# Coherent collisional decoherence

Leonardo Badurina<EMAIL_ADDRESS>Walter Burke Institute for Theoretical
Physics, California Institute of Technology, Pasadena, CA 91125, USA  Clara
Murgui<EMAIL_ADDRESS>Departament de Física, Universitat Autonoma de
Barcelona, 08193 Bellaterra, Barcelona, Spain  Ryan Plestid
<EMAIL_ADDRESS>Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

We study the decoherence of a system of $N$ non-interacting heavy particles
(atoms) due to coherent scattering with a background gas. We introduce a new
framework for computing the induced phase shift and loss of contrast for
arbitrary preparations of $N$-particle quantum states. We find new phase
shifts that are inherently $(N\geq 2)$-body effects and may be searched for in
future experiments. We analyze simple setups, including a two-mode
approximation of an interferometer. We study fully entangled $N00N$ states,
which resemble the correlated positions in a matter interferometer, as well as
totally uncorrelated product states that are representative of a typical state
in an atom interferometer. We find that the extent to which coherent
enhancements increase the rate of decoherence depends on the observable of
interest, state preparation, and details of the experimental design.

††preprint: CALT-TH/2024-005

Introduction: Decoherence is a pervasive quantum phenomenon which underlies
much of what we understand about the quantum to classical transition [1, 2].
It is universal, and can occur in kinematic regimes in which no other
measurable effect would be induced. For example, collisional decoherence
arises from the limit of ultra-soft scattering in which the positions of
single atoms in a quantum system of $N$ non-interacting atoms are left
essentially unchanged as a result of a momentum transfer $\mathbf{q}$ [3, 4].
Despite remaining static, the object’s position is resolved within a region of
size $1/{|\mathbf{q}|}$, its quantum state decoheres, and position is
einselected as the preferred classical label [5]. This phenomenon is generic,
and can be induced by almost any scattering process [6, 7, 8, 9, 10]. The
effect is calculable, being directly related to the scattering cross section
[3, 4].

Collisional decoherence is therefore intimately tied to the limit of soft
scattering. This has lead Riedel to propose, in a series of papers [11, 12,
13], that measurements of decoherence can be used as a sensitive probe of soft
scattering induced by sub-GeV dark matter. This idea has been further pursued
in more recent literature [14, 15, 16]. In the limit of low momentum transfer,
it is natural to consider enhanced sensitivity that would arise due to
coherent scattering rates that scale as $N^{2}$ rather than $N$. Such
enhancements are commonplace in scattering off nuclei [17], and offer order of
magnitude improvements in sensitivity for direct detection experiments when
considering models of spin-independent dark matter [18]. In the context of
matter and atom interferometers, coherent enhancements are quite dramatic
since $N\sim 10^{6}-10^{10}$ in some experiments [19, 20, 21, 22]. It is
therefore crucial to reliably establish how coherent scattering enters into
the derivation of, and potentially modifies, formulae for collisional
decoherence.

Surprisingly no such formalism currently exists. The original papers on
collisional decoherence [6, 3] and subsequent publications [4, 23] all
explicitly worked on an atom-by-atom basis. As we will see in what follows, a
consistent derivation of decoherence for an $N$-body system differs from $N$
iterations of a single-body experiment. Our results are of interest for
projected sensitivity to models of dark matter, and to any process that can be
coherently enhanced at small momentum transfer. This may occur due to a
macroscopic charge (as is relevant for coherent elastic neutrino scattering),
or due to light mediators (as arise in some models of dark matter).

Before proceeding to technical details, let us sketch a brief motivation as to
why $N$-particle coherent scattering may differ from $N$ independent single-
particle experiments. Consider an interferometer, with a left arm and a right
arm. Suppose an initial state is prepared such that every atom will end up in
a superposition of left and right arms at a later time, i.e.
${\ket{\Psi}=\bigotimes_{i=1}^{N}\tfrac{1}{\sqrt{2}}(\ket{L_{i}}+\ket{R_{i}})}$.
The wavefunction will have many branches with varying numbers of particles in
the left and right arms of the interferometer, respectively $N_{L}$ and
$N_{R}$. Coherent scattering with the background gas will modify the
coefficients that multiply each branch of the wavefunction in a manner that
depends on $N_{L}$ and $N_{R}$. Thus the scattered branch of the wavefunction
will no longer be a product state and will be entangled with the environment
in an $N_{L}$-dependent way. When considering measurements of only the atoms,
one may trace out the environment. Since the environment and atoms are
entangled in an $N$-dependent fashion, one expects the rate of decoherence to
depend on $N$. The purpose of this paper is to provide a formalism for
computing these effects.

In what follows we compute collisional decoherence for a non-interacting
$N$-body system. The formalism is general and flexible, and can easily
interpolate between different limits. As we will see below, a clear
identification of when coherent scattering does or does not influence
decoherence emerges.

Collisional decoherence: A “probe” $\ket{\pi}$ is incident and scatters upon a
gas of $N$ atoms prepared in a state $\rho_{A}$. Prior to scattering, the
system and probe are described in terms of the total density matrix
$\rho=\rho_{A}\otimes\rho_{\pi}$; we will assume $\rho_{\pi}$ is a mixed state
diagonalized in momentum space. The dynamics of collisions between the atoms
and probe particles are captured by the (unitary) scattering operator $S$.
Assuming that the probe gas is sufficiently dilute, so that the time between
collisions is long compared to the duration of a collision, we model the
effects of collisions by a two-particle scattering operator
$\rho\rightarrow\rho^{\prime}=S\rho S^{\dagger}$. Using the standard
definition of the $T$-matrix, $S=1+\mathrm{i}\,T$, we may write the change in
the density matrix due to scattering, $\Delta\rho=\rho^{\prime}-\rho$ as

$\begin{split}\Delta\rho&=\tfrac{\mathrm{i}}{2}\quantity[T+T^{\dagger},\rho]+\tfrac{\mathrm{i}}{2}\quantity{T-T^{\dagger},\rho}+T\rho
T^{\dagger}\\\
&=\tfrac{\mathrm{i}}{2}\quantity[T+T^{\dagger},\rho]-\tfrac{1}{2}\quantity{T^{\dagger}T,\rho}+T\rho
T^{\dagger}~{},\end{split}$ (1)

where in going to the second line we used the optical theorem,
$\mathrm{i}\,(T-T^{\dagger})=-T^{\dagger}T$. Our goal is to compute

$\begin{split}\rho_{A}^{\prime}&={\rm
Tr}_{\pi}\rho^{\prime}~{}=\int\frac{\differential^{3}p}{(2\pi)^{3}}\bra{\pi(\mathbf{p})}\rho^{\prime}\ket{\pi(\mathbf{p})}~{},\end{split}$
(2)

where $\mathbf{p}$ is the three-momentum of the probe and ${\rm Tr}_{\pi}$ is
the trace over environmental degrees of freedom. In the static limit (defined
by $|\mathbf{q}|\ll M_{A}$, where $M_{A}$ is the atomic mass), atomic position
is conserved in the scattering process and
$\ket{\\{\mathbf{x}\\}}=\bigotimes_{i=1}^{N}\ket{\mathbf{x}_{i}}$ acts as a
good pointer basis [5, 3, 24]. At the level of the $T$-matrix, this manifests
as

$T\left(\ket{\\{\mathbf{x}\\}}\otimes\ket{\pi}\right)=\ket{\\{\mathbf{x}\\}}\otimes\quantity(T_{\\{\mathbf{x}\\}}\ket{\pi})~{}.$
(3)

We refer to $T_{\\{\mathbf{x}\\}}$ as the induced $T$-matrix; it depends on
all of the atomic spatial coordinates. Whenever the probe interacts in the
same way with all of the atoms, $T_{\\{\mathbf{x}\\}}$ is invariant under
permutations of atomic positions. It acts on the probe Hilbert space and has
matrix elements normalized as is appropriate for potential scattering i.e.
$\matrixelement{\pi(\mathbf{p}^{\prime})}{T_{\\{\mathbf{x}\\}}}{\pi(\mathbf{p})}=(2\pi)\delta(E^{\prime}-E)\mathcal{M}_{\\{\mathbf{x}\\}}(\mathbf{p}^{\prime},\mathbf{p})$
[25]. It may be computed by treating each atom as a background potential
centered at $\mathbf{x}_{i}$ and summing the resulting amplitudes.

We may expand the scattering matrix elements perturbatively using the Born
series. At first order we have

$\mathcal{M}_{\\{\mathbf{x}\\}}^{(1)}(\mathbf{p}^{\prime},\mathbf{p})=\sum_{i}\widetilde{V}(\mathbf{q})\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{x}_{i}}~{},$
(4)

where $\widetilde{V}(\mathbf{q})$ is the Fourier transform of the scattering
potential $V(\mathbf{x})$ with $\mathbf{q}=\mathbf{p}^{\prime}-\mathbf{p}$. At
second order we have

$\begin{split}\mathcal{M}_{\\{\mathbf{x}\\}}^{(2)}(\mathbf{p}^{\prime},\mathbf{p})=\sum_{ij}\int&\frac{\differential^{3}q_{1}}{(2\pi)^{3}}\frac{\differential^{3}q_{2}}{(2\pi)^{3}}\mathrm{e}^{\mathrm{i}\mathbf{q}_{1}\cdot\mathbf{x}_{j}}\mathrm{e}^{\mathrm{i}\mathbf{q}_{2}\cdot\mathbf{x}_{i}}\\\
&\widetilde{V}(\mathbf{q}_{2})G_{\pi}(\mathbf{p}+\mathbf{q}_{1})\widetilde{V}(\mathbf{q}_{1})\\\
&\hskip
43.36464pt(2\pi)^{3}\delta^{(3)}(\mathbf{q}_{1}+\mathbf{q}_{2}-\mathbf{q})~{}\,,\end{split}$
(5)

where the probe propagator appears as $G_{\pi}$.

