# Disordered collective motion in dense assemblies of persistent particles

Yann-Edwin Keta  Laboratoire Charles Coulomb (L2C), Université de Montpellier,
CNRS, 34095 Montpellier, France  Robert L. Jack  Yusuf Hamied Department of
Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United
Kingdom  Department of Applied Mathematics and Theoretical Physics, University
of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom  Ludovic
Berthier  Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS,
34095 Montpellier, France  Yusuf Hamied Department of Chemistry, University of
Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

###### Abstract

We explore the emergence of nonequilibrium collective motion in disordered
non-thermal active matter when persistent motion and crowding effects compete,
using simulations of a two-dimensional model of size polydisperse self-
propelled particles. In stark contrast with monodisperse systems, we find that
polydispersity stabilizes a homogeneous active liquid at arbitrary large
persistence times, characterized by remarkable velocity correlations and
irregular turbulent flows. For all persistence values, the active fluid
undergoes a nonequilibrium glass transition at large density. This is
accompanied by collective motion, whose nature evolves from near-equilibrium
spatially heterogeneous dynamics at small persistence, to a qualitatively
different intermittent dynamics when persistence is large. This latter regime
involves a complex time evolution of the correlated displacement field.

Active matter constitutes a prominent class of nonequilibrium systems
Marchetti  _et al._ (2013). Examples in biological systems range from tissues
Angelini  _et al._ (2011) to bird flocks Cavagna and Giardina (2014) and human
crowds Bain and Bartolo (2019). Synthetic active systems include robot swarms
Rubenstein  _et al._ (2014) and self-catalytic colloids, which have inspired
numerous theoretical studies. In particular, self-propelled particles are
paradigmatic models for active matter, where varying a few parameters leads to
rich collective behaviour Fodor and Marchetti (2018).

This work focuses on the interplay of motility and crowding in such models,
which is relevant in both synthetic Klongvessa  _et al._ (2019a) and living
systems Angelini  _et al._ (2011). For systems with aligning interactions or
non-spherical particles, collective motion occurs at large density Vicsek and
Zafeiris (2012); Bär  _et al._ (2020), as predicted theoretically Dunkel  _et
al._ (2013) and observed in experiments Wensink  _et al._ (2012); Angelini
_et al._ (2011) and simulations Wensink and Löwen (2012); Mandal  _et al._
(2017). For isotropic particles, it is well-known that persistent motility can
trigger phase separation (MIPS) Cates and Tailleur (2015); it also leads to
nonequilibrium velocity correlations Szamel and Flenner (2021), possibly
connected to active turbulence Mandal  _et al._ (2020); Kuroda  _et al._
(2022). A different set of phenomena are associated with crowding effects in
systems with modest persistence times: they undergo nonequilibrium glass
transitions at high density Berthier  _et al._ (2019); Janssen (2019). This
transition shares many features with its equilibrium counterpart Berthier and
Biroli (2011), such as two-step relaxation dynamics and slow, spatially
correlated motion. It has been thoroughly studied using theory Berthier and
Kurchan (2013); Nandi  _et al._ (2018); Reichert  _et al._ (2021), simulations
Berthier (2014); Flenner  _et al._ (2016); Berthier  _et al._ (2017, 2019),
and experiments Klongvessa  _et al._ (2019a, b).

In the following we present numerical simulations of isotropic particles that
unify this active glass phenomenology with the effects of crowding at much
larger persistence times, where recent exploratory works have suggested the
emergence of intermittent plasticity Mandal  _et al._ (2020), analogies to
yielding transitions Liao and Xu (2018); Morse  _et al._ (2021); Villarroel
and Düring (2021), and a breakdown of equipartition in the disordered solid
Henkes  _et al._ (2020); Caprini and Marini Bettolo Marconi (2020). We
intentionally introduce size polydispersity, to ensure the stability of
disordered states in crowded systems. Hence, our results (e.g. Fig. 1)
parallel the similar programme that was recently completed for monodisperse
particles, which display ordered phases at large density Redner  _et al._
(2013); Digregorio  _et al._ (2018); Omar  _et al._ (2021); Caprini  _et al._
(2020a, b).

Figure 1: Disordered active states when packing fraction $\phi$ and
persistence time $\tau_{p}$ are varied at constant $D_{0}=1$. The MIPS region,
enclosed by red circles, corresponds to phase separation. The homogeneous
active liquid experiences a nonequilibrium glass transition at large packing
fraction (blue squares). Figure 2: (a, b) Mean squared displacements for
different packing fractions $\phi$, at persistence time (a) $\tau_{p}=10^{-2}$
and (b) $\tau_{p}=10^{2}$, for $N=1024$. (c) Density evolution of the scaled
diffusion constant $D/D_{0}$ for different persistence times $\tau_{p}$.
Dashed lines correspond to algebraic fits.

At large persistence, we find that polydispersity stabilizes a homogeneous
liquid Fily  _et al._ (2014); Paoluzzi  _et al._ (2021) which solidifies into
a disordered glass at large packing fraction (see Fig. 1). This liquid
displays inherently active characteristics, including correlated velocities
and displacements, with similarities to active turbulence. Close to dynamic
arrest, its relaxation is intermittent, characterized by rapid rearrangements
between transient mechanical equilibria Mandal and Sollich (2020, 2021). This
behavior is distinct from active glasses at lower persistence, which tend to
resemble passive glasses Berthier  _et al._ (2019); Janssen (2019).

We simulate $N$ athermal self-propelled particles in a square box of linear
size $L$ with periodic boundary conditions, with overdamped dynamics:

$\xi\dot{\bm{r}}_{i}=-\sum_{j\neq i}\nabla_{i}U(r_{ij})+\bm{p}_{i},$ (1)

where $\bm{r}_{i}$ is the position of particle $i$,
$r_{ij}=|{\bm{r}}_{i}-{\bm{r}}_{j}|$, $\xi$ a viscous damping, and
$\bm{p}_{i}$ the propulsion force. Particles interact via a repulsive Weeks-
Chandler-Andersen potential
$U=4\varepsilon\left[(\sigma_{ij}/r_{ij})^{12}-(\sigma_{ij}/r_{ij})^{6}+1/4\right]\Theta(2^{1/6}\sigma_{ij}-r_{ij})$,
where $\Theta$ is the Heaviside function,
$\sigma_{ij}=\frac{1}{2}(\sigma_{i}+\sigma_{j})$, and $\sigma_{i}$ the
diameter of particle $i$. Diameters are drawn from a uniform distribution of
mean $\sigma=\overline{\sigma_{i}}$ and polydispersity 20%. The packing
fraction is $\phi=2^{1/3}\pi N\overline{\sigma_{i}^{2}}/(4L^{2})$, $\sigma$
sets the unit length, $\varepsilon$ the unit energy, and
$\xi\sigma^{2}/\varepsilon$ the unit time. We measure the positions
$\bm{r}_{i}$ and velocities $\bm{v}_{i}$ in the center-of-mass frame.

Different self-propulsion models differ by the dynamics of $\bm{p}_{i}$. For
active Brownian and run-and-tumble particles Solon  _et al._ (2015),
$|\bm{p}_{i}|$ is fixed. We consider active Ornstein-Uhlenbeck particles
Szamel (2014); Koumakis  _et al._ (2014):

$\tau_{p}\dot{\bm{p}}_{i}=-\bm{p}_{i}+\sqrt{2D_{0}}\bm{\eta}_{i},$ (2)

with $\bm{\eta}_{i}$ a Gaussian white noise of zero mean and variance
$\left<\bm{\eta}_{i}(t)\bm{\eta}_{j}(0)\right>=\mathbbm{1}\,\delta_{ij}\,\delta(t)$,
and $\tau_{p}$ the persistence time. The amplitude and correlation time of
$\bm{p}_{i}$ can be varied independently so there are three control parameters
$(\phi,D_{0},\tau_{p})$. The parameter space reduces to $(\phi,\tau_{p})$ for
self-propelled hard disks Berthier (2014), while adding a thermal noise in Eq.
(1) would instead make the phase diagram four-dimensional Paoluzzi  _et al._
(2021). From Eqs. (1, 2), we see that $D_{0}$ is the diffusion constant of a
free particle, the typical amplitude of the propulsion is
$\sqrt{\left<|{\bm{p}}_{i}|^{2}\right>}=\sqrt{2D_{0}/\tau_{p}}\equiv v_{0}$,
and the persistence length $\ell_{p}=\sqrt{D_{0}\tau_{p}}$. When $\tau_{p}\to
0$, equilibrium Brownian dynamics at temperature $T_{\rm eff}=D_{0}$ is
recovered. We have numerically explored the full three-dimensional phase
diagram of the model and found that its main features are understood by
keeping $D_{0}$ constant and varying the persistence time $\tau_{p}$. We show
results for $D_{0}=1$, so that equilibrium at $T_{\rm eff}=1$ is recovered
when $\tau_{p}=0$, but the system is out of equilibrium in the entire
$(\tau_{p},\phi)$ phase diagram in Fig. 1.

Figure 3:  (a) Velocity distribution $P(v_{x,y})$ for different packing
fractions $\phi$ measured along the $x$ and $y$ directions. The dashed line
corresponds to the Maxwell distribution. (b) Spatial correlations of the
velocities
$C_{\bm{v}\bm{v}}(r)=\left<\bm{v}_{i}(0)\cdot\bm{v}_{j}(0)\delta(r-r_{ij})\right>/v(\phi)^{2}\left<\delta(r-r_{ij})\right>$
with $r_{ij}=|\bm{r}_{i}(0)-\bm{r}_{j}(0)|$. (c) Example of a velocity field
for $N=4096$, $\tau_{p}=10^{3}$, $\phi=0.84$. (d) Time autocorrelations of the
velocity
$C_{\bm{v}\bm{v}}(t)=\left<\bm{v}_{i}(t)\cdot\bm{v}_{i}(0)\right>/v(\phi)^{2}$.
(e) Mean-squared displacements for $\tau_{p}=10^{3}$ and $N=4096$ with
ballistic, superdiffusive, and diffusive behaviours indicated. (f)
Displacement field after a time $t=0.11\tau_{p}$ (the MSD is $\Delta
r^{2}(t)=0.16\sigma^{2}$) starting from the velocity field shown in (c) at
$t=0$.

