# Imprints from a Riemann-Cartan space-time on the energy levels of Dirac
spinors

Francisco Cabral ID<EMAIL_ADDRESS>Instituto de Astrofísica e Ciências do
Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo
Grande, P-1749-016 Lisbon, Portugal  Francisco S. N. Lobo ID<EMAIL_ADDRESS>Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da
Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Diego Rubiera-Garcia ID<EMAIL_ADDRESS>Departamento de Física Teórica and
IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

(August 27, 2024)

###### Abstract

In this work, we investigate the effects of the torsion-fermionic interaction
on the energy levels of fermions within a Riemann-Cartan geometry using a
model-independent approach. We consider the case of fermions minimally coupled
to the background torsion as well as non-minimal extensions via additional
couplings with the vector and axial fermionic currents which include parity-
breaking interactions. In the limit of zero-curvature, and for the cases of
constant and spherically symmetric torsion, we find a Zeeman-like effect on
the energy levels of fermions and anti-fermions depending on whether they are
aligned/anti-aligned with respect to the axial vector part of the torsion (or
to specific combination of torsion quantities), and determine the
corresponding fine-structure energy transitions. We also discuss non-minimal
couplings between fermionic fields and torsion within the Einstein-Cartan
theory and its extension to include the (parity-breaking) Holst term. Finally
we elaborate on the detection of torsion effects related to the splitting of
energy levels in astrophysics, cosmology and solid state physics using current
capabilities.

##  I Introduction

Non-Riemannian geometries naturally appear in gravitation in the gauging
(localizing) of space-time symmetries Blagojevic:2002du ; Blagojevic:2012bc ;
Cabral:2020fax . A particularly interesting case is the Riemann-Cartan (RC)
geometry, linked to the gauging of the Poincaré group, where in addition to
curvature, torsion (the antisymmetric part of the affine connection) is also
present. The simplest implementation of the RC geometry is given by the
Einstein-Cartan (EC) theory Hehl:2007bn , where the corresponding
gravitational action is formally given by the Einstein-Hilbert of General
Relativity (GR), but the new freedom encoded in the affine connection allows
for new couplings in the matter fields, which are not present when torsion is
absent. Analogously to the energy-momentum tensor being the source of
curvature, in EC theory the spin tensor feeds the torsion effects into the
metric field equations, that become relevant at scales given by Cartan’s
density. The effects of torsion in bosonic and fermionic fields can be
implemented via the covariant derivatives present in the corresponding
Lagrangian densities, and manifest at scales different than the Cartan’s
threshold Cabral:2019gzh ; Cabral:2020mst ; Cabral:2020mzw , while non-minimal
couplings can also be considered.

Most research carried out in the corresponding literature so far takes
advantage of these new matter couplings to look for new phenomenology in those
environments where the spin density is strong enough so as to excite the new
dynamics fed by torsion. In this manner, the spin density of fermions
translate into modifications of the background space-time metric as compared
to the GR predictions. As a consequence, many applications, such as the
astrophysics of compact objects or the early and late-time cosmological
dynamics have been widely explored and characterized Boehmer:2008ah ;
Poplawski:2011jz ; Poplawski:2012ab ; Vakili:2013fra ; Bronnikov:2016xvj ;
Ivanov:2016xjm ; Cembranos:2016xqx ; Cembranos:2017pcs ; Unger:2018oqo ;
Mehdizadeh:2018smu . The main outcome of this analysis is to find new
predictions of these theories with torsion that can be compared with their GR
counterparts, and that could constitute observational discriminators for the
existence of new physics in the strong gravitational field regime. Through
this process, these models and their predictions can subsequently be
constrained within the newly born field of multimessenger astronomy
Barausse:2020rsu .

The main aim of this work is to look for more direct physical effects induced
by torsion. Indeed, instead of the effects in the space-time background where
the torsion fields have been effectively removed in favour of other
observables, here we are interested in the behaviour of fermionic fields
themselves when propagating on a RC geometry. Let us recall that, in general,
the affine connection can be split into its curvature, torsion, and non-
metricity pieces (what is known as a metric-affine geometry), allowing for
geometrically alternative interpretations but physically equivalent
implementations of GR when the lowest scalar object built upon any of these
terms is included in the action of the theory (see e.g. BeltranJimenez:2019tjy
for an interesting discussion on this topic). The interaction of fermionic
fields with the space-time geometry in gravitational phenomena is of utmost
relevance in this respect, as it can provide a breaking of the degeneracy
between these different theories for gravity and space-time Cabral:2020fax .
Indeed, the different pieces of the affine connection (if non-vanishing) can
have direct observable physical effects.

The existence of observable imprints on the modifications to the energy levels
on free-falling one-electron atoms induced by curvature in an arbitrary
gravitational field are known since the seminal works by Parker Parker:1980hlc
; Parker:1980kw ; Parker:1982nk . On the other hand, the presence of non-
metricity has been recently found to yield measurable effects via new
4-fermion contact interactions that can be used to put constraints on the
scale of non-metricity, for instance, in electron-positron scattering in
particle accelerators Latorre:2017uve ; Delhom:2019wir . In this paper, we
show that both minimal and non-minimal couplings of torsion to Dirac fermions
in different backgrounds also yield measurable effects, via a splitting in the
energy levels of fermions/antifermions (driven by the axial vector part of the
torsion tensor in minimal couplings and also by the trace-vector part for
specific non-minimal extensions), depending on the relative orientation
between spin and torsion vector/pseudo-vector quantities, which resembles a
Zeeman-like effect, allowing for transitions between such different
levels111For further discussions on the experimental manifestations of torsion
in the interaction between spinors, torsion, and electromagnetic fields see
e.g. Shapiro1 ; Shapiro2 ; Shapiro3 ; Shapiro4 . .

This work is organized as follows: In Sec. II, we establish the main equations
of fermions minimally coupled to the torsion field, and find that they only
interact via the axial vector part of the torsion tensor. In Sec. III, we find
the energy levels for fermions/anti-fermions in the simplified regime of zero-
curvature, taking the ansatze of (static) constant and spherically symmetric
torsion, respectively, and find the frequency of the corresponding
transitions. These results are extended in Sec. IV to the case of non-minimal
couplings between the fermionic currents (vector and axial currents) and the
torsion (the axial and trace-vector irreducible parts) where some interaction
terms break the parity symmetry. In Sec. V we briefly discuss the topic of
non-minimal couplings within the EC theory and its generalization to include
the parity-breaking Holst term. We conclude in Sec. VI with an extended
discussion on different types of physical effects connected to the splitting
of fermionic energy levels driven by torsion in astrophysics and cosmology.

##  II Torsion-fermions couplings

###  II.1 Dirac fermions in a flat space-time

Let us start our analysis, for self-consistency and self-completeness, by
reviewing the corrections to the non-relativistic Schrödinger equation induced
by the Dirac equation. Accordingly, consider the Dirac Lagrangian in curved
space-time and minimally coupled to the electromagnetic field, given by222From
now on hatted quantities will denote computation with the standard Levi-Civita
connection of the curvature (i.e., the Christoffel symbols of the metric).

$\mathcal{\tilde{L}}_{\rm
Dirac}=\dfrac{i\hbar}{2}\left(\bar{\psi}\gamma^{\mu}\tilde{D}_{\mu}\psi-(\tilde{D}_{\mu}\bar{\psi})\gamma^{\mu}\psi\right)-m\bar{\psi}\psi+j^{\lambda}A_{\lambda}\
,$ (1)

where $\psi$ denotes spinors and $\bar{\psi}=\psi^{+}\gamma^{0}$ its adjoint,
while $j^{\lambda}=q\bar{\psi}\gamma^{\lambda}\psi$ is the $U(1)$ charge
current density vector. The (Fock-Ivanenko) covariant derivatives in this
space-time are given by

$\displaystyle\tilde{D}_{\mu}\psi$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi+\dfrac{1}{2}\tilde{w}_{ab\mu}\sigma^{ab}\psi\,,$
(2) $\displaystyle\tilde{D}_{\mu}\bar{\psi}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\bar{\psi}-\dfrac{1}{2}\tilde{w}_{ab\mu}\bar{\psi}\sigma^{ab}\,,$
(3)

while the matrices
$\sigma^{ab}\equiv\tfrac{1}{4}\left[\gamma^{a},\gamma^{b}\right]=\tfrac{1}{2}\gamma^{[a}\gamma^{b]}$
are the Lorentz group generators in the spinorial representation. The effect
of space-time curvature is encoded in the Levi-Civita 1-form spin connection,
$\tilde{w}_{ab\mu}$, of the Riemann geometry.

The corresponding Dirac equation is obtained by varying the action
$S=\tfrac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}\mathcal{\tilde{L}}_{\rm Dirac}$
of the Dirac Lagrangian (1) with respect to $\bar{\psi}$ and reads as

$i\hbar\gamma^{\mu}\tilde{D}_{\mu}\psi+(q\gamma^{\mu}A_{\mu}-m)\psi=0\ .$ (4)

In a flat (Minkowski) space-time, and taking the quasi non-relativistic limit
(leaving only terms up to $(v/c)^{2}$), one finds the time independent
equation in the static external electromagnetic potential
$A_{\mu}=(\phi,\vec{A}\,)$ (here we reinsert the speed of light $c$ for
convenience):

$\displaystyle\left[\dfrac{1}{2m}\left(\hat{\vec{p}}-q\vec{A}\right)^{2}-\dfrac{\hat{p}^{4}}{8m^{3}c^{2}}+q\phi+\dfrac{q\hbar^{2}}{4m^{2}c^{2}}\dfrac{1}{r}\partial_{r}\phi\;\hat{\vec{S}}\cdot\hat{\vec{L}}\right.$
$\displaystyle\left.\hskip
28.45274pt-\dfrac{q\hbar}{m}\hat{\vec{S}}\cdot\hat{\vec{B}}-\dfrac{q\hbar^{2}}{4m^{2}c^{2}}\partial_{r}\phi\,\partial_{r}-E\right]\psi(\vec{r})=0\,,$
(5)

with $E\ll mc^{2}$ and $q\phi\ll mc^{2}$, and spherical symmetry,
$\phi=\phi(r)$, is assumed. The solution to these equations gives the four-
spinor $\psi=\psi(\vec{r})e^{-iEt/\hbar}$, which corresponds to the
eigenfunction of the Hamiltonian with energy $E$. In the expression above,
$\hat{\vec{S}}\sim\hbar\vec{\sigma}/2$ is the intrinsic angular momentum
(spin), and $\vec{\sigma}=(\sigma^{1},\sigma^{2},\sigma^{3})$ is the Pauli
matrices spatial vector. As is well known, the second term on the left-hand
side of Eq. (5) is a relativistic correction to the three-momentum, the forth
and the fifth terms give the spin-orbit and Zeeman-effect magnetic energy,
respectively, and the sixth term is the so-called Darwin term correction.

