# Bremsstrahlung of Light through Spontaneous Emission of Gravitational Waves

Charles H.-T. Wang  Department of Physics, University of Aberdeen, King’s
College, Aberdeen AB24 3UE, United Kingdom  Melania Mieczkowska  Department of
Physics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United
Kingdom

(11 May 2021)

###### Abstract

Zero-point fluctuations are a universal consequence of quantum theory. Vacuum
fluctuations of electromagnetic field have provided crucial evidence and
guidance for QED as a successful quantum field theory with a defining gauge
symmetry through the Lamb shift, Casimir effect, and spontaneous emission. In
an accelerated frame, the thermalisation of the zero-point electromagnetic
field gives rise to the Unruh effect linked to the Hawking effect of a black
hole via the equivalence principle. This principle is the basis of general
covariance, the symmetry of general relativity as the classical theory of
gravity. If quantum gravity exists, the quantum vacuum fluctuations of the
gravitational field should also lead to the quantum decoherence and
dissertation of general forms of energy and matter. Here we present a novel
theoretical effect involving the spontaneous emission of soft gravitons by
photons as they bend around a heavy mass and discuss its observational
prospects. Our analytic and numerical investigations suggest that the
gravitational bending of starlight predicted by classical general relativity
should also be accompanied by the emission of gravitational waves. This in
turn redshifts the light causing a loss of its energy somewhat analogous to
the bremsstrahlung of electrons by a heavier charged particle. It is suggested
that this new effect may be important for a combined astronomical source of
intense gravity and high-frequency radiation such as X-ray binaries and that
the proposed LISA mission may be potentially sensitive to the resulting sub-Hz
stochastic gravitational waves.

Published in the Special Issue of Symmetry: New Frontiers in Quantum Gravity

DOI: 10.3390/sym13050852

Keywords: quantum gravity, quantum gravity phenomenology, gauge symmetry,
quantum vacuum, spacetime fluctuations, gravitational decoherence,
gravitational bremsstrahlung, gravitational waves, gravitational astronomy,
X-ray binary.

##  1 Introduction

The quantum vacuum is a remarkable consequence of the quantum field theory
(QFT). To be sure, the quantum electrodynamics (QED) as the first successful
QFT has received crucial guidance and support through its quantum vacuum
effects including the Lamb shift, Casimir effect, and spontaneous emission.

Although the physical reality of the quantum vacuum seems to contradict the
void classical vacuum, it in fact forges essential links between classical and
quantum dynamics. The general agreement between the classical emission rate
and quantum spontaneous emission rate of electromagnetic (EM) dipole
radiations have been well-known at atomic scales (see e.g., [1]). Such an
agreement is also clear in the classical cyclotron radiation and the quantum
spontaneous emission of the Landau levels [2], in the context of detecting
Unruh radiation as a quantum vacuum effect in non-inertial frames [3].

At present, a fully quantised theory of gravity is still to be reached (for
some recent developments, see e.g., [4, 5, 6] and references therein).
Nevertheless, the effective QFT for linearised general relativity is expected
to yield satisfactory physical descriptions at energies sufficiently lower
than the Planck scale [7, 8, 9]. Indeed, the spontaneous emission rate of
gravitons for a nonrelativistic bound system due to the zero-point
fluctuations of spacetime in linearised quantum gravity has been recently
shown [10, 11] to agree with the quadrupole formula of gravitational wave
radiation in general relativity [12]. The preservation of the local
translational symmetry of linearised gravity is crucial in the theoretical
steps of establishing this agreement through the gauge invariant Dirac
quantisation technique [8].

Based on this development, the next challenge would be the spontaneous
emission of gravitons from a relativistic and unbound system, which we will
address in this paper. The possible gravitational radiation by photons has
long been a subject of interest and has been considered by many researchers
with various approaches [13, 14, 15, 16, 17, 18, 19, 20, 21]. The obtained
size of the effect has generally been quite small.

We therefore seek an amplified effect in the astronomical context involving
the deflection of starlight by a celestial body or distribution of mass. We
show that soft gravitons are spontaneously emitted resulting in scattering
modes of incident photons to decay into lower energy scattering modes in the
fashion of the bremsstrahlung of electrons by ions [22, 23, 24, 25]. Our
preliminary estimates of such effects suggest they may be important for high
frequency photons deflected by a compact heavy mass.

Under weak gravity, the polarisations of light subject to gravitational
bending are expected to be negligible. Therefore, as a first approximation,
the effect of spin of photon is neglected similar to neglecting spins in
standard descriptions of the bremsstrahlung of electrons.

Throughout this paper, we adopt units in which the speed of light $c$ equals
unity, unless otherwise stated. We also write $\log_{10}=\log$ and use Greek
indices $\mu,\nu\ldots=0,1,2,3$ and Latin indices $i,j\ldots=1,2,3$ for
spacetime $(t,x,y,z)$ and spatial $(x,y,z)$ coordinates, respectively.

##  2 Light Modelled as Massless Scalar Field in a Weak Central Gravitational
Field

As alluded to in the introduction section, in what follows, we shall model
photons as massless scalar particles with a linearised metric

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta_{\mu\nu}+h_{\mu\nu}$ (1)

where $\eta_{\mu\nu}=$ diag$(-1,+1,+1,+1)$ is the Minkowski metric and
$h_{\mu\nu}$ is the metric perturbation arising from a spherical gravitational
source with mass $M_{\star}$ so that

$\displaystyle h_{00}$ $\displaystyle=$ $\displaystyle
h_{11}=h_{22}=h_{33}=-2\Phi$ (2)

in terms of the Newtonian potential [12]

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-\frac{GM_{\star}}{r}$ (3)

with $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and the gravitational constant $G$. Fig. 1
illustrates the schematic physical and geometrical configurations under
consideration.

Figure 1: An illustration of the key physical and geometrical features of an
astronomical configuration for the gravitational bremsstrahlung of light
involving an X-ray binary and a possible detection concept with LISA. Here the
mass of the compact object is assumed to be dominant for simplicity.

Treating the above source as a gravitational lens, its effective refractive
index is given approximately by $n=1-2\Phi$ [26]. This gives rise to the
approximate dispersion relation

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle(1+2\Phi)p$ (4)

for a real massless scalar field $\phi$ having a frequency $\omega$ and wave
vector $\boldsymbol{p}$ with wave number $p=|\boldsymbol{p}|$. We will
continue to denote wave vectors associated with the scalar field by
$\boldsymbol{p}$ and $\boldsymbol{q}$ and wave vectors associated with the
gravitons by ${\boldsymbol{k}}$, unless otherwise stated.

To capture the salient physical effects carried by the light frequency and to
simplify our technical derivations, we consider the wave number of the scalar
to peak around some fixed value $p_{\circ}$. Then, it follows from Eq. (4) to
leading contributions, that

$\displaystyle\omega^{2}$ $\displaystyle=$ $\displaystyle p^{2}+v(r)$ (5)

where

$\displaystyle v(r)$ $\displaystyle=$
$\displaystyle-4p_{\circ}^{2}\frac{GM_{\star}}{r}.$ (6)

Using Eq. (6), the Lagrangian density of the scalar field

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\sqrt{-g}\,g_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$

reduces to

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\eta^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}v(r)\phi^{2}$
(7)

with $v(r)$ as the effective external scalar potential [27, 28].

The stress-energy tensor of the scalar field is given by

$\displaystyle T^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\nu\alpha}-\eta^{\mu\nu}\eta^{\alpha\beta})\phi_{,\alpha}\phi_{,\beta}-\frac{1}{2}\eta^{\mu\nu}v\phi^{2}.$
(8)

The field equation follows as

$\displaystyle\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi-v\phi$
$\displaystyle=$ $\displaystyle 0.$ (9)

To solve this field equation, one naturally invokes the separable ansatz

$\displaystyle\phi(\boldsymbol{r},t)=\psi(\boldsymbol{r})e^{-i\omega t}$ (10)

so that the general solutions are the real parts of the linear combinations of
Eq. (10). Substituting the above ansatz into Eq. (9), we see that this field
equation is equivalent to

$\displaystyle-\nabla^{2}\psi+v\psi=\omega^{2}\psi$ (11)

taking the form of a time-independent Schrödinger equation.

It follows that the solutions to the field equation (9) representing the
deflection of light with an incident wave vector $\boldsymbol{p}$ and
frequency $\omega=p$ can be obtained from the solutions of the Schrödinger
equation (11) describing a scattering problem involving a Coulomb-type, i.e.
$1/r$, central potential as the “scattering wave functions” of the form

$\displaystyle\psi_{\boldsymbol{p}}(\boldsymbol{r})$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}4\pi\,i^{l}w_{l}(\eta,\rho)Y_{l}^{m}(\hat{\boldsymbol{r}})Y_{l}^{m*}(\hat{\boldsymbol{p}})$
(12)

where $\hat{\boldsymbol{p}}=\boldsymbol{p}/p$, $\boldsymbol{r}=(x,y,z)$,
$Y_{l}^{m}(\boldsymbol{r})$ are spherical harmonics, and $w_{l}(\eta,\rho)$
are the Coulomb wave functions satisfy the wave equation (see e.g. Ref. [29])

$\displaystyle\Big{\\{}\frac{\text{d}^{2}}{\text{d}\rho^{2}}+\Big{[}1-\frac{2\eta}{\rho}-\frac{l(l+1)}{\rho^{2}}\Big{]}\Big{\\}}w_{l}(\eta,\rho)$
$\displaystyle=$ $\displaystyle 0$ (13)

using the dimensionless variable $\rho=p\,r$ and dimensionless parameter
$\eta=-{\nu}/{p}$ with

$\displaystyle\nu$ $\displaystyle=$ $\displaystyle
2\,GM_{\star}p_{\circ}^{2}.$ (14)

By virtue of the orthogonality of $w_{l}(\eta,\rho)$, we can choose the
normalisation of $w_{l}(\eta,\rho)$ so that
$\psi_{\boldsymbol{p}}(\boldsymbol{r})$ satisfy the following orthonormality

$\displaystyle\int\text{d}^{3}r\,\psi_{\boldsymbol{p}^{\prime}}^{*}(\boldsymbol{r})\psi_{\boldsymbol{p}}(\boldsymbol{r})$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{p}^{\prime}),$ (15)
$\displaystyle\int{\text{d}^{3}p}\,\psi_{\boldsymbol{p}}^{*}(\boldsymbol{r}^{\prime})\psi_{\boldsymbol{p}}(\boldsymbol{r})$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}).$ (16)

It is useful to introduce the “momentum representation” scattering wave
functions [30] $\psi_{\boldsymbol{q}}(\boldsymbol{p})$ of the above “position
representation” scattering wave functions
$\psi_{\boldsymbol{q}}(\boldsymbol{r})$ given by

$\displaystyle\psi_{\boldsymbol{q}}(\boldsymbol{p})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\int\text{d}^{3}r\,e^{-i\boldsymbol{p}\cdot\boldsymbol{r}}\psi_{\boldsymbol{q}}(\boldsymbol{r}),$
(17) $\displaystyle\psi_{\boldsymbol{q}}(\boldsymbol{r})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\int\text{d}^{3}p\,e^{i\boldsymbol{p}\cdot\boldsymbol{r}}\psi_{\boldsymbol{q}}(\boldsymbol{p}).$
(18)

The orthogonality of $\psi_{\boldsymbol{q}}(\boldsymbol{p})$ follows
immediately from Eqs. (15), (16), (18) and (17) to be

$\displaystyle\int\text{d}^{3}q\,\psi_{\boldsymbol{q}}^{*}(\boldsymbol{p}^{\prime})\psi_{\boldsymbol{q}}(\boldsymbol{p})$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{p}^{\prime}),$ (19)
$\displaystyle\int\text{d}^{3}q\,\psi_{\boldsymbol{p}^{\prime}}^{*}(\boldsymbol{q})\psi_{\boldsymbol{p}}(\boldsymbol{q})$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{p}^{\prime}).$ (20)

For a weak interaction with the central potential where $|\eta|\ll 1$, the
first order Born approximation yields

$\displaystyle\psi_{\boldsymbol{p}}(\boldsymbol{q})$ $\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{q})+\frac{\nu}{\pi^{2}(p^{2}-q^{2})|\boldsymbol{p}-\boldsymbol{q}|^{2}}$
(21)

Note that the first term of Eq. (21) corresponds to the first term of Eq. (22)
which represents the incident (asymptotically) free particle. This term does
not contribute to Eq. (31) under Markov approximation of the gravitational
master equation as discussed in Ref. [9].