Figure 1 shows the three disordered phases of polydisperse self-propelled
disks. The homogeneous fluid appears for intermediate packing fraction and
persistence time $\tau_{p}\lesssim 10^{2}$; it phase separates in the MIPS
region at larger persistence. The MIPS boundaries (red circles in Fig. 1) were
established by locating the two peaks of the probability distribution function
of the local packing fraction Digregorio  _et al._ (2018). We only sketch the
critical point of MIPS, its precise characterisation requiring a more detailed
analysis Maggi  _et al._ (2021). Unlike monodisperse particles, whose
relaxation relies on the movement of defects Briand and Dauchot (2016), the
dense phase of MIPS is fully disordered and flows easily: it corresponds to an
active liquid phase which exists up to arbitrarily large persistence in
between MIPS and dynamic arrest.

For all $\tau_{p}$, the active liquid undergoes dynamic arrest at large $\phi$
and transforms into an active glass Berthier  _et al._ (2019). To characterize
the liquid-glass boundary, in Fig. 2 we analyze the mean-squared displacement
(MSD)

$\Delta
r^{2}(t)=\frac{1}{N}\sum_{i}\left<|\bm{r}_{i}(t)-\bm{r}_{i}(0)|^{2}\right>,$
(3)

where brackets indicate an ensemble average. In the liquid, the MSD is
ballistic at short times, $\Delta r^{2}\sim(vt)^{2}$, and diffusive at long
times, $\Delta r^{2}\sim 4Dt$. These limits define the mean-squared velocity,
$v^{2}(\phi)\equiv\left<|\bm{v}_{i}|^{2}\right>$, and the self-diffusion
constant, $D(\phi)$, which respectively reduce to $v_{0}^{2}$ and $D_{0}$ in
the dilute limit, $\phi\to 0$.

In Figs. 2(a,b), we contrast the $\phi$-dependent MSDs at small and large
$\tau_{p}$. In both cases, $D(\phi)$ decreases by several orders of magnitude
when $\phi$ varies over a modest range, see Fig. 2(c). As in equilibrium
systems, the determination of a critical packing fraction for dynamic arrest
is ambiguous Brambilla  _et al._ (2009): we fit $D(\phi)$ to an algebraic form
inspired by mode-coupling theory, $D\sim(\phi_{c}-\phi)^{\gamma}$, to estimate
a liquid-glass boundary $\phi_{c}(\tau_{p})$ (blue squares in Fig. 1). The MSD
data in Figs. 2(a,b) both show the emergence at large $\phi$ of a two-step
relaxation process separated by a subdiffusive plateau. Despite these
similarities, the remainder of this work demonstrates several important
differences between the highly-persistent case and weakly persistent or
equilibrium glasses.

In equilibrium systems, velocities are spatially uncorrelated and obey the
Maxwell distribution with a mean-squared velocity $v^{2}$ independent of
$\phi$. The situation is very different in persistent systems, where
$v^{2}(\phi)$ decreases sharply with $\phi$, see Fig. 2(b). This already
signals the importance of nonequilibrium effects on the velocities of
interacting self-propelled particles Fodor and Marchetti (2018); Marconi  _et
al._ (2016); Caprini and Marini Bettolo Marconi (2020).

In Fig. 3, we analyze particle dynamics in a highly persistent liquid
($\tau_{p}=10^{3}$) whose density $\phi$ is between MIPS and dynamic arrest.
The velocity distributions in Fig. 3(a) are strongly non-Maxwellian with much
broader tails. Since a Gaussian distribution is expected both for interacting
equilibrium particles and for non-interacting Ornstein-Uhlenbeck particles,
the measured distributions reveal the non-trivial influence of many-body
interactions in persistent liquids Marconi  _et al._ (2016); Caprini and
Marini Bettolo Marconi (2020).

Even more interestingly, persistent propulsions in the homogeneous liquid
produce spatial velocity correlations Szamel  _et al._ (2015); Flenner  _et
al._ (2016); Henkes  _et al._ (2020); Szamel and Flenner (2021); Caprini  _et
al._ (2020b, a), which we quantify in Fig. 3(b) via the velocity correlation
$C_{\bm{v}\bm{v}}(r)$. The data clearly reveal the existence of correlations
extending over a length scale which grows with $\phi$ and $\tau_{p}$. The
corresponding real-space correlations extend over several particle diameters,
as illustrated in Fig. 3(c). Such correlated velocity patterns have no
equilibrium analog. They can be rationalised via the coupling of collective
elastic modes and highly-persistent active forces Henkes  _et al._ (2020).
Excitation of these modes by thermal fluctuations affects displacements in
passive glasses Shiba  _et al._ (2016); Illing  _et al._ (2017): the
Supplemental Material (SM) sup  shows that their effects are weak here.

Figure 4: (a) Time series of the kinetic energy
$\sum_{i}|\bm{v}_{i}(t)|^{2}=|\nabla U_{\mathrm{eff}}(t)|^{2}$ rescaled by its
time average, with $N=1024$ for two dense systems at
$\tau_{p}=10^{-2},10^{2}$. (b) Non-Gaussian parameter $\alpha_{2}(t)$ for
$\tau_{p}=10^{2}$, $N=1024$ and different packing fractions. (c) Velocity
field for $N=4096$, $\tau_{p}=10^{2}$, $\phi=0.8825$. (d) Displacement field
after a time $t=200\tau_{p}$ (The MSD is $\Delta r^{2}(t)=0.13\sigma^{2}$)
starting from the velocity field shown in (a) at $t=0$ (e) Dynamical
susceptibility $\chi_{4}(t,a)$ with $N=1024$, $\tau_{p}=10^{2}$, $\phi=0.88$.
(f) Dynamical susceptibility $\chi_{4}(t,a)$ with $N=1024$,
$\tau_{p}=10^{-2}$, $\phi=1.055$.

We use the velocity autocorrelation function $C_{\bm{v}\bm{v}}(t)$ to
characterize time fluctuations. Fig. 3(d) shows that these decay over a
timescale that depends strongly on the persistence time $\tau_{p}$ (see also
SM sup  Fig. S2), and weakly on $\phi$. Because particle motion is not
hindered by crowding in the active liquid, large particle displacements easily
occur along the spatially correlated velocity field, which maintains its
structure over long timescales $\sim\tau_{p}\gg 1$. This is confirmed in Fig.
3(e) which shows that the MSD is superdiffusive for times up to $\tau_{p}$,
corresponding in real space to a spatially correlated displacement field [Fig.
3(f)] which resembles the velocity field at $t=0$. The many vortices and
abrupt changes of this velocity field, alongside its large-scale correlations,
bear strong similarities with the family of active systems described as
‘turbulent’ Wensink  _et al._ (2012); Nishiguchi and Sano (2015);
Doostmohammadi  _et al._ (2017); Alert  _et al._ (2022) (see SuppMovie1.mov
and SuppMovie2.mov sup ): remarkably, this occurs here in an active liquid
without explicit aligning interactions.

On increasing density, the system becomes crowded, and diffusive motion sets
in over a large timescale $\tau_{\alpha}(\phi)\propto D^{-1}$ that increases
rapidly with $\phi$. At equilibrium, particle dynamics is triggered by
infrequent thermally activated relaxation events characterized by a broad
distribution of energy barriers, reflecting a rugged energy landscape Berthier
and Biroli (2011). This physical picture survives for modest values of the
persistence times, as in Fig. 2(a), except that activated dynamics is now
driven by a nonequilibrium colored noise, as recently studied in simpler
active situations Woillez  _et al._ (2019, 2020). Therefore, glassy dynamics
for weak persistence qualitatively resembles passive systems Flenner  _et al._
(2016); Berthier  _et al._ (2019).

The physics is radically different when the persistence time is large, see
Fig. 4. Since particles are always in contact at these high densities, the
natural time scale for relaxation is $\xi\sigma^{2}/\varepsilon=1$. This opens
a time window, $1\ll t\ll\tau_{p}$, where particle dynamics is nearly arrested
and $|\bm{v}_{i}|\ll|\bm{p}_{i}|$, see Fig. 2(b). The forces stemming from
particle interactions then nearly balance the self-propulsion forces, and the
system is close to mechanical equilibrium. Such configurations correspond to
local minima of an effective potential energy Mandal and Sollich (2020, 2021),
$U_{\rm eff}(\bm{r};\bm{p})\equiv
U(\bm{r})-\sum_{i}\bm{p}_{i}\cdot\bm{r}_{i}$, which depends on both positions
and propulsions. Because $U_{\rm eff}$ evolves slowly over a timescale
$\tau_{p}\gg 1$, a mechanical equilibrium at time $t$ may be unstable at time
$t+\tau_{p}$, because the propulsion forces will have significantly evolved
via Eq. (2). The loss of mechanical equilibrium triggers fast particle
rearrangements Mandal  _et al._ (2020) (on a timescale $\sim 1$), as the
system relaxes towards a new minimum of $U_{\rm eff}$. This effect is
illustrated in Fig. 4(a), which shows the squared gradient of $U_{\rm eff}$ as
a function of time. The succession of spikes correspond to rearrangement
events. These results demonstrate the intermittent dynamics at large
$\tau_{p}$, which differs qualitatively from the low-persistence case.