Without loss of generality we can write any atomic density matrix in terms of
its position eigenstates

$\rho_{A}=\int[\differential\\{\mathbf{x}\\}][\differential\\{\mathbf{x}^{\prime}\\}]~{}\rho_{A}\quantity(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})~{}\ket{\\{\mathbf{x}\\}}\bra{\\{\mathbf{x}^{\prime}\\}}~{}\,,$
(6)

where $[\differential\\{\mathbf{x}\\}]=\prod_{i=1}^{N}\differential^{3}x_{i}$.
The behavior of the states $\ket{\\{\mathbf{x}\\}}$ as a pointer basis allows
us to evolve each of these matrix elements independently of one another. The
matrix elements of $\Delta\rho_{A}$ then satisfy

$\begin{split}\Delta\rho_{A}\quantity(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})=\rho_{A}\quantity(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})\int\frac{\differential^{3}p}{(2\pi)^{3}}~{}\rho_{\pi}(\mathbf{p})~{}&\bigg{(}\tfrac{\mathrm{i}}{2}\bra{\mathbf{p}}\big{(}T_{\\{\mathbf{x}\\}}+T_{\\{\mathbf{x}\\}}^{\dagger}-T_{\\{\mathbf{x}^{\prime}\\}}-T_{\\{\mathbf{x}^{\prime}\\}}^{\dagger}\big{)}\ket{\mathbf{p}}\\\
&\hskip
32.52017pt-\tfrac{1}{2}\bra{\mathbf{p}}\quantity(T_{\\{\mathbf{x}\\}}T_{\\{\mathbf{x}\\}}^{\dagger}+T_{\\{\mathbf{x}^{\prime}\\}}T_{\\{\mathbf{x}^{\prime}\\}}^{\dagger})\ket{\mathbf{p}}\\\
&\hskip
114.90903pt+\bra{\mathbf{p}}T_{\\{\mathbf{x}^{\prime}\\}}^{\dagger}T_{\\{\mathbf{x}\\}}\ket{\mathbf{p}}\bigg{)}~{}.\end{split}$
(7)

This is our major result, and what follows are simple applications of this
formula. Notice that Eq. 7 is manifestly traceless and Hermitian and, by the
invariance of $T_{\\{\mathbf{x}\\}}$ under particle relabeling, there also
exist off-diagonal entries in Eq. 7 that vanish. The first line of Eq. 7
corresponds to the forward scattering phase, and contains non-trivial
contributions at $O(V^{2})$ while the second- and third-lines appear at first
order in the Born series i.e., $T\sim O(V)$. Whenever
$T_{\\{\mathbf{x}\\}}\neq T_{\\{\mathbf{x}^{\prime}\\}}$ the forward
scattering phase is observable and we comment on its impact on specific
measurements in what follows.

Equation 7 reduces to the standard result for a single atom. The first line
has only forward scattering amplitudes and vanishes for a single atom, as
shown in Ref. [4]. However, it does not vanish for $N\geq 2$ and therefore
represents a new phase shift which is a bona fide $N$-particle effect, and
which can survive even for an isotropic density matrix for probe states.

Interpreting each unitary $S$-matrix as inducing a change in the state over a
small interval of time $\Delta t$, the above equations can be re-written in
differential form,

$\derivative{t}\rho_{A}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})=-\lambda(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})\rho_{A}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\}).$
(8)

The solution of this differential equation is trivial being given by
$\rho_{A}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})\times\exp\quantity[-\int_{0}^{t}\differential\tau\lambda(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\},\tau)]$.
The function $\lambda$ is calculable and can be decomposed into a unitary,
$\lambda_{U}$, and dissipative component, $\lambda_{D}$, which are identified
by their appearance in the first line and subsequent lines of Eq. 7,
respectively. At leading order in the Born series, the expressions for
$\lambda_{U}$ and $\lambda_{D}$ are given via Eqs. 4 and 5 by

$\displaystyle\lambda_{U}$
$\displaystyle=\int\frac{\differential^{3}p}{(2\pi)^{3}}~{}\rho_{\pi}(\mathbf{p})\int\frac{\differential^{3}q^{\prime}}{(2\pi)^{3}}~{}\widetilde{V}(-\mathbf{q^{\prime}})\quantity[G_{\pi}(\mathbf{p}+\mathbf{q^{\prime}})+G_{\pi}^{\dagger}(\mathbf{p}+\mathbf{q^{\prime}})]\widetilde{V}(\mathbf{q^{\prime}})\bigg{[}\frac{\mathrm{i}}{2}\,\sum_{ij}\mathrm{e}^{\mathrm{i}\mathbf{q^{\prime}}\cdot(\mathbf{x}_{i}^{\prime}-\mathbf{x}_{j}^{\prime})}-\mathrm{e}^{\mathrm{i}\mathbf{q^{\prime}}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}\bigg{]}~{},$
(9) $\displaystyle\lambda_{D}$
$\displaystyle=\int\frac{\differential^{3}p}{(2\pi)^{3}}~{}\rho_{\pi}(\mathbf{p})\int\frac{\differential^{3}q}{(2\pi)^{3}}~{}(2\pi)\delta(\Sigma
E)|\widetilde{V}(\mathbf{q})|^{2}\bigg{[}\frac{1}{2}\sum_{ij}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}+\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}^{\prime}_{i}-\mathbf{x}^{\prime}_{j})}-2\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j}^{\prime})}\bigg{]}~{},$
(10)

where $\mathbf{q^{\prime}}$ is not to be confused with the momentum
transferred by the probe. Equations 10 and 9 factorize into a real function
$\omega_{U,D}$ that depends solely on the kinematics [i.e., after averaging
over $\rho_{\pi}(\mathbf{p})$], and a kernel (in square brackets)
$K_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x^{\prime}}\\})$. For definiteness, in
what follows we will treat the $N$-particles as distinguishable; this is
justified for atomic de Broglie wavelengths that are short relative to the
interparticle spacing.

Toy model of a two-arm interferometer: The parametric scaling of Eqs. 10 and 9
with $N$ can be most easily understood in a two-mode interferometer with $N$
atoms, for which the Hilbert space is $2^{N}$-dimensional. This approximates a
realistic two-arm interferometer when momenta transfers are sufficiently small
that the cloud appears point-like. We will label the number of atoms in the
left arm of $\ket{\\{\mathbf{x}\\}}$ by $N_{L}$ and the number of atoms in the
left arm of $\bra{\\{\mathbf{x}^{\prime}\\}}$ by $N_{L}^{\prime}$, and define
$n=N_{L}-N_{L}^{\prime}$ as the atom asymmetry. With this convention, we may
evaluate the kernels for any $N$-tuple $\\{\mathbf{x}\\}$ and
$\\{\mathbf{x^{\prime}}\\}$

$\displaystyle K_{U}$
$\displaystyle=\mathrm{i}\,n(n+N-2N_{L})\quantity[1-\cos(\mathbf{q^{\prime}}\cdot\Delta\mathbf{x})]~{},$
(11) $\displaystyle K_{D}$
$\displaystyle=n^{2}\quantity[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]-\mathrm{i}Nn\sin(\mathbf{q}\cdot\Delta\mathbf{x})\,,$
(12)

where $-N\leq n\leq N$ and $0\leq N_{L}\leq N$, and $|\Delta\mathbf{x}|$ is
the separation between the interferometer arms. Notice that both kernels are
invariant under permutations of the atom’s labels and therefore vanish for
$n=0$. This also agrees with the information-theoretic understanding of
decoherence [26]: the larger the asymmetry between states, the more “which-
path” information is gathered by a single scattering event. Pairs of
$N$-particle states for which $n=0$ (e.g., $\ket{LR}\bra{RL}$ for $N=2$ or
$\ket{LLR}\bra{LRL}$ for $N=3$) are therefore decoherence-free [27].

Product states: Let us now consider a two-arm interferometer in which the
atoms are initially uncorrelated and each prepared in a superposition of the
left and right arm, $\frac{1}{\sqrt{2}}(\ket{L}+\ket{R})$. This many-particle
wavefunction can be written as a product state
$\ket{\Psi}=\bigotimes_{i=1}^{N}\frac{1}{\sqrt{2}}(\ket{L_{i}}+\ket{R_{i}})$.
This may be considered as a toy model of an atom interferometer, such as MAGIS
[20] and AION [28], employing a dilute atom cloud of radius
$r_{c}\ll\Delta\mathbf{x},1/|\mathbf{q}|$. In this case the density matrix is
not sparse, and one must consider all entries in $\rho_{A}$, for which the
decoherence kernel is given by Eq. 12.

Equation 12 shows that density matrix elements with $n\sim O(N)$ have
coherently enhanced rates of decoherence. A typical measurement in an atom
interferometer, however, is a one-body measurement [29], i.e., the observable
is represented by an operator $\mathcal{O}_{\rm
1B}=\sum_{i=1}^{N}\mathcal{O}_{i}$ where $i$ labels each atom in the gas. In
this case, only terms with $|n|=1$ in $\rho_{A}$ contribute to the expectation
value of the observable. For example, the contrast loss $V$ and phase shift
$\varphi$ are typically inferred using
$\mathcal{O}_{i}=\ket{+_{i}}\bra{+_{i}}$ with
$\ket{+_{i}}=\frac{1}{\sqrt{2}}(\ket{L_{i}}+\ket{R_{i}})$ [29, 30]. The
expectation value of $\mathcal{O}_{+}=\sum_{i=1}^{N}\mathcal{O}_{i}$ is then
given by

$\left\langle\mathcal{O}_{+}\right\rangle=\mathrm{Tr}(\rho_{A}\mathcal{O}_{+})=\tfrac{N}{2}\big{(}1+V\cos\varphi\big{)}~{},$
(13)

where the trace is over the $2^{N}$-dimensional Hilbert space. The measured
contrast and phase shift are related to the one-body reduced density matrix
$\rho_{1}={\rm Tr}_{N-1}(\rho_{A}$) via

$V\cos\varphi=2\,\mathrm{Re}(\matrixelement{L}{\rho_{1}}{R})\,.$ (14)

After accumulating a dynamical phase-shift $\phi$, the off-diagonal element of
$\rho_{1}$ can be written as