The corresponding asymptotic scattering wave function of the position
$\boldsymbol{r}$ is given by

$\displaystyle\psi_{\boldsymbol{p}}(\boldsymbol{r})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\Big{[}\,e^{i\boldsymbol{p}\cdot\boldsymbol{r}}+f(\theta)\frac{e^{ipr}}{r}\,\Big{]}$
(22)

with the scattering amplitude

$\displaystyle f(\theta)=\frac{\nu}{2p^{2}\sin^{2}(\theta/2)}$ (23)

where $\theta=\measuredangle(\boldsymbol{r},\boldsymbol{p})$ is the scattering
angle.

Just as with the Coulomb potential, so does the infinitely long range of the
Newtonian potential imply a divergent total scattering cross-section. However,
to account for the realistic limited dominance of this potential due to other
influences beyond a range distance $R_{\text{rng}}=1/\epsilon$ which can be
conveniently incorporated by modifying the Newtonian potential with an
additional exponential-decay factor of $e^{-\epsilon r}$ as a long-range
regularisation. On the other hand, the finite extension with a radius
$R_{\star}=1/\delta$ of the gravity source means the need for a compensating
short-range potential within this radius.

These considerations lead to the following phenomenological Yukawa
regularisation

$\displaystyle\frac{1}{r}\to\frac{e^{-\epsilon r}}{r}-\frac{e^{-\delta r}}{r}$
(24)

with $\epsilon$ and $\delta$ as long and short range regularisation parameters
respectively.

Accordingly, we find the regularised scattering wave function to be

$\displaystyle\psi_{\boldsymbol{q}}(\boldsymbol{p})$ $\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{q})+\frac{\nu}{\pi^{2}[p^{2}-(q+i\epsilon)^{2}][|\boldsymbol{p}-\boldsymbol{q}|^{2}+\epsilon^{2}]}$
(25)
$\displaystyle-\frac{\nu}{\pi^{2}[p^{2}-(q+i\delta)^{2}][|\boldsymbol{p}-\boldsymbol{q}|^{2}+\delta^{2}]}$

with $p_{\circ}\gg\delta>\epsilon$.

##  3 Quantisation of the Scalar Field in the Regularised Potential

Using scattering wave function $\psi_{\boldsymbol{p}}(\boldsymbol{r})$ derived
in the preceding section, we can now perform the so-called second quantisation
of the scalar field $\phi$ into a quantum field operator in the Heisenberg
picture as follows

$\displaystyle\phi(\boldsymbol{r},t)$ $\displaystyle=$
$\displaystyle\int\text{d}^{3}p\,\sqrt{\frac{\hbar}{2\omega_{p}}}\;\Big{[}a_{\boldsymbol{p}}\psi_{\boldsymbol{p}}(\boldsymbol{r})e^{-i\omega_{\boldsymbol{p}}t}+a_{\boldsymbol{p}}^{\dagger}\psi_{\boldsymbol{p}}^{*}(\boldsymbol{r})e^{i\omega_{\boldsymbol{p}}t}\Big{]}$
(26)

where $\hbar$ is the reduced Planck constant and the creation and annihilation
operators $a_{\boldsymbol{p}}$ and $a_{\boldsymbol{p}}^{\dagger}$ satisfy the
standard nontrivial canonical commutation relation

$\displaystyle\big{[}\,a_{\boldsymbol{p}},a_{\boldsymbol{p}^{\prime}}^{\dagger}\big{]}$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{p}^{\prime}).$ (27)

The associated field momentum is given by $\pi=\partial\phi/\partial
t=\dot{\phi}$, which satisfies the equal time field commutation relation

$\displaystyle[\phi(\boldsymbol{r},t),\pi(\boldsymbol{r}^{\prime},t)]$
$\displaystyle=$ $\displaystyle
i\hbar\,\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime})$

following from Eqs. (16) and (27).

Substituting Eq. (18) into Eq. (26), we can write $\phi$ in terms of the
momentum representations of the scattering wave functions as follows

$\displaystyle\phi(\boldsymbol{r},t)$ $\displaystyle=$
$\displaystyle\int\text{d}^{3}q\,\text{d}^{3}p\,\sqrt{\frac{\hbar}{2(2\pi)^{3}{q}}}\;\Big{[}a_{\boldsymbol{q}}\psi_{\boldsymbol{q}}(\boldsymbol{p})e^{i\boldsymbol{p}\cdot\boldsymbol{r}}e^{-i{q}t}+a_{\boldsymbol{q}}^{\dagger}\psi_{\boldsymbol{q}}^{*}(\boldsymbol{p})e^{-i\boldsymbol{p}\cdot\boldsymbol{r}}e^{i{q}t}\Big{]}.$
(28)

The coupling of $\phi$ to the metric fluctuations due to low energy quantum
gravity in addition to the metric perturbation Eq. (2) due to the lensing mass
$M_{\star}$ is through the transverse-traceless (TT) part of its stress-energy
tensor Eq. (8) to be $\tau_{ij}:=T_{ij}^{\text{TT}}$ [8] given by

$\displaystyle\tau_{ij}(\boldsymbol{r},t)$ $\displaystyle=$ $\displaystyle
P_{ijkl}\phi_{,k}(\boldsymbol{r},t)\phi_{,l}(\boldsymbol{r},t),$ (29)

with the Fourier transform

$\displaystyle\tau_{ij}({\boldsymbol{k}},t)$ $\displaystyle=$
$\displaystyle\int\text{d}^{3}r\,\tau_{ij}(\boldsymbol{r},t)\,e^{-i{\boldsymbol{k}}\cdot\boldsymbol{r}}$
(30) $\displaystyle=$
$\displaystyle\int\text{d}^{3}r\,e^{-i{\boldsymbol{k}}\cdot\boldsymbol{r}}P_{ijkl}({\boldsymbol{k}})\phi_{,k}(\boldsymbol{r},t)\phi_{,l}(\boldsymbol{r},t)$

where $P_{ijkl}$ is the TT projection operator [8, 31].

Using Eq. (28) and applying normal orders and neglecting
$a_{\boldsymbol{p}}a_{\boldsymbol{q}}$ and
$a_{\boldsymbol{p}}^{\dagger}a_{\boldsymbol{q}}^{\dagger}$ terms through the
rotating wave approximation [32], we see that Eq. (30) becomes

$\displaystyle\tau_{ij}({\boldsymbol{k}},t)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\,P_{ijkl}({\boldsymbol{k}})\int\text{d}^{3}p\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\frac{p_{k}p_{l}}{\sqrt{{p^{\prime}}{p^{\prime\prime}}}}$
(31)
$\displaystyle\Big{[}a_{\boldsymbol{p}^{\prime\prime}}^{\dagger}a_{\boldsymbol{p}^{\prime}}\psi_{\boldsymbol{p}^{\prime}}(\boldsymbol{p})\psi_{\boldsymbol{p}^{\prime\prime}}^{*}(\boldsymbol{p}-{\boldsymbol{k}})e^{-i({p^{\prime}}-{p^{\prime\prime}})t}+a_{\boldsymbol{p}^{\prime}}^{\dagger}a_{\boldsymbol{p}^{\prime\prime}}\psi_{\boldsymbol{p}^{\prime}}^{*}(\boldsymbol{p})\psi_{\boldsymbol{p}^{\prime\prime}}(\boldsymbol{p}+{\boldsymbol{k}})e^{i({p^{\prime}}-{p^{\prime\prime}})t}\Big{]}.$

From Eq. (31) we see that

$\displaystyle\tau_{ij}(-{\boldsymbol{k}},t)$ $\displaystyle=$
$\displaystyle\tau_{ij}^{\dagger}({\boldsymbol{k}},t)$ (32)

as a useful property for later derivations.

##  4 Coupling to the Gravitational Quantum Vacuum

To induce the spontaneous emission of the photon lensed by the regularised
Newtonian potential, we now include an additional TT gravitational wave-like
metric perturbation $h^{\text{TT}}_{\mu\nu}$ into Eq. (1) which carries
spacetime fluctuations at zero temperature [8]. The photon is assumed to
travel a sufficiently long distance and time past the gravitational lens (see
justifications below) so that we can neglect any memory effects in its
statistical interactions with spacetime fluctuations. Additionally, we
consider the energy scale to be low enough for the self interaction of the
photon to be negligible. This leads to the Markov quantum master equation

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\hbar}\int\\!\frac{\text{d}^{3}k}{2(2\pi)^{3}k}\int_{0}^{\infty}\text{d}s\,e^{-iks}[\tau^{\dagger}_{ij}({\boldsymbol{k}},t),\,\tau_{ij}({\boldsymbol{k}},t-s)\rho(t)]+\text{H.c.}$
(33)

in the interaction picture, where $\kappa=8\pi G$,
$\tau_{ij}({\boldsymbol{k}},t)$ is given by Eq. (31), and H.c. denotes the
Hermitian conjugate of a previous term, for the reduced density operator, i.e.
density matrix, $\rho(t)$ of the photon by averaging, i.e. tracing, out the
degrees of freedom in the noisy gravitational environment [8, 10].

It is convenient to express Eq. (31) in the form

$\displaystyle\tau_{ij}({\boldsymbol{k}},t)=\int\text{d}^{3}p^{\prime}\text{d}^{3}p^{\prime\prime}\,\Big{[}\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})e^{-i({p^{\prime}}-{p^{\prime\prime}})t}+\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})e^{i({p^{\prime}}-{p^{\prime\prime}})t}\Big{]}$
(34)

in terms of

$\displaystyle\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})=\frac{\hbar}{2}\,P_{ijkl}({\boldsymbol{k}})\int\text{d}^{3}p\,\frac{p_{k}p_{l}}{\sqrt{{p^{\prime}}{p^{\prime\prime}}}}\,a_{\boldsymbol{p}^{\prime\prime}}^{\dagger}a_{\boldsymbol{p}^{\prime}}\psi_{\boldsymbol{p}^{\prime}}(\boldsymbol{p})\psi_{\boldsymbol{p}^{\prime\prime}}^{*}(\boldsymbol{p}-{\boldsymbol{k}})$
(35)

From Eqs. (34) and (32) we also have

$\displaystyle\tau_{ij}^{\dagger}({\boldsymbol{k}},t)=\int\text{d}^{3}q^{\prime}\text{d}^{3}q^{\prime\prime}\,\Big{[}\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime})e^{-i({q^{\prime}}-{q^{\prime\prime}})t}+\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime})e^{i({q^{\prime}}-{q^{\prime\prime}})t}\Big{]}$
(36)

Substituting Eq. (34) into Eq. (33) we have

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\hbar}\int\\!\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}}{2(2\pi)^{3}k}\int_{0}^{\infty}\text{d}s\,e^{-i(k-{p^{\prime}}+{p^{\prime\prime}})s}e^{-i({p^{\prime}}-{p^{\prime\prime}})t}\big{\\{}[\tau^{\dagger}_{ij}({\boldsymbol{k}},t),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}$
(37)
$\displaystyle-\frac{\kappa}{\hbar}\int\\!\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}}{2(2\pi)^{3}k}\int_{0}^{\infty}\text{d}s\,e^{-i(k+{p^{\prime}}-{p^{\prime\prime}})s}e^{i({p^{\prime}}-{p^{\prime\prime}})t}\big{\\{}[\tau^{\dagger}_{ij}({\boldsymbol{k}},t),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}.$

We then apply the Sokhotski-Plemelj theorem

$\int_{0}^{\infty}\text{d}s\,e^{-i\epsilon
s}=\pi\delta(\epsilon)-i\mathbf{P}\frac{1}{\epsilon}$ (38)

where $\mathbf{P}$ is the Cauchy principal value. Since this Cauchy principal
value contributes to a renormalised energy which can be absorbed in physical
energies [32], we can neglect it.

The remaining part of Eq. (37) on account of Eq. (36) is

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k-({p^{\prime}}-{p^{\prime\prime}}))e^{-i[k+({q^{\prime}}-{q^{\prime\prime}})]t}$
(39)
$\displaystyle\;\;\times\;\big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k-({p^{\prime}}-{p^{\prime\prime}}))e^{-i[k-({q^{\prime}}-{q^{\prime\prime}})]t}$
$\displaystyle\;\;\times\;\big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k+({p^{\prime}}-{p^{\prime\prime}}))e^{-i[k+({q^{\prime}}-{q^{\prime\prime}})]t}$
$\displaystyle\;\;\times\;\big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k+({p^{\prime}}-{p^{\prime\prime}}))e^{-i[k-({q^{\prime}}-{q^{\prime\prime}})]t}$
$\displaystyle\;\;\times\;\big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho(t)]+\text{H.c.}\big{\\}}.$

Due to the weakness of interactions, the density matrix will evolve only
slightly over time so that

$\displaystyle\rho(t)=\rho_{0}+\Delta\rho(t)$ (40)

where $\rho_{0}=\rho(t=0)$ is the initial density matrix and the components of
$\Delta\rho(t)$ are small compared to the components of $\rho_{0}$.