Intermittent dynamics between mechanical equilibria superficially resembles
plasticity in slowly sheared amorphous solids Maloney and Lemaître (2006). In
this regime, the spatially correlated velocity field at $t=0$ does not dictate
the displacements associated with structural relaxation. To see this, note the
weak correlations between Figs. 4(c,d); these snapshots should be contrasted
with Figs. 3(c,f) for the persistent liquid, where velocities and
displacements are strongly correlated. (The time scales were chosen such that
the MSD is similar in both cases.) In fact, the displacement field in the
glassy case [Fig. 4(d)] results from the accumulation of smaller particle
rearrangements. This mechanism is distinct from the turbulent flows seen in
the persistent liquid, and from activated relaxation events at weak
persistence.

While it might be desirable to analyze relaxation events individually,
$\tau_{p}$ is finite in our simulations, so changes in $U_{\rm eff}$ are not
quasistatic. This makes individual rearrangements hard to characterize, a
problem which is familiar from equilibrium supercooled liquids Berthier  _et
al._ (2011). Hence we analyze the dynamics using tools from that context, and
start with the non-Gaussian parameter Kob and Andersen (1995); Kob  _et al._
(1997)

$\alpha_{2}(t)=\frac{N^{-1}\sum_{i}\left<|\bm{r}_{i}(t)-\bm{r}_{i}(0)|^{4}\right>}{2(\Delta
r^{2}(t))^{2}}-1,$ (4)

which is zero when the distribution of particle displacements is Gaussian. In
the persistent liquid ($\phi\lesssim 0.87$) the dominant sources of
heterogeneity are the correlated velocities which decay on the time scale
$\tau_{p}$. Hence $\alpha_{2}(t)$ decays monotonically from the non-Gaussian
velocity distribution at $t=0$ to the Gaussian diffusive limit at
$t\gg\tau_{p}$. For larger $\phi$, $\alpha_{2}(t)$ starts off similar to the
active liquid, but it increases again for $t>\tau_{p}$, leading to a maximum
at a much longer timescale $\tau_{\alpha}\gg\tau_{p}$. This peak shows that
the slow dynamics are strongly non-Gaussian, supporting the picture of
intermittent transitions Chaudhuri  _et al._ (2007) between minima of $U_{\rm
eff}$.

To quantify the collective nature of the dynamics we study the dynamic
susceptibility Berthier  _et al._ (2011)

$\chi_{4}(t,a)=N\left[\langle Q^{2}(t,a)\rangle-\langle
Q(t,a)\rangle^{2}\right],$ (5)

where $Q(t,a)=N^{-1}\sum_{i}\Theta(a-|\bm{r}_{i}(t)-\bm{r}_{i}(0)|)$ is a
dynamic overlap and $a$ a length scale, see Fig. 4(e). In dense passive
systems, $\chi_{4}(t,a)$ is only sizeable if $t\sim\tau_{\alpha}$ and $a$
close to the typical cage size Lačević  _et al._ (2003), thus capturing the
cooperative nature of activated relaxation. Here, $\chi_{4}(t,a)$ is large at
all times up to $\tau_{\alpha}$, and is maximized for a given $t$ for
$a\sim\sqrt{\Delta r^{2}(t)}$. These data quantitatively confirm that particle
displacements are spatially correlated over a broad range of time scales in
very persistent glassy systems. This contrasts with results at low persistence
[Fig. 4(f)], where the short-time correlations are absent, similar to the
passive case.

We showed that strongly persistent systems support active liquid states with
turbulent nonequilibrium flows and superdiffusive motion and that at high
density, the dynamic arrest of these liquids is accompanied by complex spatio-
temporal correlations, spanning a range of length and time scales, whose
origin differs qualitatively from active glasses analysed so far. While slow
dynamics are ubiquitous in crowded systems (both active and passive), a simple
replacement of thermal noises by highly-persistent propulsions dramatically
changes the mechanisms by which a liquid explores its energy landscape. The
resulting dynamical processes are unusual and raise several questions for
fiture work. Can the intermittent dynamics of the persistent active glass be
understood by analysing the idealised limit of large $\tau_{p}$ Mandal and
Sollich (2021)? How general are active turbulent states in systems without
explicit aligning interactions Mandal  _et al._ (2020); Kuroda  _et al._
(2022)? Can existing theories of velocity correlations be extended to such
states Szamel and Flenner (2021); Marconi  _et al._ (2021)?

###### Acknowledgements.

We thank L. Cugliandolo, J. Tailleur, and F. van Wijland for discussions. This
work was supported by a grant from the Simons Foundation (#454935 L.
Berthier).

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# Torsion fields generated by the quantum effects of macro-bodies

Da-Ming Chen1,2<EMAIL_ADDRESS>1National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China;
2School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We generalize Einstein’s General Relativity (GR) by assuming that all matter
(including macro-objects) has quantum effects. An appropriate theory to
fulfill this task is Gauge Theory Gravity (GTG) developed by the Cambridge
group. GTG is a “spin-torsion” theory, according to which, gravitational
effects are described by a pair of gauge fields defined over a flat Minkowski
background spacetime. The matter content is completely described by the Dirac
spinor field, and the quantum effects of matter are identified as the spin
tensor derived from the spinor field. The existence of the spin of matter
results in the torsion field defined over spacetime. Torsion field plays the
role of Bohmian quantum potential which turns out to be a kind of repulsive
force as opposed to the gravitational potential which is attractive. The
equivalence principle remains and essential in this theory so that GR is
relegated to a locally approximate theory wherein the quantum effects
(torsion) are negligible. As a toy model, we assume that the macro matter
content can be described by the covariant Dirac equation and apply this theory
to the simplest radially symmetric and static gravitational systems.
Consequently, by virtue of the cosmological principle, we are led to a static
universe model in which the Hubble redshifts arise from the torsion fields.

cosmology:theory

##  I Introduction

In physics, when contradictions between the predictions of a fundamental
theory and the credible experimental observations are unavoidable, one may
modify the theory or assume some new types of matter, and both approaches must
be verified by experiments. It is now well established that when Einstein’s
General Relativity (GR) is applied to galaxies (always in Newtonian form) we
need some mysterious matter called dark matter, and when applied to cosmology
we need even more mysterious element called dark energy in addition to the
dark matter. While more and more experimental instruments are constructed to
probe the direct or indirect evidences of dark matter and dark energy, it is
equally important for physicists to investigate the theories of gravity
alternative to GR. In such a theory, we hope that there is no need to assume
the existence of dark matter or dark energy or both. For example, it was shown
by Lovelock (Lovelock, 1971, 1972) that a polynomial form of the Lagrangian is
possible leading to higher order curvature corrections to GR known as the
Lovelock theory of gravity. Specifically, second order Lovelock or Einstein-
Gauss-Bonnet gravity has been investigated recently (Boulware and Deser, 1985;
Kobayashi, 2005; Oikonomou, 2021; Brassel et al., 2022). Another important
extension to GR was achieved by constructing the so-called nonlinear
Lagrangians, which are some arbitrary but well-defined functions of the Ricci
scalar (Buchdahl, 1970), and now this theory has been developed into the well-
known $f(R)$ gravity (e.g., Goswami et al. (2014)). In this paper, however, we
attempt to extend GR in a quite different way. Note the fact that GR has been
tested with high precision only for the gravitational systems with spatial
size equal to or smaller than the solar system (e.g., compact objects) when
gravity is relatively strong. Therefore, it is possible that the astronomical
observations for larger gravitational systems (galaxies, cluster of galaxies
until the whole universe) may suggest that GR should be extended or modified
when the gravitational field is weak (except for the very early universe). We
first assume that macro bodies have observable quantum effects on large scales
and then attempt to extend GR to describe them (which are usually believed to
be negligibly small in actual practice), and we require that GR is recovered
on small scales.

Up to date, it is well known that quantum effects are believed to be the
exclusive properties for microparticles which are the subjects of quantum
mechanics and quantum field theory. When gravity is present, there are so-
called quantum gravity and quantum field theory in curved spacetime, and all
such theories are significant only under the extreme conditions like very
strong gravity and very high energy, as in the cases when we study the
phenomena near the horizons of black holes and in very early universe.
Although all macro bodies come down to microparticles as their final
compositions, physicists believe that the quantum effects of macro bodies are
“averaged out” and thus negligibly small compared with other kinematic
quantities. Note, however, that the preceding conclusions are based on an
implicit assumption that the quantum effects of macro bodies arise only from
their micro constituents, while the macro bodies themselves remain classical.
We thus ask a question: what if we assume that the quantum effects of macro
bodies arise from their own quantum randomness which are independent of their
micro constituents? In order to understand our assumption, we need to know
what the quantum effects of microparticles may imply. The quantum effects of
electrons, for example, can be attributed to their quantum randomness (or
called intrinsic randomness), this distinguishes remarkably from classical
randomness (or called apparent randomness) of which the Brownian motions is a
typical example. It is well known that the random motions of pollen grains in
water result from the frequent collisions of the molecules of water according
to Newtonian dynamics, while quantum random motion is the intrinsic property
of a particle itself that cannot be attributed to any external reasons and
thus Newtonian dynamics fails to describe. Our assumption is that, like
electrons, all macro bodies (including galaxies if we treat them as
“particles”) have the quantum random motions and thus should exhibit quantum
effects.

