for the charge of ions *q<sup>i</sup>* = *e*. The electromagnetic vector and scalar potentials *A<sup>p</sup>* = *A*(*rp*) and *φ* = *φ*(*rp*) are taken at the positions *r<sup>p</sup>* of the *p*-th particle, and the same applies to the external magnetic *B<sup>p</sup>* = *B*(*rp*) and the torsion *T***ˇ** *<sup>p</sup>* = *T***ˇ**(*rp*) fields. The last terms in (25) and (26) manifest the inertial contributions in the rotating reference frame with the angular velocity *ω*.

As compared to the standard case of a system in an external electromagnetic field, the many-particle Hamiltonian (23) includes the torsion effects, encoded in the second term <sup>∼</sup> *<sup>σ</sup>* · *<sup>T</sup>***<sup>ˇ</sup>** *<sup>p</sup>*, that has the same form as the Zeeman energy in the magnetic field. In addition, this Hamiltonian includes Mashhoon's spin-rotation contribution ∼ *σ* · *ω*, see [57–59].

The state of the system is characterized by the concentration of particles in the neighborhood of a point *r* in the physical space as:

$$m(r,t) = \int d\mathcal{R} \sum\_{p=1}^{N} \delta(r - r\_p) \psi\_s^\*(\mathcal{R}, t) \psi\_s(\mathcal{R}, t) = \langle \psi^\dagger \hat{n} \psi \rangle. \tag{27}$$

Here the integration measure reads *dR* = ∏*<sup>p</sup> d* 3 *rp*. The function *n*(*r*, *t*) is thus determined as the quantum average of the concentration operator *n*ˆ = ∑*<sup>p</sup> δ*(*r* − *rp*) in the coordinate representation. The spin density vector of fermions is determined in a similar way:

$$\mathcal{S}(r,t) = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(r - r\_p) \psi\_s^\*(\mathbf{R}, t) (\mathfrak{s}\_p)\_{s\prime} \psi\_{s'}(\mathbf{R}, t) = \langle \psi^\dagger \mathfrak{s} \psi \rangle,\tag{28}$$

as the quantum average of the spin operator s = ∑*<sup>p</sup> <sup>δ</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>r</sup>p*)*s***ˆ***p*, with *<sup>s</sup>***ˆ***<sup>p</sup>* <sup>=</sup> *<sup>h</sup>*¯ 2 *σp*.

The continuity equation for the concentration of the particles *n*(*r*, *t*) can be derived by taking the time derivative of the definition (27) and making use of the many-particle Pauli-like Equation (21):

$$
\partial\_t n(\mathbf{r}, t) + \nabla \cdot \mathbf{J}(\mathbf{r}, t) = 0,\tag{29}
$$

where the current density is defined as the microscopic average *J*(*r*, *t*) = <sup>1</sup> 2 h*ψ* †J*<sup>ψ</sup>* <sup>+</sup> c.c.<sup>i</sup> of the operator

$$\mathfrak{J} = \sum\_{p} \delta(\mathfrak{r} - \mathfrak{r}\_{p}) \frac{\mathfrak{A}\_{p}}{m\_{p}}.\tag{30}$$

Here the generalized momentum operator is defined by (26).

### *3.1. Spin Density Evolution*

In a similar way, the dynamical equation for the spin density can be obtained by differentiating the definition (28) with respect to time and making use of the many-particle Pauli-like Equation (21):

$$
\partial\_t \mathcal{S}^a(\mathbf{r}, t) + \partial\_b \Lambda^{ba}(\mathbf{r}, t) = \varepsilon^{abc} \Omega^b \mathcal{S}^c(\mathbf{r}, t). \tag{31}
$$

Here the spin precession angular velocity is defined as:

$$
\boldsymbol{\Omega} = -\boldsymbol{\omega} - \frac{q}{m} \mathbf{B} - \frac{c}{2} \boldsymbol{\Upsilon}, \tag{32}
$$

cf. (24), whereas the spin current density tensor is introduced as a microscopic average Λ*ba*(*r*, *t*) = <sup>1</sup> 2 h*ψ* †L *ba<sup>ψ</sup>* <sup>+</sup> c.c.<sup>i</sup> of the operator

$$
\mathfrak{L}^{ba} = \sum\_{p} \delta(\mathfrak{r} - \mathfrak{r}\_{p}) \frac{\hat{\pi}\_{p}^{a} \hat{s}\_{p}^{b}}{m\_{p}}.\tag{33}
$$