Then Eq. (39) yields

$\displaystyle\Delta\rho(t)$ $\displaystyle=$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k-({p^{\prime}}-{p^{\prime\prime}}))\int_{0}^{t}\text{d}s\,e^{-i[k+({q^{\prime}}-{q^{\prime\prime}})]s}$
(41)
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k-({p^{\prime}}-{p^{\prime\prime}}))\int_{0}^{t}\text{d}s\,e^{-i[k-({q^{\prime}}-{q^{\prime\prime}})]s}$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k+({p^{\prime}}-{p^{\prime\prime}}))\int_{0}^{t}\text{d}s\,e^{-i[k+({q^{\prime}}-{q^{\prime\prime}})]s}$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\delta(k+({p^{\prime}}-{p^{\prime\prime}}))\int_{0}^{t}\text{d}s\,e^{-i[k-({q^{\prime}}-{q^{\prime\prime}})]s}$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}.$

For $t\to\infty$, physically corresponding to $t$ greater than the effective
interaction time of the system, we can again apply the Sokhotski-Plemelj
theorem Eq. (38) to Eq. (41) and neglect the Cauchy principal value terms to
obtain

$\displaystyle\Delta\rho$ $\displaystyle:=$
$\displaystyle\Delta\rho(t\to\infty)$ (42) $\displaystyle=$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\,\delta(k+({q^{\prime}}-{q^{\prime\prime}}))\,\delta(k-({p^{\prime}}-{p^{\prime\prime}}))$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\,\delta(k-({q^{\prime}}-{q^{\prime\prime}}))\,\delta(k-({p^{\prime}}-{p^{\prime\prime}}))$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\,\delta(k+({q^{\prime}}-{q^{\prime\prime}}))\,\delta(k+({p^{\prime}}-{p^{\prime\prime}}))$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}(-{\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}$
$\displaystyle-\frac{\pi\kappa}{\hbar}\int\frac{\text{d}^{3}k\,\text{d}^{3}p^{\prime}\,\text{d}^{3}p^{\prime\prime}\,\text{d}^{3}q^{\prime}\,\text{d}^{3}q^{\prime\prime}}{2(2\pi)^{3}k}\,\delta(k-({q^{\prime}}-{q^{\prime\prime}}))\,\delta(k+({p^{\prime}}-{p^{\prime\prime}}))$
$\displaystyle\;\;\times\;\Big{\\{}[\tau_{ij}^{\dagger}({\boldsymbol{k}},\boldsymbol{q}^{\prime},\boldsymbol{q}^{\prime\prime}),\,\tau_{ij}^{\dagger}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\rho_{0}]+\text{H.c.}\Big{\\}}.$

More precisely, the limit $t\to\infty$, means $t$ exceeds the effective
interaction duration between the photon and the lensing mass with an effective
force range $\sim 1/\epsilon$, and this is satisfied if the photon is measured
at a large distance with $t\gg 1/\epsilon$. Considering the time integrals in
Eq. (41), the validity of the foregoing limit is further justified through the
numerical simulations described towards the end of this paper where $k$ and
$|p-q|$ are found to peak around $\epsilon$.

Adopting the Born approximation (25) and keeping up to the first orders in
$\nu$, we see that Eq. (35) becomes

$\displaystyle\tau_{ij}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
$\displaystyle=$
$\displaystyle\tau_{ij}^{(0)}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})+\nu\,\tau_{ij}^{(1)}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
(43)

in terms of

$\displaystyle\tau_{ij}^{(0)}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2}\,\frac{P_{ijkl}({\boldsymbol{k}})p_{k}p_{l}}{\sqrt{{p}{p^{\prime}}}}\,\delta(\boldsymbol{p}-\boldsymbol{p}^{\prime}-{\boldsymbol{k}})\,a_{\boldsymbol{p}^{\prime}}^{\dagger}a_{\boldsymbol{p}}$
(44)
$\displaystyle\tau_{ij}^{(1)}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\hbar}{2\pi^{2}}\,\frac{P_{ijkl}({\boldsymbol{k}})}{\sqrt{{p}{p^{\prime}}}[|\boldsymbol{p}-\boldsymbol{p}^{\prime}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]}\,\Big{[}\frac{p_{k}p_{l}}{|\boldsymbol{p}-{\boldsymbol{k}}|^{2}-(p^{\prime}-i\epsilon)^{2}}+\frac{p^{\prime}_{k}p^{\prime}_{l}}{|\boldsymbol{p}^{\prime}+{\boldsymbol{k}}|^{2}-(p+i\epsilon)^{2}}\Big{]}a_{\boldsymbol{p}^{\prime}}^{\dagger}a_{\boldsymbol{p}}$
(45) $\displaystyle-(\epsilon\to\delta)$

satisfying

$\displaystyle\tau_{ij}^{(0){\dagger}}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
$\displaystyle=$
$\displaystyle\tau_{ij}^{(0)}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p})$
(46)
$\displaystyle\tau_{ij}^{(1){\dagger}}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})$
$\displaystyle=$
$\displaystyle\tau_{ij}^{(1)}(-{\boldsymbol{k}},\boldsymbol{p}^{\prime},\boldsymbol{p})$
(47)

which are consistent with Eq. (32).

Substituting Eqs. (43) into Eq. (42) and repetitively using the same argument
leading to free particles suffering no Markovian gravitational decoherence, we
obtain the 2nd order (in $\nu$) asymptotic change of the density matrix

$\displaystyle\Delta\rho$ $\displaystyle=$
$\displaystyle-\frac{2\,G\nu^{2}}{\pi\hbar}\int\text{d}^{3}k\,\text{d}^{3}p\,\text{d}^{3}p^{\prime}\,\text{d}^{3}q\,\text{d}^{3}q^{\prime}\,\frac{1}{k}\,\delta(k+({q}-{q^{\prime}}))\,\delta(k-({p}-{p^{\prime}}))$
(48) $\displaystyle\hskip
8.0pt\;\;\times\;[\tau_{ij}^{(1)}(-{\boldsymbol{k}},\boldsymbol{q},\boldsymbol{q}^{\prime}),\,\tau_{ij}^{(1)}({\boldsymbol{k}},\boldsymbol{p},\boldsymbol{p}^{\prime})\rho_{0}]+\text{H.c.}$

On account of Eqs. (14) and (47), Eq. (48) takes a more explicit form as
follows

$\displaystyle\Delta\rho$ $\displaystyle=$
$\displaystyle-\zeta\int\text{d}^{3}k\,\text{d}^{3}p\,\text{d}^{3}p^{\prime}\,\text{d}^{3}q\,\text{d}^{3}q^{\prime}\,\,\delta(k-{p}+{p^{\prime}})\,\delta(k+{q}-{q^{\prime}})\frac{P_{ijkl}({\boldsymbol{k}})}{k\sqrt{{p}{p^{\prime}}{q}{q^{\prime}}}}$
(49)
$\displaystyle\;\;\times\;\Big{\\{}\frac{1}{[|\boldsymbol{p}-\boldsymbol{p}^{\prime}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]}\Big{[}\frac{p_{i}p_{j}}{|\boldsymbol{p}-{\boldsymbol{k}}|^{2}-(p^{\prime}-i\epsilon)^{2}}+\frac{p^{\prime}_{i}p^{\prime}_{j}}{|\boldsymbol{p}^{\prime}+{\boldsymbol{k}}|^{2}-(p+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}$
$\displaystyle\;\;\times\;\Big{\\{}\frac{1}{[|\boldsymbol{q}-\boldsymbol{q}^{\prime}+{\boldsymbol{k}}|^{2}+\epsilon^{2}]}\Big{[}\frac{q_{k}q_{l}}{|\boldsymbol{q}+{\boldsymbol{k}}|^{2}-(q^{\prime}-i\epsilon)^{2}}+\frac{q^{\prime}_{k}q^{\prime}_{l}}{|\boldsymbol{q}^{\prime}-{\boldsymbol{k}}|^{2}-(q+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}$
$\displaystyle\;\;\times\;\big{[}a_{\boldsymbol{q}^{\prime}}^{\dagger}a_{\boldsymbol{q}}\,,a_{\boldsymbol{p}^{\prime}}^{\dagger}a_{\boldsymbol{p}}\,\rho_{0}\big{]}+\text{H.c.}$

where

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\frac{2\hbar\,G^{3}M_{\star}^{2}p_{\circ}^{4}}{\pi^{5}}$ (50)

is a dimensionless parameter.

The action of Eq. (49) can be obtained from its action on an initial basis
matrix element of the form

$\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle|\boldsymbol{p}_{0}\rangle\langle\boldsymbol{q}_{0}|=a_{\boldsymbol{p}_{0}}^{\dagger}|0\rangle\langle
0|a_{\boldsymbol{q}_{0}}.$ (51)

Substituting this form (51) into Eq. (49) and using the relation

$\displaystyle\big{[}a_{\boldsymbol{q}^{\prime}}^{\dagger}a_{\boldsymbol{q}}\,,a_{\boldsymbol{p}^{\prime}}^{\dagger}a_{\boldsymbol{p}}\,\rho_{0}\big{]}$
$\displaystyle=$
$\displaystyle\delta(\boldsymbol{p}-\boldsymbol{p}_{0})\delta(\boldsymbol{q}-\boldsymbol{p}^{\prime})|\boldsymbol{q}^{\prime}\rangle\langle\boldsymbol{q}_{0}|-\delta(\boldsymbol{p}-\boldsymbol{p}_{0})\delta(\boldsymbol{q}^{\prime}-\boldsymbol{q}_{0})|\boldsymbol{p}^{\prime}\rangle\langle\boldsymbol{q}|$

obtained from Eq. (27), we can usefully write the resulting $\Delta\rho$ as

$\displaystyle\Delta\rho$ $\displaystyle=$
$\displaystyle\Delta\varrho^{\natural}+\Delta\varrho^{\flat}+\text{H.c.}$ (52)

where

$\displaystyle\Delta\varrho^{\natural}$ $\displaystyle=$
$\displaystyle-\zeta\int\text{d}^{3}k\,\text{d}^{3}p\,\text{d}^{3}q\,\,\delta(k+q-p_{0})\,\delta(k+q-p)\frac{P_{ijkl}({\boldsymbol{k}})\,|\boldsymbol{p}\rangle\langle\boldsymbol{q}_{0}|}{kq\sqrt{pp_{0}}}$
(53)
$\displaystyle\;\;\times\;\Big{\\{}\frac{1}{|\boldsymbol{p}_{0}-\boldsymbol{q}-{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{p_{0i}p_{0j}}{|\boldsymbol{p}_{0}-{\boldsymbol{k}}|^{2}-(q-i\epsilon)^{2}}+\frac{q_{i}q_{j}}{|\boldsymbol{q}+{\boldsymbol{k}}|^{2}-(p_{0}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}$
$\displaystyle\;\;\times\;\Big{\\{}\frac{1}{|\boldsymbol{q}-\boldsymbol{p}+{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{q_{k}q_{l}}{|\boldsymbol{q}+{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{k}p_{l}}{|\boldsymbol{p}-{\boldsymbol{k}}|^{2}-(q+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}$

does not lose energy and

$\displaystyle\Delta\varrho^{\flat}$ $\displaystyle=$
$\displaystyle\zeta\\!\\!\\!\int\displaylimits_{\begin{subarray}{c}k<\min\\\
(p_{0},q_{0})\end{subarray}}\\!\\!\\!\text{d}^{3}k\,$
$\displaystyle\times\;P_{mnij}({\boldsymbol{k}})\int\text{d}^{3}p\,\Big{\\{}\frac{\delta(k+p-p_{0})}{\sqrt{p_{0}kp}\,[|\boldsymbol{p}_{0}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]}\Big{[}\frac{p_{0i}p_{0j}}{|\boldsymbol{p}_{0}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{i}p_{j}}{|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{0}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}|\boldsymbol{p}\rangle$
$\displaystyle\times\;P_{mnkl}({\boldsymbol{k}})\int\text{d}^{3}q\,\Big{\\{}\frac{\delta(k+q-q_{0})}{\sqrt{q_{0}kq}\,[|\boldsymbol{q}_{0}-\boldsymbol{q}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]}\Big{[}\frac{p_{0k}p_{0l}}{|\boldsymbol{q}_{0}-{\boldsymbol{k}}|^{2}+(q+i\epsilon)^{2}}+\frac{q_{k}q_{l}}{|\boldsymbol{q}+{\boldsymbol{k}}|^{2}-(q_{0}-i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}\langle\boldsymbol{q}|$

is responsible for dissipating photon energy by emitting bremsstrahlung
gravitons.