An immediate doubt about our assumption is: why we have never observed such
quantum effects of macro bodies? The answer is simple: if the macro bodies and
an observer (also an “macro body”) are located near the same (but arbitrary)
point of spacetime manifold, then they have approximately the same randomness
(due to gravity) and quantum effects cannot be observed. The locally
“classical” behavior of macro bodies is a familiar scientific fact, however,
this fact does not necessarily require the macro bodies to behave classically
on arbitrary large scales. It is believed that gravity makes disorders order,
and order is fundamental for space and time and spacetime, or as we can say,
spacetime is nothing else but continuous order of events. Thus no gravity, no
spacetime. In any local inertial frames, the gravity is still present but the
net gravity force is canceled by inertial force, so order or flat spacetime
remains. In special relativity, rigid rods are employed to form coordinate
lattices so that any event has a position at any given time. According to GR,
the global structure of curved spacetime can still be described by arbitrary
curved rigid rods (depending on the matter distributions) with arbitrary
length. However, quantum mechanics tells us that microparticles can escape the
order depicted by the spacetime and thus get out of control of gravity.
Therefore, if we extend the quantum randomness owned by microparticles to
macro bodies, gravity of one body (or system) cannot make remote bodies in
order. So we should give up the averaged background curved spacetime
determined by all matter in the universe. Instead, we assume a spacetime
formed by sewing together infinitely many local pieces of spacetime depicted
by GR. That is to say, if the observed macro bodies and an observer are
separated by a finite spatial distance, the quantum effects appear, and the
strength of the quantum effects increases with the increasing distances
between them. So the rods are rigid only approximately on small scales (i.e.,
rigid locally), but are flexible on large scales. Consequently, GR is correct
only approximately in a sufficiently small local region around any point of
the spacetime manifold, wherein quantum effects can be ignored. For large
systems, quantum effects appear among different parts, and thus GR should be
replaced by a new theory of gravity that can account for them. As a summary,
we aim to extend GR to a theory of gravity based on the following hypotheses:

1\. Equivalence principle: frames of reference undergoing acceleration and
frames of reference in gravitational fields are equivalent.

2\. The principle of general covariance: any physical predictions are
independent of the frames of references employed, i.e, any physical theory can
be written in a covariant form.

3\. All matter has quantum random motion irrespective of its mass.

Clearly, compared with GR, the only added hypothesis is the quantum effects of
macro matter, since the quantum effects of micromatter have been verified and
well known. At the present stage, we emphasize that our extra hypothesis is
compatible with Einstein’s equivalence principle in the sense that, at any
local point, a freefall macro body has no quantum effects at its own rest
inertial frame, so that its trajectory is well-defined locally.

We cannot go further until we have an optimal mathematical language to
describe our assumptions in a natural way. We find that geometric algebra (GA)
is just what we seek. GA is rediscovered and greatly promoted by David
Hestenes (Hestenes, 1966; Hestenes and Sobczyk, 1986; Hestenes, 2002), and
among other things, it provides a unified mathematical language for relativity
(both special and general theories) and quantum mechanics (Hestenes, 2003a, b;
Doran et al., 2003). We summarize the main results of spacetime algebra and
Dirac theory for free particles in Appendix A. Fortunately, we have a theory
of gravity based on GA as we need at hand. More than twenty years ago, a new
gauge theory of gravity constructed on flat spacetime alternative to GR was
developed by Cambridge group (Lasenby, Doran, and Gull) called gauge theory
gravity (GTG) (Lasenby et al., 1998). When applying GTG for our purpose, we
still use the Dirac spinor to describe the macro matter content. The only
modification to GTG is that, when gravity is absent, we require that the gauge
invariant equations for Dirac spinor are reduced to a conservation law for
“classical ” (i.e., no spin) pressure free ideal fluid, rather than to the
usual Dirac equation in flat spacetime (which is required for electrons).
Unfortunately, up to date, we do not know the spinor field equations that
macro bodies may satisfy. So as a first try, we still employ the Dirac
equation for free particles as a toy model for our macro bodies. We outline
the main idea and the prime processes followed by GTG for the matter content
that can be described by Dirac equation in Appendix B.

In what follows, we first apply GTG to cosmology and the brief conclusions and
discussions are given at the end of the paper. We employ natural units
($G=\hslash=c=1$) throughout except stated otherwise.

##  II Cosmology: static universe model and Hubble redshift

The simplest application of our theory that can be tested with credible
observational data is the study of cosmology. The standard $\Lambda$CDM
cosmology model is based on GR, in which the universe is expanding
(accelerating at the present stage), the quantum effects are important only in
very early universe (Challinor et al., 1997). However, according to the
assumption in this paper, macro bodies also have quantum effects, we expect
that the quantum pressure (torsion field, or spin field) will balance the
gravity, so that the whole universe is in a state of equilibrium all the time.
So we should try a static universe model, in which $\bar{h}$ field is
independent of time, and the average matter density $\rho_{m}=Nm\rho$ is a
constant parameter, where $N$ is the number of macro bodies of mass $m$ for a
finite system and is suppressed in our subsequent calculations. Thus the Dirac
spinor $\psi(r)$ is a function of $r$ only, although the probability density
$\rho=\psi\tilde{\psi}$ is also a constant due to the constant value of
$\rho_{m}$. As for the magnitude of the spin density $S$, we assume that it is
proportional to $\rho_{m}$, and since it has dimension of $\hbar$, we simply
take the proportionality coefficient to be unity to match the covariant Dirac
equation and the matter stress-energy tensor (see below).

We start by defining a set of spherical coordinates. From the position vector
of the flat spacetime

$x=t\gamma_{0}+r\sin\theta(\cos\phi\gamma_{1}+\sin\phi\gamma_{2})+r\cos\theta\gamma_{3}$
(1)

we obtain

$\begin{split}e_{t}=\partial_{t}x=&\gamma_{0},\\\
e_{r}=\partial_{r}x=&\sin\theta(\cos\phi\gamma_{1}+\sin\phi\gamma_{2})+\cos\theta\gamma_{3},\\\
e_{\theta}=\partial_{\theta}x=&r\cos\theta(\cos\phi\gamma_{1}+\sin\phi\gamma_{2})-r\sin\theta\gamma_{3},\\\
e_{\phi}=\partial_{\phi}x=&r\sin\theta(-\sin\phi\gamma_{1}+\cos\phi\gamma_{2}).\end{split}$
(2)

From these vectors, the corresponding reciprocal basis
$e^{t},e^{r},e^{\theta},e^{\phi}$ can be easily obtained. Since $e_{\theta}$
and $e_{\phi}$ are not unit, we define

$\hat{\theta}\equiv e_{\theta}/r,\;\;\;\hat{\phi}\equiv
e_{\phi}/(r\sin\theta).$ (3)

With these unit vectors, we further define the unit bivectors (relative basis
vectors for $e_{t}=\gamma_{0}$)

$\begin{split}\sigma_{r}\equiv&e_{r}e_{t},\\\
\sigma_{\theta}\equiv&\hat{\theta}e_{t},\\\
\sigma_{\phi}\equiv&\hat{\phi}e_{t}.\end{split}$ (4)

These bivectors satisfy

$\sigma_{r}\sigma_{\theta}\sigma_{\phi}=e_{t}e_{r}\hat{\theta}\hat{\phi}=i.$
(5)

For our purpose, we next try a form of $\bar{h}$ field that satisfies a
static, radially symmetric matter distribution. We assume that it takes the
form (Lasenby et al., 1998; Doran et al., 2003)

$\displaystyle\bar{h}(e^{t})=f_{1}e^{t}+f_{2}e^{r},$
$\displaystyle\bar{h}(e^{r})=g_{1}e^{r}+g_{2}e^{t},$ (6)
$\displaystyle\bar{h}(e^{\theta})=\alpha e^{\theta},$
$\displaystyle\bar{h}(e^{\phi})=\alpha e^{\phi},$

where $f_{1}$, $f_{2}$, $g_{1}$, $g_{2}$ and $\alpha$ are all functions of $r$
only. From (81) and (82), we write $\omega^{\prime}(a)$ as (Lasenby et al.,
1998)

$\displaystyle\omega^{\prime}(a)=$ $\displaystyle(a\cdot e_{t}G-a\cdot
e_{r}F)e_{r}e_{t}-a\cdot\hat{\theta}X\hat{\theta}e_{t}$ (7)
$\displaystyle-a\cdot\hat{\theta}(Y-\frac{\alpha}{r})e_{r}\hat{\theta}-a\cdot\hat{\phi}X\hat{\phi}e_{t}$
$\displaystyle-a\cdot\hat{\phi}(Y-\frac{\alpha}{r})e_{r}\hat{\phi},$

where new functions $G$, $F$, $X$ and $Y$ are also all functions of $r$ only,
their introduction circumvents the unnecessary complexity when derived
directly in terms of the functions $f_{1}$, $f_{2}$ etc. With the $\omega$
function at hand, the curvature tensor can be calculated directly from (72),
the torsion free part is (Lasenby et al., 1998)

$\begin{split}R^{\prime}(B)=&\alpha_{1}\sigma_{r}B\cdot\sigma_{r}+(\alpha_{2}\sigma_{\theta}+\alpha_{3}i\sigma_{\phi})B\cdot\sigma_{\theta}\\\
&+(\alpha_{2}\sigma_{\phi}-\alpha_{3}i\sigma_{\theta})B\cdot\sigma_{\phi}+\alpha_{6}\sigma_{r}(B\wedge\sigma_{r})\\\
&+(\alpha_{4}\sigma_{\theta}-\alpha_{5}i\sigma_{\phi})(B\wedge\sigma_{\theta})\\\
&+(\alpha_{4}\sigma_{\phi}+\alpha_{5}i\sigma_{\theta})(B\wedge\sigma_{\phi}),\end{split}$
(8)

where $B$ is a bivector, and $\alpha_{1},\ldots,\alpha_{6}$ are given by

$\begin{array}[]{lll}\alpha_{1}=L_{r}G-L_{t}F+G^{2}-F^{2},&\alpha_{2}=-L_{t}X+GY-X^{2},\\\
\alpha_{3}=L_{t}Y+XY-XG,&\alpha_{4}=L_{r}Y+Y^{2}-FX,\\\ \alpha_{5}=L_{r}X+XY-
FY,&\alpha_{6}=-X^{2}+Y^{2}-(\frac{\alpha}{r})^{2}.\end{array}$ (9)