## *3.2. Equation of Motion*

Along the same lines, the derivation of the equation of motion of a hydrodynamic system in an external electromagnetic and torsion fields is based on differentiating the expression for the current density *J*(*r*, *t*) with respect to time and using the Pauli-like Equation (21) with the Hamiltonian (23). The result reads:

$$2m\partial\_t f^a(\mathbf{r}, t) + \partial\_b \Pi^{ab}(\mathbf{r}, t) = qnE^a(\mathbf{r}, t) + q\varepsilon^{abc}f^b(\mathbf{r}, t)B^c(\mathbf{r}, t) - S^b(\mathbf{r}, t)\partial^a \Omega^b + F^a\_{\text{inner}} \tag{34}$$

where the momentum flux tensor appears in fluid dynamics as a quantum average Π*ab*(*r*, *t*) = <sup>1</sup> 2 h*ψ* †P*ab<sup>ψ</sup>* <sup>+</sup> c.c.<sup>i</sup> of the operator

$$\mathfrak{P}^{ab} = \sum\_{p} \delta(\mathbf{r} - \mathbf{r}\_p) \frac{\mathfrak{R}\_p^{(a} \mathfrak{R}\_p^{b)}}{m\_p}. \tag{35}$$

The equation of motion (34) describes the influence of the external electromagnetic and torsion fields on the fermion matter in terms of the Lorentz and the Stern-Gerlach forces.

As a next step, one can move from the microscopic representation of the particle current density and the spin current density to their corresponding macroscopic variables by making use of an explicit representation of the spinor wave function. Such an explicit representation of the wave function is known as the Madelung decomposition.

### **4. Madelung Decomposition**

The microscopic many-particle wave function or the Madelung decomposition [60] of the *N*-particle wave function can be represented in terms of the amplitude *a*(*R*, *t*), the phase *ξ*(*R*, *t*) and the local spinor Z(*R*,*r*, *t*), defined in the local rest frame and normalized so that <sup>Z</sup>†<sup>Z</sup> <sup>=</sup> 1:

$$
\psi(\mathbf{R},t) = a(\mathbf{R},t)\,\varphi(\mathbf{R},\mathbf{r},t), \qquad \varphi(\mathbf{R},\mathbf{r},t) = e^{\frac{i}{\hbar}\tilde{\xi}(\mathbf{R},t)}\,\mathcal{Z}(\mathbf{R},\mathbf{r},t). \tag{36}
$$

Applying the decomposition (36) to the *p*-th particle, we can introduce a microscopic velocity and a microscopic spin as *v<sup>p</sup>* := <sup>1</sup> *mp ϕ* † *π***ˆ** *<sup>p</sup> ϕ* and *s<sup>p</sup>* := *ϕ* † *s***ˆ***<sup>p</sup> ϕ* = *<sup>h</sup>*¯ 2 *ϕ* † *σ<sup>p</sup> ϕ*, respectively. Explicitly, we then find:

$$\sigma\_p(\mathbf{R}, \mathbf{r}, t) \quad = \quad \frac{1}{m\_p} \left( \nabla\_p \xi - q \mathbf{A}\_p - i \hbar \mathcal{Z}^\dagger \nabla\_p \mathcal{Z} - m\_p \boldsymbol{\omega} \times \mathbf{r}\_p \right) \tag{37}$$

$$\mathbf{s}\_p(\mathbf{R}, \mathbf{r}, t) \quad = \quad \frac{\hbar}{2} \mathcal{Z}^\dagger \sigma\_p \mathcal{Z}. \tag{38}$$