If we consider the case of an electron in the Coulomb potential
$\phi=-Ze^{2}/r$, then the corresponding energy levels of this system are
given by

$\displaystyle E$ $\displaystyle=$ $\displaystyle
mc^{2}\left[1-\dfrac{Z^{2}\alpha^{2}}{2n^{2}}-\dfrac{Z^{4}\alpha^{4}}{2n^{4}}\left(\dfrac{n}{j+1/2}-\dfrac{3}{4}\right)\right.$
(6) $\displaystyle\qquad\qquad\left.+\mathcal{O}(Z^{6}\alpha^{6})\right]\,,$

where $n$ is the principal quantum number, $j$ is the total angular momentum
quantum number, and $\alpha$ is the fine structure constant. The first term
correction inside the brackets is the relativistic correction of the energy
associated to the mass of the electron, the second term corresponds to the
Bohr energy levels, while the next term is the fine structure (spin-orbit)
correction. As an example, the fine structure between the energy levels
($nl_{j}$) $2P_{3/2}$ and $2P_{1/2}$ corresponds to an energy difference
$|\Delta E|=mc^{2}Z^{2}\alpha^{4}/32$.

If one considers, instead of the Minkowski limit, the full curved space-time
background of a Riemannian geometry, then there will be gravitational metric-
induced corrections to the energy levels Parker:1982nk , which should become
non-negligible for strong gravitational fields. It is thus natural to wonder
whether new physical effects will manifest if one generalizes the Riemann
geometry to include torsion in a RC space-time. As we will see, in the minimal
coupling scenario a clear analogy with the Zeeman-effect term can be
recognized.

###  II.2 Fermions minimally coupled to torsion

Let us consider a free Dirac fermionic field minimally coupled to the RC
space-time geometry (for a more detailed analysis of fermions in RC and
metric-affine geometries see for example Audretsch:1981xn ; Obukhov:2014fta ;
Obukhov:2018zmz ; Adak:2002pq ; Cembranos:2018ipn ; Bahamonde:2020fnq ). The
torsion tensor of the affine connection $\Gamma_{\mu\nu}^{\lambda}$ is defined
by
$T_{\mu\nu}\equiv\tfrac{1}{2}(\Gamma_{\mu\nu}^{\lambda}-\Gamma_{\nu\mu}^{\lambda})$,
with the irreducible components
$T^{\lambda}_{\;\mu\nu}=\bar{T}^{\lambda}_{\;\mu\nu}+\tfrac{2}{3}\delta^{\lambda}_{[\nu}T_{\mu]}+g^{\lambda\sigma}\epsilon_{\mu\nu\sigma\rho}\breve{T}^{\rho}$,
and the traceless tensor obeying $\bar{T}^{\lambda}_{\;\mu\lambda}=0$ and
$\epsilon^{\lambda\mu\nu\rho}\bar{T}_{\mu\nu\rho}=0$, while $T_{\mu}$ is the
trace vector and

$\breve{T}^{\lambda}\equiv\frac{1}{6}\epsilon^{\lambda\alpha\beta\gamma}T_{\alpha\beta\gamma}\
,$ (7)

is the pseudo-trace (axial) vector, which will play a key role in our work.

The minimally-coupled fermionic (Dirac) piece, Eq. (77), in presence of
torsion, is given by

$\mathcal{L}_{\rm
Dirac}=\dfrac{i\hbar}{2}\left(\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(D_{\mu}\bar{\psi})\gamma^{\mu}\psi\right)-m\bar{\psi}\psi\
,$ (8)

which is formally equal to Eq. (1), but where the generalized Fock-Ivanenko
covariant derivatives of spinors are now defined as

$D_{\mu}\psi=\partial_{\mu}\psi+\dfrac{1}{2}w_{ab\mu}\sigma^{ab}\psi=\tilde{D}_{\mu}\psi+\dfrac{1}{4}K_{ab\mu}\gamma^{[a}\gamma^{b]}\psi\
,$ (9)

and

$D_{\mu}\bar{\psi}=\partial_{\mu}\bar{\psi}-\dfrac{1}{2}w_{ab\mu}\bar{\psi}\sigma^{ab}=\tilde{D}_{\mu}\bar{\psi}-\dfrac{1}{4}K_{ab\mu}\bar{\psi}\gamma^{[a}\gamma^{b]}\
,$ (10)

where the Lorentzian spin connection ($w_{ab\nu}=-w_{ba\nu}$) of the RC space-
time can be written as the spin connection of the Riemann geometry plus the
so-called contortion tensor $K_{\alpha\mu\nu}\equiv
T_{\alpha\mu\nu}+2T_{(\mu\nu)\alpha}$, that is

$w_{ab\mu}=\tilde{w}_{ab\mu}+K_{ab\mu}\ .$ (11)

Replacing the expressions of the covariant derivatives (9) and (10) in the
Lagrangian density (8) yields

$\mathcal{L}_{\rm Dirac}=\tilde{\mathcal{L}}_{\rm
Dirac}+\dfrac{i\hbar}{8}K_{ab\mu}\bar{\psi}\\{\gamma^{\mu},\gamma^{a}\gamma^{b}\\}\psi\
,$ (12)

where $\tilde{\mathcal{L}}_{\rm Dirac}$ is given by Eq. (1).

Using the canonical properties of the tetrads we can write the contractions
$K_{ab\mu}=\vartheta^{c}_{\,\,\mu}K_{abc}$ and
$\gamma^{\mu}=e^{\,\,\mu}_{d}\gamma^{d}$, so that the Lagrangian density above
can also be written as

$\mathcal{L}_{\rm Dirac}=\tilde{\mathcal{L}}_{\rm
Dirac}+\dfrac{i\hbar}{8}K_{abc}\bar{\psi}\\{\gamma^{c},\gamma^{a}\gamma^{b}\\}\psi\
.$ (13)

Next, by using the identities
$\\{\gamma^{c},\gamma^{a}\gamma^{b}\\}=2\gamma^{[c}\gamma^{a}\gamma^{b]}=-2i\epsilon^{cabd}\gamma_{d}\gamma^{5}$
we can rewrite it as

$\mathcal{L}_{\rm Dirac}=\tilde{\mathcal{L}}_{\rm
Dirac}+\dfrac{i\hbar}{4}K_{abc}\bar{\psi}\gamma^{[c}\gamma^{a}\gamma^{b]}\psi\
,$ (14)

and noting that $K_{[\alpha\beta\gamma]}=T_{\alpha\beta\gamma}$, we arrive at
the final expression

$\mathcal{L}_{\rm Dirac}=\tilde{\mathcal{L}}_{\rm
Dirac}+3\breve{T}^{\lambda}\breve{s}_{\lambda}\ ,$ (15)

where

$\breve{s}^{\lambda}\equiv\dfrac{\hbar}{2}\bar{\psi}\gamma^{\lambda}\gamma^{5}\psi\
,$ (16)

is the Dirac axial spin vector current. Note that in this expression we have
reinserted the space-time (holonomic) indices. This simple expression, which
is valid for any Dirac field minimally coupled to a RC space-time geometry
(regardless of the gravitational theory chosen) means that, in the minimal
coupling case, Dirac fermionic fields only interact with the axial vector part
of torsion.

The axial vector $\breve{s}^{\lambda}$ in Eq. (16) can be understood as
representing the density of fermionic spin (spin/volume or energy/area, in
$c=1$ units). To see this more explicitly, let us consider the
$\gamma^{a}\gamma^{5}$ matrices for $a=0,1,2,3$, that is

$\gamma^{a}\gamma^{5}=\Bigg{\\{}\begin{pmatrix}0&I\\\
-I&0\end{pmatrix},\begin{pmatrix}\sigma^{i}&0\\\
0&-\sigma^{i}\end{pmatrix}\Bigg{\\}}\ ,$ (17)

(with $i=1,2,3$), respectively, and with $I$ representing the $2\times 2$
identity matrix.

Since the eigenvalues of the Pauli matrices are $\lambda=\pm 1$ for the spin
up/down configurations, and using the fact that in the usual Pauli-Dirac
representation the $\sigma^{3}$ matrix is already diagonal, we can use this
direction as the one relative to which we define the up and down spin states.
Then one can show that

$|\breve{s}^{3}|\sim\dfrac{\hbar}{2}n\ ,$ (18)

where $n$ is a normalization constant giving the number of particles (or anti-
particles) per volume.

The torsion-spin interaction term in Eq. (15) actually resembles a Zeeman-like
effect with the axial spin vector playing the role of an external magnetic
field:

$\mathcal{L}_{\rm ts}\sim\breve{\mathbb{T}}\cdot\breve{\mathbb{s}}\,.$ (19)

where bold letters indicate a product of two (axial) vectors.