From (8), the torsion free part of Ricci tensor and Ricci scalar are given by

$\begin{split}R^{\prime}(a)=&[(\alpha_{1}+2\alpha_{2})a\cdot
e_{t}+2\alpha_{5}a\cdot e_{r}]e_{t}\\\ &+[2\alpha_{3}a\cdot
e_{t}-(\alpha_{1}+2\alpha_{4})a\cdot e_{r}]e_{r}\\\
&-(\alpha_{2}+\alpha_{4}+\alpha_{6})a\cdot\hat{\theta}\hat{\theta}\\\
&-(\alpha_{2}+\alpha_{4}+\alpha_{6})a\cdot\hat{\phi}\hat{\phi},\end{split}$
(10) $R^{\prime}=2\alpha_{1}+4\alpha_{2}+4\alpha_{4}+2\alpha_{6}.$ (11)

In order to solve Einstein equation (78), we need write out $T(a)$ given by
(80) explicitly, which is

$\begin{split}T(a)&=\langle a\cdot D\psi i\gamma_{3}\tilde{\psi}\rangle_{1}\\\
&=\langle a\cdot\bar{h}(e^{\mu})\partial_{\mu}\psi
i\gamma_{3}\tilde{\psi}+\frac{1}{2}\omega(a)\psi
i\gamma_{3}\tilde{\psi}\rangle_{1}\\\
&=a\cdot(g_{1}e^{r}+g_{2}e^{t})\langle\partial_{r}\psi
i\gamma_{3}\tilde{\psi}\rangle_{1}+\omega(a)\cdot S.\end{split}$ (12)

The term $\langle\partial_{r}\psi i\gamma_{3}\tilde{\psi}\rangle_{1}$ in the
last equation can be calculated straightforward from Dirac equation (79). We
obtain

$\begin{split}T(a)=&\frac{g_{2}a\cdot e_{t}-g_{1}a\cdot
e_{r}}{g_{1}^{2}-g_{2}^{2}}\\{(\rho_{m}-\frac{3}{2}\kappa
S^{2})(g_{1}e_{r}-g_{2}e_{t})\\\
&+[g_{2}(G+2(Y-\frac{\alpha}{r}))-g_{1}(F+2X)](e_{r}e_{t})\cdot
S\\}.\end{split}$ (13)

Now set $a=e_{t}$ and $a=e_{r}$ in (78) respectively, we obtain the following
set of equations:

$\begin{split}2\alpha_{4}+\alpha_{6}+\frac{3}{4}\kappa^{2}S^{2}=&\frac{g_{2}^{2}}{g_{1}^{2}-g_{2}^{2}}\kappa(\rho_{m}-\frac{3}{2}\kappa
S^{2})\\\
\alpha_{3}=\alpha_{5}=&\frac{g_{1}g_{2}}{2(g_{1}^{2}-g_{2}^{2})}\kappa(\rho_{m}-\frac{3}{2}\kappa
S^{2})\\\
2\alpha_{2}+\alpha_{6}+\frac{3}{4}\kappa^{2}S^{2}=&-\frac{g_{1}^{2}}{g_{1}^{2}-g_{2}^{2}}\kappa(\rho_{m}-\frac{3}{2}\kappa
S^{2}).\end{split}$ (14)

Recall that the functions $G(r)$, $F(r)$, $X(r)$ and $Y(r)$ are introduced
only for the torsion-free parts $\omega^{\prime}$ of $\omega$ function, they
thus should be related to the functions $f_{1}$ etc in $\bar{h}$ field under
the same conditions (i.e., torsion free). Note also that
$L_{t}=e_{t}\cdot\hat{h}(e^{\mu})\partial_{\mu}=g_{2}\partial_{r}$ and
$L_{r}=e_{r}\cdot\hat{h}(e^{\mu})\partial_{\mu}=g_{1}\partial_{r}$, we further
adopt the suggestions in Lasenby et al. (1998) that $f_{2}=0$, $\alpha=1$,
$g_{1}=L_{r}r=Yr$ and $g_{2}=L_{t}r=Xr$. Applying all these relations to the
equations in (14), and defining
$M_{Q}=-\frac{1}{2}r^{3}\alpha_{6}=\frac{r}{2}(g_{2}^{2}-g_{1}^{2}+1)$, we
find

$L_{r}M_{Q}=\frac{3}{8}g_{1}\kappa^{2}S^{2}r^{2},\quad\text{or}\quad\partial_{r}M_{Q}=\frac{3}{8}\kappa^{2}S^{2}r^{2}.$
(15)

Since $S=\frac{1}{2}\psi i\gamma_{3}\tilde{\psi}$ and due to our special
choice of $\psi=\rho^{1/2}R$, we have $S^{2}=-\frac{1}{4}\rho^{2}<0$. For a
static homogeneous universe with constant matter density $\rho_{m}=\rho m$,
$S^{2}$ is also a minus constant. By integrating (15) we get

$M_{Q}=\frac{1}{8}\kappa^{2}S^{2}r^{3}=\frac{r}{2}(g_{2}^{2}-g_{1}^{2}+1),$
(16)

where we have set the integration constant to zero since we require $M_{Q}=0$
when $r=0$. Remarkably, if we interpret $M_{Q}$ as some kind of “mass” within
$r$, then it provides a repulsive force! In our gauge choice, we can define a
Bohmian quantum potential
$\Phi_{Q}(r)=\frac{M_{Q}(r)}{r}=\frac{1}{8}\kappa^{2}S^{2}r^{2}$. To keep the
whole universe static, this potential should balance the usual matter
gravitational potential $\Phi_{m}(r)=\frac{4\pi\rho_{m}}{3}r^{2}$. We thus
have

$\rho_{m}=-\frac{3}{4}\kappa S^{2}.$ (17)

As mentioned, we should relate $G$ to the functions $f_{1}$, $g_{1}$ in
$\bar{h}$ field under torsion-free condition. In this case, the torsion
equation (77) is reduced to the “Wedge equation”
$D^{\prime}\wedge\bar{h}(a)=0$. Let $a=e_{t}$ in this equation we obtain (the
same result is obtained by Lasenby et al. (1998) in a different way)

$f_{1}=e^{-\int\frac{G}{g_{1}}dr}.$ (18)

From (14), we get (assuming $g_{2}=0$)

$-\frac{G}{g_{1}}=\frac{\kappa(\rho_{m}-\kappa
S^{2})r}{2(1-\kappa^{2}S^{2}r^{2}/4)}.$ (19)

Substituting this expression into (18) and applying (17) we get

$\begin{split}f_{1}(r)&=\left(1-\frac{1}{4}\kappa^{2}S^{2}r^{2}\right)^{\frac{\rho_{m}-\kappa
S^{2}}{-\kappa S^{2}}}\\\
&=\left(1+H_{0}^{2}r^{2}\right)^{\frac{7}{4}},\end{split}$ (20)

where

$H_{0}^{2}=\frac{8\pi\rho_{m}}{3}$ (21)

is the Hubble constant. We are now ready to study the redshifts of light
signals emitted from some source at distance $r$. According to the equivalence
principle, the line element of events given by (69) is still valid in our
theory. From (64), $g^{t}=\bar{h}(e^{t})=f_{1}e^{t}$, we thus get
$g_{t}=\underline{h}^{-1}(e_{t})=f_{1}^{-1}e_{t}$. Then from (69), the period
of the light signal at the source is

$\Delta
t(r)=f_{1}\Delta\tau=\left(1+H_{0}^{2}r^{2}\right)^{\frac{7}{4}}\Delta\tau,$
(22)

where $\Delta\tau$ is the invariant proper value of the period. So the
redshift we observed is

$1+z(r)=\frac{\Delta t(r)}{\Delta
t(0)}=\left(1+H_{0}^{2}r^{2}\right)^{\frac{7}{4}}.$ (23)

Clearly, the redshifts derived in this manner arise purely from the time
dilations induced by the “quantum potential”
$\Phi_{Q}(r)=\frac{1}{8}\kappa^{2}S^{2}r^{2}$ rather than the gravitational
potential or expanding of the universe, since the matter distribution is
homogeneous and the whole universe is static. According to quantum mechanics,
however, the result we obtained here is not a surprising. As indicated by
David Hestenes (Hestenes, 1997), when negative muons are captured in atomic
$s$-states their lifetimes are increased by a time dilation factor
corresponding to the Bohr velocity. Clearly, such a time dilation results from
quantum potential, or quantum pressure or zero-point energy so to speak. We
emphasize that all the quantum effects of matter are the same irrespective of
the mass of the matter (i.e., no matter the matter is macro or micro). So an
observer located at any place in the universe would observe the redshifted
light signals emitted from the sources located far away from the observer in
arbitrary directions. This explains the observations of Hubble redshifts. It
should be also pointed out that the concepts of quantum potential or quantum
force originally proposed by David Bohm correspond to the “quantum pressure”
or “zero-point energy” usually referred to by the major physicists according
to their “standard” viewpoint of quantum mechanics. It is not helpful for us
to indulge in the controversies between the two different interpretations of
quantum mechanics, it suffices to know the fact that the causal or particle
(trajectories or histories) interpretations of quantum effects of matter are
the very natural results if one employs GA as the mathematical language: the
spinor $\psi=\rho^{\frac{1}{2}}R$ determines a unique family of matter
trajectories and all other conclusions follow.

##  III Summary and discussion

We attempt to generalize Einstein’s GR by adding the third hypothesis
concerning the quantum effects of macro bodies to form a new theory of
gravity. This must be a “spin-torsion” theory of gravity, except that we
require the torsion field (or the spin of source matter) to vanish locally
according to the equivalence principle when only macro matter is involved. GTG
is such a theory in the sense that it can define the gravitational strength
corresponding to $\omega$ field self-consistently both from a Dirac spinor
field $\psi(x)$ or a multivector field $M(x)$. Consequently, the obtained
curvature tensor $R(a\wedge b)$ may or may not contain the torsion depending
on whether or not the source matter having spins. Employing spinor field in
the form $\psi(x)=\rho^{\frac{1}{2}}(x)R(x)$ to represent the macro matter
distributions is essential to our theory. The reason is that this form of
spinor, obtained by setting $\beta=0$ in the more general canonical form
$\psi=(\rho e^{i\beta})^{1/2}R$, naturally ensures the neutral matter current
in the free falling frames. Although GTG incorporates quantum effects in
gravity only for microparticles, and the minimal coupling procedure (gauge
principle, which results in or replaces Einstein’s equivalence principle)
ensures that the minimally coupled Dirac action yield the minimally coupled
Dirac equation, one can manage to derive some spinor equations such that
torsion vanishes naturally in an approximate way in any sufficiently small
local regions. This is equivalent to say that the quantum spin of macro matter
cannot be observed locally. Therefore, in our new theory of gravity, GR is
valid only approximately in local regions or on small scales.