The velocity field of the *p*-th particle can be decomposed *vp*(*R*,*r*, *t*) = *v*(*r*, *t*) + *ηp*(*R*,*r*, *t*) into a sum of the macroscopic average *v*(*r*, *t*) and the thermal fluctuations part *ηp*(*R*,*r*, *t*) of the velocity. In a similar way, the spin of the *p*-th particle can be represented as the sum *sp*(*R*,*r*, *t*) = *s*(*r*, *t*) + *τp*(*R*,*r*, *t*) of the macroscopic average *s*(*r*, *t*) and the thermal fluctuations part *τp*(*R*,*r*, *t*) of the spin. By definition, the averages of the fluctuations vanish, h*a* <sup>2</sup>*ηp*<sup>i</sup> <sup>=</sup> 0 and <sup>h</sup>*<sup>a</sup>* <sup>2</sup>*τp*<sup>i</sup> <sup>=</sup> 0. We assume that the particle system is closed and not placed in a thermostat. Recalling that the temperature is the average kinetic energy of the chaotic motion of the particles of the system, we consider deviations of the velocity and spin of quantum particles from the local average values, which correspond to the ordered motion of the particles.

Combining Equations (27), (30) and (28) with (36)–(38), we can derive the macroscopic concentration, the macroscopic current density and the macroscopic spin density from the corresponding microscopic variables:

$$m(r,t) = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \, a^2(\mathbf{R}, t),\tag{39}$$

$$\mathbf{J}(\mathbf{r},t) = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \, a^2(\mathbf{R}, t) \, \mathbf{v}\_p(\mathbf{R}, \mathbf{r}, t) = n(\mathbf{r}, t) \, \mathbf{v}(\mathbf{r}, t), \tag{40}$$

$$\mathbf{S}(\mathbf{r},t) = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \, a^2(\mathbf{R}, t) \, \mathbf{s}\_p(\mathbf{R}, \mathbf{r}, t) = n(\mathbf{r}, t) \, \mathbf{s}(\mathbf{r}, t). \tag{41}$$

After the Madelung decomposition procedure for the basic physical variables in the microscopic representation, the spin current density (33) and the momentum flux (35) can be recast in terms of the fluid variables into:

$$\Delta^{ba}(\mathbf{r},t) \quad = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \left( a^2 s\_p^a v\_p^b - \frac{a^2}{m\_p} \varepsilon^{acd} s\_p^c \partial\_p^b s\_p^d \right) \tag{42}$$

$$\begin{split} \Pi^{ab}(\mathbf{r},t) &= \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r}-\mathbf{r}\_{p}) \left( \frac{\hbar^{2}}{2m\_{p}} \left( \partial\_{p}^{a}a \partial\_{p}^{b}a - a \partial\_{p}^{a} \partial\_{p}^{b}a \right) \right. \\ &\left. + m\_{p}a^{2}v\_{p}^{a}v\_{p}^{b} + \frac{a^{2}}{m\_{p}} \partial\_{p}^{a}s\_{p}^{c} \partial\_{p}^{b}s\_{p}^{c} \right), \end{split} \tag{43}$$

respectively. We are now ready to write down the complete set of dynamical equations for the quantum system of spinning particles explicitly in terms of the fluid variables. This set encompasses the continuity equation

$$
\partial\_t \mathfrak{n} + \nabla \cdot (\mathfrak{n} \mathfrak{v}) = 0,\tag{44}
$$

and the momentum balance equation

$$\begin{aligned} \left(\partial\_t + v^b \partial\_b\right) v^a &= \quad \frac{q}{m} E^a + \frac{q}{m} (\mathbf{v} \times \mathbf{B})^a - \frac{1}{n} \partial\_b p^{ab} + \frac{\hbar^2}{2m^2} \partial^a \left(\frac{\Delta \sqrt{n}}{\sqrt{n}}\right) \\ &+ \frac{1}{2m^2} \partial^a (\partial^b \mathbf{s} \cdot \partial^b \mathbf{s}) - \frac{1}{m} \mathbf{s} \cdot \partial^a \mathbf{\hat{D}} + f^a\_{\text{iner}} - \frac{1}{m} Q^a\_{\text{therm}} \end{aligned} \tag{45}$$

whereas the spin evolution Equation (31) reads:

$$\left(\partial\_t + v^b \partial\_b\right)\mathbf{s} = \mathbf{\hat{M}} \times \mathbf{s} - \Theta\_{\text{therm}\prime} \tag{46}$$

where the spin precession angular velocity is modified, cf. (32),

$$
\hat{\mathbf{M}} = -\boldsymbol{\omega} - \frac{q}{m} \mathbf{B} - \frac{c}{2} \check{\mathbf{T}} - \frac{1}{mn} \partial\_b (n \hat{\boldsymbol{\sigma}}^b \mathbf{s}).\tag{47}
$$