The setup above allows us to consider a wave packet profile

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\int\text{d}^{3}p\,\psi(\boldsymbol{p})|\boldsymbol{p}\rangle$
(55)

as an initial normalised state with

$\displaystyle\int\text{d}^{3}p\,|\psi(\boldsymbol{p})|^{2}$ $\displaystyle=$
$\displaystyle 1$ (56)

and a mean wave number vector $\boldsymbol{p}_{\circ}$ so that

$\displaystyle\int\text{d}^{3}p\,\boldsymbol{p}\,|\psi(\boldsymbol{p})|^{2}$
$\displaystyle=$ $\displaystyle\boldsymbol{p}_{\circ}.$ (57)

This can be used to construct an initial density matrix

$\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle|\psi\rangle\langle\psi|=\int\text{d}^{3}p\int\text{d}^{3}q\,\psi(\boldsymbol{p})\psi^{*}(\boldsymbol{q})\,|\boldsymbol{p}\rangle\langle\boldsymbol{q}|.$
(58)

Then from Eqs. (58) and (LABEL:dvrhm), we have

$\displaystyle\Delta\varrho^{\flat}$ $\displaystyle=$
$\displaystyle\zeta\int\text{d}^{3}k\,|p_{ij}\rangle\langle
p_{ij}|=\zeta\int\text{d}^{3}k\,\big{(}|p_{+}\rangle\langle
p_{+}|+|p_{\times}\rangle\langle p_{\times}|\big{)}$ (59)

where

$\displaystyle|p_{ij}\rangle$ $\displaystyle=$
$\displaystyle|p_{ij}\rangle({\boldsymbol{k}})$ (60) $\displaystyle=$
$\displaystyle
P_{ijkl}({\boldsymbol{k}})\int\text{d}^{3}p^{\prime}\int\text{d}^{3}p\,\psi(\boldsymbol{p}^{\prime})\,\delta(k+p-p^{\prime})\,\frac{1}{\sqrt{p^{\prime}kp}}$
$\displaystyle\times\;\Big{\\{}\frac{1}{|\boldsymbol{p}^{\prime}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{p^{\prime}_{k}p^{\prime}_{l}}{|\boldsymbol{p}^{\prime}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{k}p_{l}}{|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p^{\prime}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}|\boldsymbol{p}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{p_{\circ}^{3}}\int\displaylimits_{p^{\prime}>k}\text{d}^{3}p^{\prime}\int\displaylimits_{p=p^{\prime}-k}\text{d}\Omega_{p}\,A_{ij}|\boldsymbol{p}\rangle$

with the dimensionless TT amplitude

$\displaystyle A_{ij}$ $\displaystyle=$ $\displaystyle
A_{ij}({\boldsymbol{k}},\boldsymbol{p}^{\prime},\Omega_{p})$ $\displaystyle:=$
$\displaystyle
P_{ijkl}({\boldsymbol{k}})\,\psi(\boldsymbol{p}^{\prime})\,\frac{p_{\circ}^{3}p^{3/2}}{\sqrt{p^{\prime}k}}$
$\displaystyle\times\;\Big{\\{}\frac{1}{|\boldsymbol{p}^{\prime}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{p^{\prime}_{k}p^{\prime}_{l}}{|\boldsymbol{p}^{\prime}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{k}p_{l}}{|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p^{\prime}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}\Big{|}_{p=p^{\prime}-k}.$

##  5 Gravitational Bremsstrahlung with a Single Momentum Initial State

For simplicity, we now restrict to single momentum initial state, deferring
the more general wave-packet initial states to a future investigation.

Such a state is obtained by setting
$\boldsymbol{q}_{0}=\boldsymbol{p}_{0}=\boldsymbol{p}_{\circ}$ in Eq. (51) so
that Eq. (LABEL:dvrhm) takes the form

$\displaystyle\Delta\varrho^{\flat}$ $\displaystyle=$
$\displaystyle\zeta\int_{k<p_{\circ}}\text{d}^{3}k\,|p_{ij}\rangle\langle
p_{ij}|=\zeta\int_{k<p_{\circ}}\text{d}^{3}k\,\big{(}|p_{+}\rangle\langle
p_{+}|+|p_{\times}\rangle\langle p_{\times}|\big{)}$ (61)

where

$\displaystyle|p_{ij}\rangle$ $\displaystyle=$
$\displaystyle|p_{ij}\rangle({\boldsymbol{k}})=p_{\circ}^{-3/2}\int\displaylimits_{p=p_{\circ}-k}\text{d}\Omega_{p}\,A_{ij}|\boldsymbol{p}\rangle$
(62)

with the dimensionless TT amplitude

$\displaystyle A_{ij}$ $\displaystyle=$ $\displaystyle
A_{ij}({\boldsymbol{k}},\Omega_{p})$ $\displaystyle:=$
$\displaystyle\frac{p_{\circ}p^{3/2}P_{ijkl}({\boldsymbol{k}})\,}{\sqrt{k}}\Big{\\{}\frac{1}{\boldsymbol{p}_{\circ}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{p_{0k}p_{0l}}{|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{k}p_{l}}{|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}\Big{|}_{p=p_{\circ}-k}.$

There are two orthogonal parts of $A_{ij}$:

$\displaystyle A_{ij}$ $\displaystyle=$ $\displaystyle
A_{ij}^{+}+A_{ij}^{\times}$

corresponding to the two ($+$ and $\times$) polarisations of the gravitational
waves [31], so that the gravitational wave square amplitude decomposes
accordingly as

$\displaystyle|A|^{2}$ $\displaystyle:=$ $\displaystyle
A_{ij}A_{ij}^{*}=A_{ij}^{+}A_{ij}^{+*}+A_{ij}^{\times}A_{ij}^{\times*}.$ (63)

For numerical evaluations, we can conveniently choose
$\boldsymbol{p}_{\circ}=p_{\circ}\hat{\boldsymbol{z}}$, then we have

$\displaystyle A_{ij}$ $\displaystyle=$
$\displaystyle\frac{p_{\circ}p^{3/2}}{\sqrt{k}}\Big{\\{}\frac{1}{\boldsymbol{p}_{\circ}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}}\Big{[}\frac{p_{\circ}^{2}\delta_{3k}\delta_{3l}}{|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}}+\frac{p_{k}p_{l}}{|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2}}\Big{]}-(\epsilon\to\delta)\Big{\\}}\Big{|}_{p=p_{\circ}-k}.$

For example, in the limit of a point source of gravity with effective radius
$1/\delta\to 0$, Eq. (63) yields

$\displaystyle|A|^{2}_{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{p_{\circ}^{2}p^{3}}{k\,[|\boldsymbol{p}_{\circ}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]^{2}}\bigg{\\{}\frac{p_{\circ}^{4}P_{3333}({\boldsymbol{k}})}{\big{|}|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}\big{|}^{2}}$
(64)
$\displaystyle+\frac{2\,p_{\circ}^{2}p^{2}P_{33ij}({\boldsymbol{k}})\hat{p}_{i}\hat{p}_{j}}{\Re\big{[}(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p+i\epsilon)^{2})(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2})\big{]}}+\frac{p^{4}P_{ijkl}({\boldsymbol{k}})\,\hat{p}_{i}\hat{p}_{j}\hat{p}_{k}\hat{p}_{l}}{\big{|}|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2}\big{|}^{2}}\bigg{\\}}\bigg{|}_{p=p_{\circ}-k}$

where

$\displaystyle\big{|}|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p-i\epsilon)^{2}\big{|}^{2}$
$\displaystyle=$
$\displaystyle(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-p^{2}+\epsilon^{2})^{2}+4\epsilon^{2}p^{2},$
$\displaystyle\Re\big{[}(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-(p+i\epsilon)^{2})(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2})\big{]}$
$\displaystyle=$
$\displaystyle(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-p^{2}+\epsilon^{2})(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-p_{\circ}^{2}+\epsilon^{2})-4\epsilon^{2}p_{\circ}p,$
$\displaystyle\big{|}|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-(p_{\circ}+i\epsilon)^{2}\big{|}^{2}$
$\displaystyle=$
$\displaystyle(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-p_{\circ}^{2}+\epsilon^{2})^{2}+4\epsilon^{2}p_{\circ}^{2},$

and

$\displaystyle P_{3333}({\boldsymbol{k}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-\hat{k}_{3}^{2})^{2},$ $\displaystyle
P_{33ij}({\boldsymbol{k}})\hat{p}_{i}\hat{p}_{j}$ $\displaystyle=$
$\displaystyle\hat{p}_{3}^{2}-2(\hat{k}_{i}\hat{p}_{i})\hat{k}_{3}\hat{p}_{3}+\frac{1}{2}\big{[}(\hat{k}_{i}\hat{p}_{i})^{2}(\hat{k}_{3}^{2}+1)+\hat{k}_{3}^{2}-1\big{]},$
$\displaystyle
P_{ijkl}({\boldsymbol{k}})\hat{p}_{i}\hat{p}_{j}\hat{p}_{k}\hat{p}_{l}$
$\displaystyle=$
$\displaystyle\big{[}1-(\hat{k}_{i}\hat{p}_{i})^{2}\big{]}^{2}.$

It is also useful to introduce
$|A|^{2}_{\delta}:=|A|^{2}_{\epsilon\to\delta}$. For simplicity, let us adopt
as a reasonable first order-of-magnitude estimate of the gravitational wave
square amplitude for a source of gravity with an effective range $1/\epsilon$
and radius $1/\delta$ to be

$\displaystyle|A|^{2}_{\epsilon,\delta}:=|A|^{2}_{\epsilon}-|A|^{2}_{\delta}.$
(65)

The total dissipated outgoing energy corresponding to Eqs. (52) and (61) then
follows as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\frac{2\zeta}{p_{\circ}^{3}}\int\text{d}^{3}k\,\int\text{d}\Omega_{p}\,\hbar
k\,|A|^{2}_{\epsilon,\delta}({\boldsymbol{k}},\Omega_{p})$ (66)
$\displaystyle=$
$\displaystyle\frac{2\zeta\hbar}{p_{\circ}^{3}}\int_{0}^{p_{\circ}}\text{d}k\,k^{3}\int\text{d}\Omega_{k}\int\text{d}\Omega_{p}\,|A|^{2}_{\epsilon,\delta}({\boldsymbol{k}},\Omega_{p}).$

To obtain an expression in term of dimensionless quantities, we express
$\boldsymbol{p}_{\circ},\boldsymbol{p},{\boldsymbol{k}},\epsilon$ as
dimensionless quantities in units of $p_{\circ}$.

Then, by replacing $\boldsymbol{p}_{\circ}\to
p_{\circ}\boldsymbol{p}_{\circ},\boldsymbol{p}\to
p_{\circ}\boldsymbol{p},{\boldsymbol{k}}\to
p_{\circ}{\boldsymbol{k}},\epsilon\to p_{\circ}\epsilon,\delta\to
p_{\circ}\delta$ and using Eq. (50) with the mass-energy of the gravity source
$E_{\star}=M_{\star}$, we arrive at the fractional energy loss $\mu:=\Delta
E/E_{\circ}=\Delta E/(\hbar p_{\circ})$ given by

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{4}{\pi^{5}}\,\hbar\,G^{3}E_{\star}^{2}p_{\circ}^{4}\,\Big{(}\frac{S_{\epsilon}}{\epsilon}-\frac{S_{\delta}}{\delta}\Big{)}$
(67) $\displaystyle=$
$\displaystyle\frac{4}{\pi^{5}}\,\frac{E_{\star}^{2}E_{\circ}^{4}}{E_{\text{P}}^{\,6}}\,\Big{(}\frac{S_{\epsilon}}{\epsilon}-\frac{S_{\delta}}{\delta}\Big{)}$

where $E_{\text{P}}$ is the Planck energy and

$\displaystyle S_{\epsilon}$ $\displaystyle=$
$\displaystyle\epsilon\int_{0}^{1}\text{d}k\,k^{3}\int\text{d}\Omega_{k}\int\text{d}\Omega_{p}\,|A|^{2}_{\epsilon}(k,\Omega_{k},\Omega_{p})$
(68) $\displaystyle=$
$\displaystyle\int_{0}^{1}\text{d}k\,\int\text{d}\theta_{k}\int\text{d}\phi_{k}\int\text{d}\theta_{p}\int\text{d}\phi_{p}\,F_{\epsilon}(k,\theta_{k},\phi_{k},\theta_{p},\phi_{p})$

with $|A|^{2}_{\epsilon}$ obtained from Eq. (64) for $p_{\circ}\to 1$ and

$\displaystyle F_{\epsilon}(k,\theta_{k},\phi_{k},\theta_{p},\phi_{p})$
$\displaystyle=$ $\displaystyle\frac{\epsilon
k^{2}p^{3}\sin\theta_{k}\sin\theta_{p}}{[|\boldsymbol{p}_{\circ}-\boldsymbol{p}-{\boldsymbol{k}}|^{2}+\epsilon^{2}]^{2}}\bigg{\\{}\frac{\frac{1}{2}(1-\hat{k}_{3}^{2})^{2}}{(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-p^{2})^{2}+2\epsilon^{2}(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}+p^{2})+\epsilon^{4}}$
$\displaystyle+\frac{2\,p^{2}\Big{[}\hat{p}_{3}^{2}-2(\hat{k}_{i}\hat{p}_{i})\hat{k}_{3}\hat{p}_{3}+\frac{1}{2}\big{[}(\hat{k}_{i}\hat{p}_{i})^{2}(\hat{k}_{3}^{2}+1)+\hat{k}_{3}^{2}-1\big{]}\Big{]}}{(|\boldsymbol{p}_{\circ}-{\boldsymbol{k}}|^{2}-p^{2}+\epsilon^{2})(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-1+\epsilon^{2})-4\epsilon^{2}p}+\frac{p^{4}\big{[}1-(\hat{k}_{i}\hat{p}_{i})^{2}\big{]}^{2}}{(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}-1)^{2}+2\epsilon^{2}(|\boldsymbol{p}+{\boldsymbol{k}}|^{2}+1)+\epsilon^{4}}\bigg{\\}}\bigg{|}_{p=1-k}.$