$\matrixelement{L}{\rho_{1}}{R}=\frac{1}{2}\cos^{N-1}(\tau)~{}\mathrm{e}^{-s+\mathrm{i}\quantity[\phi-N\gamma]}\,,$
(15)

where,

$\displaystyle s$ $\displaystyle=\int\differential
t\int\frac{\differential^{3}q}{(2\pi)^{3}}~{}~{}\omega_{D}(\mathbf{q},t)\quantity[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]~{},$
(16) $\displaystyle\gamma$ $\displaystyle=\int\differential
t\int\frac{\differential^{3}q}{(2\pi)^{3}}~{}~{}\omega_{D}(\mathbf{q},t)\sin(\mathbf{q}\cdot\Delta\mathbf{x})~{},$
(17) $\displaystyle\tau$ $\displaystyle=\int\differential
t\int\frac{\differential^{3}q^{\prime}}{(2\pi)^{3}}~{}~{}\omega_{U}(\mathbf{q^{\prime}},t)\quantity[1-\cos(\mathbf{q^{\prime}}\cdot\Delta\mathbf{x})]~{},$
(18)

are real functions which scale as $|\widetilde{V}(\mathbf{q})|^{2}$ and
parameterize previously-known ($s$ and $\gamma$) [3, 4] and novel ($\tau$)
collision-induced effects. The phase $\tau$, which originates from forward
scattering, is only observable for $N\geq 2$ and does not vanish even for a
rotationally invariant $\rho_{\pi}(\mathbf{p})$ (e.g. a Maxwell-Boltzmann
gas). The cosine in Eq. 15 arises from (destructive) interference between
different terms in the partial trace. Both $\gamma$ and $\tau$ receive
$N$-enhancements while $s$ does not, in agreement with Eqs. 12 and 11 for
$|n|=1$. Importantly, none of these terms scale with $N^{2}$. Hence, the
decoherence rate of a one-body observable does not receive a coherent
enhancement.

Statistical fluctuations of one-body operators: Higher-order moments of one-
body operators offer a natural probe of enhanced decoherence rates. For
example consider counting atoms in a port “$+$”. After $\aleph$ runs of the
experiment the collected data can be written as the sequence
$\\{n_{+}^{(1)},n_{+}^{(2)},n_{+}^{(3)},\ldots,n_{+}^{(\aleph)}\\}$. The
fluctuations of $n_{+}^{(i)}$ are sensitive to enhancements that scale like
$n^{2}$. As a simple illustration let us consider the variance (second moment)
estimated using $\frac{1}{\aleph}\sum_{a=1}^{\aleph}[n_{+}^{(a)}-\langle
n_{+}\rangle]^{2}$, where $\langle
n_{+}\rangle=\frac{1}{\aleph}\sum_{a=1}^{\aleph}n_{+}^{(a)}$ is the estimator
of the mean.

The counting operation corresponds to the one-body operator $\mathcal{O}_{+}$
previously defined. The mean number of counts in this port is
$\langle\mathcal{O}_{+}\rangle={\rm Tr}\quantity[\rho\,\mathcal{O}_{+}]$,
while the variance is given by
$\sigma_{+}^{2}=\langle\mathcal{O}_{+}^{2}\rangle-\langle\mathcal{O}_{+}\rangle^{2}={\rm
Tr}\quantity[\rho\,\mathcal{O}_{+}^{2}]-\langle\mathcal{O}_{+}\rangle^{2}$.
The two-body operator $\mathcal{O}_{+}^{2}$ can be expanded as

$\mathcal{O}_{+}^{2}=\sum_{i=1}^{N}\sum_{j\neq
i}^{N}\mathcal{O}_{i}\mathcal{O}_{j}~{}+~{}\sum_{i=1}^{N}\mathcal{O}_{i}~{}.$
(19)

To evaluate $\langle\mathcal{O}_{+}^{2}\rangle$, one must use $\rho_{2}={\rm
Tr}_{N-2}\rho_{A}$ see e.g., Ref. [31]. The two-body reduced density matrix,
$\rho_{2}$, will have two entries which decohere with a rate enhanced by
$(n=2)^{2}$. We show this schematically below for the elements of $\rho_{2}$
labeled (left to right, and top to bottom) by states $\ket{LL}$, $\ket{LR}$,
$\ket{RL}$, and $\ket{RR}$

$\rho_{2}=\begin{pmatrix}\circ~{}&~{}\blacksquare~{}&~{}\blacksquare~{}&~{}\bigstar\\\
\blacksquare~{}&~{}\circ~{}&~{}\circ~{}&~{}\blacksquare\\\
\blacksquare~{}&~{}\circ~{}&~{}\circ~{}&~{}\blacksquare\\\
\bigstar~{}&~{}\blacksquare~{}&~{}\blacksquare~{}&~{}\circ\end{pmatrix}~{}.$
(20)

Entries with $\circ$ have $n=0$, entries with $\blacksquare$ have $|n|=1$, and
entries with $\bigstar$ have $n=2$. The variance
${\sigma_{+}^{2}=\langle\mathcal{O}_{+}^{2}\rangle-\langle\mathcal{O}_{+}\rangle^{2}}$
can be constructed using

$\displaystyle\langle\mathcal{O}_{+}^{2}\rangle=\langle\mathcal{O}_{+}\rangle+\frac{N(N-1)}{4}\bigg{[}\frac{3}{2}$
$\displaystyle+4{\rm Re}\matrixelement{L}{\rho_{1}}{R}\hfill$ (21)
$\displaystyle+2{\rm Re}\matrixelement{LL}{\rho_{2}}{RR}\bigg{]}~{}.$

The corner-entry of $\rho_{2}$ may be written as

$\begin{split}\matrixelement{LL}{\rho_{2}}{RR}=&\frac{1}{4}\cos^{N-2}(2\tau)~{}\mathrm{e}^{-4s+2\mathrm{i}\quantity[\phi-N\gamma]}~{},\end{split}$
(22)

for $N\geq 2$. Equation 22 exhibits enhanced decoherence relative to Eq. 15
due to $\matrixelement{LL}{\rho_{2}}{RR}$ having $n=2$. More generally, the
$\eta^{\rm th}$-moment of $\mathcal{O}_{+}$ is computed using
$\rho_{\eta}={\rm Tr}_{N-\eta}\rho_{A}$ whose corner entry has $n=\eta$. Thus,
for $\eta\sim O(N)$ one can construct observables which experience rates of
decoherence enhanced by $O(N)$.

Examining Eq. 20 one finds another interesting phenomenon that occurs in the
limit of $\mathrm{e}^{-s}\rightarrow 0$. Although the mean follows what would
be expected from a conventional classical state of coin flipping,
$\langle\mathcal{O}_{+}\rangle=1/2$, we find that the variance remarkably
scales as $N^{2}$ instead of $N$,

$\sigma^{2}_{+}=\frac{N(N+1)}{8}~{}.$ (23)

The statistics follow a super-Poissonian distribution. This is a consequence
of the non-zero off-diagonal entries ($\circ$) with $n=0$ in Eq. 20 which
would vanish for a maximally mixed state. These entries do not decohere
because of the permutation symmetry of the atomic positions $\\{\mathbf{x}\\}$
and $\\{\mathbf{x}^{\prime}\\}$, cf. Eq. 7.

Entangled states: Before concluding, let us consider a simple example that
illustrates the $N^{2}$-enhancements. Suppose that the $N$-atom system is
prepared in the maximally-entangled ($N00N$) state
${\ket{\Psi}=\frac{1}{\sqrt{2}}(\bigotimes_{i=1}^{N}\ket{L_{i}}+\bigotimes_{i=1}^{N}\ket{R_{i}})}$.
This configuration mimics a matter-wave interferometer, which is similarly
entangled but due to intra-material forces rather than state preparation. For
such a system preparation, there are only two non-vanishing off-diagonal
entries in $\rho_{A}$. These correspond to $(n,N_{L})=(N,N)$ and
$(n,N_{L})=(-N,0)$. Using Eqs. 11 and 12 one finds $K_{U}=0$ and

$\displaystyle
K_{D}(\mathbf{q},\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})=N^{2}(\mathrm{e}^{\pm\mathrm{i}\mathbf{q}\cdot(\mathbf{x}-\mathbf{x}^{\prime})}-1)~{},$
(24)

equivalent to the replacement $|\widetilde{V}(\mathbf{q})|^{2}\rightarrow
N^{2}|\widetilde{V}(\mathbf{q})|^{2}$ in the single atom formula. This shows
that existing approaches in the literature [11, 13, 14, 15] for computing
decoherence for matter-wave interferometers, such as MAQRO [22], are reliable.
An atom interferometer employing $N00N$ states and measuring
$\mathcal{O}=\ket{\psi}\bra{\psi}$ is therefore susceptible to coherent
enhancements.

Discussion and conclusions: In this work we have computed the collisional
decoherence of a system of $N$ non-interacting atoms due to scattering with a
background gas. Specifically, Eq. 7 can be used to compute the collisional
decoherence of a generic $N$-particle system. Focusing on the case when the
probe states scatter coherently off of the atoms, we find that a careful
treatment of the $N$-body system is required to properly characterize both the
phase shift and loss of visibility arising from collisional decoherence cf.
Eq. 13. Indeed, there exist new inherently $(N\geq 2)$-body effects that can
be searched for in future two-arm interferometers. Importantly, the
observation of this phase shift and of coherently-enhanced decoherence rates
depends on: i) the initial state preparation, ii) the dynamics of the
interferometer, and iii) the final measurement being performed.

Although we have focused on a toy-model of an interferometer, our qualitative
conclusions have immediate consequences for dark matter direct detection [11,
13, 14, 15], and atom interferometers more generally. While we find that one-
body observables are less sensitive to coherent enhancements than one would
naively expect, it is clear that some coherent effects can be used as a
resource for the discovery of dark matter. For example atom interferometers
employing $N00N$ states and matter-wave interferometers naturally exhibit
coherently enhanced decoherence. Similarly, statistical fluctuations of
counting measurements in atom interferometers offer a new probe of coherent
enhancements using uncorrelated initial states. We leave a more detailed
investigation into optimal strategies for measuring decoherence and modeling
of proposed and existing experimental configurations to future work.