We can now find the Dirac equation corresponding to the Lagrangian density
(15) for spinors and adjoint spinors as

$\displaystyle i\hbar\gamma^{\mu}\tilde{D}_{\mu}\psi-m\psi$ $\displaystyle=$
$\displaystyle-\dfrac{3\hbar}{2}\breve{T}^{\lambda}\gamma_{\lambda}\gamma^{5}\psi\,,$
(20) $\displaystyle i\hbar(\tilde{D}_{\mu}\bar{\psi})\gamma^{\mu}+m\bar{\psi}$
$\displaystyle=$
$\displaystyle-\dfrac{3\hbar}{2}\breve{T}^{\lambda}\bar{\psi}\gamma_{\lambda}\gamma^{5}\,,$
(21)

respectively. In the next section we shall study specific solutions of this
system in order to determine the energy levels for fermions and anti-fermions.

##  III Imprints of torsion upon fermion/anti-fermion energy levels

###  III.1 Constant background axial torsion

We start our analysis by taking again the zero-curvature limit and, moreover,
we specify an axial torsion vector along one specific direction (for example
the $z$ axis of a cartesian coordinate system). Under these conditions, the
Dirac equation (20) reads

$i\hbar\gamma^{\alpha}\partial_{\alpha}\psi=m\psi-\dfrac{3\hbar}{2}\breve{T}^{3}\gamma_{3}\gamma^{5}\psi\,.$
(22)

More explicitly, using the expressions of the Pauli matrices and recalling
that $\gamma^{3}=-\gamma_{3}$, one can recast this equation into the dynamical
system

$\displaystyle
i\hbar\Big{(}\partial_{t}\psi^{I}+\sigma^{k}\partial_{k}\psi^{II}\Big{)}$
$\displaystyle=$
$\displaystyle\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\sigma^{3}\right)\psi^{I},$
(23)
$\displaystyle-i\hbar\Big{(}\partial_{t}\psi^{II}+\sigma^{k}\partial_{k}\psi^{I}\Big{)}$
$\displaystyle=$
$\displaystyle\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\sigma^{3}\right)\psi^{II},$
(24)

where we have introduced the notation $\psi^{I}=(\psi^{1},\psi^{2})$ and
$\psi^{II}=(\psi^{3},\psi^{4})$, while $k=1,2,3$, and the $2\times 2$ identity
matrix $I$ is implicit in the first terms of the left-hand side and in the
first (mass) terms on the right-hand side.

From these equations one can see that the axial-axial interaction between the
fermionic spin density and the background space-time torsion gives a spin-
dependent energy (depending on the relative orientation between the axial spin
vector and the background space-time torsion). Therefore, an electron or any
massive free fermion in a well defined momentum (eigen)state will have two
possible energy levels depending on the alignment/anti-alignment between its
spin and the axial torsion vector, which is analogous to the Zeeman effect.
Moreover, if we assume that $\breve{T}^{3}>0$, then the anti-alignment is
preferred for the fermion as it corresponds to the lower energy level and the
same result occurs for the anti-fermion.

To make our analysis more concrete, let us assume the simpler case of a
static, constant torsion field. Consider then a 4-spinor
$\psi=\psi(\vec{r}\,)e^{-iEt/\hbar}$, corresponding to the eigenfunction of a
well-defined energy state. After substituting in Eq. (22), we obtain the time-
independent equation

$-i\hbar\gamma^{k}\partial_{k}\psi+\Big{(}m-\dfrac{3\hbar}{2}\breve{T}^{3}\gamma_{3}\gamma^{5}\Big{)}\psi(\vec{r})=\gamma^{0}E\psi(\vec{r})\
.$ (25)

In terms of their components this equation reads

$\displaystyle-i\hbar\sigma^{k}\partial_{k}\psi^{II}$ $\displaystyle=$
$\displaystyle\left(E-m-\dfrac{3\hbar}{2}\breve{T}^{3}\sigma^{3}\right)\psi^{I},$
(26) $\displaystyle-i\hbar\sigma^{k}\partial_{k}\psi^{I}$ $\displaystyle=$
$\displaystyle\left(E+m-\dfrac{3\hbar}{2}\breve{T}^{3}\sigma^{3}\right)\psi^{II}\,.$
(27)

Moreover, taking into account the harmonic solution $\psi(\vec{r})=\chi
e^{i\vec{k}\cdot\vec{r}}=\chi e^{i\vec{p}\cdot\vec{r}/\hbar}$, corresponding
to a well defined momentum state, where $\chi$ is a constant 4-spinor, we get
the system of equations for the $\chi$ components as

$\displaystyle p_{1}\chi^{4}-ip_{2}\chi^{4}+p_{3}\chi^{3}$ $\displaystyle=$
$\displaystyle\left(E-m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{1},$
$\displaystyle p_{1}\chi^{3}+ip_{2}\chi^{3}-p_{3}\chi^{4}$ $\displaystyle=$
$\displaystyle\left(E-m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{2},$
$\displaystyle-\left(-p_{1}\chi^{2}+ip_{2}\chi^{2}-p_{3}\chi^{1}\right)$
$\displaystyle=$
$\displaystyle\left(E+m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{3},$
$\displaystyle-\left(-p_{1}\chi^{1}-ip_{2}\chi^{1}+p_{3}\chi^{2}\right)$
$\displaystyle=$
$\displaystyle\left(E+m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{4},$

respectively.

Since $\chi$ is assumed to have constant components, the background torsion
itself has to be constant too. In this static, constant background axial
torsion regime, assuming again that torsion is positively oriented,
$\breve{T}^{3}>0$, there are two independent solutions for the spinor
$\psi(\vec{r},t)=\chi e^{i\left(\vec{p}\cdot\vec{r}-Et\right)/\hbar}$,
corresponding to the free particle momentum eigenstates with spin up and spin
down. But, as opposed to Dirac theory in Minkowski space-time, in this case
the presence of torsion breaks the degeneracy in energy and these two states
have different (positive) energy values. As an example, consider the case of
motion along the $p_{1}$ direction for this eigenstate, for which we get

$\displaystyle p_{1}\chi^{4}$ $\displaystyle=$
$\displaystyle\left(E-m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{1},$
$\displaystyle p_{1}\chi^{3}$ $\displaystyle=$
$\displaystyle\left(E-m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{2},$
$\displaystyle p_{1}\chi^{2}$ $\displaystyle=$
$\displaystyle\left(E+m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{3},$
$\displaystyle p_{1}\chi^{1}$ $\displaystyle=$
$\displaystyle\left(E+m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)\chi^{4}.$

The two possible energy solutions for the particle are then given by

$E_{\pm}^{2}=p^{2}+\left(m\pm\dfrac{3\hbar}{2}\breve{T}^{3}\right)^{2},$ (28)

for the spin up/down, respectively. The independent solutions for the spin up
(aligned) state and the spin down (anti-aligned) state are given by

$N\begin{pmatrix}1\\\ 0\\\ 0\\\
\dfrac{p}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\end{pmatrix},\quad
N\begin{pmatrix}0\\\ 1\\\
\dfrac{p}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\ 0\end{pmatrix},$

respectively, where $N$ is a normalization constant (typically chosen to
satisfy $\psi^{\dagger}\psi=2E$) given in this case by $N=\sqrt{E+(m\pm
3\hbar\,\breve{T}^{3}/2)}$ for the spin up/down (aligned/anti-aligned) state.
From this discussion, we see that not only the axial-axial torsion-spin
interaction is analogous to a Zeeman effect but also the equations reveal that
one could think of the fermion state with the spin aligned with torsion as
being slightly more massive than the fermion state with the spin anti-aligned
to the axial torsion. In the coupling to the space-time structure, torsion is
therefore providing an effective mass to fermions that distinguishes between
spin states.

Let us also note that in this regime of static constant background torsion
there are two more independent solutions for the spinor $\psi(\vec{r})=\chi
e^{-i\left(\vec{p}\cdot\vec{r}-Et\right)/\hbar}$, corresponding to the free
anti-particle momentum eigenstates with spin down or spin up, respectively. In
this case, we obtain

$N\begin{pmatrix}0\\\
\dfrac{p}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\ 1\\\
0\end{pmatrix},\quad
N\begin{pmatrix}\dfrac{p}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\
0\\\ 0\\\ 1\end{pmatrix},$ (29)

respectively, with $N=\sqrt{E+(m\mp 3\hbar\,\breve{T}^{3}/2)}$ for the spin
down/up (anti-aligned/aligned) states, respectively. The two corresponding
energy levels are

$E^{2}=p^{2}+\left(m\mp\dfrac{3\hbar}{2}\breve{T}^{3}\right)^{2}\ ,$ (30)

for the spin down/up states.

For completeness, let us also mention that in the general case of motion along
any direction, with $\vec{p}=(p_{1},p_{2},p_{3})$, then we would reach similar
conclusions with the spin up and spin down solutions for particles:

$N\begin{pmatrix}1\\\ 0\\\
\dfrac{p_{3}}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\
\dfrac{p_{1}+ip_{2}}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\end{pmatrix},\quad
N\begin{pmatrix}0\\\ 1\\\
\dfrac{p_{1}-ip_{2}}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\
\dfrac{-p_{3}}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\end{pmatrix},$
(31)

and those for anti-particles:

$N\begin{pmatrix}\dfrac{p_{1}-ip_{2}}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\
\dfrac{-p_{3}}{E+\left(m+\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\ 0\\\
1\end{pmatrix},\quad
N\begin{pmatrix}\dfrac{p_{3}}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\
\dfrac{p_{1}+ip_{2}}{E+\left(m-\dfrac{3\hbar}{2}\breve{T}^{3}\right)}\\\ 1\\\
0\end{pmatrix}\ .$ (32)

In all cases the energy of the anti-aligned state is lower than the aligned
state.