We also have analogous constructions for $S_{\delta}$ and $F_{\delta}$.
Numerical evaluations of $F_{\epsilon}$ and $F_{\delta}$ as functions of
$(k,\theta_{k},\phi_{k},\theta_{p},\phi_{p})$ show sharp but finite peaks
around small $k$, $\theta_{k}$ and $\theta_{p}$ for small $\epsilon,\delta$.
This leads to finite numerical integrations with $S_{\epsilon}\approx
S_{\delta}\approx 100$ for $\epsilon,\delta\ll 1$. Therefore, for a gravity
source with a characteristic radius $1/\delta$ and effective range
$1/\epsilon$ in units of $1/p_{\circ}$, from Eq. (67) we have

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{E_{\star}^{2}E_{\circ}^{4}}{E_{\text{P}}^{\,6}}\,\Big{(}\frac{1}{\epsilon}-\frac{1}{\delta}\Big{)}.$
(70)

approximately, corresponding to the rough spread of the photon impact
parameter to be from the surface of the gravitational lens to one radius from
the surface.

The energy loss rate above can be interpreted $\mu\approx\Gamma\;\tau$ where
$\Gamma={E_{\star}^{2}E_{\circ}^{4}}/{E_{\text{P}}^{\,6}}$ is the effective
graviton emission transition rate by the photon, and
$\tau={1}/{\epsilon}-{1}/{\delta}$ is the effective interaction time between
the photon and the lensing mass $M_{\star}$ with the corresponding effective
interaction range $R_{\text{rng}}=1/\epsilon$. See Fig. 1. This is consistent
with the long travel time or distance assumption stated in section 4.

In full physical units, Eq. (70) becomes

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle{\frac{\hbar\,G^{3}}{c^{12}}\,\Big{(}R_{\text{rng}}-R_{\star}\Big{)}M_{\star}^{2}\omega_{\circ}^{5}}$
(71)

where the speed of light $c$ has now been reinstated.

##  6 Conclusions and Discussion

Based on the gravitational quantum vacuum, which has recently been shown to
lead to gravitational decoherence [8] and gravitational spontaneous radiation
that recovers the well-established quadrupole radiation [10], in this paper we
have presented, to our knowledge, the first approach to the spontaneous
bremsstrahlung of light due to the combined effects of gravitational lensing
and spacetime fluctuations. Our present work yields a new quantum
gravitational mechanism whereby starlight emits soft gravitons and becomes
partially redshifted. This effect may contribute to the stochastic
gravitational wave background [11, 33, 34]. We also note that while the term
(53) for the outgoing light does not undergo photon to graviton energy
conversion, it exhibits a type of recoherence of photons [35].

Figure 2: The fractional energy loss $\mu$ given by Eq. (71) as a function of
the photon frequency $f_{\circ}$ (Hz) in logarithmic scale along the
horizontal axis evaluated with a chosen range of the effective gravity range
parameter $d=(R_{\text{rng}}-R_{\star})/R_{\star}$ for the X-ray binary Cygnus
X-1. Above, $\log\mu$ is plotted in (a) as a projected height onto the $(\log
f_{\circ},d)$ plane and in (b) as a line for 3 representative values of $d$.
It is obvious that as the photon frequency $f_{\circ}$ increases from the
optical spectrum, the photon-to-graviton energy conversion “chips in” at
around $f_{\circ}=3\times 10^{17}$ Hz at the start of the soft X-ray spectrum,
so that $\mu\sim 0.01$ for $d=4$ corresponding to $R_{\text{rng}}\sim
100R_{\odot}$. Although here $\mu$ appears to carry on increasing with
$f_{\circ}$, the validity of Eq. (71) is strictly limited to small $\mu$ due
to the Born approximation with weak interactions we have assumed in Eq. (21)
used to derive $\mu$ in Eq. (71). Nonetheless, the ascending trend of $\mu$
with the photon frequency $f_{\circ}$ suggests that starting from the soft
X-ray frequencies, the gravitational bremsstrahlung of light in the vicinity
of Cygnus X-1 may be important.

Our work naturally raises the prospect of potential detection of the released
stochastic gravitational waves. Addressing this question in detail requires a
further investigation which is currently underway by the authors. It is
however of interest at the present stage to envisage a plausible observation
scenario. To this end, let us take the well studied strong X-ray binary Cygnus
X-1 [36] because this system has both a large gravitational field and a strong
X-ray source similar to that illustrated in Fig. 1. However, both the compact
object/black hole and the companion supergiant star in the Cygnus X-1 binary
system are massive, with a total mass $M_{\star}\approx 40M_{\odot}$ and an
orbiting radius $\approx 20R_{\odot}$, which we will take as the effective
$R_{\star}$. This would make $R_{\text{rng}}$ to be in the region of
$100R_{\odot}$ and so from the discussions in Sec. 5, the spontaneously
emitted stochastic gravitational waves would have a mean wavelength in the
same region having a mean frequency $f\sim 0.01$ Hz. Using these parameters,
the fractional energy loss $\mu$ in Eq. (71) is plotted in Fig. 2 with the
effective gravity range parameter $d:=(R_{\text{rng}}-R_{\star})/R_{\star}$
chosen to be $0\leq d\leq 4$, corresponding to the rough spread of the photon
impact parameter to be from the surface of the gravitational lens to
$R_{\text{rng}}\approx 100R_{\odot}$ discussed above. The detection of sub-Hz
gravitational waves is a unique strength of the proposed LISA mission, which
is expected to reach a corresponding characteristic strain sensitivity close
to $h\sim 10^{-22}$ [37], as shown in Fig. 3.

From the X-ray luminosity of Cygnus X-1 $L_{X}\approx 4\times 10^{37}$ erg s-1
[38, 39] and its distance $D\approx 6100$ light years to the Earth, one gets
the arriving X-ray energy flux to be $S_{X}=L_{X}/(4\pi D^{2})\approx 10^{-7}$
erg cm-2s-1. Let us suppose that the total gravitational wave luminosity of
Cygnus X-1 arises from $L_{\text{GW}}=\mu_{\text{eff}}L_{X}$, in terms of an
effective photon-to-graviton energy transfer rate $0<\mu_{\text{eff}}<1$. We
can then estimate the characteristic strain $h$ of the gravitational waves
with energy density expression $U_{\text{GW}}=c^{2}\omega^{2}h^{2}/(32\pi G)$
where $h^{2}=h_{+}^{2}+h_{\times}^{2}$ with $\omega=2\pi f$ and the energy
flux $S_{\text{GW}}=c\,U_{\text{GW}}$ so that
$S_{\text{GW}}=\mu_{\text{eff}}S_{X}$. For example, if
$\mu_{\text{eff}}=0.1\%$ and $f=0.01$ Hz, then $S_{\text{GW}}\approx 10^{-10}$
erg cm-2s-1 yielding $h\approx 10^{-22}$.

As shown in Fig. 3, we choose moderate effective transfer rate values
$0.01\%\leq\mu_{\text{eff}}\leq 1\%$ across the gravitational wave frequencies
$10^{-5}$ Hz $\leq f\leq$ 10 Hz. Indeed, for $\mu_{\text{eff}}\gtrsim 10^{-3}$
at around $f\sim$ 0.01 Hz, the characteristic strain of the bremsstrahlung
gravitational waves from Cygnus X-1 could be above the sensitivity level of
LISA and hence potentially detectable. Furthermore, the buildup of similar
sources could also contribute to an overall stochastic gravitational wave
background. Based on the initial results and estimates reported in this work,
we plan to analyse and quantify the properties of the bremsstrahlung
gravitational waves from intense astronomical sources of light and gravity in
a more realistic setting for future publication.

Figure 3: Estimated characteristic strain $h$ of the gravitational waves with
effective photon-to-graviton energy transfer rate $\mu_{\text{eff}}$ for
Cygnus X-1 with $0.01\%\leq\mu_{\text{eff}}\leq 1\%$ against the gravitational
wave frequency $f$ in Hz compared with the LISA characteristic strain
sensitivity curve [37].

## Acknowledgements

C.W. is indebted to John S. Reid for stimulating discussions and grateful to
the Cruickshank Trust for financial support. M.M. would like to thank the
Student Awards Agency Scotland (SAAS) for funding.

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Some results on Nilpotency, Solvability and Frattini Theory for Bicommutative
and for Novikov algebras

David A. Towers

Department of Mathematics, Lancaster University

Lancaster LA1 4YF, England

Abstract

This paper starts by showing that for algebras in a certain class the concepts
of weak nilpotency and nilpotency coincide. It goes on to describe some
solvability and nilpotency properties of bicommutative algebras and of Novikov
algebras and to introduce a Frattini theory for both of them. A description is
also given for semisimple bicommutative algebras over any field.

Mathematics Subject Classification 2020: 17A32, 17B05, 17B20, 17B30, 17B50.
Key Words and Phrases: Bicommutative algebra, Novikov algebra, solvable,
nilpotent, semisimple, Frattini ideal.

##  1 Introduction

###### Definition 1

A nonassociative algebra $A$ is called right (respectively, left) commutative
if $(ab)c=(ac)b$ (respectively, $a(bc)=b(ac)$) for all $a,b,c\in A$; it is
bicommutative if it is both right and left commutative.

One-sided commutative algebras first appeared in a paper of Cayley in 1857
([5]). The study of bicommutative algebras was initiated by Dzhumadil’daev and
Tulenbaev in 2003 ([8]), where, in particular, they proved that the derived
algebra of such an algebra is commutative and associative, and hence that any
simple algebra is a field. They were also studied by Burde, Dekimpe and
Deschamps, who called them $LR$-algebras, in 2009 ([4]). They arise in a
number of settings, including the study of affine actions on nilpotent Lie
groups. Low dimensional nilpotent bicommutative algebras have been classified
algebraically and geometrically in [1, 12, 13].

###### Definition 2

A nonassociative algebra $A$ is called a (left) Novikov algebra if
$x(yz)-(xy)z=y(xz)-(yx)z$ (left symmetry)$(xy)z=(xz)y$ (right commutativity)
for all $x,y,z\in A$. there is also an opposite (right) version satisfying
right symmetry and left commutativity.

These algebras were introduced independently by Gelfand and Dorfman in 1979 as
an algebraic approach to the Hamiltonian operator in mechanics ([10]) and by
Balinskii and Novikov in 1985 in relation to hydrodynamics ([2]). In [2], a
question was raised about the simple algebras in this class and this was
answered in the same year by Zelmanov who showed that, over a field of
characteristic zero, the simple Novikov algebras are fields ([17]). Low-
dimensional Novikov algebras were classified in [3, 11].

Both of these classes of algebras have attracted the attention of many authors
and have a rich theory. They are both Lie admissible algebras; that is,
commutator multiplication on them gives a Lie algebra. In certain areas they
have some similarities (such as the simple algebras being commutative
associative algebras), but in others they differ significantly, as we shall
see later. For a fuller bibliography, see [7] for bicommutative algebras and
[14] for Novikov algebras. The purpose of this paper is to study some further
solvability and nilpotency properties and to introduce a Frattini theory for
them.

In section 2 we consider some concepts of nilpotency in general nonassociative
algebras. In particular, we show that, in a class of algebras $\mathfrak{X}$
which is factor algebra and subalgebra closed and, for every
$A\in\mathfrak{X}$, $IJ$ is an ideal of $A$ whenever $I,J$ are ideals of $A$,
a weaker version of nilpotency is equivalent to nilpotency. Section 3 is
devoted to bicommutative algebras. We show that the Frattini ideal of such an
algebra is nilpotent, that minimal ideals are zero algebras or are simple, and
produce a decomposition of semisimple such algebras as a direct sum of fields
extended by a zero subalgebra (one whose square is zero). It is also shown
that, unlike Novikov algebras, the sum of two zero subalgebras need not be
solvable; in fact, it can be semisimple.

