# Path integral action for a resonant detector of gravitational waves in the
generalized uncertainty principle framework

$\mathbf{Soham}$ $\mathbf{Sen}^{a}$, $\mathbf{Sukanta}$
$\mathbf{Bhattacharyya}^{b}$ and $\mathbf{Sunandan}$
$\mathbf{Gangopadhyay}^{a}$
sensohomhary@gmail.comsukanta706<EMAIL_ADDRESS>a
Department of Astrophysics and High Energy Physics,
S.N. Bose National Centre for Basic Sciences
JD Block, Sector III, Salt Lake, Kolkata 700106, India
b Department of Physics,
West Bengal State University, Barasat, Kolkata 700126, India

###### Abstract

The Heisenberg uncertainty principle gets modified by the introduction of an
observer independent minimal length. In this work we have considered the
resonant gravitational wave detector in the modified uncertainty principle
framework where we have used the position momentum uncertainty relation with a
quadratic order correction only. We have then used the path integral approach
to calculate an action for the bar detector in presence of a gravitational
wave and then derived the Lagrangian of the system leading to the equation of
motion for the configuration-space position coordinate in one dimension. We
then find a perturbative solution for the coordinate of the detector for a
circularly polarized gravitational wave leading to a classical solution of the
same for given initial conditions. Using this classical form of the coordinate
of the detector, we finally obtain the classical form of the on-shell action
describing the harmonic oscillator-gravitational wave system. Finally, we have
obtained the free particle propagator containing the quantum fluctuation term
considering gravitational wave interaction.

## 1 Introduction

Quantum mechanics and general relativity are the two most successful theories
explaining the phenomena at two most fundamental length scales of the
universe. While quantum mechanics explains the intricacies of the atomic
length scale, general relativity [1, 2] sheds light on the large-scale
structure of the universe. In order to understand the fundamental mysteries of
the universe, we need a quantum theory of gravity, explaining the analytical
structure of the gravitational interaction at the quantum length scale.
Theories like loop quantum gravity[3, 4], string theory[5, 6], and
noncommutative geometry[7] have provided a convincing theoretical framework
explaining the Planck-scale nature of gravity but none of them have compelling
experimental evidence to support their claim of providing an exact description
of the quantum nature of gravity. Although all of them prescribe the existence
of an observer-independent minimal length which can be incorporated by the
modification of the standard Heisenberg uncertainty principle (HUP), also
known as the generalized uncertainty principle (GUP). The first few attempts
to improvise an integral relation between minimal length scale and gravity was
shown in [8, 9], followed by [10]. We also get a strong evidence of the
existence of this fundamental length scale from the various gedanken
experiments in quantum gravity phenomenology as well. This GUP framework has
been used to investigate several areas of theoretical physics, like black hole
physics and its thermodynamics[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21],
various quantum systems like particle in a box, simple harmonic oscillator[22,
23], optomechanical systems[24, 25, 26], and gravitational wave bar
detectors[27, 28]. There have been several recent studies involving the path
integral formalism of a non-relativistic particle moving in an arbitrary
potential in the generalized uncertainty principle framework[27, 29, 30]. The
simplest form the modified HUP can be written in the following form[31]

$\Delta q_{i}\Delta p_{i}\geq\frac{\hbar}{2}\left[1+\gamma\left(\Delta
p^{2}+\left<p\right>^{2}\right)+2\gamma\left(\Delta
p_{i}^{2}+\left<p_{i}\right>^{2}\right)\right];~{}i=1,2,3$ (1)

where $p^{2}=\sum_{k=1}^{3}p_{k}p_{k}$ and $q_{k},p_{k}$ are the phase space
position and its conjugate momenta. In eq.(1) the GUP parameter $\gamma$ in
terms of the dimensionless parameter $\gamma_{0}$ can be recast as follows

$\gamma=\frac{\gamma_{0}}{m_{p}^{2}c^{2}}$ (2)

where $m_{p}$ is the Planck mass and $c$ is the speed of light. It is quite
natural to realize that the order of magnitude of the GUP parameter will play
a significant role in providing an understanding of the GUP effects. There
have been several studies to find a bound on the GUP parameter itself[28, 17,
22, 32, 33, 34, 35, 36, 37, 38].