Let us denote by $m_{\breve{T}}$ the mass correction due to the spin-torsion
interaction, and consider the two possible energy levels $E_{1}$ and $E_{2}$,
with $E_{2}>E_{1}$. We therefore get the expression for the energy transition

$h\nu=E_{2}-E_{1}=\dfrac{4mm_{\breve{T}}}{\tilde{p}_{+}+\tilde{p}_{-}}\ ,$
(33)

where $\tilde{p}^{2}_{\pm}=p^{2}+(m\pm m_{\breve{T}})^{2}$, and in the
reference frame of the particle we obtain

$h\nu=E_{2}-E_{1}=\dfrac{1}{2}m_{\breve{T}}=\dfrac{3\hbar}{4}\breve{T}\ .$
(34)

Therefore, reinserting the speed of light in vacuum, we get

$\nu=\frac{3c}{8\pi}\breve{T}$ (35)

If we consider, for instance, $\breve{T}\sim 10^{-16}m^{-1}$, then we end up
with the prediction of a transition in the $\nu\sim$nHz regime.

###  III.2 Spherically symmetric torsion background

In this section, we analyze the case of a static, spherically symmetric
torsion background, which is relevant for astrophysical applications. The
Dirac equations in this case are still given by Eqs. (20) and (21). To
estimate the effect of this scenario on the energy levels we consider the
following axial torsion ansatz around some astrophysical source:

$\breve{T}^{\mu}(r)=b^{\mu}f(r)\ ,$ (36)

where $b^{\mu}$ is a constant (axial) 4-vector. If we neglect the effect of
curvature, the limit of the generalized Dirac equation above is

$i\hbar\gamma^{\alpha}\partial_{\alpha}\psi=m\psi-\dfrac{3\hbar}{2}\breve{T}^{\lambda}(r)\gamma_{\lambda}\gamma^{5}\psi\,.$
(37)

The torsion-spin interaction can be seen as a small perturbation to an
(unperturbed) time-independent Hamiltonian. Using perturbation theory to first
order, we have then

$E\simeq E_{(0)}+\left<\psi_{(0)}\right|\hat{U}_{ts}\left|\psi_{(0)}\right>\
,$ (38)

where $\psi_{(0)}$ are the eigenstates of the unperturbed Hamiltonian
associated to the eigenvalue $E_{(0)}$. Again, taking the 4-spinor
$\psi=\psi(\vec{r})e^{-iEt/\hbar}$, corresponding to the eigenfunction of a
well-defined energy state, we obtain the time-independent equation

$-i\hbar\gamma^{k}\partial_{k}\psi+\Big{(}m-\dfrac{3\hbar}{2}\breve{T}^{\lambda}(r)\gamma_{\lambda}\gamma^{5}\Big{)}\psi=\gamma^{0}E\psi\,,$
(39)

such that the torsion-spin operator $\hat{U}_{ts}$ reads

$\hat{U}_{ts}=-\dfrac{3\hbar}{2}\hat{\breve{T}}^{\lambda}(r)\gamma_{\lambda}\gamma^{5}.$
(40)

Now, consider the 4-spinor state

$\displaystyle\left|\psi_{(0)}\right>$ $\displaystyle=$
$\displaystyle\left|\psi_{(0)}^{1}\right>\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix}+\left|\psi_{(0)}^{2}\right>\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix}$ (41)
$\displaystyle+\left|\psi_{(0)}^{3}\right>\begin{pmatrix}0\\\ 0\\\ 1\\\
0\end{pmatrix}+\left|\psi_{(0)}^{4}\right>\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix}\ ,$

solution to the unperturbed Hamiltonian. In configuration space, this
expression becomes

$\displaystyle\left<\vec{r}\;\big{|}\psi_{(0)}\right>$ $\displaystyle=$
$\displaystyle\psi_{(0)}^{1}(r)\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix}+\psi_{(0)}^{2}(r)\begin{pmatrix}0\\\ 1\\\ 0\\\ 0\end{pmatrix}$
(42) $\displaystyle+\psi_{(0)}^{3}(r)\begin{pmatrix}0\\\ 0\\\ 1\\\
0\end{pmatrix}+\psi_{(0)}^{4}(r)\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix}\,.$

If we assume the motion to take place along a specific direction, then there
are four independent solutions, two for the particle states (up/down):

$N\begin{pmatrix}1\\\ 0\\\ 0\\\
\dfrac{p}{E_{(0)}+m}\end{pmatrix}e^{i\vec{p}\cdot\vec{r}/\hbar},\quad
N\begin{pmatrix}0\\\ 1\\\ \dfrac{p}{E_{(0)}+m}\\\
0\end{pmatrix}e^{i\vec{p}\cdot\vec{r}/\hbar},$ (43)

and two for the anti-particle states (down/up)

$N\begin{pmatrix}0\\\ \dfrac{p}{E_{(0)}+m}\\\ 1\\\
0\end{pmatrix}e^{-i\vec{p}\cdot\vec{r}/\hbar},\quad
N\begin{pmatrix}\dfrac{p}{E_{(0)}+m}\\\ 0\\\ 0\\\
1\end{pmatrix}e^{-i\vec{p}\cdot\vec{r}/\hbar},$ (44)

with $N=\sqrt{E_{(0)}+m}$ and $E_{(0)}^{2}=p^{2}+m^{2}$.

Next, we need to compute from Eq. (40) the following quantity

$\displaystyle\left<\psi_{(0)}\right|\hat{U}_{ts}\left|\psi_{(0)}\right>$
$\displaystyle=$
$\displaystyle-\dfrac{3\hbar}{2}\int\psi^{\dagger}_{0}(\vec{r})\hat{\breve{T}}^{\lambda}(r)\gamma_{\lambda}\gamma^{5}\psi_{0}(\vec{r})d^{3}r$
$\displaystyle=$ $\displaystyle-\dfrac{3\hbar b_{0}}{2}\int
f(r)\psi^{\dagger}_{0}(\vec{r})\gamma_{0}\gamma^{5}\psi_{0}(\vec{r})d^{3}r$
(45) $\displaystyle+$ $\displaystyle\dfrac{3\hbar}{2}\sum_{i=1}^{3}b^{i}\int
f(r)\psi^{\dagger}_{0}(\vec{r})\gamma^{i}\gamma^{5}\psi_{0}(\vec{r})d^{3}r\ ,$

which, taking into account the quite useful general relations

$\displaystyle\gamma_{0}\gamma^{5}\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{pmatrix}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}z_{3}\\\
z_{4}\\\ -z_{1}\\\
-z_{2}\end{pmatrix},\qquad\gamma^{1}\gamma^{5}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\\\ z_{4}\end{pmatrix}=\begin{pmatrix}z_{2}\\\ z_{1}\\\ -z_{4}\\\
-z_{3}\end{pmatrix},$
$\displaystyle\gamma^{2}\gamma^{5}\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{pmatrix}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}-iz_{2}\\\
iz_{1}\\\ iz_{4}\\\
-iz_{3}\end{pmatrix},\qquad\gamma^{3}\gamma^{5}\begin{pmatrix}z_{1}\\\
z_{2}\\\ z_{3}\\\ z_{4}\end{pmatrix}=\begin{pmatrix}z_{1}\\\ -z_{2}\\\
-z_{3}\\\ z_{4}\end{pmatrix},$

and assuming the particle with the spin up configuration in Eq. (43), with
$z_{1}=Ne^{i\vec{p}\cdot\vec{r}/\hbar}$,
$z_{4}=N\frac{p}{E+m}e^{i\vec{p}\cdot\vec{r}/\hbar}$ and $z_{2}=z_{3}=0$, we
arrive at

$\left<\psi_{(0)}\right|\hat{U}_{ts}\left|\psi_{(0)}\right>=\dfrac{3\hbar
b^{3}}{2}N^{2}\left[1+\dfrac{p^{2}}{(E_{(0)}+m)^{2}}\right]F(r)\ ,$ (46)

where $F(r)=b^{3}\int f(r)d^{3}r$ represents the geometrical factor coming
from a spherically symmetric torsion function integrated over the relevant
volume of the spatial 3-dimensional hypersurfaces, for a specific space-time
foliation. Doing the same exercise with the spin down state with
$z_{2}=Ne^{i\vec{p}\cdot\vec{r}/\hbar}$,
$z_{3}=N\frac{p}{E_{(0)}+m}e^{i\vec{p}\cdot\vec{r}/\hbar}$ and
$z_{1}=z_{4}=0$, we also get Eq. (46) but with a global minus sign. The energy
difference between these two states thus becomes

$\delta E=3\hbar b^{3}N^{2}\left(1+\frac{p^{2}}{(E_{(0)}+m)^{2}}\right)F(r)\
,$ (47)

which corresponds to the frequency

$\nu=\dfrac{\delta E}{h}=\dfrac{3E_{(0)}}{\pi}F(r)\ ,$ (48)

where we have used again the conventional normalization
$\psi^{\dagger}_{0}\psi_{0}=2E_{0}$. Note that the same result for the energy
levels would have been obtained if we had considered all the components of the
3-momentum.