In section 4, Novikov algebras $A$ are introduced. It is shown that, if the
field has characteristic $p$, where $p=0$ or $p>\dim A+1$, then $\phi(A)$ is
right nilpotent. Also, that if $R$ is the solvable radical, then $AR$ is a
nilpotent ideal of $A$. In the final section, algebras with trivial Frattini
ideal are considered. It is shown that, if $A$ is a bicommutative algebra over
any field or a Novikov algebra over a field of characteristic $p$, where $p=0$
or $p>\dim A+1$, then $A$ is $\phi$-free if and only if it splits over the sum
of its zero ideals. Decomposition results are found for both classes of
algebras, over a general field for bicommutative algebras, and over a field of
characteristic zero for Novikove algebras with nilpotent radical. A
consequence of the latter result is that, over a field of characteristic zero,
the Frattini ideal of a Novikov agebra is nilpotent. The section finishes with
two results concerning maximal subalgebras.

Throughout, $A$ will denote a finite-dimensional nonassociative algebra over a
field $F$. Algebra direct sume will be denoted by $\oplus$, whereas direct
sume of the vector space structure alone will be denoted by $\dot{+}$. We will
use $\subseteq$ to indicate inclusion, and $\subset$ for strict inclusion.

##  2 Some General Results

###### Definition 3

If $B$ is a subalgebra of an algebra $A$, the idealiser of $B$ in $A$ is
$I_{A}(B)=\\{a\in A\mid aB+Ba\subseteq B\\}$. The annihilator of $I$ in $A$ is
$Ann_{A}(I)=\\{a\in A\mid aI=Ia=0\\}$. We will write $Ann(A)$ for
$Ann_{A}(A)$.

###### Definition 4

For any algebra $A$, the derived series of subalgebras $A^{(0)}\supseteq
A^{(1)}\supseteq A^{(2)}\supseteq\ldots$ of $A$ is obtained by defining
$A^{(0)}=A$, $A^{(i+1)}=(A^{(i)})^{2}$ for all $i\geq 0$. $A$ is called
solvable if $A^{(n)}=0$ for some $n$. Every algebra $A$ has a unique solvable
ideal which we will call the solvable radical of $A$ and denote by $R(A)$. We
will call $A$ semisimple if $R(A)=0$. (Note that this is different from the
definition used by some authors.)

###### Definition 5

Put $A^{1}=$ ${}^{1}A=A^{[1]}=A$, $A^{n+1}=A^{n}A$, ${}^{n+1}A=A(^{n}A)$,
$A^{[n+1]}=\sum_{i+j=n+1}A^{[i]}A^{[j]}$. We call $A$ right nilpotent if
$A^{n}=0$, left nilpotent if ${}^{n}A=0$, weakly nilpotent if it is both right
and left nilpotent, and nilpotent if $A^{[n]}=0$ for some $n\geq 1$.

Clearly, an algebra is nilpotent if and only if there is an $n$ such that
every product of $n$ elements is zero. It follows that $A^{[n]}$ is an ideal
of $A$ for every $n\geq 1$

###### Proposition 2.1

Let $A$ be a nilpotent algebra and let $B$ be a subalgebra of $A$. Then
$B\subset I_{A}(B)$; in particular, all maximal subalgebras of $A$ are ideals
of $A$.

Proof. Since $A$ is nilpotent, there is a $k$ such that $A^{[k]}\not\subseteq
B$, but $A^{[k+1]}\subseteq B$. Then $B\subset A^{[k]}+B\subseteq I_{A}(B)$.
$\Box$

###### Definition 6

The Frattini subalgebra, $F(A)$, of $A$ is the intersection of the maximal
subalgebras of $A$; the Frattini ideal, $\phi(A)$, is the largest ideal of $A$
contained in $F(A)$.

###### Corollary 2.2

([15, Theorem 6]) If $A$ is a nilpotent algebra, then $\phi(A)=F(A)=A^{2}$.

We also have the following characterisation of nilpotent algebras.

###### Lemma 2.3

For an algebra $A$, the following are equivalent:

  * (i)

$A$ is nilpotent;

  * (ii)

there is a chain of ideals of $A$

$0=A_{(0)}\subset A_{(1)}\subset\ldots\subset A_{(n)}=A,$

where $\dim A_{(i)}=i$ and $AA_{(i)}+A_{(i)}A\subseteq A_{(i-1)}$.

Proof. This is part of [15, Theorem 3]. $\Box$

In general, weakly nilpotent algebras need not be nilpotent, as is shown in
[15]. However, we have the following results.

###### Lemma 2.4

Let $A$ be an algebra in which $IJ$ is an ideal of $A$ for every pair of
ideals $I,J\subseteq A$. Let $N$ be a right (respectively, left) nilpotent
ideal of $A$ and let $B/C$ be a chief factor of $A$. Then $BN\subseteq C$
(respectively $NB\subseteq C$).

Proof. Then $BN+C$ is an ideal of $A$ and $C\subseteq BN+C\subseteq B$. Hence
$BN\subseteq C$ or $BN+C=B$. Suppose the latter holds. Then

$B=BN+C\subseteq(BN)N+C\subseteq((BN)N)N+C=\ldots=C,$

since $N$ is right nilpotent. Hence $BN\subseteq C$. Similarly, if $N$ is left
nilpotent then $NB\subseteq C$. $\Box$

###### Theorem 2.5

Let $A$ be a weakly nilpotent algebra in which $IJ$ is an ideal of $A$ for
every pair of ideals $I,J\subseteq A$. Then $A$ is nilpotent.

Proof. Let $B/C$ be a chief factor of $A$. Then $BA+AB\subseteq C$, by Lemma
2.4. It follows that $\dim B/C=1$.

Now let

$0=A_{(0)}\subset A_{(1)}\subset\ldots\subset A_{(n)}=A,$

be a chief series for $A$. Then $A$ is nilpotent by Lemma 2.3. $\Box$

###### Definition 7

We will call a class of algebras $\mathfrak{X}$ natural if it is factor
algebra and subalgebra closed and, for every $A\in\mathfrak{X}$, $IJ$ is an
ideal of $A$ whenever $I,J$ are ideals of $A$.

###### Corollary 2.6

Let $A\in\mathfrak{X}$, where $\mathfrak{X}$ is a natural class of algebras.
Then every weakly nilpotent subalgebra of $A$ is nilpotent.

###### Corollary 2.7

Let $A\in\mathfrak{X}$, where $\mathfrak{X}$ is a natural class of algebras.
Then $A$ has a maximal right nilpotent/ left nilpotent/nilpotent ideal.

Proof. It suffices to show that, if $N_{1}$ and $N_{2}$ are two right
nilpotent ideals of $A$, then so is $N_{1}+N_{2}$. Now, $N_{1}+N_{2}$ is
certainly a solvable ideal of $A$. Let $0=B_{0}\subset
B_{1}\subset\ldots\subset B_{n}=N_{1}+N_{2}$ be a chief series for
$N_{1}+N_{2}$. Then
$B_{n}^{2}=B_{n}(N_{1}+N_{2})=B_{n}N_{1}+B_{n}N_{2}\subseteq B_{n-1}$, by
Lemma 2.4. A straightforward induction proof then shows that $B_{n}^{n+1}=0$.
It follows that $N_{1}+N_{2}$ is right nilpotent. Left nilpotency follows
similarly. If $N_{1},N_{2}$ are nilpotent, then $N_{1}+N_{2}$ is weakly
nilpotent, and so nilpotent, by Theorem 2.5. $\Box$

###### Definition 8

We will call the maximal right nilpotent (respectively, left nilpotent,
nilpotent) ideal given in Corollary 2.7, the right nilradical (respectively,
left nilradical, nilradical) of $A$, and denote it by $N_{r}(A)$
(respectively, $N_{\ell}(A)$, $N(A)$). Then $N(A)\subseteq
N_{r}(A),N_{\ell}(A)\subseteq R(A)$, by [6, Proposition 3.1].

##  3 Bicommutative algebras

###### Proposition 3.1

If $A$ is a bicommutative algebra and $I,J$ are ideals of $A$, then $IJ$ is an
ideal of $A$.

Proof. Simply note that $A(IJ)\subseteq I(AJ)\subseteq IJ$ and
$(IJ)A\subseteq(IA)J\subseteq IJ$. $\Box$

Note that it follows from the above Corollary that the class of bicommutative
algebras is natural and so has a nilradical.

We will make use of the following result of Dzhumadil’daev and Tulenbaev which
has some useful consequences..

###### Theorem 3.2

([8, Theorem 1]) Let $A$ be a bicommutative algebra. Then $A^{2}$ is
commutative and associative.

###### Corollary 3.3

If $A$ is a solvable bicommutative algebra, $A^{2}$ is nilpotent.

###### Corollary 3.4

If $A$ is a bicommutative algebra which is either right or left nilpotent,
then $A^{2}$ is nilpotent, and so $A$ is solvable.

###### Corollary 3.5

Let $A$ be a bicommutative algebra and let R be its solvable radical. Then
$RA$ and $AR$ are nilpotent ideals of $A$.

###### Definition 9

Let $A$ be an algebra and let $a\in A$. Define $\rho_{a}:A\rightarrow
A:x\mapsto xa$. Put $E_{A}^{r}(a)=\\{x\in A\mid\rho_{a}^{n}(x)=0$ for some
$n\\}$. Similarly, define $\lambda_{a}:A\rightarrow A:x\mapsto ax$. Put
$E_{A}^{l}(a)=\\{x\in A\mid\lambda_{a}^{n}(x)=0$ for some $n\\}$.

###### Lemma 3.6

If $A$ is a right (respectively, left) commutative algebra and $a\in A$, then
$E_{A}^{r}(a)$ (respectively, $E_{A}^{l}(a)$) is a subalgebra of $A$..

Proof. Let $x,y\in E_{A}^{r}(a)$. Then there are $m,n$ such that
$\rho_{a}^{m}(x)=\rho_{a}^{n}(y)=0$. Without loss of generality we can assume
that $m=n$. But now $\rho_{a}^{n}(xy)=(\rho_{a}^{n}(x))y=0$, by repeated use
of right commutativity, so $xy\in E_{A}^{r}(a)$.

The left version is similar. $\Box$

###### Definition 10

Let $a\in A$. We call $a$ right nil (respectively, left nil) if
$\rho_{a}^{n}(a)=0$ (respectively, $\lambda_{a}^{n}(a)=0$) for some $n\geq 0$.
We will say that an ideal $B$ of $A$ is right nil (respectively, left nil) if
every element of $B$ is right nil (respectively, left nil).

###### Theorem 3.7

Let $B$ be a left (respectively, right) subideal of a right (respectively,
left) commutative algebra $A$, and let $C$ be an ideal of $B$ with
$C\subseteq\phi(L)$. If $B/C$ is right (respectively, left) nil, then so is
$B$.

Proof. Let $b\in B$ and $B=B_{0}\subseteq B_{1}\subseteq\ldots\subseteq
B_{r}=A$ be a chain of subalgebras of $A$ with $B_{i}$ a left ideal of
$B_{i+1}$ for $0\leq i\leq r-1$. then $\rho_{b}^{r}(A)\subseteq B$. Since
$B/C$ is right nilpotent, there exists $s$ such that $\rho_{b}^{s}(B)\subseteq
C$. Hence $\rho_{b}^{r+s}(A)\subseteq\phi(A)$. But
$\rho_{b}^{r+s}(A)+E_{A}^{r}(b)=A$, by Fitting’s Lemma, so $E_{A}^{r}(b)=A$
and $B$ is right nil.

Again, the left version is similar. $\Box$

###### Corollary 3.8

If $A$ is a right (respectively, left) commutative algebra, then $\phi(A)$ is
right (respectively, left) nil.

###### Corollary 3.9

If $A$ is a bicommutative algebra, then $\phi(A)$ is nilpotent.

Proof. This follows from Corollary 3.8 and Theorem 3.2, since
$\phi(A)\subseteq A^{2}$. $\Box$

The following result concerning minimal ideals will be useful.

###### Theorem 3.10

Let $B$ be a minimal ideal in a bicommutative algebra $A$. If $R$ is the
solvable radical of $A$, then, either $RB=0$ and $BR^{2}=0$, or $BR=0$ and
$R^{2}B=0$.