This modification arises from the interaction of spin with the surrounding spin-texture of the fluid, and one can formally interpret this in terms of an effective magnetic field defined as a sum of an external magnetic field and the emergent field:

$$
\hat{\mathbf{B}} = \mathbf{B} + \frac{1}{qn} \partial\_b (n \partial^b \mathbf{s}). \tag{48}
$$

In fact, one can also view the first term on the right-hand side of (47), which is due to Mashhoon's spin-rotation coupling term and describes the Barnett effect, and the third term on the right-hand side of (47), representing the spin-torsion coupling, as the two additional contributions to the "effective" magnetic field

$$\mathbf{B}\_{\omega} = \frac{m}{q} \,\omega, \qquad \mathbf{B}\_{T} = \frac{mc}{2q} \,\mathbf{T}. \tag{49}$$

The dynamical Equation (46) describes the precession of spin under the action of the torque produced by the external magnetic field and the emergent fields, leading to the Zeeman type effect. The additional torque in (46) arises from thermal-spin interactions:

$$\Theta\_{\rm thermal} = \partial\_{\rm b} \int dR \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \left( a^2 \eta\_p^b \mathbf{s}\_p - \frac{a^2}{m\_p} \mathbf{s}\_p \times \partial\_p^b \mathbf{r}\_p \right) . \tag{50}$$

that is also responsible for the last force term in the momentum balance Equation (45)

$$Q\_{\rm thermal}^{a} = -\frac{1}{n}\partial\_{\boldsymbol{b}} \int d\boldsymbol{R} \sum\_{p=1}^{N} \delta(\boldsymbol{r} - \boldsymbol{r}\_{p}) \frac{a^{2}}{m\_{p}} \left(\partial\_{p}^{b}\boldsymbol{\tau}\_{p} \cdot \partial\_{p}^{a}\boldsymbol{s}\_{p} + \partial\_{p}^{b}\boldsymbol{s}\_{p} \cdot \partial\_{p}^{a}\boldsymbol{\tau}\_{p} - \partial\_{p}^{a}\boldsymbol{\tau}\_{p} \cdot \partial\_{p}^{b}\boldsymbol{\tau}\_{p}\right)$$

$$-\partial^{a} \left\{\frac{1}{n}\,\partial\_{b} \left[n\,\partial^{b} \left(\frac{1}{n}\int d\boldsymbol{R} \sum\_{p=1}^{N} \delta(\boldsymbol{r} - \boldsymbol{r}\_{p})\,\frac{a^{2}}{2m\_{p}}\,\boldsymbol{\tau}\_{p}\cdot\boldsymbol{\tau}\_{p}\right)\right]\right\}. \tag{51}$$

Analysing the structure of the equation of motion (45), we identify the first two terms on the right hand side with the Lorenz force determined by the external electric and magnetic fields *E* and *B*, while the third term is the divergence of the kinetic pressure tensor

$$p^{ab}(\mathbf{r},t) = \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \, a^2 \, m\_p \, \eta\_p^a \, \eta\_p^b. \tag{52}$$

The fifth term on the right hand side of the Equation (45) represents the effect of spin–spin interactions inside the fluid, the interaction of the spin with the spin background texture. The sixth term describes the Stern–Gerlach force that characterizes the influence of the nonuniform effective magnetic and the torsion field. In the non-inertial frame, an additional contribution encompasses the Coriolis force, the centrifugal force and Euler force field:

$$f\_{\text{iner}} = -2\,\omega \times \boldsymbol{\nu} - \omega \times (\boldsymbol{\omega} \times \mathfrak{R}) - \frac{\partial \boldsymbol{\omega}}{\partial t} \times \mathfrak{R} \,\tag{53}$$

where the vector of center of mass is defined as:

$$\mathfrak{R}(\mathbf{r},t) = \frac{1}{n} \int d\mathbf{R} \sum\_{p=1}^{N} \delta(\mathbf{r} - \mathbf{r}\_p) \psi\_s^\*(\mathbf{R},t) \mathbf{r}\_p \psi\_s(\mathbf{R},t). \tag{54}$$

Our derivations are consistent with the earlier analysis [54].