In the 1969, the first proposition to detect gravitational waves was made by
J. Weber[39] which was followed by a subsequent paper in 1982 by Ferrari et
al.[40]. Bar detectors currently have a sensitivity $\frac{\Delta L}{L}\sim
10^{-19}$[41], where $\Delta L$ is fractional variation of the length $L$
($\sim 1$ meter) of the bar detector. A historical perspective of these
resonant detectors is given in [42]. The detection of gravitational waves by
the LIGO[43, 44] and Virgo[45] detectors have unveiled a new realm of quantum
gravity phenomenology. There have been several recent investigations regarding
the traces of quantum gravitational effects in these gravitational wave
detectors. A lot of investigation has been done to check if any signature of
this fundamental Planck length, whether it is noncommutativity[46, 47, 48, 49,
50, 51] or GUP[52, 28], is visible in GW bar detectors. We would like to point
out that till date there has not been a successful detection of gravitational
waves in resonant bar detectors. However, there is a good hope that the
sensitivity of the detectors would increase in future enabling detection of
these waves. The AURIGA (Antenna Ultracriogenica Risonante per l’Indagine
Gravitazionale Astronomica) detector at INFN, Italy is probably the only
functional bar detector. These bar detectors are sensitive to frequencies of
the order of $1$kHz along with a strain sensitivity of the order
$10^{-19}$[53]. In case of astrophysical events, collapsing and bouncing cores
of supernova can be a source to huge intensities of gravitational waves having
frequencies in the vicinity of $1-3$kHz. The value of the strain sensitivity
can be calculated using Thorne’s formula[54]. The strain sensitivity ($h$),
according to this formula, is given by

$h=2.7\times 10^{-20}\left[\frac{\Delta
E_{GW}}{M_{s}c^{2}}\right]^{\frac{1}{2}}\left[\frac{1kHz}{f}\right]^{\frac{1}{2}}\left[\frac{10Mpc}{d}\right]$
(3)

where $\Delta E_{GW}$ is the energy converted to gravitational waves, $f$ is
the characteristic frequency of the burst, $M_{s}$ is the solar mass and $d$
is the distance of the burst source from earth. A possible value of fraction
of energy converted to gravitational waves for supernova events is around
$7\times 10^{-4}$. Now for $h\sim 3\times 10^{-19}$ and $f\sim 0.9kHz$, the
distance $d$ has the value around $25kpc$. The occurrence of such a supernova
event of the required magnitude at this distance from the earth would
definitely result in the detection of gravitational waves by the bar
detectors. An effort to increase the sensitivity of these detectors to $h\sim
10^{-20}$ is presently being carried out, and achieving this sensitivity would
increase the distance of the supernova event from the earth to $250kpc$, which
is more likely to occur. The main motivation to work with a gravitational wave
bar detector is that it is a very resourceful and economic alternative to the
LIGO/VIRGO detectors.

In this work we investigate the path integral formalism of a resonant
gravitational wave bar detector interacting with the gravitational wave
emitted from a distant source in the GUP framework. The incoming gravitational
waves interact with the elastic matter in the resonant bar detector causing
tiny vibrations called phonons. Physically we can describe these detectors as
a quantum mechanical gravitational wave-harmonic oscillator (GW-HO) system
because we call these vibrations; the quantum mechanical forced harmonic
oscillator. To calculate the perturbative solution to the system we use the
gravitational wave and generalized uncertainty modifications as perturbations.
Our study presents a path integral approach to look at such a system and is
the first work using path integral. The advantage of working with path
integrals is that the effective action describing the system can be easily
read off from the structure of the configuration space path integral[55].

## 2 The gravitational wave resonant detector interaction model

To begin the discussion, we need to write down the Hamiltonian for the
resonant bar detector in the presence of a gravitational wave in the
generalized uncertainty principle framework. The modified commutation relation
following from eq.(1) takes the following from[31]

$[\hat{q}_{i},\hat{p}_{j}]=i\hbar\left[\delta_{ij}+\gamma\delta_{ij}\hat{p}^{2}+2\gamma\hat{p}_{i}\hat{p}_{j}\right]$
(4)

where $i,j=1,2,3$. The modified position and momentum operators $\hat{q}_{i}$
and $\hat{p}_{i}$ in terms of the usual variables $\hat{q}_{0i}$ and
$\hat{p}_{0i}$ read

$\hat{q}_{i}=\hat{q}_{0i}~{},~{}\hat{p}_{i}=\hat{p}_{0i}\left(1+\gamma\hat{p}_{0}^{2}\right)~{}.$
(5)