Proof. We have that $RB$ is an ideal of $A$, by Proposition 3.1, so $RB=0$ or
$RB=B$. Similarly, $BR=0$ or $BR=B$. Suppose that $RB=BR=B$. Then

$B=RB=R(BR)\subseteq BR^{2}\hbox{ and }B=BR=(RB)R\subseteq R^{2}B.$

Suppose that $B\subseteq BR^{(k)}$ and $B\subseteq R^{(k)}B$. Then

$\displaystyle B$ $\displaystyle\subseteq R^{(k)}B\subseteq
R^{(k)}(BR^{(k)})\subseteq BR^{(k+1)}\hbox{ and }$ $\displaystyle B$
$\displaystyle\subseteq BR^{(k)}\subseteq(R^{(k)}B)R^{(k)}\subseteq
R^{(k+1)}B.$

Since $R$ is solvable and $B\neq 0$, either $RB=0$ or $BR=0$. If $RB=0$, then
$BR^{2}\subseteq R(BR)=0$. Similarly, $R^{2}B=0$ if $BR=0$. $\Box$

###### Proposition 3.11

Let $A$ be a right (respectively, left) nilpotent bicommutative algebra. Then
every maximal subalgebra of $A$ is a left (respectively, right) ideal of $A$
and $A^{3}\subseteq\phi(A)$ (respectively, ${}^{3}A\subseteq\phi(A)$).

Proof. Let $A$ be a right nilpotent bicommutative algebra and let $M$ be a
maximal subalgebra of $A$. Then there exists $k$ such that $A^{k}\not\subseteq
M$ but $A^{k+1}\subseteq M$ and $A=M+A^{k}$. Hence $AM=M^{2}+A^{k}M\subseteq
M$ and $M$ is a left ideal of $A$. Also, $A^{3}=A^{2}A\subseteq
A^{2}M+A^{2}A^{k}\subseteq M+(AA^{k})A\subseteq M+A^{k+1}\subseteq M$.

The resuilt for a left nilpotent bicommutative algebra follows similarly.
$\Box$

###### Proposition 3.12

Let $A$ be a bicommutative algebra and let $B$ be a subalgebra of $A$. Then
$I_{A}(B)$ and $Ann_{A}(B)$ are subalgebras of $A$. If $B$ is an ideal of $A$
then so is $Ann_{A}(B)$

Proof. Let $B$ be a subalgebra of $A$ and let $a_{1},a_{2}\in I_{A}(B)$. Then,
for all $b\in B$, $(a_{1}a_{2})b=(a_{1}b)a_{2}\in B$ and
$b(a_{1}a_{2})=a_{1}(ba_{2})\in B$. The argument for $Ann_{A}(B)$ is similar.

So now suppose that $B$ is an ideal of $A$ and let $b\in B$, $k\in
K=Ann_{A}(B)$, $a\in A$. Then $(ak)b=(ab)k\in BK=0$, $(ka)b=(kb)a=0$,
$b(ak)=a(bk)=0$ and $b(ka)=k(ba)\in KB=0$, whence $Ann_{A}(B)$ is an ideal of
$A$. $\Box$

The following property of minimal ideals of bicommutative algebras is shared
by Novikov algebras.

###### Theorem 3.13

If $B$ is a minimal ideal of a bicommutative algebra $A$, then either
$B^{2}=0$ or $B$ is a simple algebra.

Proof. Assume that $B^{2}\neq 0$, so $B^{2}=B$. Then $B\not\subseteq
Ann_{A}(B)$, so $B\cap Ann_{A}(B)=0$ by Proposition 3.12. Let $C$ be a proper
ideal of $B$. Then

$BC=B^{2}C\subseteq(BC)B\subseteq CB\hbox{ and }CB=CB^{2}\subseteq
B(CB)\subseteq BC,$

so $BC=CB$. (Alternatively, simply note that $B,C\subseteq A^{2}$ which is
commutative.) But now

$(BC)A\subseteq(BA)C\subseteq BC\hbox{ and }A(BC)=A(CB)\subseteq
C(AB)\subseteq CB=BC,$

so $BC$ is an ideal of $A$. Since $BC\subseteq C\neq B$ we have that $BC=0$.
Also $CB=BC=0$, whence $C\subseteq B\cap Ann_{A}(B)=0$. $\Box$

###### Definition 11

If $B,C$ are ideals of an algebra $A$ with $B\subset C$, a chief series of $A$
from $B$ to $C$ is a series $B=B_{0}\subset B_{1}\subset\ldots\subset
B_{r}=C$, where the $B_{i}$ are ideals of $A$ and $B_{i+1}/B_{i}$ is a minimal
ideal of $A/B_{i}$ for $0\leq i\leq r-1$.

We can now give the follwing result, which leads to a characterisation of
semisimple bicommutative algebras.

###### Theorem 3.14

Let $S$ be a semisimple bicommutative algebra over any field $F$. Then every
ideal of $S$ inside $S^{2}$ is a direct sum of simple minimal ideals of $S$.

Proof. Let $B$ be an ideal of $S$ inside $S^{2}$. We use induction on the
length of a chief series of $S$ from $0$ to $B$. If $B$ is a minimal ideal of
$S$ then it is simple, by Theorem 3.13. So suppose the result holds when the
length of such a chief series is less than $r+1$, and let the length from $0$
to $B$ be $r+1$. Let $C$ be an ideal of $S$ such that $B/C$ is a chief factor
of $S$, so $C=S_{1}\oplus\ldots\oplus S_{r}$ where $S_{i}$ is a simple minimal
ideal of $S$ for $1\leq i\leq r$.

Now there is a subalgebra $U$ of $S$ such that $S=C+U$ and $C\cap
U\subseteq\phi(U)$, by [16, Lemma 7.1]. But $(S_{i}\phi(U))S_{i}\subseteq
S_{i}^{2}\phi(U)=S_{i}\phi(U)$ and $S_{i}(S_{i}\phi(U))\subseteq
S_{i}\phi(U)$, since $S^{2}$ is associative and $\phi(U)\subseteq
U^{2}\subseteq S^{2}$, by Theorem 3.2. Hence, $S_{i}\phi(U)$ is an ideal of
$S_{i}$ and so is $S_{i}$ or $0$, by Theorem 3.13. But a straightforward
induction proof shows that if $S_{i}\phi(U)=S_{i}$, then
$S_{i}=S_{i}(\phi(U))^{k}$, whence $S_{i}=0$, by Corollary 3.9. It follows
that $S_{i}\phi(U)=\phi(U)S_{i}=0$ for each $1\leq i\leq r$. Thus, $\phi(U)$
is an ideal of $S$ and so $\phi(U)=0$, since $S$ is semisimple and $\phi(U)$
is nilpotent, whence $S=C\dot{+}U$ and $B=C\dot{+}B\cap U$.

Now, if $R(B)$ is the radical of $B$, we have that $R(B)S_{i}=S_{i}R(B)=0$, by
the same argument as for $\phi(U)$. Hence $R(B)\subseteq Ann_{S}(C)\cap
S^{2}$, which is an ideal of $S$, by Proposition 3.12. Clearly,
$Ann_{S}(C)\cap S^{2}=0$ or $B=C\oplus Ann_{S}(C)\cap S^{2}$. The former is
impossible, since then $R(B)=0$, so $B$ is semisimple, whence $B=C\oplus
S_{r+1}$ and $S_{r+1}\subseteq Ann_{S}(C)\cap S^{2}$. We conclude that
$B=C\oplus Ann_{S}(C)\cap S^{2}$ and $S_{r+1}=Ann_{S}(C)\cap S^{2}$ is a
minimal ideal of $S$ and so must be simple. $\Box$

###### Corollary 3.15

Let $S$ be a semisimple bicommutative algebra over any field. Then
$S=S^{2}\dot{+}U$ where $S^{2}=S_{1}\oplus\ldots\oplus S_{n}$, the $S_{i}$ are
simple ideals of $S$, $U^{2}=0$ and either $S_{i}U=0$ or $US_{i}=0$ for each
$1\leq i\leq n$..

Proof. The fact that $S^{2}=S_{1}\oplus\ldots\oplus S_{n}$ follows from
Theorem 3.14 above. There is a subalgebra $U$ of $S$ such that $S=S^{2}+U$ and
$U\cap S^{2}\subseteq\phi(U)$, by [16, Lemma 7.1]. But $\phi(U)=0$ as in the
proof of Theorem 3.14. Clearly, $U^{2}\subseteq U\cap S^{2}=0$. Also
$US_{i}=S_{i}$ or $0$. If $US_{i}=S_{i}$, then $S_{i}U=(US_{i})U\subseteq
U^{2}S_{i}=0$. $\Box$

In the above result, $S^{2}$ is a commutative associative algebra, and so the
$S_{i}$ are extension fields of the base field $F$, but $S$ need not be equal
to $S^{2}$ as the following easy example shows.

###### Example 3.1

Let $A$ be spanned by $x,y$ where $x^{2}=x$, $xy=x$ and all other products are
zero. It is easy to check that $A$ is a semisimple bicommutative algebra that
is neither commutative nor associative.

In [14] it is shown that, for Novikov algebras, the sum of two solvable
subalgebras is solvable, and the sum of two zero subalgebras is metabelian.
Neither of these results hold for bicommutative algebras, as the above example
also shows: note that $A=F(x-y)+Fy$ is its decomposition as a sum of two zero
subalgebras.

##  4 Novikov algebras

The following three results were proved in [14, Lemma 2.1, Theorem 3.3 and
Corollary 3.6].

###### Lemma 4.1

If $I,J$ are ideals of a Novikov algebra $A$, then so is $IJ$. In particular,
$A^{n}$, ${}^{n}A$, $A^{(n)}$ and $A^{[n]}$ are ideals of $A$ for all $n\geq
1$. Moreover, the annihilator of $I$ in $A$ is also an ideal of $A$.

It follows from the above Lemma that the class of Novikov algebras is natural.

###### Theorem 4.2

Let $A$ be a Novikov algebra. Then the following are equivalent:

  * (i)

$A$ is right nilpotent;

  * (ii)

$A^{2}$ is nilpotent; and

  * (iii)

$A$ is solvable.

###### Corollary 4.3

Every left nilpotent Novikov algebra is nilpotent.

###### Corollary 4.4

If $A$ is a solvable Novikov algebra, then $\phi(A)$ is nilpotent.

Proof. This follows from the fact that $\phi(A)\subseteq A^{2}$. $\Box$

###### Proposition 4.5

Let $A$ be a Novikov algebra over a field of characterictic $p$, where $p=0$
or $p>\dim A+1$. Then $\phi(A)$ is right nilpotent.

Proof. This follows from [9, Theorem 1] and Theorem 4.2. $\Box$

###### Proposition 4.6

Let $A$ be a Novikov algebra with solvable radical $R$. Then $AR$ is a
nilpotent ideal of $A$.

Proof. We proceed as in Corollary 3.5. We use induction on the maximum length
$k$ of a chief series of $A$ from $0$ to $R$. If $R$ is a minimal ideal of
$A$, then $R^{2}=0$, so $(AR)^{2}\subseteq R^{2}=0$ and ${}^{2}(AR)\subseteq
R^{2}=0$, so the result holds for $k=1$. So, suppose it holds whenever $k\leq
n$ ($n\geq 1$), and let $A$ be such that $k=n+1$. Let $B$ be a minmal ideal of
$A$ with $B\subseteq R$. Then $R/B$ is the solvable radical of $A/B$ and so
$\frac{A}{B}\frac{R}{B}$ is nilpotent. Hence, there is an $r$ such that
${}^{r}(AR)\subseteq B$. But now $BR=0$, since $R$ is right nilpotent, and so

${}^{r+1}(AR)\subseteq(AR)B\subseteq(AB)R\subseteq BR=0.$

Thus, $AR$ is left nilpotent, and so is nilpotent, by Corollary 4.3. The
result follows by induction. $\Box$

###### Proposition 4.7

Let $A$ be a Novikov algebra and let $B$ be a subalgebra of $A$. Then
$I_{A}(B)$ and $Ann_{A}(B)$ are subalgebras of $A$. If $B$ is an ideal of $A$
then so is $Ann_{A}(B)$

Proof. Let $B$ be a subalgebra of $A$ and let $a_{1},a_{2}\in I_{A}(B)$. Then,
for all $b\in B$, $(a_{1}a_{2})b=(a_{1}b)a_{2}\in B$ and
$b(a_{1}a_{2})=(ba_{1})a_{2}+a_{1}(ba_{2})-(a_{1}b)a_{2}\in B$. The argument
for $Ann_{A}(B)$ is similar.

So now suppose that $B$ is an ideal of $A$. Then $Ann_{A}(B)$ is an ideal of
$A$ is shown in [14, Lemma 2.1]. $\Box$

The following was proved by Zelmanov in [17, Proposition 2].

###### Proposition 4.8

Let $S$ be a semisimple Novikov algebra over a field of characteristic zero.
Then $S$ is a direct sum of fields.