It would be interesting to test the new features of decoherence that are
inherent to ($N\geq 2$)-particle systems in the lab. This could be achieved
even for a system with a modest number of atoms (e.g., $N\sim 10$) and a
background gas which has a large coherent cross section with atoms in the
limit of low momentum transfer. One could study higher order moments of
single-body operators as described above, and search for coherently enhanced
rates of decoherence and/or the influence of the forward scattering phase.
Ideally, to match onto the two-mode approximation employed in this work, the
radial size $r_{c}$ of the $N$-atom system (not to be confused with the
separation between the arms of the interferometer) would have to satisfy
$r_{c}\ll 1/|\mathbf{q}_{\rm max}|$ with $|\mathbf{q}_{\rm
max}|\approx\sqrt{2mT}$ with $T$ and $m$ the temperature and mass of the
background gas, respectively. Furthermore, while Eq. 7 applies generally to
any $N$-atom system, the expressions in Eqs. 9 and 10 rely on the Born
approximation. Therefore, if these analytic results are to be used, one should
have a dilute gas satisfying $na^{3}\ll 1$ where $n$ is the number density and
$a$ is the probe-atom scattering length [32, 33, 34]. We leave a more detailed
modeling of effects (e.g., system substructure, finite temperature, etc.) that
go beyond the idealized limit discussed above to future work.

The formalism presented here can be tested in future experiments, including
searches for the forward scattering phase, and coherently enhanced rates of
decoherence. This would both shed light on the quantum to classical
transition, and aid in the design and optimization of future interferometers.
The forward scattering phase, $\tau$, and the decoherence phase, $\gamma$, are
enhanced by $O(N)$, and may be an important background (or new signal) when
searching for anomalous phases shifts in atom interferometers. Theoretical
control over coherently enhanced effects will, in turn, benefit dark matter
searches, gravitational wave detection, and other applications to fundamental
physics (see Refs. [35, 36] and references therein).

Acknowledgements: We thank Klaus Hornberger, Duncan O’Dell and John E. Sipe
for feedback during the completion of this work. We also thank Daniel Carney
for discussions on the validity of the Born approximation and comments on an
early version of this manuscript. We are especially grateful to Kathryn M.
Zurek for many stimulating discussions and comments on the manuscript.

C.M. is supported by “Ayuda Beatriz Galindo Junior” from the Spanish
“Ministerio de Universidades”, grant BG22/00155. C.M. thanks Caltech for its
hospitality during the completion of this work. R.P. & L.B. are supported by
the U.S. Department of Energy, Office of Science, Office of High Energy
Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for
Theoretical Physics. R.P. is supported by the Neutrino Theory Network under
Award Number DEAC02-07CHI11359.

##  Appendix A Scattering theory conventions

For a self-contained discussion, we provide here our conventions for the
normalization of states and the definition of the scattering cross section. We
take our states to be relativistically normalized

$\innerproduct{p^{\prime}}{p}=2E_{p}(2\pi)^{3}\delta^{(3)}(\mathbf{p}-\mathbf{p}^{\prime})~{}.$
(25)

Scattering matrix elements are defined in terms of the $T$-matrix,
$S=1+\mathrm{i}T$,

$\matrixelement{f}{T}{i}=(2\pi)^{4}\delta^{(4)}(\Sigma p)\mathcal{M}_{fi}~{},$
(26)

where $\Sigma p=\Sigma p_{f}-\Sigma p_{i}$, and $p_{i,f}$ are the four momenta
of the initial and final states, respectively. Scattering cross sections
involve $2\rightarrow n$ processes, and are given by

$\sigma_{fi}=\frac{1}{\mathcal{F}}\\!\int\\!\differential\Pi_{1}\ldots\\!\differential\Pi_{n}(2\pi)^{4}\delta^{(4)}(\Sigma
p)|\mathcal{M}_{fi}|^{2}~{},$ (27)

where $\differential\Pi=\differential^{3}p/(2E_{p})/(2\pi)^{3}$, and
$\mathcal{F}=2\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})}$ is a Lorentz invariant
flux normalization with $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$ the
Källen triangle function. The masses of the incoming particles are $m_{1}$ and
$m_{2}$. Using the above expression it is straightforward to derive

$\derivative{\sigma}{\Omega}=\int\frac{\differential^{3}q}{(2\pi)^{3}}(2\pi)\delta(\Sigma
E)|\mathcal{M}|^{2}~{},$ (28)

where $\Sigma E=\Sigma E_{f}-\Sigma E_{i}$, for $2\rightarrow 2$ scattering in
the static limit where $s-m_{1}^{2}\ll m_{1}^{2}$.

It is also useful to introduce the relevant normalization for potential
scattering. These are frame dependent since the static background field
implicitly selects a preferred frame i.e., the rest frame of the target.
Scattering in a background field is a $1\rightarrow 1$ process and in this
case the normalization of $T$-matrix elements is

$\matrixelement{p^{\prime}}{T}{p}=(2\pi)\delta(E^{\prime}-E)\mathcal{M}(p^{\prime},p)~{}.$
(29)

The differential cross section is written in terms of the potential-scattering
matrix element as

$\sigma v=\frac{1}{2E}\int\differential\Pi~{}(2\pi)\delta(\Sigma
E)~{}|\mathcal{M}|^{2}~{},$ (30)

where $v=|\mathbf{p}|/E$ is the velocity of the incident particle in the lab
frame. When scattering from a background potential, the matrix element at
first order in the Born approximation is related to the Fourier transform of
the potential $\mathcal{M}=(2E)\widetilde{V}(\mathbf{q})$. The factor of
$(2E)$ is related to the relativistic normalization of states.

##  Appendix B Potential scattering derivation

In this section we explain how the background potential approximation emerges
from the analysis of Feynman diagrams. This helps resolve certain conceptual
ambiguities. For example, if one considers $3\rightarrow 3$ scattering
involving atom $i$, atom $j$, and the probe $\pi$, the associated Feynman
diagram appears as a tree-level graph. However, in the potential scattering
calculation, this diagram is second order in the Born series, and one must
integrate over momentum transfers. This “paradox” arises because we consider
atoms in position eigenstates, rather than momentum eigenstates. We discuss
this point in detail in what follows.

Let us work out the scattering of a probe on a single particle at a fixed
position,

$\ket{\mathbf{x}}=\int\frac{\differential^{3}k}{(2\pi)^{3}}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\ket{\mathbf{k}}~{}.$
(31)

As a short-hand we will use $[\differential k]=\differential^{3}k/(2\pi)^{3}$.
We are interested in

$\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{x},\mathbf{p}}=\int[\differential
k]\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{k},\mathbf{p}}~{},$
(32)

where $T$ is the $T$-matrix appearing in the definition of the $S$-matrix,
i.e., $S=1+\mathrm{i}\,T$. Inserting a complete set of atomic momentum
eigenstates,

$\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{x},\mathbf{p}}=\int[\differential
k][\differential
k^{\prime}]\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\ket{\mathbf{k}^{\prime}}\bra{\mathbf{p}^{\prime},\mathbf{k}^{\prime}}T\ket{\mathbf{k},\mathbf{p}}~{}.$
(33)

Using the definition of the scattering matrix element

$\begin{split}\bra{\mathbf{p}^{\prime},\mathbf{k}^{\prime}}T\ket{\mathbf{k},\mathbf{p}}&=(2\pi)^{4}\delta^{(4)}(\Sigma
p)~{}\mathrm{i}\mathcal{M}(v\cdot q,q^{2})~{},\end{split}$ (34)

where we assume that the matrix element depends only on the Lorentz invariants
$v\cdot q$ and $q^{2}$, with $v$ the atom’s four velocity111This approximation
is related to the hierarchy of scales $|\mathbf{q}|\ll M_{A}$. For an atom at
rest, $v_{\mu}=(1,0,0,0)$., and $q_{\mu}=p^{\prime}_{\mu}-p_{\mu}$. To first
order (i.e., the probe scattering off of an atom once), the corresponding
Feynman diagram for the $i$-the atom is (trivially),

$\begin{gathered}\fmffile{diagram_{s}oft}\fmfstraight\fmfgraph*(20,7)\fmfleft{i1,i2}\fmfright{o1,o2}\fmf{double}{i1,b1,o1}\fmf{plain}{i2,t1,o2}\fmf{dashes,tension=0,lab=$\mathbf{q}$}{b1,t1}\fmflabel{$\pi$}{i2}\fmflabel{$\mathbf{x}_i$}{i1}\end{gathered}$
(35)

At second order mediators can talk to the same atom twice. For the $i$-th
atom, this corresponds to a two-body one-loop graph,

$\begin{gathered}\fmffile{diagram_{s}oft_{1}}\fmfstraight\fmfgraph*(20,9)\fmfleft{i1,i2,fi,i3}\fmfright{o1,o2,fo,o3}\fmf{double}{i2,b1,b2,o2}\fmf{plain}{i3,t1,t2,o3}\fmf{dashes,tension=0,lab=$\mathbf{q}_1$}{b1,t1}\fmf{dashes,tension=0,lab=$\mathbf{q}_2$}{b2,t2}\fmflabel{$\pi$}{i3}\fmflabel{$\mathbf{x}_i$}{i2}\end{gathered}$
(36)

The two momentum transfers $\mathbf{q_{1}}$ and $\mathbf{q_{2}}$ are
constrained by $\mathbf{q_{1}}+\mathbf{q_{2}}=\mathbf{q}$. Using the delta
function to integrate over $\mathbf{k^{\prime}}$, Eq. 34 takes the form

$\begin{split}\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{x},\mathbf{p}}=&\int[\differential
k]\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\ket{\mathbf{k}+\mathbf{q}}\\\
&\hskip 65.04034pt\times(2\pi)\delta(\Sigma E)\mathrm{i}\mathcal{M}(v\cdot
q,q^{2})~{}.\end{split}$ (37)

Next, writing
$\ket{\mathbf{k}+\mathbf{q}}=\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\hat{\mathbf{x}}}\ket{\mathbf{k}}$,
and then using the fact that the matrix element and energy conserving delta
function do not depend on $\mathbf{k}$, we find,

$\begin{split}\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{x},\mathbf{p}}&=(2\pi)\delta(\Sigma
E)\mathrm{i}\mathcal{M}(v\cdot
q,q^{2})\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\hat{\mathbf{x}}}\ket{\mathbf{x}}\\\
&=(2\pi)\delta(\Sigma E)\mathrm{i}\mathcal{M}(v\cdot
q,q^{2})\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{x}}\ket{\mathbf{x}}~{},\end{split}$
(38)

where in the last equality we have acted with the position operator on the
position eigenstate. We may re-write this as

$T\left(\ket{\mathbf{x}}\otimes\ket{\mathbf{p}}\right)=\ket{\mathbf{x}}\otimes(T_{\mathbf{x}}\ket{\mathbf{p}})~{}.$
(39)

Notice the that the induced $T$-matrix is normalized as is appropriate for
potential scattering [25].