##  IV Non-minimally coupled fermions to torsion

Let us study now a fermionic Dirac Lagrangian non-minimally coupled to the RC
geometry. Consider the vector and axial vector fermionic currents
$j^{\lambda}\equiv\bar{\psi}\gamma^{\lambda}\psi$ and
$a^{\lambda}\equiv\bar{\psi}\gamma^{\lambda}\gamma^{5}\psi$, which are coupled
to torsion via the matter Lagrangian density

$\mathcal{L}_{\rm fermions}=\tilde{\mathcal{L}}_{\rm
Dirac}+\alpha_{1}\mathbb{T}\cdot\mathbb{j}+\alpha_{2}\breve{\mathbb{T}}\cdot\mathbb{a}\
,$ (49)

where $T^{\lambda}\equiv T^{\nu\lambda}_{\;\;\;\nu}$ is the trace vector part
of torsion and $\alpha_{1},\alpha_{2}$ some constants. For $\alpha_{1}=0$ and
$\alpha_{2}=3\hbar/2$ we recover the case of the minimal coupling to torsion
analyzed in the previous section. The extended Dirac equation in this case
reads

$i\hbar\gamma^{\mu}\tilde{D}_{\mu}\psi-m\psi=-\alpha_{1}T^{\lambda}\gamma_{\lambda}\psi-\alpha_{2}\breve{T}^{\lambda}\gamma_{\lambda}\gamma^{5}\psi\,.$
(50)

This model for the free fermion in a RC space-time geometry has parity
symmetry, and can be extended into a family of parity-breaking models. To
include parity-breaking terms (for a more detailed account of parity violation
in the general framework of Poincare theories of gravity see the recent work
Obukhov:2020zal ) in the Lagrangian density (which are expected to be relevant
in the early Universe due to the matter-antimatter asymmetry), we consider the
additional couplings $\mathbb{T}\cdot\mathbb{a}$ and
$\mathbb{\breve{T}}\cdot\mathbb{j}$, which yields the new Lagrangian density

$\mathcal{L}_{\rm fermions}=\tilde{\mathcal{L}}_{\rm
Dirac}+(\alpha_{1}\mathbb{T}+\beta_{2}\breve{\mathbb{T}})\cdot\mathbb{j}+(\alpha_{2}\breve{\mathbb{T}}+\beta_{1}\mathbb{T})\cdot\mathbb{a}\
,$ (51)

with new coefficients $\beta_{1},\beta_{2}$. The corresponding generalized
Dirac equation is given by

$\displaystyle i\hbar\gamma^{\mu}\tilde{D}_{\mu}\psi-m\psi$ $\displaystyle=$
$\displaystyle-\left(\alpha_{1}T^{\lambda}+\beta_{2}\breve{T}^{\lambda}\right)\gamma_{\lambda}\psi$
(52)
$\displaystyle-\left(\alpha_{2}\breve{T}^{\lambda}+\beta_{1}T^{\lambda}\right)\gamma_{\lambda}\gamma^{5}\psi\,,$

for spinors and

$\displaystyle i\hbar(\tilde{D}_{\mu}\bar{\psi})\gamma^{\mu}+m\bar{\psi}$
$\displaystyle=$
$\displaystyle-\left(\alpha_{1}T^{\lambda}+\beta_{2}\breve{T}^{\lambda}\right)\bar{\psi}\gamma_{\lambda}$
(53)
$\displaystyle-\left(\alpha_{2}\breve{T}^{\lambda}+\beta_{1}T^{\lambda}\right)\bar{\psi}\gamma_{\lambda}\gamma^{5}\
,$

for the adjoint spinors.

To estimate the new physics involved in this model we will take again the
zero-curvature limit in order to identify the effects of torsion and have a
qualitative notion of its consequences in the context of beyond the standard
model of particle physics interactions. In this limit, the Dirac equation (52)
becomes

$\displaystyle i\hbar\gamma^{\mu}\partial_{\mu}\psi-m\psi$ $\displaystyle=$
$\displaystyle-\left(\alpha_{1}T^{\lambda}+\beta_{2}\breve{T}^{\lambda}\right)\gamma_{\lambda}\psi$
(54)
$\displaystyle-\left(\alpha_{2}\breve{T}^{\lambda}+\beta_{1}T^{\lambda}\right)\gamma_{\lambda}\gamma^{5}\psi\
,$

As in the minimally coupled case, one can in principle consider different
ansatze for the background torsion depending on whether one is interested, for
instance, in gravitational wave astronomy (setting a dynamic, harmonic
torsion), or in simple models of the RC geometry around spherical compact
objects (setting a spherically symmetric ansatz). For free fermionic spinors
we try again the solutions
$\psi=\psi(\vec{r})e^{-iEt/\hbar}=e^{-i(Et-\vec{p}\cdot\vec{r})/\hbar}$.
Accordingly, we have the following time-independent Dirac equation

$\displaystyle-i\hbar\gamma^{k}\partial_{k}\psi+\Big{[}m-\left(\alpha_{1}T^{\lambda}+\beta_{2}\breve{T}^{\lambda}\right)\gamma_{\lambda}$
$\displaystyle\qquad-\Big{(}\alpha_{2}\breve{T}^{\lambda}+\beta_{1}T^{\lambda}\Big{)}\gamma_{\lambda}\gamma^{5}\Big{]}\psi(\vec{r})=\gamma^{0}E\psi(\vec{r})\
.$ (55)

Using the properties of the matrices $\gamma_{k}=-\gamma^{k}$,
$\gamma_{k}\gamma^{5}$ and $\gamma^{0}$, we can also write this equation via
the Hamiltonian matrix

$\hat{H}\psi(\vec{r})=E\psi(\vec{r})\ ,$ (56)

which explicitly reads as

$\displaystyle\begin{pmatrix}m-(t^{0}-\vec{\tau}\cdot\vec{\sigma})&\;\;\vec{\sigma}\cdot\hat{\vec{p}}-(\tau^{0}-\vec{t}\cdot\vec{\sigma})\\\
-\vec{\sigma}\cdot\hat{\vec{p}}+(\tau^{0}-\vec{t}\cdot\vec{\sigma})&\;\;m+(t^{0}-\vec{\tau}\cdot\vec{\sigma})\end{pmatrix}\begin{pmatrix}\psi_{I}\\\
\psi_{II}\end{pmatrix}$ $\displaystyle\qquad\qquad\qquad=\begin{pmatrix}E&0\\\
0&-E\end{pmatrix}\begin{pmatrix}\psi_{I}\\\ \psi_{II}\end{pmatrix}\ ,$ (57)

where $\hat{\vec{p}}=-i\hbar\vec{\nabla}$ is the 3-momentum operator, and we
have introduced the following notation for the torsion quantities

$t^{\nu}\equiv\alpha_{1}T^{\nu}+\beta_{2}\breve{T}^{\nu},\qquad\tau^{\lambda}\equiv\alpha_{2}\breve{T}^{\lambda}+\beta_{1}T^{\lambda}\
.$ (58)

Alternatively, this system can also be written in the more convenient way

$\displaystyle\begin{pmatrix}m-E&\;\;\vec{\sigma}\cdot\hat{\vec{p}}\\\
-\vec{\sigma}\cdot\hat{\vec{p}}&\;\;m+E\end{pmatrix}\begin{pmatrix}\psi_{I}\\\
\psi_{II}\end{pmatrix}$
$\displaystyle\qquad=\begin{pmatrix}t^{0}-\vec{\tau}\cdot\vec{\sigma}&\;\;\tau^{0}-\vec{t}\cdot\vec{\sigma}\\\
-\tau^{0}+\vec{t}\cdot\vec{\sigma}&\;\;-t^{0}+\vec{\tau}\cdot\vec{\sigma}\end{pmatrix}\begin{pmatrix}\psi_{I}\\\
\psi_{II}\end{pmatrix}\ ,$ (59)

which highlights the fact that the matrix on the right-hand side contains the
geometrical effects due to torsion, including spin-torsion interactions of
both parity-breaking and parity-preserving types.

The eigenvalue problem above is a system of two coupled equations for the
2-spinors $\psi_{I}$ and $\psi_{II}$. To solve it we use the general form of
the spinor $\psi(\vec{r})=\chi e^{i\vec{p}\cdot\vec{r}/\hbar}$ and the
properties of Pauli matrices, so that the first of these equations can be
written as

$\displaystyle\begin{pmatrix}p_{3}+t^{3}-\tau^{0}&\;\;p_{1}+t^{1}-i(p_{2}+t^{2})\\\
p_{1}+t^{1}+i(p_{2}+t^{2})&\;\;-p_{3}-t^{3}-\tau^{0}\end{pmatrix}\begin{pmatrix}\chi^{II}_{1}\\\
\chi^{II}_{2}\end{pmatrix}$
$\displaystyle\quad=\begin{pmatrix}E-m+t^{0}-\tau^{3}&-(\tau^{1}-i\tau^{2})\\\
-(\tau^{1}+i\tau^{2})&E-m+t^{0}+\tau^{3}\end{pmatrix}\begin{pmatrix}\chi^{I}_{1}\\\
\chi^{I}_{2}\end{pmatrix}.$ (60)

Now let us consider the two orthogonal spin up/down solutions for the
particle: $\chi^{I}=\begin{pmatrix}1\\\ 0\end{pmatrix}$ and
$\chi^{I}=\begin{pmatrix}0\\\ 1\end{pmatrix}$, and obtain the corresponding
4-spinor solutions. In the first (spin up) case, we get the system of
equations

$\displaystyle(p_{3}+t^{3}-\tau^{0})\chi^{II}_{1}+(p_{1}+t^{1}-i(p_{2}+t^{2}))\chi^{II}_{2}$
$\displaystyle\qquad\qquad\qquad=E-m+t^{0}-\tau^{3},$ (61)
$\displaystyle(p_{1}+t^{1}+i(p_{2}+t^{2}))\chi^{II}_{1}+(-p_{3}-t^{3}-\tau^{0})\chi^{II}_{2}$
$\displaystyle\qquad\qquad\qquad=-(\tau^{1}+i\tau^{2}),$ (62)

and therefore we find the solution $\chi=\begin{pmatrix}1\\\ 0\\\
\chi^{II}_{1}\\\ \chi^{II}_{2}\end{pmatrix}$, with

$\chi^{II}_{1}=\dfrac{(E-m+t^{0}-\tau^{3})(-p_{3}-t^{3}-\tau^{0})+(p_{1}+t^{1}-i(p_{2}+t^{2}))(\tau^{1}+i\tau^{2})}{(p_{3}+t^{3}-\tau^{0})(-p_{3}-t^{3}-\tau^{0})-(p_{1}+t^{1}-i(p_{2}+t^{2}))(p_{1}+t^{1}+i(p_{2}+t^{2}))},$
(63)
$\chi^{II}_{2}=\dfrac{-(\tau^{1}+i\tau^{2})(p_{3}+t^{3}-\tau^{0})-(p_{1}+t^{1}+i(p_{2}+t^{2}))(E-m+t^{0}-\tau^{3})}{(p_{3}+t^{3}-\tau^{0})(-p_{3}-t^{3}-\tau^{0})-(p_{1}+t^{1}-i(p_{2}+t^{2}))(p_{1}+t^{1}+i(p_{2}+t^{2}))}\
.$ (64)