Before finding out a correct spinor equation for macro matter , we are eager
to know the possible results for practical applications. Dirac equation has
been well-studied when gravity is present, although we know that it does not
meet our demand. As a toy model, we have employed the covariant Dirac equation
for free particles to describe the spherically symmetric and static
gravitational systems. Remarkably, we find that such a system has a negative
mass determined by the spin density. We then applied the results to cosmology
and find that the repulsive force provided by the negative mass can balance
the gravity and we achieve a static universe model. We have arrived at an
expression for the Hubble redshift as a function of the distance to the light
source. Interestingly, from this expression we can take the constant average
density of the universe $\rho_{m}$ as a fundamental physical constant (like
gravitational constant $G$, the speed of light $c$, etc). The disadvantages of
the toy model are obvious. The procedure we followed to define the mass of the
system is similar to the torsion-free ideal fluid model, however, the obtained
mass is negative, which is nothing else but the spin of the system.
Surprisingly, the real (positive) mass is “lost” in our calculations, so the
balance condition (17) has to be put by hand! Further more, for $z\ll 1$, our
formula gives $z\sim r^{2}$, which is not linear of distance $r$ as usually
declared by astronomers. These disadvantages are the evidence that Dirac
equation is not the correct one we are seeking.

From a spinor field we can always form vector currents $\rho
e_{\mu}=\psi\gamma_{\mu}\tilde{\psi}$. In any local free-falling frames, we
require these currents to be constant, so that if we identify
$e_{3}=R\gamma_{3}\tilde{R}$ as the spin axis vector $s/|s|$ and $e_{1}\wedge
e_{2}$ as the spin plane, then the angular velocity in the spin plane is zero,
and thus no spin can be observed, which is just what we need for macro matter.
This may suggest a equation $\nabla\psi=0$, but its solutions do not satisfy
the Dirac equation unless $m=0$. This can partly explain why we cannot employ
the Dirac equation directly in our new theory. On the other hand, when gravity
is present, covariant derivative of spinor fields would produce non-zero spin
or torsion fields. Like the torsion-free Riemann curvature fields, torsion
fields constructed in this way would ensure that the redshift or relative time
dilation originated from macro quantum effects increases with increasing
distances between observers and light sources.

###### Acknowledgements.

The author is very grateful to the anonymous referee for the good evaluation
of the paper and the helpful suggestions for improvements. This work is
supported by the NSFC grant (No. 11988101) and the K.C.Wong Education
Foundation. The author would like to thank Liang Gao in NAOC for his strong
financial support for many years.

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##  Appendix A Spacetime algebra and Dirac theory

The geometric algebra (GA) that is generated by a 4-dimensional Minkowski
vector space is called spacetime algebra (STA). The inner and outer products
of the four orthonormal basis vectors in Minkowski vector space
{$\gamma_{\mu},\mu=0\ldots 3$} are defined to be

$\displaystyle\gamma_{\mu}\cdot\gamma_{\nu}\equiv\frac{1}{2}(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu})$
$\displaystyle\equiv$ $\displaystyle\eta_{\mu\nu}=\text{diag}(+---)$
$\displaystyle\gamma_{\mu}\wedge\gamma_{\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}).$
(24)

A full basis for the STA is

$1,\;\\{\gamma_{\mu}\\},\;\\{\sigma_{k},i\sigma_{k}\\},\;\\{i\gamma_{\mu}\\},\;i$
(25)

where $\sigma_{k}\equiv\gamma_{k}\gamma_{0},k=1\ldots 3$, and
$i=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}=\sigma_{1}\sigma_{2}\sigma_{3}$.
The STA is a linear space of dimension $1+4+6+4+1=2^{4}=16$. We call the
general elements of STA multivectors, and each multivector decomposes into a
sum of elements of different grades. The grade-$r$ multivectors $A$ are
denoted by $A_{r}$. We call grade-$0$ multivectors scalars, grade-$1$ vectors,
grade-$2$ bivectors and grade-$3$ trivectors. The geometric product of a
grade-$r$ multivector $A_{r}$ with a grade-$s$ multivector $B_{s}$ is defined
simply by $A_{r}B_{s}$, which decomposes into

$A_{r}B_{s}=\langle A_{r}B_{s}\rangle_{|r-s|}+\langle
A_{r}B_{s}\rangle_{|r-s|+2}+...+\langle A_{r}B_{s}\rangle_{r+s},$ (26)

where $\langle X\rangle_{r}$ denotes the projection onto the grade-$r$ part of
$X$. The grade-$0$ (scalar) part of $X$ is written $\langle X\rangle$. We
employ “$\cdot$” and “$\wedge$” symbols to denote the lowest-grade and
highest-grade terms in (26), so that

$\displaystyle A_{r}\cdot B_{s}$ $\displaystyle=$ $\displaystyle\langle
A_{r}B_{s}\rangle_{|r-s|},$ (27) $\displaystyle A_{r}\wedge B_{s}$
$\displaystyle=$ $\displaystyle\langle A_{r}B_{s}\rangle_{r+s},$ (28)

which are called inner and outer products respectively. The simple example is
the geometric product of two vectors $a$ and $b$

$ab=a\cdot b+a\wedge b.$ (29)

We define the reverse of a geometric product $AB$ by
$(AB)^{\sim}=\tilde{B}\tilde{A}$, so that for vectors $a_{1},a_{2},...,a_{r}$,
we have $\tilde{a}_{1}=a_{1}$ and

$(a_{1}a_{2}...a_{r})^{\sim}=a_{r}a_{r-1}...a_{1}.$ (30)

It is easy to show that

$\tilde{A}_{r}=(-1)^{r(r-1)/2}A_{r}.$ (31)

Thus suppose $r\leq s$, the inner and outer products satisfy the symmetry
properties

$\displaystyle A_{r}\cdot B_{s}$ $\displaystyle=$
$\displaystyle(-1)^{r(s-1)}B_{s}\cdot A_{r},$ (32) $\displaystyle A_{r}\wedge
B_{s}$ $\displaystyle=$ $\displaystyle(-1)^{rs}B_{s}\wedge A_{r}.$ (33)

The scalar product is defined by

$A\ast B=\langle AB\rangle.$ (34)

From (27), $A_{r}\ast B_{s}$ is nonzero only if $r=s$, thus the scalar product
(34) is commutative

$\langle AB\rangle=\langle BA\rangle.$ (35)

This commutative property of the scalar product is very useful in later
calculations. We further define the commutator product

$A\times B=\frac{1}{2}(AB-BA),$ (36)

which satisfies the Jacobi identity

$A\times(B\times C)+B\times(C\times A)+C\times(A\times B)=0.$ (37)

The advantages of STA are that it enables coordinate-free representation and
computation of physical systems and processes, and it incorporates the spinors
of quantum mechanics along with the tensors of classical field theory.

Geometric calculus (GC) is the extension of a geometric algebra (like STA) to
include differentiation and integration. Let multivector $F$ be an arbitrary
function of some multivector argument $X$, then the derivative of $F(X)$ with
respective to $X$ in the $A$ direction is defined by

$A\ast\partial_{X}F(X)\equiv\lim_{\tau\rightarrow 0}\frac{F(X+\tau
A)-F(X)}{\tau},$ (38)

where the multivector partial derivative $\partial_{X}$ inherits the
multivector properties of its argument $X$. We have

$\partial_{X}\langle XA\rangle=P_{X}(A),$ (39)

where $P_{X}(A)$ is the projection of $A$ onto the grades contained in $X$.
For vector argument $x$ and constant vector $a$, (38) and (39) give

$a\cdot\partial_{x}x=a=\partial_{x}(x\cdot a).$ (40)

For a vector variable
$a=a^{\mu}\gamma_{\mu}=a\cdot\gamma_{\mu}\gamma^{\mu}=a\cdot\gamma^{\mu}\gamma_{\mu}$,
where $\gamma^{\mu}$ constitute the reciprocal basis and satisfy
$\gamma_{\mu}\cdot\gamma^{\nu}=\delta_{\mu}^{\nu}$, the vector derivative can
be defined as

$\partial_{a}\equiv\gamma^{\mu}\frac{\partial}{\partial a^{\mu}},$ (41)

For the derivative with respect to a spacetime position vector $x$ we use the
symbol $\nabla\equiv\partial_{x}=\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}$, if $x=x^{\mu}\gamma_{\mu}$. From (40) and (41) we can obtain useful
results

$\partial_{a}=\partial_{b}b\cdot\partial_{a}=\gamma^{\mu}\gamma_{\mu}\cdot\partial_{a}.$
(42)

Once again, one great advantage of GC is that it eliminates unnecessary
conceptual barriers between classical, quantum and relativistic physics.