What if the probe scatters off of two different atoms? In this case, the
problem at hand is $3\rightarrow 3$ scattering. The leading order Feynman
diagram appears as a tree-level graph,

$\begin{gathered}\fmffile{diagram_{s}oft_{2}}\fmfstraight\fmfgraph*(20,9)\fmfleft{i1,i2,fi,i3}\fmfright{o1,o2,fo,o3}\fmf{double}{i1,b1,f1,o1}\fmf{double}{i2,f2,b2,o2}\fmf{plain}{i3,t1,t2,o3}\fmf{dashes,tension=0,lab=$\mathbf{q}_1$}{b1,t1}\fmf{dashes,tension=0,lab=$\mathbf{q}_2$}{b2,t2}\fmflabel{$\pi$}{i3}\fmflabel{$\mathbf{x}_i$}{i2}\fmflabel{$\mathbf{x}_j$}{i1}\end{gathered}\vspace{6pt}$
(40)

where $i\neq j$. Performing the exact same analysis as above in terms of
position eigenstates and $T$-matrices, one arrives at

$\begin{split}\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{p}}=&(2\pi)\delta(\Sigma
E)\times\\\ &\int[\differential
q_{1}]\mathrm{i}\mathcal{M}(\mathbf{q}_{1},\mathbf{q})\\\ &\hskip
43.36464pt\mathrm{e}^{\mathrm{i}\mathbf{q}_{1}\cdot\mathbf{x}_{1}}\mathrm{e}^{\mathrm{i}(\mathbf{q}-\mathbf{q}_{1})\cdot\mathbf{x}_{2}}\ket{\mathbf{x}_{1},\mathbf{x}_{2}}~{}.\end{split}$
(41)

It follows that
$T\left(\ket{\mathbf{x}_{1},\mathbf{x}_{2}}\otimes\ket{\mathbf{p}}\right)=\ket{\mathbf{x}_{1},\mathbf{x}_{2}}\otimes(T_{(\mathbf{x}_{1},\mathbf{x}_{2})}\ket{\mathbf{p}})$.
Eq. 41 is the same amplitude that would be obtained using potential scattering
Feynman rules for two potentials located at $\mathbf{x}_{1}$ and
$\mathbf{x}_{2}$. i.e., the second order term in the Born series. Note that
the matrix element is normalized with a single energy conserving delta
function, as is appropriate for potential scattering.

For decoherence, in which one typically considers small momentum transfers, it
is most efficient to formulate the problem in the language of potential
scattering. The probe state may be relativistic or non-relativistic. Iteration
generates higher loop order diagrams from potential scattering such that the
induced $S$-matrix corresponds to potential scattering at arbitrary order in
perturbation theory.

We have focused on a non-interacting dilute gas. In the instance where
interactions among particles are important, the Feynman diagram approach here
can be used to incorporate interactions between atoms. Such a program is often
considered in the context of nuclear scattering where one-pion exchange graphs
generate two-body potentials [37, 38]. One could similarly consider an
interacting gas of probe states via the same approach.

##  Appendix C Non-vanishing forward scattering phase

The forward scattering phase does not vanish upon averaging over all angular
directions of the probe’s momentum $\mathbf{p}$. To see this explicitly,
consider the Feynman diagrams in Eqs. 40 and 36 for
$\mathbf{q}=\mathbf{p^{\prime}-p}=0$. Recall that we assume the clouds to be
point-like, i.e., the distance between interferometer arms is large, but the
interparticle separation is small. Any term with $\mathbf{x}_{i}$ and
$\mathbf{x}_{j}$ in different arms of an interferometer can be dropped due to
their rapidly oscillating exponentials. For the remaining terms, with
$\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ in the same arm, we can make the
replacement
$\mathrm{e}^{\mathrm{i}\mathbf{q^{\prime}}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}\rightarrow
1$ in the loop integral. We are then left with the integral,

$\mathcal{I}(\mathbf{p})=\int\frac{\differential^{3}\mathbf{q^{\prime}}}{(2\pi)^{3}}\quantity|\widetilde{V}(\mathbf{q^{\prime}})|^{2}~{}\quantity[G_{\pi}(\mathbf{p}+\mathbf{q^{\prime}})+G_{\pi}^{\dagger}(\mathbf{p}+\mathbf{q^{\prime}})]~{}.$
(42)

Assuming $\widetilde{V}(\mathbf{q^{\prime}})$ is rotationally invariant, there
is no additional reference vector in the problem except for $\mathbf{p}$. This
implies that the integral written above can depend only on $\mathbf{p}^{2}$,
$\mathcal{I}(\mathbf{p})=\mathcal{I}(\mathbf{p}^{2})$. Therefore the average
over initial particle directions does not vanish even for isotropic probe
momentum distributions,

$\int[\differential
p]~{}\rho_{\pi}(\mathbf{p})~{}\mathcal{I}(\mathbf{p}^{2})\neq 0~{}.$ (43)

##  Appendix D Combinatorics

In the limit of coherent scattering, the momenta transfers are sufficiently
small that the cloud in each arm of the experimental appears point-like.
Hence, we may assume that an atom in the left arm is located at position
$\mathbf{x}_{L}$, while an atom in the right arm is located at position
$\mathbf{x}_{R}$. Let us now suppose that our state $\ket{\\{\mathbf{x}\\}}$
has $N_{L}$ atoms in the left arm and $N_{R}$ atoms in the right arm.
Similarly let us take $\ket{\\{\mathbf{x}^{\prime}\\}}$ to have
$N_{L}^{\prime}$ atoms in the left arm and $N_{R}^{\prime}$ atoms in the right
arm.

Since Eqs. 10 and 9 are manifestly invariant under particle relabeling, let us
order the particle labels such that
$\mathbf{x}_{1}\ldots\mathbf{x}_{N_{L}}=\mathbf{x}_{L}$ and
$\mathbf{x}_{N_{L}+1}\ldots\mathbf{x}_{N}=\mathbf{x}_{R}$ (and similarly for
the primed case). It is convenient to align the kets in this form:

$\begin{split}\ket{\\{\mathbf{x}\\}}=&\ket{L,L,\ldots,\ldots,L,L}\overbrace{\ket{R,\ldots,R}}^{N_{R}}\\\
\ket{\\{\mathbf{x}^{\prime}\\}}=&\underbrace{\ket{L,\ldots,L}}_{N_{L}^{\prime}}\ket{R,R,\ldots,\ldots,R,R}~{}.\end{split}$
(44)

We can now ask what the possible outcomes of the terms in the sum above above.
It is clear that every exponential can evaluate to one of three numbers:

  1. 1.

$1$.

  2. 2.

$\mathrm{e}^{+\mathrm{i}\mathbf{q}\cdot\Delta\mathbf{x}}$.

  3. 3.

$\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\Delta\mathbf{x}}$.

We define the difference in path length via
$\Delta\mathbf{x}=\mathbf{x}_{L}-\mathbf{x}_{R}$. Therefore, the decoherence
kernel [i.e., term between squared brackets in Eq. 10] is given by

$\begin{split}&\frac{1}{2}\bigg{(}\sum_{ij}2\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j}^{\prime})}-\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}-\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot(\mathbf{x}^{\prime}_{i}-\mathbf{x}^{\prime}_{j})}\bigg{)}\\\
=&\frac{1}{2}\bigg{(}A_{1}+A_{+}\mathrm{e}^{+\mathrm{i}\mathbf{q}\cdot\Delta\mathbf{x}}+A_{-}\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\Delta\mathbf{x}}\bigg{)}~{}.\end{split}$
(45)

Evaluating the coefficients $A_{1}$, $A_{+}$, and $A_{-}$ is a straightforward
combinatorics problem. We may without loss of generality, focus on the case
where $n\geq 0$,

$\displaystyle\begin{split}A_{1}&=2N_{L}^{\prime}N_{L}+2N_{R}^{\prime}N_{R}\\\
&~{}~{}-(N_{L}^{2}+N_{R}^{2}+N_{L}^{\prime 2}+N_{R}^{\prime
2})~{},\end{split}$ (46)
$\displaystyle\begin{split}A_{+}&=2N_{L}N_{R}^{\prime}-(N_{L}N_{R}+N_{L}^{\prime}N_{R}^{\prime})~{},\end{split}$
(47)
$\displaystyle\begin{split}A_{-}&=2N_{L}^{\prime}N_{R}-(N_{L}N_{R}+N_{L}^{\prime}N_{R}^{\prime})~{}.\end{split}$
(48)

As discussed in the main text, it is convenient to introduce the atom
asymmetry

$n=N_{L}-N_{L}^{\prime}=N_{R}-N_{R}^{\prime}~{},$ (49)

which characterizes how different the state $\ket{\\{\mathbf{x}\\}}$ is from
the state $\ket{\\{\mathbf{x}^{\prime}\\}}$, so that we may usefully express
the previous expressions in terms of $n$,

$\begin{split}A_{1}&=-2n^{2}~{},\\\ A_{+}&=n(n-N)~{},\\\
A_{-}&=n(n+N)~{}.\end{split}$ (50)

It is important to note that all of these corrections vanish as $n\rightarrow
0$. Adding everything together we get the following expression for the
decoherence kernel,