Here $\hat{p}_{0}^{2}=\sum_{k=1}^{3}\hat{p}_{0k}\hat{p}_{0k}$ and
$[\hat{q}_{0i},\hat{p}_{0j}]=i\hbar\delta_{ij}$. In order to write the
Hamiltonian of the system we start by analyzing the background metric as a
superposition of a small perturbation on the flat background metric. The
background metric is taken as follows

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (6)

where $\eta_{\mu\nu}=\text{diag}\\{1,-1,-1,-1\\}$ and $|h_{\mu\nu}|\ll 1$. We
now consider a two dimensional harmonic oscillator with mass $m$ and intrinsic
frequency $\varpi$. The geodesic deviation equation for the aforementioned
system in the proper detector frame is given as follows[56]

$\begin{split}m\ddot{q}^{k}&=-mR^{k}_{~{}0l0}q^{l}-m\varpi^{2}q^{k}\\\
\implies\ddot{q}^{k}&=\frac{d\Gamma^{k}_{~{}0l}}{dt}q^{l}-\varpi^{2}q^{k}~{};~{}k=1,2\end{split}$
(7)

where $R^{k}_{~{}0l0}$ in terms of the background perturbation is given by

$R^{k}_{~{}0l0}=-\frac{d\Gamma^{k}_{~{}0l}}{dt}=-\frac{\ddot{h}_{kl}}{2}~{}.$
(8)

Note that here we are using the transverse traceless gauge to get rid of the
unphysical degrees of freedom. The Lagrangian from which eq.(8) can be
obtained reads

$L=\frac{1}{2}m\dot{q}_{k}^{2}-m\Gamma^{k}_{~{}0l}\dot{q}_{k}q^{l}-\frac{1}{2}m\varpi^{2}q_{k}^{2}~{}.$
(9)

The Hamiltonian corresponding to the Lagrangian in eq.(9) reads

$H=\frac{1}{2m}\left(p_{k}+m\Gamma^{k}_{~{}0l}q^{l}\right)^{2}+\frac{1}{2}m\varpi^{2}q_{l}^{2}~{}.$
(10)

To write the Hamiltonian in eq.(10) in quantum mechanical description we just
elevate $q$ and $p$ to the operator prescription. Therefore, the Hamiltonian
in terms of the position and momentum operators can be expressed as follows

$\hat{H}=\frac{1}{2m}\left(\hat{p}_{k}+m\Gamma^{k}_{~{}0l}\hat{q}^{l}\right)^{2}+\frac{1}{2}m\varpi^{2}\hat{q}_{l}^{2}~{}.$
(11)

Using the representation of the position and momentum operators in eq.(5), the
Hamiltonian (11) of the GW-HO system in presence of GUP can be written as
follows

$\hat{H}=\left(\frac{\hat{p}_{0k}^{2}}{2m}+\frac{1}{2}m\varpi^{2}\hat{q}_{0k}^{2}\right)+\frac{\gamma}{m}\hat{p}_{0k}^{2}\hat{p}_{0}^{2}+\frac{1}{2}\Gamma^{k}_{~{}0l}\left(\hat{p}_{0k}\hat{q}^{0l}+\hat{q}^{0l}\hat{p}_{0k}\right)+\frac{\gamma}{2}\Gamma^{k}_{~{}0l}\left(\hat{p}_{0k}\hat{p}_{0}^{2}\hat{q}^{0l}+\hat{q}^{0l}\hat{p}_{0k}\hat{p}_{0}^{2}\right)~{}.$
(12)

Now a typical bar is a cylinder of length $L\equiv 3m$ and radius $R\equiv
30cm$[56]. Hence in a first approximation, we can treat the GW detector in
presence of GUP as a one dimensional HO. The Hamiltonian in eq.(12) can be
recast in one dimension as follows

$\hat{H}=\frac{p^{2}}{2m}+\frac{1}{2}m\varpi^{2}q^{2}+\gamma\frac{p^{4}}{m}+\frac{1}{2}\Gamma^{1}_{~{}01}(pq+qp)+\frac{\gamma}{2}\Gamma^{1}_{~{}01}(p^{3}q+qp^{3})$
(13)

where for notational simplicity we have used $\hat{p}_{01}=p$ and
$\hat{q}_{01}=q$. In the next section we will proceed to construct the path
integral formalism of the GW-HO system in presence of the GUP and calculate
the propagation kernel for that system.