Note that, in the vanishing-torsion (Minkowski) limit we obtain
$\chi_{1}^{II}=\tfrac{p_{3}}{E+m}$, $\chi_{2}^{II}=\tfrac{p_{1}+ip_{2}}{E+m}$,
which is exactly the 4-spinor solution corresponding to the free fermion, spin
up state, with $E^{2}=p^{2}+m^{2}$. As for the second (spin down) case, we
obtain the system

$\displaystyle(p_{3}+t^{3}-\tau^{0})\chi^{II}_{1}+(p_{1}+t^{1}-i(p_{2}+t^{2}))\chi^{II}_{2}$
$\displaystyle\qquad\qquad\qquad=-(\tau^{1}-i\tau^{2}),$ (65)
$\displaystyle(p_{1}+t^{1}+i(p_{2}+t^{2}))\chi^{II}_{1}+(-p_{3}-t^{3}-\tau^{0})\chi^{II}_{2}$
$\displaystyle\qquad\qquad\qquad=E-m+t^{0}+\tau^{3}\ ,$ (66)

and therefore we find the solution $\chi=\begin{pmatrix}0\\\ 1\\\
\chi^{II}_{1}\\\ \chi^{II}_{2}\end{pmatrix}$, with

$\chi^{II}_{1}=\dfrac{-(\tau^{1}-i\tau^{2})(-p_{3}-t^{3}-\tau^{0})-(p_{1}+t^{1}-i(p_{2}+t^{2}))(E-m+t^{0}+\tau^{3})}{(p_{3}+t^{3}-\tau^{0})(-p_{3}-t^{3}-\tau^{0})-(p_{1}+t^{1}-i(p_{2}+t^{2}))(p_{1}+t^{1}+i(p_{2}+t^{2}))}$
(67)
$\chi^{II}_{2}=\dfrac{(E-m+t^{0}+\tau^{3})(p_{3}+t^{3}-\tau^{0})+(p_{1}+t^{1}+i(p_{2}+t^{2}))(\tau^{1}-i\tau^{2})}{(p_{3}+t^{3}-\tau^{0})(-p_{3}-t^{3}-\tau^{0})-(p_{1}+t^{1}-i(p_{2}+t^{2}))(p_{1}+t^{1}+i(p_{2}+t^{2}))}\
.$ (68)

Again we have the correct Minkowski limit,
$\chi_{1}^{II}=\tfrac{p_{1}-ip_{2}}{E+m}$, and
$\chi_{2}^{II}=\tfrac{-p_{3}}{E+m}$, describing the free particle, spin down
state. Note that, proceeding in a similar manner, we could derive the
corresponding expressions for the 4-spinor solutions associated to the anti-
fermion in the spin up/down states.

To simplify further our analysis let us consider the ansatz for the torsion
components $t^{\mu}=(0,t^{1},t^{2},t^{3})$, and $\tau^{\mu}=(0,0,0,\tau)$ in
Eq. (58). The spin up particle solution is then given by

$\psi=\begin{pmatrix}1\\\ 0\\\ \dfrac{p^{\rm eff}_{3}}{E+m_{\rm eff}}\\\
\dfrac{p_{1}^{\rm eff}+ip_{2}^{\rm eff}}{E+m_{\rm
eff}}\end{pmatrix}e^{i(\vec{p}\cdot\vec{r}-Et)/\hbar}\ ,$ (69)

where

$E^{2}=p^{2}_{\rm eff}+m^{2}_{\rm eff}\ ,$ (70)

with the definitions

$\displaystyle p_{k}^{\rm eff}$ $\displaystyle\equiv$ $\displaystyle
p_{k}+t^{k}$ (71) $\displaystyle p^{2}_{\rm eff}$ $\displaystyle\equiv$
$\displaystyle(p_{1}+t^{1})^{2}+(p_{2}+t^{2})^{2}+(p_{3}+t^{3})^{2}$ (72)
$\displaystyle m_{\rm eff}$ $\displaystyle\equiv$ $\displaystyle m+\tau\ .$
(73)

Analogously, for the spin down particle we get the solution

$\psi=\begin{pmatrix}0\\\ 1\\\ \dfrac{p^{\rm eff}_{1}+ip^{\rm
eff}_{2}}{E+m_{\rm eff}}\\\ \dfrac{-p^{\rm eff}_{3}}{E+m_{\rm
eff}}\end{pmatrix}e^{i(\vec{p}\cdot\vec{r}-Et)/\hbar}\ ,$ (74)

where Eqs. (70)–(72) still hold but now the effective mass in Eq. (73) becomes
$m_{\rm eff}\equiv m-\tau$. Therefore, two different energy levels are
obtained for the spin up and spin down states. The energy of the anti-aligned
state with respect to the direction of $\vec{\tau}$ is lower than the aligned
state. These two possible energy states, $E_{2}^{2}=p_{\rm
eff}^{2}+(m+\tau)^{2}$ and $E_{1}^{2}=p_{\rm eff}^{2}+(m-\tau)^{2}$,
correspond to the energy transition

$h\nu=E_{2}-E_{1}=\dfrac{4m\tau}{E_{1}+E_{2}}\ ,$ (75)

which in the reference frame of the particle reads

$h\nu=\dfrac{4m\tau}{\left[\vec{t}^{2}+(m+\tau)^{2}\right]^{1/2}+\left[\vec{t}^{2}+(m-\tau)^{2}\right]^{1/2}}\
,$ (76)

where $\vec{t}^{2}\equiv(t^{1})^{2}+(t^{2})^{2}+(t^{3})^{2}$ can be written
simply as $t^{2}$ assuming that $\vec{t}$ is aligned in any of the spatial
axis directions of the reference system of coordinates. We recall that the
torsion functions $t^{\mu}$ and $\tau^{\mu}$ are constructed from the torsion
trace vector and axial vectors and depend on the (parity-preserving)
$(\alpha_{1},\alpha_{2})$ and (parity-breaking) $(\beta_{1},\beta_{2})$
coupling parameters.

The bottom line of this section is that parity-breaking effects can arise in a
RC space-time from the interaction of fermions with a background torsion field
via non-minimal couplings. These effects include the prediction of well-
defined frequencies that a free fermion can absorb or emit in order to make
transitions between the predicted two energy levels that arise depending on
the spin orientation with respect to external torsion quantities. The
signature of parity breaking might also be present in the radiated field
itself.

##  V Non-minimal couplings in the Einstein-Cartan theory

This framework could be further specified by considering the EC theory. The
corresponding Lagrangian then reads

$S_{\rm EC}=\dfrac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}R(\Gamma)+\int
d^{4}x\sqrt{-g}\,\mathcal{L}_{\rm fermions}\,,$ (77)

where the Ricci scalar of the independent connection, $R(\Gamma)$, with
$\Gamma\equiv\Gamma_{\mu\nu}^{\lambda}$, can be related to the one constructed
with the metric-compatible connection, $\tilde{R}(\tilde{\Gamma})$, via an
expression of the form

$R\sim\tilde{R}-4\tilde{\nabla}_{\alpha}T^{\alpha}-\dfrac{1}{3}T^{\lambda}T_{\lambda}+\dfrac{1}{24}\breve{T}^{\lambda}\breve{T}_{\lambda}+\dfrac{1}{2}\bar{T}_{\mu\nu\rho}\bar{T}^{\mu\nu\rho}.$
(78)

Inserting this in the action (77) and taking the fermionic Lagrangian defined
in Eq. (49), the corresponding Cartan equations become

$T^{\mu}\sim\kappa^{2}\alpha_{1}j^{\mu}\,,\qquad\breve{T}^{\mu}\sim\kappa^{2}\alpha_{2}a^{\mu}\
.$ (79)

Re-inserting these expressions in the (49) we obtain effective vector-vector
contact interactions besides the usual well-known axial-axial (spin-sin)
interaction (Hehl-Data term) as

$\mathcal{L}_{\rm fermions}\sim\tilde{\mathcal{L}}_{\rm
Dirac}+\dfrac{\kappa^{2}}{3}(\alpha_{1})^{2}{\bf j}\cdot{\bf
j}-\dfrac{\kappa^{2}}{24}(\alpha_{2})^{2}{\bf a}\cdot{\bf a}\ .$ (80)

The corresponding Dirac equation can be written as

$\displaystyle i\hbar\gamma^{\mu}\tilde{D}_{\mu}\psi-m\psi$ $\displaystyle=$
$\displaystyle\dfrac{\kappa^{2}\alpha_{2}^{2}}{12}(\bar{\psi}\gamma^{\lambda}\gamma^{5}\psi)\gamma_{\lambda}\gamma^{5}\psi$
(81)
$\displaystyle-\kappa^{2}\dfrac{2\alpha_{1}}{3}(\bar{\psi}\gamma^{\lambda}\psi)\gamma_{\lambda}\psi\,.$

As in the usual Dirac-Hehl-Data equation, under charge conjugation operation
$\psi\rightarrow\psi^{ch}$ one obtains different dynamics for the $\psi^{ch}$
representing anti-fermions. If we use instead the Lagrangian in (51), then the
Cartan equations are

$T^{\mu}\sim\kappa^{2}(\zeta_{1}j^{\mu}+\zeta_{2}a^{\mu})\,,\qquad\breve{T}^{\mu}\sim\kappa^{2}(\theta_{1}j^{\mu}+\theta_{2}a^{\mu})\
,$ (82)

where $\zeta_{i},\theta_{i}$ ($i=1,2$) are constants, and the resulting Dirac
equation, after substitution in (52), includes parity-breaking and C-breaking
cubic terms.