$K_{D}=n^{2}\quantity[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]-\mathrm{i}nN\sin(\mathbf{q}\cdot\Delta\mathbf{x})~{}.$
(51)

The unitary kernel [i.e., term between squared brackets in Eq. 9] contains
only a subset of the terms above. We find for this case,

$\begin{split}&\frac{\mathrm{i}}{2}\bigg{(}\sum_{ij}\mathrm{e}^{\mathrm{i}\mathbf{q^{\prime}}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}-\mathrm{e}^{\mathrm{i}\mathbf{q^{\prime}}\cdot(\mathbf{x}^{\prime}_{i}-\mathbf{x}^{\prime}_{j})}\bigg{)}\\\
=&\frac{\mathrm{i}}{2}\bigg{(}B_{1}+B_{+}\mathrm{e}^{+\mathrm{i}\mathbf{q^{\prime}}\cdot\Delta\mathbf{x}}+B_{-}\mathrm{e}^{-\mathrm{i}\mathbf{q^{\prime}}\cdot\Delta\mathbf{x}}\bigg{)}~{}.\end{split}$
(52)

These constants can be taken from the relevant terms above,

$\begin{split}B_{1}&=N_{L}^{2}+N_{R}^{2}-N_{L}^{\prime 2}-N_{R}^{\prime
2}~{},\\\ B_{+}&=N_{L}N_{R}-N_{L}^{\prime}N_{R}^{\prime}~{},\\\
B_{-}&=N_{L}N_{R}-N_{L}^{\prime}N_{R}^{\prime}~{},\end{split}$ (53)

which can be expressed in terms of $n$ as

$\begin{split}B_{1}&=-2n(n+N-2N_{L})~{},\\\ B_{+}&=n(n+N-2N_{L})~{},\\\
B_{-}&=n(n+N-2N_{L})~{}.\end{split}$ (54)

As for the dissipative part, these corrections vanish as $n\rightarrow 0$.
From these expression we find the following closed-form expression for the
unitary kernel,

$K_{U}=-\mathrm{i}\,n(n+N-2N_{L})\quantity[1-\cos(\mathbf{q^{\prime}}\cdot\Delta\mathbf{x})]~{}.$
(55)

From Eqs. 51 and 55, one can show that the multi-particle density matrix
remains Hermitian after scattering. Since the multi-particle density matrix
prior to scattering, $\rho_{A}$, is Hermitian by definition,

$\rho_{A}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})=\rho_{A}(\\{\mathbf{x}^{\prime}\\},\\{\mathbf{x}\\})^{*}~{}.$
(56)

Therefore, for $\rho_{A}(t)$ to remain Hermitian for $t>0$, we require that

$\lambda_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})=\lambda_{U,D}(\\{\mathbf{x}^{\prime}\\},\\{\mathbf{x}\\})^{*}~{}.$
(57)

For consistency, when mapping $n\rightarrow-n$ one must hold
$N_{L}+N_{L}^{\prime}$ fixed. Under this map we have

$\displaystyle\rho_{A}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})$
$\displaystyle\rightarrow\rho_{A}(\\{\mathbf{x}^{\prime}\\},\\{\mathbf{x}\\})~{},$
(58) $\displaystyle K_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})$
$\displaystyle\rightarrow K_{U,D}(\\{\mathbf{x}^{\prime}\\},\\{\mathbf{x}\\})$
(59) $\displaystyle\hskip
12.0pt=K_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})^{*}~{}.$

Since $\lambda_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})\propto
K_{U,D}(\\{\mathbf{x}\\},\\{\mathbf{x}^{\prime}\\})$ and all other factors in
$\lambda_{U,D}$ are real. It follows that the multi-particle density matrix
after scattering is manifestly Hermitian.

##  Appendix E Reduced density matrices

In the main body of the text we have made use of reduced density matrices. For
an $N$-body system, the $M$-body reduced density matrix is defined as
$\rho_{M}={\rm Tr}_{N-M}[\rho]$. In the case of an $N$-atom system, ${\rm
Tr}_{N-M}$ is the trace over $N-M$ single-atom Hilbert spaces.

Let us now consider the matrix elements of a reduced density matrix. Without
loss of generality, take
$\matrixelement{\mathcal{N}_{L}^{\prime}}{\rho_{\alpha}}{\mathcal{N}_{L}}$,
where the states $\bra{\mathcal{N}_{L}^{\prime}}$ and $\ket{\mathcal{N}_{L}}$
have $\mathcal{N}_{L}^{\prime}$ and $\mathcal{N}_{L}$ particles in the left
arm, and $\alpha-\mathcal{N}_{L}^{\prime}$ and $\alpha-\mathcal{N}_{L}$
particles in the right arm, respectively. The partial trace which defines
$\rho_{\alpha}$ involves a sum over states with $N_{L}=\mathcal{N}_{L}+M_{L}$
and $N_{L}^{\prime}=\mathcal{N}_{L}^{\prime}+M_{L}$ where $M_{L}$ accounts for
the number of particles in the left arm from the states being traced over.
Using the definitions of the kernels [cf. Eqs. 55 and 51], and the fact that
the unitary kernel is linear in $N_{L}$, we find

$\displaystyle\matrixelement{\mathcal{N}_{L}^{\prime}}{\rho_{\alpha}}{\mathcal{N}_{L}}$
$\displaystyle=\frac{1}{2^{N}}\mathrm{e}^{f(n)}\mathrm{e}^{\mathrm{i}2n\mathcal{N}_{L}\tau}\sum_{M_{L}=0}^{N-\alpha}{N-\alpha\choose
M_{L}}\mathrm{e}^{\mathrm{i}2nM_{L}\tau}$
$\displaystyle=\frac{1}{2^{\alpha}}\mathrm{e}^{f(n)}\quantity(\frac{1+\mathrm{e}^{\mathrm{i}2n\tau}}{2})^{N-\alpha}\mathrm{e}^{\mathrm{i}2n\mathcal{N}_{L}\tau}~{},$
(60)

for a function $f(n)$ given by,

$f(n)=-n^{2}s-\mathrm{i}nN\gamma-\mathrm{i}n(N-n)\tau\,,$ (61)

where $s$, $\gamma$ and $\tau$ are defined in Eqs. 16, 17 and 18. Importantly,
Appendices E and 61 provide compact expressions for reduced density matrix
elements.

For sufficiently weak coupling between the probe and the atoms (or
equivalently at early times), the impact of the novel phase shift $\tau$ can
be inferred from Appendix E for a particular observable. For example, consider
the expectation value of the observable $\mathcal{O}_{+}$ [cf. Eqs. 13, 14 and
15]. Since this observable is a one-body observable, it depends on a one-body
reduced density matrix [i.e., $\alpha=1$ in Appendix E]. The expansion for
$\langle\mathcal{O}_{+}\rangle$ is given by

$\begin{split}\langle\mathcal{O}_{+}\rangle&=\frac{N}{2}\bigg{(}\quantity[1+\cos\phi]-\quantity[s\cos\phi+N\gamma\sin\phi]\\\
&\hskip
21.68231pt+\bigg{[}\frac{1}{2}\cos\phi\big{(}s^{2}+N^{2}\gamma^{2}-(N-1)\tau^{2}\big{)}\\\
&\hskip 108.405pt+Ns\gamma\sin\phi\bigg{]}\bigg{)}+\ldots~{},\end{split}$ (62)

where the first bracketed expression is $O(t^{0})$, the second is $O(t^{1})$
and the remaining pieces are $O(t^{2})$. We see that $\tau$ neither imprints a
phase, nor affects decoherence at $O(t^{1})$, but influences observables at
$O(t^{2})$.

##  Appendix F Statistical fluctuations of one-body operators

In this appendix we provide completely general formulae for the statistical
fluctuations in our toy model of a two-mode interferometer. We begin with a
formulae for $\langle\mathcal{O}^{\eta}_{1B}\rangle$ with $\eta$ a positive
integer, and then describe how to construct arbitrary statistical moments.

Using the property that $\mathcal{O}_{i}^{2}=\mathcal{O}_{i}$ we find,

$\mathcal{O}_{1B}^{\eta}=\sum_{\alpha=1}^{\eta}C(\alpha)\sum_{\\{i\\}_{\rm
dist.}}\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}~{}.$
(63)

The notation $\\{i\\}_{\rm dist.}$ refers to the set of distinct indices
satisfying $i_{1}\neq i_{2}\neq...\neq i_{\alpha}$, and the product of
operators should be understood as a tensor product. Since the expectation
value $\langle\mathcal{O}_{i_{1}}\ldots\mathcal{O}_{i_{\alpha}}\rangle$ is
invariant under permutations of the indices, we can replace the sum over the
set $\\{i\\}_{\rm dist.}$ by $N(N-1)(N-2)\ldots(N-(\alpha-1))=N!/(N-\alpha)!$.
We then find

$\langle\mathcal{O}_{1B}^{\eta}\rangle=\sum_{\alpha=1}^{\eta}C(\alpha)\frac{N!}{(N-\alpha)!}\langle\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}\rangle~{}.$
(64)

The coefficients $C(\alpha)$ can be constructively determined using the
Bookkeeper’s Rule, however it is more efficient for $\eta\gtrsim 4$ to
determine them iteratively using the sum rule
$\sum_{\alpha}C(\alpha)N!/(N-\alpha)!=N^{\eta}$; for $3\leq\eta\leq 7$ the
coefficients $C(\alpha)$ are shown in Table 1. Next, when computing the
expectation value we make use of the identity

$\langle\underbrace{\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}}_{\text{all
indices distinct}}\rangle={\rm
Tr}\quantity[\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}\rho_{\alpha}]~{}.$
(65)

This gives

$\langle\mathcal{O}_{1B}^{\eta}\rangle=\sum_{\alpha=1}^{\eta}C(\alpha)\frac{N!}{(N-\alpha)!}{\rm
Tr}\quantity[\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}\rho_{\alpha}]~{}.$
(66)