We now discuss the Dirac theory in terms of STA. Recall the symbols
$\gamma^{\prime}s$ and $\sigma^{\prime}s$ in (A) and (25), which stand for the
basis vectors of 4-dimensional Minkowski space and the relative three-
dimensional space, respectively. The same symbols are used in Dirac theory of
relativistic quantum mechanics, but therein they are matrices ($4\times 4$ for
Dirac $\gamma$’s and $2\times 2$ for Pauli $\sigma$’s). This is not a
coincidence, since they satisfy exactly the same algebraic equations. In view
of STA, this correspondence reveals the geometric properties of Dirac spinor,
and leads to the causal interpretation of quantum mechanics (i.e.,
microparticles have the well-defined trajectories in spacetime), which are of
prime importance for the present work. We call the elements of the even
subalgebra of STA defined in spacetime as Dirac spinor fields, denoted by
$\psi(x)$, and can be written in the canonical form

$\psi(x)=(\rho e^{i\beta})^{1/2}R(x),$ (43)

where $\rho(x)$ is the proper probability density,
$i=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$, $\beta(x)$ is a scalar field and
$R(x)$ (called a rotor) satisfies the normalization condition $R\tilde{R}=1$.
The rotor $R(x)$ determines a Lorentz rotation of a given fixed frame vectors
$\gamma_{\mu}$ into a frame $e_{\mu}$ given by

$e_{\mu}=R\gamma_{\mu}\tilde{R},$ (44)

and according to this we write

$\rho e_{\mu}=\psi\gamma_{\mu}\tilde{\psi}.$ (45)

We identify $e_{0}=v$ as the proper velocity of a particle of which the spinor
(wave function) is $\psi$, so that $v^{2}=1$ and

$\rho v=\rho e_{0}=\psi\gamma_{0}\tilde{\psi}$ (46)

is the Dirac current. We interpret another vector field as
$e_{3}=R\gamma_{3}\tilde{R}=s/|s|$, where $|s|=\frac{\hslash}{2}$ is the
magnitude of the spin vector $s$ (we recover the Planck constant $\hslash$
from unity in this Appendix). The spin angular momentum $S(x)$ is a bivector
field related to $s(x)$ by

$S=isv=\frac{\hslash}{2}ie_{3}e_{0}=\frac{\hslash}{2}Ri\sigma_{3}\tilde{R}=\frac{\hslash}{2}R\gamma_{2}\gamma_{1}\tilde{R}.$
(47)

In STA, the Dirac equation for a free-particle of mass $m$ is (Hestenes, 1967,
1973)

$\nabla\psi i\sigma_{3}=m\psi\gamma_{0},$ (48)

which admits plane wave solutions of the form (Hestenes, 1997)

$\psi(x)=(\rho e^{i\beta})^{\frac{1}{2}}R=(\rho
e^{i\beta})^{\frac{1}{2}}R_{0}e^{-i\sigma_{3}p\cdot x/\hslash},$ (49)

where $R_{0}$ is independent of position vector $x$ and $p$ is the momentum
vector of the free particle, so the rotor $R$ has been decomposed to
explicitly exhibit its spacetime dependent in a phase factor. Inserting this
into (48) and using (40), we obtain

$p\psi=m\psi\gamma_{0}.$ (50)

Right multiplying by $\tilde{\psi}$ we get

$p=me^{i\beta}R\gamma_{0}\tilde{R}=mve^{-i\beta},$ (51)

where we have used $i\gamma_{0}=-\gamma_{0}i$ and
$iR=Ri,i\tilde{R}=\tilde{R}i$. Since $e^{i\beta}=\cos\beta+i\sin\beta$, to
ensure the momentum $p$ to be a vector, we must have $\beta=0$ or $\pi$. We
can identify these as corresponding to the electron or positron wave functions
(i.e., spinors) respectively. Since we want to use spinor field to describe
neutral macro body, this result suggests that we should set $\beta=0$ in this
paper. Thus for a free electron, $p=mv$, i.e., the momentum is collinear with
the proper velocity, which in general is not true.

We now consider the expected trajectories (streamlines) of a free electron.
The position vector is $x(\tau)$, where $\tau$ is the proper time, so we have

$v=\frac{dx}{d\tau}=\dot{x}=e_{0}=R\gamma_{0}\tilde{R}.$ (52)

Then the spinor associated to the trajectory is

$\displaystyle\psi(\tau)$ $\displaystyle=$
$\displaystyle\rho^{\frac{1}{2}}R_{0}e^{-\gamma_{2}\gamma_{1}(mv\cdot
x)/\hslash}$ (53) $\displaystyle=$
$\displaystyle\rho^{\frac{1}{2}}R_{0}e^{-\gamma_{2}\gamma_{1}m\tau/\hslash}$
$\displaystyle=$
$\displaystyle\rho^{\frac{1}{2}}R_{0}e^{-\gamma_{2}\gamma_{1}\omega\tau/2},$

where $\tau=v\cdot x$ and $\omega=2m/\hslash$ is the angular velocity in the
spin plane $e_{2}e_{1}$. Since $\psi(\tau)=\rho^{1/2}R(\tau)$, (53) implies

$R(\tau)=R_{0}e^{-\frac{1}{2}\gamma_{2}\gamma_{1}\omega\tau}.$ (54)

Clearly, the proper velocity of the free particle
$v=R_{0}\gamma_{0}\tilde{R}_{0}$ and the spin vector
$s=\frac{\hslash}{2}R_{0}\gamma_{3}\tilde{R}_{0}$ are constant, however, for
$k=1,2$,

$\displaystyle e_{k}(\tau)$ $\displaystyle=$ $\displaystyle
R(\tau)\gamma_{k}\tilde{R}(\tau)$ (55) $\displaystyle=$ $\displaystyle
R_{0}e^{-\frac{1}{2}\gamma_{2}\gamma_{1}\omega\tau}\gamma_{k}e^{\frac{1}{2}\gamma_{2}\gamma_{1}\omega\tau}\tilde{R}_{0}$
$\displaystyle=$ $\displaystyle
R_{0}\gamma_{k}e^{\gamma_{2}\gamma_{1}\omega\tau}\tilde{R}_{0}$
$\displaystyle=$ $\displaystyle
R_{0}\gamma_{k}\tilde{R}_{0}R_{0}e^{\gamma_{2}\gamma_{1}\omega\tau}\tilde{R}_{0}$
$\displaystyle=$ $\displaystyle e_{k}(0)e^{e_{2}e_{1}\omega\tau},$

where $e_{k}(0)=R_{0}\gamma_{k}\tilde{R}_{0}$. Thus, the expected trajectories
are straight lines ($v$ is constant), and as the electron moves along the
straight trajectory, the spin is constant and $e_{1}$ and $e_{2}$ rotate about
$e_{3}$ (spin axis) with angular velocity $\omega$.

We conclude that, for free electrons, the probable trajectories $x(\tau)$ are
well-defined, and along them, the proper velocity $v$ is constant and
satisfies $v^{2}=1$ and the spin vector $s$ is also constant with magnitude
$\frac{\hslash}{2}$. Further studies of this particle-interpretation of Dirac
theory show us that, quantum effects can be determined by the spins of
electrons, so that no spins, no quantum effects (Hestenes, 2003b).
Surprisingly, when gravity is present, Hestenes has applied the spinor
approach to compute the gravitational precession of a gyroscope and found that
gravitational effects on electron motion are exactly the same on the classical
rigid body motion (Hestenes, 1986). Encouragingly, these studies are very
close to our aim since we want to extend GR to describe quantum effects of
macro bodies. So it is natural for us to choose Dirac spinor field to
represent matter field

$\psi(x)=\rho^{\frac{1}{2}}(x)R(x).$ (56)

According to our hypotheses, however, in any local inertial frames, quantum
effects of macro bodies should vanish, so we cannot directly employ Dirac
equation for our spinor fields. The classical spins such as that for
gyroscopes will not be considered since they have nothing to do with quantum
effects. So what we seek is a differential equation for the spinor of the form
in (56), such that, in local inertial frames it describes a pressure-free
(classical) ideal fluid, and when gravity is present quantum effects appear
naturally. However, before finding out the correct equations, it would be
interesting to apply the Dirac equation to a system of $N$ identical macro
particles of mass $m$, so that matter density is
$\rho_{m}=Nm\psi\tilde{\psi}$. We know that Dirac equation does not meet our
demand, but as a toy model, we apply it to cosmology and see what can be
learned from this.

##  Appendix B GTG and matter

By virtual of GC, GTG is constructed such that gravitational effects are
described by a pair of gauge fields, $\bar{h}(a)=\bar{h}(a,x)$ and
$\omega(a)=\omega(a,x)$, defined over a flat Minkowski background spacetime
(Lasenby et al., 1998), where $x$ is the STA position vector and is usually
suppressed for short.

The first of them, $\bar{h}(a)$, is a position-dependent linear function
mapping the vector argument $a$ to vectors. The introduction of $\bar{h}(a)$
ensures covariance of the equations under arbitrary local displacements (or an
arbitrary remapping $x^{\prime}=f(x)$) of the matter fields in the background
spacetime. In order to understand the physical meaning of the $\bar{h}$ field,
we first define the covariant displacement transformation as

$M(x)\xrightarrow{x^{\prime}=f(x)}M^{\prime}(x)=M(x^{\prime}),$ (57)

so that the equations $A(x)=B(x)$ and $A(x^{\prime})=B(x^{\prime})$ have
exactly the same physical content. Suppose we have a vector field
$b(x)=\nabla\phi(x)$, where $\phi(x)$ is a scalar field that is already
covariant under displacement, i.e., $\phi^{\prime}(x)=\phi(x^{\prime})$. Now
can we write $b^{\prime}(x)=b(x^{\prime})$ or
$\nabla\phi^{\prime}(x)=\nabla_{x^{\prime}}\phi(x^{\prime})$? By the chain
rule we find

$a\cdot\nabla\phi^{\prime}(x)=a\cdot\nabla\phi(x^{\prime})=(a\cdot\nabla
f(x))\cdot\nabla_{x^{\prime}}\phi(x^{\prime})=\textsf{f}(a)\cdot\nabla_{x^{\prime}}\phi(x^{\prime})=a\cdot\bar{\textsf{f}}(\nabla_{x^{\prime}})\phi(x^{\prime}),$
(58)

where $\textsf{f}(a)=a\cdot\nabla f(x)$ is a linear function of $a$ and an
arbitrary function of $x$, and $\bar{\textsf{f}}(\nabla_{x^{\prime}})=\nabla
f(x)\cdot\nabla_{x^{\prime}}$, and we call $\bar{\textsf{f}}$ the adjoint of
f, satisfying $a\cdot\textsf{f}(b)=\bar{\textsf{f}}(a)\cdot b$, or
$\bar{\textsf{f}}(a)=\partial_{b}\langle\textsf{f}(b)a\rangle$. It follows
that
$\nabla\phi^{\prime}(x)=\bar{\textsf{f}}(\nabla_{x^{\prime}}\phi(x^{\prime}))$,
or