##  5 $\phi$-free algebras

###### Definition 12

An ideal $B$ of an algebra $A$ is called a zero ideal if $B^{2}=0$. The socle,
denoted $Soc(A)$ (respectively, zero socle, denoted $Zsoc(A)$) of $A$ is the
sum of the minimal ideals (respectively, minimal zero ideals) of $A$. We will
say that $A$ is $\phi$-free if $\phi(A)=0$.

###### Theorem 5.1

Let $A$ be a bicommutative algebra over any field or a Novikov algebra over a
field of characteristic $p$, where $p=0$ or $p>\dim A+1$. Then $A$ is
$\phi$-free if and only if it splits over its zero socle.

Proof. If $A$ is $\phi$-free, it splits over its zero socle, by [16, Lemma
7.2]. So suppose that $A=Zsoc(A)\dot{+}C$, where $C$ is a subalgebra of $A$
and $Zsoc(A)=Z_{1}\oplus\ldots\oplus Z_{n}$, where $Z_{i}$ is a minimal zero
ideal of $A$, and that $\phi(A)\neq 0$. Then there is a minimal ideal $Z$ of
$A$ contained in $\phi(A)$. If $A$ is bicommutative, then $\phi(A)$ is
nilpotent, by Corollary 3.9, and if $A$ is a Novikov algebra, then $\phi(A)$
is right nilpotent,by Proposition 4.5. In either case, $Z$ is a zero ideal.
Thus $\phi(A)\cap Zsoc(A)\neq 0$. However,
$M_{i}=(Z_{1}\oplus\ldots\oplus\hat{Z_{i}}\oplus\ldots\oplus Z_{n})\dot{+}C$,
where $\hat{Z_{i}}$ indicates a term that is missing, is a maximal subalgebra
of $A$ for each $1\leq i\leq n$. Hence

$\phi(A)\subseteq\cap_{i=1}^{n}M_{i}\subseteq C,$

and $\phi(A)\cap Zsoc(A)=0$, a contradiction. It follows that $\phi(A)=0$.
$\Box$

###### Theorem 5.2

Let $A$ be a solvable Novikov algebra. Then $A$ is $\phi$-free if and only if
it splits over its zero socle.

Proof. The only point at which the hypothesis on the field is used in Theorem
5.1 above is to deduce that, if $Z$ is a minimal ideal contained in $\phi(A)$,
then it is a zero ideal. Here that holds because $\phi(A)$ is nilpotent, by
Corollary 4.4. $\Box$

###### Theorem 5.3

Let $A$ be a $\phi$-free bicommutative or Novikov algebra. Then
$Zsoc(A)=N(A)=Ann_{A}(Soc(A))$.

Proof. Clearly, $Zsoc(A)\subseteq N(A)$. Let $B$ be a minimal ideal of $A$ and
let $N$ be a nilpotent ideal of $A$. Then $B\cap N=0$ or $B$. If the former
holds, then $BN+NB\subseteq B\cap N=0$, so $N\subseteq Ann_{A}(B)$. So suppose
that the latter holds. Then $B\subseteq N$ and so $B\subseteq Ann_{A}(N)$, by
Lemma 2.4, whence $BN+NB=0$ again. It follows that $N(A)\subseteq
Ann_{A}(Soc(A))$. It now suffices to show that $Ann_{A}(Soc(A))\subseteq
Zsoc(A)$.

There is a subalgebra $C$ of $A$ such that $A=Zsoc(A)\dot{+}C$, by [16, Lemma
7.2]. Now, $Ann_{A}(Soc(A))\cap C$ is an ideal of $A$ and so must contain a
minimal ideal $D$. But $D$ must be a zero ideal, since it annihilates itself,
by assumption. Hence $D\subseteq Zsoc(A)\cap C=0$ and the result is proved.
$\Box$

In the next results we seek to characterise the $\phi$-free bicommutative and
Novikov algebras.

###### Theorem 5.4

Let $A$ be a bicommutative algebra with solvable radical $R$. Then $A$ is
$\phi$-free if and only if $A=Zsoc(A)\dot{+}(D\oplus E)$, where $D$ is a zero
subalgebra of $A$, $R=Zsoc(A)\dot{+}D$, $E=E^{2}\dot{+}U$,
$E^{2}=S_{1}\oplus\ldots\oplus S_{n}$, $S_{i}$ is a simple commutative
associative ideal of $E$, $U^{2}=0$, either $S_{i}U=0$ or $US_{i}=0$ for
$1\leq i\leq n$, and $Zsoc(A)=Z_{1}\oplus Z_{2}$ where $Z_{1}R=0$, $RZ_{2}=0$.

Proof. Suppose first that $A$ is $\phi$-free. We have that
$A=Zsoc(A)\dot{+}C$, by Theorem 5.1. Now $(C\cap R)C\subseteq RA\cap
C\subseteq N(A)\cap C=0$ and $C(C\cap R)\subseteq AR\cap C\subseteq N(A)\cap
C=0$, by Corollary 3.5. Put $C\cap R=D$. If $R(C)$ is the solvable radical of
$C$, then $Zsoc(A)+R(C)$ is a solvable ideal of $A$, and so $R(C)=R\cap C=D$
and $D^{2}=0$

Clearly, $\phi(C)\subseteq R(C)=D$, by Corollary 3.9, so
$C\phi(C)=\phi(C)C=0$. Moreover, for every minimal zero ideal $Z$,

$\displaystyle(Z\phi(C))A$
$\displaystyle=(Z\phi(C))C\subseteq(ZC)\phi(C)\subseteq Z\phi(C)\hbox{ and}$
$\displaystyle A(Z\phi(C))$ $\displaystyle=C(Z\phi(C))\subseteq
Z(C\phi(C))\subseteq Z\phi(C).$

Thus, $Z\phi(C)$ is an ideal of $C$, whence $Z\phi(C)=0$ or $Z\phi(C)=Z$.
Suppose the latter holds. Then $ZR\neq 0$, so $RZ=0$, by Theorem 3.10. It
follows that $\phi(C)Z=0$. But $\phi(C)$ and $Z$ are in $A^{2}$, which is
commutative, so we have a contradiction. Hence, $Z\phi(C)=0$. Similarly,
$\phi(C)Z=0$, and so $\phi(C)$ is an ideal of $A$. This implies that
$\phi(C)\subseteq\phi(A)=0$, by [16, Lemma 1].

Since $D$ is a zero ideal of $C$, $C=D\oplus E$ where $E$ is a semisimple
subalgebra of $C$. Hence $E=E^{2}\dot{+}U$ where
$E^{2}=S_{1}\oplus\ldots\oplus S_{n}$, the $S_{i}$ are simple ideals of $E$,
$U^{2}=0$ and either $S_{i}U=0$ or $US_{i}=0$ for each $1\leq i\leq n$, by
Corollary 3.15.

Let $Z$ be a minimal ideal in $Zsoc(A)$. Then $ZR=0$ or $ZR=Z$. Suppose the
latter holds. Then

$RZ=R(ZR)\subseteq ZR^{2}=0,$

by Corollary 3.3. This completes the proof of necessity.

The converse follows from Theorem 5.1. $\Box$

###### Theorem 5.5

Let $A$ be a $\phi$-free Novikov algebra with solvable radical $R$. Then
$A=Zsoc(A)\dot{+}C$, where $C$ is a subalgebra of $A$ and $A(C\cap R)=0$.

Proof. We have that $A=Zsoc(A)\dot{+}C$, by Theorem 5.1. Also $C(C\cap
R)\subseteq AR\cap C\subseteq Zsoc(A)\cap C=0$, by Proposition 4.6. Moreover,
$Zsoc(A)(C\cap R)=Zsoc(A)R=0$, since $R$ is right nilpotent. Hence $A(C\cap
R)=0$. $\Box$

###### Corollary 5.6

Let $A$ be a Novikov algebra with solvable radical $R$. Then
$(AR)R\subseteq\phi(A)\subseteq A^{2}$.

Proof. From Theorem 5.5, $AR\subseteq Zsoc(A)$, so $(AR)R=0$. $\Box$

###### Theorem 5.7

Let $A$ be a Novikov algebra with solvable radical $R$ over a field $F$ of
characteristic zero. If $R$ is nilpotent, then $A$ is $\phi$-free if and only
if $A=R\dot{+}S$ where $R$ is a zero algebra and $S$ is a semisimple
commutative associative algebra; that is, $S=\oplus_{i=1}^{n}S_{i}$, where
$S_{i}$ is a field for $1\leq i\leq n$.

Proof. Let $A$ be $\phi$-free. Then $A=R\dot{+}S$, where $R$ is a zero algebra
and $S$ is a semisimple subalgebra of $A$. Also, $S$ has the claimed form, by
Proposition 4.8.

The converse follows from Theorem 5.1, since $R$ is an $S$-module and so is
completely reducible. $\Box$

###### Corollary 5.8

Let $A$ be a Novikov algebra with solvable radical $R$ over a field $F$ of
characteristic zero. If $R$ is nilpotent, then $A$ is $\phi$-free if and only
if its radical is a zero algebra.

###### Corollary 5.9

Let $A$ be a Novikov algebra with solvable radical $R$ over a field $F$ of
characteristic zero. Then $\phi(A)\subseteq R^{2}$ and so $\phi(A)$ is
nilpotent..

Proof. The radical of $A/R^{2}$ is a zero algebra, and so $\phi(A/R^{2})=0$,
by Corollary 5.8. It follows from [16, Corollary 4.4] that $|phi(A)\subseteq
R^{2}$. Also, $\phi(A)$ is nilpotent, by Theorem 4.2. $\Box$

###### Corollary 5.10

Let $A$ be a Novikov algebra with solvable radical $R$ over a field $F$ of
characteristic zero. If $R$ is nilpotent, then $\phi(A)=R^{2}$.

Proof. We have that $\phi(A)\subseteq R^{2}$ from Corollary 5.9. But
$R^{2}=\phi(R)\subseteq\phi(A)$, by [15, Theorem 6] and [16, Lemma 4.1].
$\Box$

###### Lemma 5.11

If $A$ be an algebra with $A^{3}=0$, then $A$ is a Novikov algebra if and only
if it is bicommutative.

Proof. Substituting $A^{3}=0$ into left symmetry yields left commutativity.
$\Box$

Finally, we have two results concerning maximal subalgebras.

###### Theorem 5.12

Let $A$ be a Novikov algebra over a field $F$, and consider the following
statements.

  * (i)

$A$ is right nilpotent;

  * (ii)

$A^{3}\subseteq\phi(A)$; and

  * (iii)

all maximal subalgebras of $A$ are left ideals of $A$.

Then $(i)\Rightarrow(ii),(iii)$, $(ii)\Rightarrow(iii)$ and, if $F$ has
characterictic $p$, where $p=0$ or $p>\dim A+1$, $(ii)\Rightarrow(i)$.

Proof. $(i)\Rightarrow(ii)$: This follows from Theorem 4.2 and Corollary 5.6.

$(i)\Rightarrow(iii)$: This follows as in Proposition 3.11.

$(ii)\Rightarrow(iii)$: This follows from Lemma 5.11 and Proposition 3.11.

$(ii)\Rightarrow(i)$: We have that $\phi(A)$ is right nil, by Corollary 3.8
and that $A/\phi(A)$ is clearly right nilpotent. Hence $A$ is right nil, by
Theorem 3.7. It follows that $A^{2}$ is nilpotent, by [9, Theorem 1], and
hence that $A$ is right nilpotent, by Theorem 4.2. $\Box$

###### Theorem 5.13

Let $A$ be a solvable bicommutative algebra. Then the following are
equivalent.

  * (i)

$A^{3}\subseteq\phi(A)$; and

  * (ii)

all maximal subalgebras of $A$ are left ideals of $A$.

Proof. $(i)\Rightarrow(ii)$: This follows from Lemma 5.11 and Theorem 5.12.

$(ii)\Rightarrow(i)$: Suppose first that $A$ is $\phi$-free. Then
$A=\oplus_{i=1}^{n}Z_{i}\dot{+}C$ where $Z_{i}$ is a minimal zero ideal of $A$
and $C^{2}=0$, by Theorem 5.4. Suppose that $Z_{k}C=Z_{k}$ for some $1\leq
k\leq n$. Then $M_{k}=Z_{1}\oplus\ldots\oplus\hat{Z_{k}}\oplus\ldots\oplus
Z_{n}+C$, where the hat indicates a term that is missing from the sum, is a
maximal subalgebra of $A$. Since it is a left ideal of $A$, we have that
$Z_{k}M_{k}=Z_{k}C\subseteq Z_{k}\cap M_{k}=0$. Hence
$A^{3}=A^{2}A=(\oplus_{i=1}^{n}Z_{i})A=0$. The result now follows by
considering $A/\phi(A)$. $\Box$

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