For $\mathcal{O}_{i}=\ket{A_{i}}\bra{A_{i}}$, where
$\ket{A_{i}}=\frac{1}{\sqrt{2}}\quantity(\ket{L_{i}}+\mathrm{e}^{\mathrm{i}\phi}\ket{R_{i}})$,
and using the binomial distribution, the trace in Eq. 66 takes the form

$\begin{split}&{\rm
Tr}\quantity[\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}\rho_{\alpha}]=\\\
&\hskip
65.04034pt\frac{1}{2^{\alpha}}\sum_{N_{L},N_{L}^{\prime}}{\alpha\choose
N_{L}}{\alpha\choose
N_{L}^{\prime}}\mathrm{e}^{\mathrm{i}n\phi}\matrixelement{N_{L}^{\prime}}{\rho_{\alpha}}{N_{L}}\,,\end{split}$
(67)

where we made use of the short-hand notation for the matrix elements of a
reduced density matrix. It is convenient to re-express this as a sum over
$n=N_{L}-N_{L}^{\prime}$,

$\begin{split}\sum_{N_{L}=0}^{\alpha}\sum_{N_{L}^{\prime}=0}^{\alpha}=\sum_{n=-\alpha}^{\alpha}\sum_{N_{L}=0}^{\alpha}\sum_{N_{L}^{\prime}=0}^{\alpha}\delta_{N_{L}-N_{L}^{\prime},n}~{},\end{split}$
(68)

We may treat $n=0$, $n<0$, and $n>0$ separately. The results at $n<0$ are just
the complex conjugate of $n>0$ and so we can further restrict to $n=0$ and
$n>0$. For $n=0$, we simply set $N_{L}=N_{L}^{\prime}$ and evaluate the sum.
For $n>0$ the sum over $N_{L}$ runs from $N_{L}=n$ to $N_{L}=N$. By making use
of Appendix E one finds

$\displaystyle{\rm
Tr}\quantity[\mathcal{O}_{i_{1}}\mathcal{O}_{i_{2}}\ldots\mathcal{O}_{i_{\alpha}}\rho_{\alpha}]=\frac{1}{2^{\alpha}}\sum_{N_{L}=0}^{\alpha}{\alpha\choose
N_{L}}^{2}$ (69)
$\displaystyle+\frac{1}{2^{\alpha}}\sum_{n=1}^{\alpha}\sum_{N_{L}=n}^{\alpha}{\alpha\choose
N_{L}}{\alpha\choose
N_{L}^{\prime}}\quantity[\mathrm{e}^{\mathrm{i}n\phi}\matrixelement{N_{L}^{\prime}}{\rho_{\alpha}}{N_{L}}+{\rm
c.c.}]~{},$

where $N_{L}^{\prime}=N_{L}-n$.

In general, it is possible to find analogous formulas for one-body observables
of the type $\ket{A_{i}}=a_{L}\ket{L_{i}}+a_{R}\ket{R_{i}}$, where
$a_{L},a_{R}\in\mathbb{C}$ and $|a_{L}|^{2}+|a_{R}|^{2}=1$. In this case,
these formulae simply involve appropriate powers of $a_{L}$ and $a_{R}$ as
would occur when expanding $(a_{L}+a_{R})^{\alpha}$ using the binomial
distribution.

$\eta$ |  $C(1)$ |  $C(2)$ |  $C(3)$ |  $C(4)$ |  $C(5)$ |  $C(6)$ |  $C(7)$
---|---|---|---|---|---|---|---
3 | 1 | 3 | 1 | – | – | – | –
4 | 1 | 7 | 6 | 1 | – | – | –
5 | 1 | 15 | 25 | 10 | 1 | – | –
6 | 1 | 31 | 90 | 65 | 15 | 1 | –
7 | 1 | 63 | 301 | 350 | 140 | 21 | 1
Table 1: Coefficients $C(\alpha)$ necessary to compute
$\langle\mathcal{O}_{\rm 1B}^{\eta}\rangle$ for $\eta\in\\{3,4,5,6,7\\}$.

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# Gauge model of the fifth force

G. A. Sardanashvily  E. G. Timoshenko Translated on 07.07.2024 by
<EMAIL_ADDRESS>from Russian: The Bulletin of Moscow State
University, ISSN 2074-6636, UDC 530.12 Series 3, Physics and astronomy, Vol.
31, No 4, pp. 70-72 (1990).

( Division of Theoretical Physics, Department of Physics, Lomonosov Moscow
State University, Russia, USSR )

###### Abstract

The Lagrangian and equations of the gauge model of the fifth fundamental
interaction are constructed and the corresponding corrections to the Newtonian
potential are obtained.

##  1 Introduction

The hypothesis of the existence of the fifth fundamental interaction is now
widely discussed in connection with certain ambiguous results of laboratory
tests of the Newton’s law of gravitation. The fifth force is broadly viewed as
the set of hypothetical effects that could generate contributions of the
“Yukawa” type to the Newtonian potential. We have proposed a model of such
interaction using the gauge fields of the translational group [1,2]. Its
peculiarity in comparison with other affine gauge theories consists in the
construction of a deformation mapping of a space–time manifold and
consideration of material fields on such a manifold.

Let $AX$ be the principal fibre bundle of affine frames with the structural
affine group $A(4,R)$ over the space–time manifold $X^{4}$. We equip its total
space $P$ with coordinates $\\{x^{\mu},u^{a},S^{\mu}_{a}\\}$. Here $x^{\mu}$
are the coordinates in $X^{4}$, $u^{a}$ are the coordinates of the translation
subgroup $T_{4}$, and $S^{\mu}_{a}$ are the coefficients of the linear
connection, $\\{St_{a}\\}$ with respect to frame $\\{\partial_{\mu}\\}$, where
$\\{t_{a}\\}$ is a fixed basis of $T_{4}$, and $S$ is an element of $GL(4,R)$.
Note that $\\{x^{\mu},u^{\mu}=S^{\mu}_{a}u^{a}\\}$ are the coordinates of the
affine tangent fibre bundle $AT(X)$. Let a generalised affine connection be
given in $AX$. Its connection form $\omega$ and the corresponding horizontal
fields $t^{h}$ on $P$ are of the form

$\omega=(S^{-1})^{a}_{\varepsilon}(dS^{\varepsilon}_{b}+\Gamma^{\varepsilon}_{\mu\alpha}(x)S^{\alpha}_{b}dx^{\mu})I^{b}_{a}+(du^{a}-B^{a}_{\mu}(x)dx^{\mu})T_{a},$
$\tau^{h}=\tau^{\mu}(x)\left(\frac{\partial}{\partial
x^{\mu}}+B^{a}_{\mu}(x)\frac{\partial}{\partial
u^{a}}-\Gamma^{\varepsilon}_{\mu\alpha}(x)S^{\alpha}_{b}\frac{\partial}{\partial
S^{\varepsilon}_{b}}\right),$ (1)

where $I^{a}_{b},\ T_{a}$ are the generators of group $A(4,R)$,
$\Gamma^{\varepsilon}_{\alpha\mu}$ are the coefficients of linear connection,
and $B^{\varepsilon}_{\mu}(x)=S^{\varepsilon}_{a}B^{a}_{\mu}(x)$ are gauge
fields of the translation group. The physical interpretation of the $B$ fields
presents some difficulties because the material fields are sections of linear
rather than of affine fibre bundle. The example of $B$ fields of 3-dimensional
translations in dislocation theory [3] indicates a way out.

In order to give the meaning to the displacement vectors, let us consider the
following mapping $\rho$ of the space $P$ onto the total space $Q$ onto the
total sub–bundle of linear frames $Lx$ over $X^{4}$ at the points
$S^{\varepsilon}_{a}u^{a}=u^{\varepsilon}(x)$:

$(x^{\mu},u^{a},S^{\mu}_{a})\rightarrow(\gamma^{\mu}(x^{\varepsilon};u^{\varepsilon}(x)-S^{\varepsilon}_{a}u^{a};1),0,S^{\mu}_{a})=(x^{\mu},0,S^{\mu}_{a}),$

where $\gamma(x;u;s)$ is a geodesic passing through $x$ in the direction $u$
and defined by the linear connection $\Gamma$, and $u(x)$ is some section of
the fibre bundle $AT(X)$. The tangent $\rho_{*}$ to $\rho$ mapping of a
tangent fibre bundle over $P$ to a tangent fibre bundle over $Q$ transforms
the fields (1) on $P$ into the fields

$\tau^{h}_{Q}=\tau^{\mu}(x)\left(\delta^{\nu}_{\mu}+\mathcal{D}_{\mu}u^{\nu}(x)\right)\left[\frac{\partial}{\partial
x^{\nu}}-\Gamma^{\varepsilon}_{\mu\alpha}(x)S^{\alpha}_{b}\frac{\partial}{\partial
S^{\varepsilon}_{b}}\right]$ (2)

on $Q$, where the covariant derivative of the field and $u(x)$ has the form

$\mathscr{D}_{\mu}u^{\varepsilon}(x)=\partial_{\mu}u^{\varepsilon}(x)+\Gamma^{\varepsilon}_{\mu\alpha}u^{\alpha}(x)-B^{\varepsilon}_{\mu}(x)=\sigma^{\varepsilon}_{\mu}(x).$
(3)

Note that it is usual to take the mapping $\beta:\
(x^{\mu},u^{a},S^{\mu}_{a})\rightarrow(x^{\mu},0,S^{\mu}_{a})$ of the space
$P$ onto $Q$ at the points $S^{\varepsilon}_{a}u^{a}=u^{\varepsilon}(x)$ in
affine gauge models. At these points $\beta=\rho$, but
$\beta_{*}\not=\rho_{*}$. Comparing this to the case $B=0,\ u(x)=0$, the
mappings $\rho,\ \rho_{*}$ can be interpreted as deformations of $X^{4}$
manifold. We should note that by a gauge transformation the translation field
$u(x)$ can always be converted to zero. Therefore, only its covariant
derivatives (3) are physically meaningful.