The EC theory can also be extended to include the parity-breaking Holst term
Holst:1995pc , encapsulating additional vector-axial (contact) self-
interactions. The extended action is thus

$\displaystyle S_{\rm EC}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}R(\Gamma)+\int
d^{4}x\sqrt{-g}\,\mathcal{L}_{\rm fermions}$ (83)
$\displaystyle+\dfrac{1}{2\gamma\kappa^{2}}\int
d^{4}x\sqrt{-g}\epsilon^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}\ ,$

where $\gamma$ is the Barbero-Immirzi parameter and the parity-breaking Holst
term $\epsilon^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}$ can be expressed as

$\epsilon^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}\sim-\tilde{\nabla}_{\alpha}\breve{T}^{\alpha}-\dfrac{1}{3}\breve{T}^{\lambda}T_{\lambda}+\dfrac{1}{2}\epsilon^{\alpha\beta\mu\nu}\bar{T}^{\lambda}_{\;\;\alpha\beta}\bar{T}_{\lambda\mu\nu}\
.$ (84)

The generalized Cartan equations become

$\displaystyle T^{\mu}$ $\displaystyle\sim$
$\displaystyle\kappa^{2}\dfrac{3\gamma}{1+\gamma^{2}}(\alpha_{1}\gamma
j^{\mu}+a^{\mu})\,,$ (85) $\displaystyle\breve{T}^{\mu}$ $\displaystyle\sim$
$\displaystyle\kappa^{2}\dfrac{3\gamma}{1+\gamma^{2}}(\alpha_{1}j^{\mu}-\alpha_{2}\gamma
a^{\mu})\,.$ (86)

Then, by choosing the fermionic Lagrangian in Eq. (49), one obtains a
generalized Dirac equation and Lagrangian with vector-vector, axial-axial and
parity-breaking vector-axial (contact) self-interactions. As in the model-
independent approach of the previous section, all these cases of non-minimal
couplings in specific gravitational models (EC and its Holst extension) yield
interesting and quite relevant physics (C and P symmetry-breaking, beyond
standard model interactions, etc) that in principle can be observationally
probed, upon the computation of the energy levels of fermionic systems and
searching for its signatures using advanced spectrographs.

##  VI Discussion and conclusion

Experimental constraints on the minimal and non-minimal coupling of spinors
with torsion using high-precision data has been a topic of interest in the
literature since quite a long time ago Kostelecky ; Heckel:2008hw ;
Lehnert:2013jsa . In this sense, the results derived in the present paper may
have a physical impact at several levels. First, by generalizing these results
to the case of bound states of electrons within atoms and molecules, and also
to bound states of nucleons within atomic nuclei, one opens up the possibility
of detecting the effects of torsion in the strong gravitational regime via the
measurement of spectral lines and searching for new fine structures, for
instance, using ultra-sensitive spectrographs. Therefore, there could be
astrophysical spectral signatures of torsion waiting to be discovered around
intense gravitational fields of neutron stars, or even in X-ray binaries,
where one of the objects is a black hole candidate surrounded by an accretion
disk. For a given bound system, a specific initial energy level could be
chosen such that the transitions from this level into the two (Zeeman-like)
lower levels due to the torsion-spin interaction could be searched for.
Indeed, this could be carried out for different values of the predicted
torsion in the emission regions according to different gravitational theories
accommodating torsion effects.

There are further examples of astrophysical interest where torsion effects
into the physics of fermions/anti-fermions could be observationally detected.
A particular case is that of polarized fermion/anti-fermion pairs produced
within quantum field theory, for instance in the space-time torsion around
black holes. After their subsequent annihilation, the generated photons would
have different energies depending on whether the spins of the fermions in the
pair are up or down with respect to the background torsion. In the particle
pair production in general, there are three possible scenarios: i) particle
and antiparticle are anti-aligned with each other, ii) both are with their
spins up (i.e. aligned to torsion), iii) both are with their spins down. These
correspond to the three energy/mass levels that the (initial) photons can
generate via pair production, with the first case having an intermediate
energy, the second the highest, and the third the lowest. In the very early
Universe, such effects would depend on the temperature of the quark-gluon-
lepton-photon plasma, while in Hawking’s radiation the outgoing and ingoing
energy flux through the event horizon would be spin state-dependent. In
particular, the (outflux) energy loss via particles (or antiparticles) aligned
with the background torsion would be more efficient.

These effects could also be sought for in the emission of gamma rays in high
energy astrophysical environments driven by strong gravitational fields. An
observable signature of the background torsion could be then obtained by
comparing the measured flux spectra with the detailed theoretical prediction
of the emission curves. In the most general case the theory suggests that the
radiated flux should result from the superposition of the three possibilities
above peaked at the characteristic nearby frequencies, corresponding to the
three possible values of the energy of the (annihilating) pair. Disregarding
complex environmental effects and significant changes in the background
torsion (in the typical scales of the emission region), the emission curves
would resemble emission lines very close together (a kind of hyperfine
structure). In the more conventional case of pair production with the fermion
and anti-fermion anti-aligned with respect to each other, the theory predicts
a specific characteristic frequency, determined by the particle’s mass, the
torsion field, and the temperature of the emission region, slightly deviated
from the corresponding predicted frequency when torsion is absent.

Beyond astrophysics and cosmology, torsion-induced effects of the kind studied
in this work might be found in the field of condensed matter physics. Such is
the case of the interaction of a Dirac fermion with the torsion of a sea of
vacuum fermion-condensates, provided that the latter has a non-zero
expectation value. This way, fermions in vacuum would have a different
effective mass according to the relative orientation of the fermionic spin
with respect to the background vacuum axial torsion, which could be tested in
laboratories, putting bounds on the predicted effects. While these tests do
not require strong gravitational fields the challenge lies on reproducing the
conditions of fermionic vacuum condensates in the laboratory.

Another possibility would be the existence of a continuous and smooth phase
transition for a Bose-Einstein system as a superconducting fluid/material in a
space-time background with torsion. Cooper pairs of anti-aligned fermions in
bound states are required in the BCS model of superconductivity and in general
Bose-Einstein condensates, and since the effective mass of the pair would
depend on the interaction with the torsion background, the effective spin-zero
bosonic field due to the ensemble of Cooper pairs would have a differential
effective mass powered by the relative strength of torsion. If such were the
case, then this would have a non-negligible impact on the superconducting and
superfluid phases in the interiors of neutron stars and hypothetical quark and
strange stars, with consequences on the macroscopic predictions of stellar
models (mass-radius relations, moment of inertia, etc). Moreover, the
stability of Cooper pairs might be strongly perturbed as the torsion increases
above a certain threshold, since the background torsion axial vector along a
well defined direction can act exactly as an external magnetic field does in
paramagnetic materials, i.e., above a certain critical value of the external
field a significant number of large clusters of “aligned” spins are developed
(and percolating the whole system) and the material is magnetized. The spin-
spin interaction that naturally exists in a system with spins is analogous to
a thermal-like interaction (increasing temperature tends to rise the entropy,
and generate a random distribution of spins), while the external field tends
to counteract the random distribution of spins, by establishing gradually a
more ordered state. Therefore, torsion can also act as an external field
driving a phase transition in a macroscopic system of microphysical components
with spin, magnetizing the material, with the emergence of a macroscopic
(intrinsic) spin. In that sense, the superfluid/superconducting phase of the
BCS models could suffer a phase transition for sufficiently strong “external”
space-time torsion, inside ultra-dense compact objects. These topics deserve a
much more careful analysis, since they evolve very complicated physics of the
interiors of neutron stars and related objects.

Further avenues of research are those involving laboratory tests of the space-
time torsion near Earth. In perfect analogy with the magnetic spin resonance,
one could design a torsion-spin resonance. In this effect, an external torsion
field generates the splitting of energy levels (Zeeman-like effect) in an
appropriate material sample, while a time-varying current produces an
electromagnetic wave which suffers a measurable absorption once the resonance
frequency is achieved matching the energy gap. Therefore, the indirect
detection of torsion would be achieved by the measurement of absorption
(decrease in intensity) of the electromagnetic wave interacting with the
material sample, once the resonance frequency is achieved. For free fermions
we saw that the predicted frequency (in the particle’s frame) does not depend
on the fermionic mass, only on the background torsion. If torsion has a
magnitude of about $10^{-16}-10^{-15}$ m-1 then we get an estimated resonance
frequency around $1-10$ nHz, which corresponds to resonance in the radio band.

Finally, regarding the non-minimal couplings within specific gravity models
such as the EC one and its extension with the Holst term, one also obtains
generalized Dirac equations and Lagrangians with vector-vector, axial-axial
and parity-breaking vector-axial (contact) self-interactions. These might be
relevant inside compact objects like neutron, quark, and strange (quark) stars
and also in the early Universe. If the coupling constants are taken to be
dynamical scalar fields, then this scenario leads naturally to the idea of
parity-breaking phase transitions for matter under extreme conditions, induced
by the torsion-fermion currents couplings. We also see that EC gravity plus
Holst with $T\cdot j$ and $\breve{T}\cdot a$ couplings can be made equivalent
to the usual EC theory with $T\cdot j$ and $\breve{T}\cdot a$ plus (parity-
breaking) $T\cdot a$ and $\breve{T}\cdot j$ couplings.

To conclude, the results obtained in this paper open up new avenues for
testing non-Riemannian geometries with torsion using splitting of energy
levels in both minimally and non-minimally coupled fermions to the background
torsion in a variety of astrophysical/cosmological environments. Further work
along these lines is currently underway.