$\nabla_{x}=\bar{\textsf{f}}(\nabla_{x^{\prime}})\ \ \ \ \text{and}\ \ \ \
\bar{\textsf{f}}^{-1}(\nabla_{x})=\nabla_{x^{\prime}},$ (59)

which shows us that $b(x)$ is not covariant under displacement. In order to
make the objects such as $b(x)$ covariant, we must introduce a position-gauge
field $\bar{\textsf{h}}(a,x)$, which is a linear function of $a$ and arbitrary
function of $x$, so that

$\bar{\textsf{h}}(a,x)\xrightarrow{x^{\prime}=f(x)}\bar{\textsf{h}}^{\prime}(a,x)=\bar{\textsf{h}}(\bar{\textsf{f}}^{-1}(a),x^{\prime}).$
(60)

Now if we redefine $b(x)=\bar{\textsf{h}}(\nabla\phi(x))$, then

$b(x)=\bar{\textsf{h}}(\nabla\phi(x))\xrightarrow{x^{\prime}=f(x)}b^{\prime}(x)=\bar{\textsf{h}}^{\prime}(\nabla\phi^{\prime}(x))=\bar{\textsf{h}}(\bar{\textsf{f}}^{-1}(\nabla\phi^{\prime}(x)))=\bar{\textsf{h}}(\nabla_{x^{\prime}}\phi(x^{\prime}))=b(x^{\prime}),$
(61)

which becomes covariant. The $\bar{h}(a)$ field plays the same role of
vierbein in the tensor calculus approach of gauge theory of gravity (Hehl et
al., 1976; Lasenby et al., 1998). We now give the relationship between
$\bar{h}(a)$ and the metric tensor $g_{\mu\nu}$ in GR. We define a position
gauge invariant directional derivative as (Hestenes, 2005)

$a\cdot\bar{h}(\nabla)=\underline{h}(a)\cdot\nabla,$ (62)

where $\underline{h}$ is the adjoint of $\bar{h}$ defined by
$\underline{h}(b)\equiv\partial_{c}(\bar{h}(c)\cdot b)$, and $a$ is an
invariant vector (i.e., for $x^{\prime}=f(x)$, $a(x)$ is transformed to
$a^{\prime}(x)=a(x^{\prime})$). So $\underline{h}$ maps tangent vectors to
tangent vectors and $\bar{h}$ maps cotangent vectors to cotangent vectors. For
a given set of coordinates {$x^{\mu},\mu=0,1,2,3$}, we introduce the basis
vectors

$e_{\mu}\equiv\frac{\partial x}{\partial x^{\mu}},\ \ \ e^{\mu}\equiv\nabla
x^{\mu},$ (63)

which satisfy $e_{\mu}\cdot e^{\nu}=\delta^{\nu}_{\mu}$. From these vectors we
further define vectors

$g_{\mu}\equiv\underline{h}^{-1}(e_{\mu}),\ \ \
g^{\mu}\equiv\bar{h}(e^{\mu}).$ (64)

These vectors satisfy the relation

$g_{\mu}\cdot
g^{\nu}=\underline{h}^{-1}(e_{\mu})\cdot\bar{h}(e^{\nu})=e_{\mu}\cdot\bar{h}^{-1}(\bar{h}(e^{\nu}))=e_{\mu}\cdot
e^{\nu}=\delta^{\nu}_{\mu}.$ (65)

The metric tensor then is given by

$g_{\mu\nu}\equiv g_{\mu}\cdot g_{\nu}.$ (66)

Let $x(\tau)$ be a time like curve (where $\tau$ is the proper time), a
mapping $f:x\rightarrow x^{\prime}=f(x)$ induce the transformation

$\dot{x}=\frac{dx}{d\tau}\rightarrow\dot{x}^{\prime}=\frac{dx^{\prime}}{d\tau}=\frac{dx}{d\tau}\cdot\nabla
f(x).$ (67)

Comparing (67) with (62), we introduce an invariant velocity $v=v(x(\tau))$ as
(Hestenes, 2005)

$\dot{x}=\underline{h}(v),\ \ \ v=\underline{h}^{-1}(\dot{x}).$ (68)

From the known formula $dx=dx^{\mu}e_{\mu}$, the invariant normalization
$v^{2}=1$ induces the invariant line element on a timelike curve in GR

$d\tau^{2}=[\underline{h}^{-1}(dx)]^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}.$ (69)

Another gauge field, $\omega(a)$, is a position-dependent linear function
mapping the vector argument $a$ to bivectors. Its introduction ensures
covariance of the equations of physics under local Lorentz rotations. Under
local Lorentz rotations, the multivector $M$ transforms as
$M^{\prime}=RM\tilde{R}$ and the spinor $\psi$ transforms as
$\psi^{\prime}=R\psi$. To ensure covariance of the quantities like
$\bar{h}(\nabla)M$ and $\bar{h}(\nabla)\psi$ under local Lorentz rotations,
$\bar{h}(\nabla)$ should be replaced by the covariant derivative $D$ which is
given by (Lasenby et al., 1998)

$DM\equiv\partial_{a}a\cdot
DM=\partial_{a}\left(a\cdot\bar{h}(\nabla)M+\omega(a)\times M\right),$ (70)

and

$D\psi\equiv\partial_{a}\left(a\cdot\bar{h}(\nabla)\psi+\frac{1}{2}\omega(a)\psi\right).$
(71)

The field strength corresponding to the $\omega(a)$ gauge field is defined by

$R(a\wedge b)\equiv L_{a}\omega(b)-L_{b}\omega(a)+\omega(a)\times\omega(b),$
(72)

where $L_{a}\equiv a\cdot\bar{h}(\nabla)$ and $a$, $b$ are constant vectors.
Ricci tensor $R(a)$, Ricci scalar $R$ and Einstein tensor $G(a)$ are defined,
respectively, as:

$\displaystyle R(a)$ $\displaystyle=$ $\displaystyle\partial_{b}\cdot
R(b\wedge a),$ (73) $\displaystyle R$ $\displaystyle=$
$\displaystyle\partial_{a}\cdot R(a),$ (74) $\displaystyle G(a)$
$\displaystyle=$ $\displaystyle R(a)-\frac{1}{2}aR.$ (75)

The overall action integral is of the form

$S=\int|d^{4}x|\det(\textsf{h})^{-1}(\frac{1}{2}R-\kappa\mathcal{L}_{m}),$
(76)

where $\mathcal{L}_{m}$ describes the matter content and $\kappa=8\pi$. As
mentioned at the end of Appendix A, due to lacking of appropriate differential
equations governing the spinor field for macro bodies, we temporally “borrow”
$\mathcal{L}_{m}$ from Dirac theory of electrons, which has been well-studied
(Lasenby et al., 1998; Doran et al., 1998; Challinor et al., 1997). From (76)
we obtain the following equations describe the field coupled self-consistently
to gravity (Doran et al., 1998):

$\displaystyle\text{torsion:}\quad D\wedge\bar{h}(a)$ $\displaystyle=$
$\displaystyle\kappa\bar{h}(a)\cdot S,$ (77)
$\displaystyle\text{Einstein:}\qquad G(a)$ $\displaystyle=$
$\displaystyle\kappa T(a)$ (78) $\displaystyle\text{Dirac:}\qquad D\psi
i\sigma_{3}$ $\displaystyle=$ $\displaystyle m\psi\gamma_{0},$ (79)

where $D\wedge\bar{h}(a)$ is the gravitational torsion which is determined by
the matter spin density, $S\equiv\frac{1}{2}\psi i\gamma_{3}\tilde{\psi}$ is
the spin trivector, $\kappa=8\pi$, and

$T(a)=\langle a\cdot D\psi i\gamma_{3}\tilde{\psi}\rangle_{1}$ (80)

is the matter stress-energy tensor. We can solve (77) for $\omega(a)$ to
obtain (Doran et al., 1998)

$\omega(a)=\omega^{\prime}(a)+\frac{1}{2}\kappa a\cdot
S=-H(a)+\frac{1}{2}a\cdot[\partial_{b}\wedge H(b)]+\frac{1}{2}\kappa a\cdot
S,$ (81)

this defines $\omega^{\prime}(a)$ as the $\omega$-function in the absence of
torsion, and

$H(a)\equiv\bar{h}(\dot{\nabla}\wedge\dot{\bar{h}}^{-1}(a))=-\bar{h}(\dot{\nabla})\wedge\dot{\bar{h}}(\bar{h}^{-1}(a)),$
(82)

where ‘overdot’ notation is employed to denote the scope of a differential
operator. It proves convenient if we employ the primed symbols to denote the
torsion free part of the curvature tensors, we obtain (Doran et al., 1998)

$\displaystyle R(a\wedge b)$ $\displaystyle=$ $\displaystyle
R^{\prime}(a\wedge b)+\frac{1}{4}\kappa^{2}[(a\wedge b)\cdot S]\cdot S$ (83)
$\displaystyle-\frac{1}{2}\kappa[(a\wedge b)\cdot D]\cdot S,$ $\displaystyle
R(a)$ $\displaystyle=$ $\displaystyle
R^{\prime}(a)+\frac{1}{2}\kappa^{2}(a\cdot S)\cdot S$ (84)
$\displaystyle-\frac{1}{2}\kappa a\cdot(D\cdot S),$ $\displaystyle R$
$\displaystyle=$ $\displaystyle R^{\prime}+\frac{3}{2}\kappa^{2}S^{2}.$ (85)