Let $\varphi$ be some tensor field on $X^{4}$ and $f_{\varphi}$ be the
corresponding tensorial function on $Q$. We shall say that $\varphi$ is
defined on a deformed manifold $X^{4}$ if its differentiation is defined in
the form $(\mathcal{D}_{\mu}\varphi)(\tau)=(df_{\varphi})(\tau^{h}_{Q})$,
where $\tau^{h}_{Q}$ is the horizontal (with respect to $\Gamma$) lift (2) of
the $\tau=\tau^{\mu}\partial_{\mu}$ field. This means that in field theory the
deformation of the manifold can be accounted for by replacing in the external
differentiation operator $dx^{\mu}\mathcal{D}_{\mu}$ of the ordinary covariant
derivatives $\mathcal{D}_{\mu}$ by
$\widetilde{\mathcal{D}_{\mu}}=(\delta^{\alpha}_{\mu}+\sigma^{\alpha}_{\mu})\mathcal{D}_{\alpha}=H^{\alpha}_{\mu}\mathcal{D}_{\alpha}$.

Let us write out, for example, the Lagrangian of a scalar field:

$L_{(\varphi)}=\frac{1}{2}[g^{\mu\nu}H^{\alpha}_{\mu}H^{\beta}_{\nu}\partial_{\alpha}\varphi\partial_{\beta}\varphi-m^{2}\varphi^{2}]$

and the action functional and the equations of motion of the point mass:

$S=-m\int[d_{\alpha\beta}H^{\alpha}_{\mu}H^{\beta}_{\nu}u^{\mu}u^{\nu}]^{1/2}ds,\quad\frac{du^{\mu}}{ds}+\widetilde{\Gamma}^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}=0,$
(4)

where the values of $\widetilde{\Gamma}$ take the form of Christoffel symbols
of the $H^{\alpha}_{\mu}H^{\beta}_{\nu}g_{\alpha\beta}$ “metric” but $ds$ is
defined by the metric $g$. The Lagrangians of the gravitational and
electromagnetic fields $L_{(g)}$ and $L_{(A)}$ are constructed in the usual
way, but from the modified tensors
$H^{\varepsilon}_{\mu}H^{\beta}_{\nu}R^{ab}_{\varepsilon\beta}$ of the
curvature and the strength $H^{\alpha}_{\mu}H^{\beta}_{\nu}F_{\alpha\beta}$.

The Lagrangian $L_{(\sigma)}$ of the gauge field of translations
$B^{\varepsilon}_{\mu}$ itself cannot be constructed by the usual rules of
Yang–Mills theory, since the Lie algebra of the affine group does not admit a
nondegenerate invariant bilinear form. But it can be constructed from the
values of $\sigma^{\nu}_{\mu}$ and
$F^{\alpha}_{\mu\nu}=\mathcal{D}_{[\nu}\sigma^{\alpha}_{\mu]}$. Under the
condition of positivity of the component $T^{00}_{(\sigma)}$ of the metric
tensor energy–momentum of the field $\sigma$, it has the form $(a_{1}>0,\
a_{2}>0,\ \mu>0,\ \lambda<\mu/4,\
\sigma_{\mu\nu}=g_{\mu\alpha}\sigma^{\alpha}_{\nu})$

$L_{(\sigma)}=\frac{1}{2}[a_{1}F^{\mu}_{\nu\mu}F^{\nu\alpha}_{\alpha}+a_{2}F_{\mu\nu\sigma}(F^{\mu\nu\sigma}-2F^{\nu\mu\sigma})-\mu\sigma^{\mu}_{\nu}\sigma^{\nu}_{\mu}+\lambda\sigma^{\mu}_{\mu}\sigma^{\nu}_{\nu}].$

The sources of the field $\sigma$ are: the shortened canonical energy–momentum
tensor of material fields

$\frac{\delta
L_{(m)}}{\delta\sigma^{\mu\nu}}=(H^{-1})_{\nu\beta}\mathscr{D}_{\mu}\varphi\frac{\partial
L_{(m)}}{\partial\mathscr{D}_{\beta}\varphi}=(H^{-1})_{\nu\beta}(t^{\beta}_{(m)\mu}+\delta^{\beta}_{\mu}L_{(m)})$

the shortened metric energy-momentum tensor of the electromagnetic field $A$
and the curvature tensor
$\kappa^{-1}H^{\nu}_{\mu}R^{\mu\gamma}_{\gamma\varepsilon}$ of the
gravitational field. But instead of the latter, by using the Einstein’s
equations, we can substitute its expression through the metric energy–momentum
tensors of matter and the field $\sigma$ itself. Then the equation for
$\sigma$ takes the form of

$\frac{\delta L_{(\sigma)}}{\delta\sigma^{\mu\nu}}=\left[-\frac{\delta
L_{(m)}}{\delta\sigma^{\mu\nu}}-(H^{-1})_{\nu\beta}\left(T^{\beta}_{(m)\mu}-\frac{1}{2}\delta^{\beta}_{\mu}T_{(m)}\right)\right]-$
$-(H^{-1})_{\nu\mu}L_{(A)}+(H^{-1})_{\nu\beta}\left(T^{\beta}_{(\sigma)\mu}-\frac{1}{2}\delta^{\beta}_{\mu}T_{(\sigma)}\right).$
(5)

Let us restrict ourselves to the case of a weak field $\sigma$, neglecting in
the left part of Eq. (5) the interaction of $\sigma$ with the gravitational
field and torsion, and in the right–hand side by the members of $\sigma$. We
obtain

$\frac{\delta
L_{(\sigma)}}{\delta\sigma^{\mu\nu}}=a_{1}(\eta_{\mu\nu}\partial^{\varepsilon}F_{\alpha\varepsilon}^{\alpha}-\partial_{\mu}F_{\alpha\nu}^{\alpha})+2a_{2}\partial^{\varepsilon}(F_{\mu\nu\varepsilon}+F_{\varepsilon\mu\nu}-F_{\nu\mu\varepsilon})-\mu\sigma_{\mu\nu}+\lambda\eta_{\mu\nu}\sigma_{\alpha}^{\alpha}.$

Let us consider equation (5) in the void. Given the condition

$\partial^{\nu}\frac{\delta
L_{(\sigma)}}{\delta\sigma^{\mu\nu}}=-\mu\partial^{\nu}\sigma_{\mu\nu}+\lambda\partial_{\mu}\sigma=0$
(6)

it splits into equations for the antisymmetric
$\omega_{\mu\nu}=(1/2)\sigma_{[\mu\nu]}$ and the symmetric
$\varepsilon_{\mu\nu}=(1/2)\sigma_{(\mu\nu)}$ $(e=\sigma^{\alpha}_{\alpha})$
parts of the field $\sigma$:

$4a_{2}\partial^{\varepsilon}(\omega_{\mu\varepsilon,\nu}+\omega_{\nu\mu,\varepsilon}-\omega_{\nu\varepsilon,\mu})+2a_{1}\omega_{\alpha[\nu,\mu]\alpha}-\mu\omega_{\mu\nu}=0,$
(7)
$a_{1}(\lambda/\mu-1)[\eta_{\mu\nu}\,\Box\,e-e_{,\mu\nu}]+2a_{1}\omega_{\alpha(\nu,\mu)\alpha}-\mu
e_{\mu\nu}+\lambda\eta_{\mu\nu}e=0.$ (8)

As a solution to equation (7) we choose $\omega=0$ and as for equation (8), by
using convolution by indices $\mu,\nu$, we write it in the form of

$\Box\,e+m^{2}e=0,\quad m^{2}=\frac{\mu(\mu-4\lambda)}{3a_{1}(\mu-\lambda)},$
$e_{\mu\nu}=\frac{\mu-\lambda}{3\mu}\left(\eta_{\mu\nu}e-m^{-2}\left(\frac{\mu-4\lambda}{\mu-\lambda}\right)e_{,\mu\nu}\right).$

It admits plane–wave solutions.

In the presence of sources, expression (6) is not equal to zero and, in
general, is not a gradient, so $\mu\not=0$. For example, in the case of a
point mass $M$, the source in the right-hand side of equation (5) is
$-(1/2)\eta_{\mu\nu}M\delta(r)$ and its stationary centrally symmetric
solution has the form

$e_{rr}=-(\mu-\lambda)^{-1}\left(3\lambda
e_{00}+\frac{1}{2}M\delta(r)\right),\quad e_{\theta\theta}=-e_{00}r^{2},\quad
e_{\varphi\varphi}=-e_{00}r^{2}\sin^{2}\theta,$
$\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}\frac{\partial}{\partial
r}e_{00}-m^{2}e_{00}=-\frac{1}{6}\frac{\mu}{a_{1}(\mu-\lambda)}M\delta(r),\quad
e_{00}=-\frac{\mu M}{6a_{1}(\mu-\lambda)r}e^{-mr}.$

Substituting this solution into the equation of motion of the test particle
(4), we obtain the correction to the Newtonian gravitational potential:

$\varphi^{\prime}=\varphi+e_{00}=-\frac{M}{8\pi
r}\left(\varkappa-\frac{\mu}{3a_{1}(\mu-\lambda)}e^{-mr}\right).$ (9)

Note that the fifth interaction, which gives a contribution to the
gravitational effects, must be as universal as gravitation. For this, in
particular, its material source can be mass or other components of the
energy–momentum tensor of matter and it must be described by a massive field,
albeit with an unusually small mass. The above gauge model satisfies these
conditions. Thus, the mass $m$ is expressed through the constants $\mu$ and
$\lambda$ in the Lagrangian $L_{(\sigma)}$, which have the sense of the
“elasticity” coefficients of the space–time manifold and can be considered
small enough, but $\mu\not=0,\ \lambda<\mu/4$.

## References

  * [1] G. Sardanashvily, M. Gogberashvily // Mod. Phys. Lett. 1987. A8. P. 609.
  * [2] G.A. Sardanashvily, M.Ya. Gogberashvily // Russian Physics Journal [Izvesiya Vuzov. Fizika] 1988\. No 3. P. 71.
  * [3] A. Kadic, D. G. B. Edelen // A Gauge Theory of Dislocations and Disclinations. 1983. Springer-Verlag, Berlin. ISBN 978-3-540-11977-7.

Manuscript (Brief Communications, Theoretical and Mathematical Physics)
received in the editorial office on: 29.09.89.