## Acknowledgments

We thank A. Delhom for useful comments and insights. FC is funded by the
Fundação para a Ciência e a Tecnologia (FCT, Portugal) doctoral grant
No.PD/BD/128017/2016. FSNL acknowledges support from the FCT Scientific
Employment Stimulus contract with reference CEECIND/04057/2017. DRG is funded
by the _Atracción de Talento Investigador_ programme of the Comunidad de
Madrid (Spain) No. 2018-T1/TIC-10431, and acknowledges further support from
the Ministerio de Ciencia, Innovación y Universidades (Spain) project No.
PID2019-108485GB-I00/AEI/10.13039/501100011033, the Spanish project No.
FIS2017-84440-C2-1-P (MINECO/FEDER, EU), the project PROMETEO/2020/079
(Generalitat Valenciana), and the Edital 006/2018 PRONEX (FAPESQ-PB/CNPQ,
Brazil) Grant No. 0015/2019. The authors also acknowledge funding from FCT
Projects No. UID/FIS/04434/2020, No. CERN/FIS-PAR/0037/2019 and No. PTDC/FIS-
OUT/29048/2017. FC thanks the hospitality of the Department of Theoretical
Physics and IPARCOS of the Complutense University of Madrid, where part of
this work was carried out. This article is based upon work from COST Action
CA18108, supported by COST (European Cooperation in Science and Technology).

## References

  * (1) M. Blagojevic, “Gravitation and gauge symmetries,” (Institute of Physics Publishing, 2002).
  * (2) M. Blagojevic and F. W. Hehl, “Gauge Theories of Gravitation,” [arXiv:1210.3775 [gr-qc]].
  * (3) F. Cabral, F. S. N. Lobo and D. Rubiera-Garcia, “Fundamental Symmetries and Spacetime Geometries in Gauge Theories of Gravity: Prospects for Unified Field Theories,” Universe 6 (2020), 238 [arXiv:2012.06356 [gr-qc]].
  * (4) F. W. Hehl and Y. N. Obukhov, “Elie Cartan’s torsion in geometry and in field theory, an essay,” Annales Fond. Broglie 32 (2007), 157-194 [arXiv:0711.1535 [gr-qc]].
  * (5) F. Cabral, F. S. N. Lobo and D. Rubiera-Garcia, “Einstein–Cartan–Dirac gravity with $U(1)$ symmetry breaking,” Eur. Phys. J. C 79 (2019) no.12, 1023 [arXiv:1902.02222 [gr-qc]].
  * (6) F. Cabral, F. S. N. Lobo and D. Rubiera-Garcia, “Cosmological bounces, cyclic universes, and effective cosmological constant in Einstein-Cartan-Dirac-Maxwell theory,” Phys. Rev. D 102 (2020) no.8, 083509 [arXiv:2003.07463 [gr-qc]].
  * (7) F. Cabral, F. S. N. Lobo and D. Rubiera-Garcia, “The cosmological principle in theories with torsion: The case of Einstein-Cartan-Dirac-Maxwell gravity,” JCAP 10 (2020), 057 [arXiv:2004.13693 [gr-qc]].
  * (8) C. G. Boehmer and J. Burnett, “Dark spinors with torsion in cosmology,” Phys. Rev. D 78 (2008), 104001 [arXiv:0809.0469 [gr-qc]].
  * (9) N. J. Poplawski, “Nonsingular, big-bounce cosmology from spinor-torsion coupling,” Phys. Rev. D 85 (2012), 107502 [arXiv:1111.4595 [gr-qc]].
  * (10) N. J. Poplawski, “Big bounce from spin and torsion,” Gen. Rel. Grav. 44 (2012), 1007-1014 [arXiv:1105.6127 [astro-ph.CO]].
  * (11) B. Vakili and S. Jalalzadeh, “Signature transition in Einstein-Cartan cosmology,” Phys. Lett. B 726 (2013), 28-32 [arXiv:1308.2877 [gr-qc]].
  * (12) K. A. Bronnikov and A. M. Galiakhmetov, “Wormholes and black universes without phantom fields in Einstein-Cartan theory,” Phys. Rev. D 94 (2016) no.12, 124006 [arXiv:1607.07791 [gr-qc]].
  * (13) A. N. Ivanov and M. Wellenzohn, “Einstein-Cartan Gravity with Torsion Field Serving as an Origin for the Cosmological Constant or Dark Energy Density,” Astrophys. J. 829 (2016) no.1, 47 [arXiv:1607.01128 [gr-qc]].
  * (14) J. A. R. Cembranos, J. Gigante Valcarcel and F. J. Maldonado Torralba, “Singularities and n-dimensional black holes in torsion theories,” JCAP 04 (2017), 021 [arXiv:1609.07814 [gr-qc]].
  * (15) J. A. R. Cembranos and J. Gigante Valcarcel, “Extended Reissner–Nordström solutions sourced by dynamical torsion,” Phys. Lett. B 779 (2018), 143-150 [arXiv:1708.00374 [gr-qc]].
  * (16) G. Unger and N. Popławski, “Big bounce and closed universe from spin and torsion,” Astrophys. J. 870 (2019) no.2, 78 [arXiv:1808.08327 [gr-qc]].
  * (17) M. R. Mehdizadeh and A. H. Ziaie, “Charged Wormhole Solutions in Einstein-Cartan gravity,” Phys. Rev. D 99 (2019) no.6, 064033 [arXiv:1811.03364 [gr-qc]].
  * (18) E. Barausse, E. Berti, T. Hertog, S. A. Hughes, P. Jetzer, P. Pani, T. P. Sotiriou, N. Tamanini, H. Witek and K. Yagi, et al. “Prospects for Fundamental Physics with LISA,” Gen. Rel. Grav. 52 (2020) no.8, 81 [arXiv:2001.09793 [gr-qc]].
  * (19) J. B. Jiménez, L. Heisenberg and T. S. Koivisto, “The Geometrical Trinity of Gravity,” Universe 5 (2019) no.7, 173 [arXiv:1903.06830 [hep-th]].
  * (20) L. Parker, “One-Electron Atom in Curved Space-Time,” Phys. Rev. Lett. 44 (1980) no.23, 1559.
  * (21) L. Parker, “One electron atom as a probe of space-time curvature,” Phys. Rev. D 22 (1980), 1922-1934.
  * (22) L. Parker and L. O. Pimentel, “Gravitational perturbation of the hydrogen spectrum,” Phys. Rev. D 25 (1982), 3180-3190.
  * (23) A. Delhom, V. Miralles and A. Peñuelas, “Effective interactions in Ricci-Based Gravity below the non-metricity scale”, Eur. Phys. J. C 80 (2020) no.4, 340 [arXiv:1907.05615 [hep-th]].
  * (24) A. D. I. Latorre, G. J. Olmo and M. Ronco, “Observable traces of non-metricity: new constraints on metric-affine gravity,” Phys. Lett. B 780 (2018), 294-299 [arXiv:1709.04249 [hep-th]].
  * (25) V. G. Bagrov, I. L. Buchbinder, I .L. Shapiro, “Possible experimental manifestations of the torsion field”, Sov. J. Phys., 35 (1992) n3, p.5-12.
  * (26) L. H. Ryder, I.L. Shapiro, “On the interaction of massive spinor particles with external electromagnetic and torsion fields”, Physics Letters A 247 (1998) 21-26.
  * (27) B. Goncalves, Y. N. Obukhov, I. L. Shapiro, “On the exact Foldy-Wouthuysen transformation for a Dirac spinor in torsion and other CPT and Lorentz violating backgrounds”, Phys. Rev. D 80 (2009) 125034 [arXiv:0908.0437].
  * (28) I. L. Shapiro, “Physical Aspects of the Space-Time Torsion”, Physics Reports 357 (2002) 113-213 [hep-th/0103093].
  * (29) J. Audretsch, “Dirac Electron in Space-times With Torsion: Spinor Propagation, Spin Precession, and Nongeodesic Orbits,” Phys. Rev. D 24 (1981) 1470.
  * (30) Y. N. Obukhov, A. J. Silenko and O. V. Teryaev, “Spin-torsion coupling and gravitational moments of Dirac fermions: theory and experimental bounds, Phys. Rev. D 90 (2014) no.12, 124068.
  * (31) Y. N. Obukhov, A. J. Silenko and O. V. Teryaev, “Dirac particle in Riemann–Cartan spacetimes,” Phys. Part. Nucl. 49 (2018) no.1, 9-10.
  * (32) M. Adak, T. Dereli and L. H. Ryder, “Dirac equation in space-times with nonmetricity and torsion,” Int. J. Mod. Phys. D 12 (2003), 145-156 [arXiv:gr-qc/0208042 [gr-qc]].
  * (33) J. A. R. Cembranos, J. G. Valcarcel and F. J. Maldonado Torralba, “Fermion dynamics in torsion theories,” JCAP 04 (2019), 039 [arXiv:1805.09577 [gr-qc]].
  * (34) S. Bahamonde and J. G. Valcarcel, “New models with independent dynamical torsion and nonmetricity fields,” JCAP 09 (2020), 057 [arXiv:2006.06749 [gr-qc]].
  * (35) Y. N. Obukhov, “Parity violation in Poincaré gauge gravity,” [arXiv:2010.16276 [gr-qc]].
  * (36) S. Holst, “Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action,” Phys. Rev. D 53 (1996), 5966-5969 [arXiv:gr-qc/9511026 [gr-qc]].
  * (37) V. A. Kostelecky, N. Russell and J. Tasson, “New Constraints on Torsion from Lorentz Violation,” Phys. Rev. Lett. 100, 111102 (2008) [arXiv:0712.4393 [gr-qc]].
  * (38) B. R. Heckel, E. G. Adelberger, C. E. Cramer, T. S. Cook, S. Schlamminger and U. Schmidt, “Preferred-Frame and CP-Violation Tests with Polarized Electrons,” Phys. Rev. D 78 (2008), 092006 [arXiv:0808.2673 [hep-ex]].
  * (39) R. Lehnert, W. M. Snow and H. Yan, “A First Experimental Limit on In-matter Torsion from Neutron Spin Rotation in Liquid ${}^{4}He$,” Phys. Lett. B 730 (2014), 353-356 [erratum: Phys. Lett. B 744 (2015), 415-415] [arXiv:1311.0467 [nucl-ex]].