## 3 Path integral and the propagation kernel

In this section we will use the Hamiltonian in eq.(13) to calculate the
propagation kernel via the path integral approach. We consider the initial and
the final state of the Hamiltonian in eq.(13) at initial time $t_{i}$ and
final time $t_{f}$ as $\left|q_{i},t_{i}\right>$ and
$\left|q_{f},t_{f}\right>$ respectively. The general form of the propagation
kernel can be written as follows

$\begin{split}\left<q_{f},t_{f}|q_{i},t_{i}\right>&=\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}dq_{N-1}\ldots
dq_{1}\left<q_{f},t_{f}|q_{N-1},t_{N-1}\right>\left<q_{N-1},t_{N-1}|q_{N-2},t_{N-2}\right>\ldots\left<q_{1},t_{1}|q_{i},t_{i}\right>\\\
&=\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\left<q_{f}\right|e^{-\frac{i\hat{H}(t_{f}-t_{N-1})}{\hbar}}\left|q_{N-1}\right>\ldots\left<q_{1}\right|e^{-\frac{i\hat{H}(t_{1}-t_{i})}{\hbar}}\left|q_{i}\right>\\\
&=\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\prod\limits_{\beta=0}^{N-1}\left<q_{\beta+1}\right|e^{-\frac{i\hat{H}(t_{\beta+1}-t_{\beta})}{\hbar}}\left|q_{\beta}\right>\end{split}$
(14)

where $t_{f}=t_{N}$, $t_{i}=t_{0}$ and $t_{N}-t_{N-1}=\Delta t$. Now we will
introduce the complete set of momentum eigenstates
$\left(\int_{-\infty}^{+\infty}dp\left|p\right>\left<p\right|=1\right)$ in the
following way

$\begin{split}\left<q_{f},t_{f}|q_{i},t_{i}\right>=&\lim\limits_{N\rightarrow\infty}\int\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\prod\limits_{\beta=0}^{N-1}\int~{}dp_{\beta}\left<q_{\beta+1}|p_{\beta}\right>\left<p_{\beta}|q_{\beta}\right>\exp\left(-\frac{iH(q_{\beta},p_{\beta})(t_{\beta+1}-t_{\beta})}{\hbar}\right)\\\
=&\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\prod\limits_{\beta=0}^{N-1}\int_{-\infty}^{+\infty}\frac{dp_{\beta}}{2\pi\hbar}\exp\biggr{[}\frac{i\Delta
t}{\hbar}\sum\limits_{\beta=0}^{N-1}\biggr{[}\frac{p_{\beta}(q_{\beta+1}-q_{\beta})}{\Delta
t}-\biggr{(}\frac{p_{\beta}^{2}}{2m}+\frac{1}{2}m\varpi^{2}q_{\beta}^{2}+\frac{\gamma
p_{\beta}^{4}}{m}\\\
&+\frac{p_{\beta}q_{\beta}(h_{\beta+1}-h_{\beta})}{2\Delta t}+\frac{\gamma
p_{\beta}^{3}q_{\beta}({h}_{\beta+1}-h_{\beta})}{2\Delta
t}\biggr{)}\biggr{]}\biggr{]}\end{split}$ (15)

where we have used $h_{11}=h$ . The final form of eq.(15) in the $\Delta
t\rightarrow 0$ limit can be recast as follows

$\left<q_{f},t_{f}|q_{i},t_{i}\right>=\int\mathcal{D}q\mathcal{D}p\exp\left(\frac{i}{\hbar}\mathcal{S}\right)$
(16)

where $\mathcal{S}$ is the phase space action. The phase space action is given
as follows

$\mathcal{S}=\int_{t_{i}}^{t_{f}}dt\left[p\dot{q}-\left(\frac{p^{2}}{2m}+\frac{\dot{h}_{11}}{2}pq+\frac{1}{2}m\varpi^{2}q^{2}+\frac{\gamma
p^{4}}{m}+\frac{\gamma\dot{h}_{11}}{2}p^{3}q\right)\right]~{}.$ (17)

To obtain the configuration space Lagrangian we will simplify eq.(15) as
follows

$\begin{split}\left<q_{f},t_{f}|q_{i},t_{i}\right>&\cong\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\prod\limits_{\beta=0}^{N-1}\int_{-\infty}^{+\infty}\frac{dp_{\beta}}{2\pi\hbar}\left[1-\frac{i\gamma\Delta
t}{m\hbar}\left(p_{\beta}^{4}+\frac{h_{\beta+1}-h_{\beta}}{2\Delta
t}p_{\beta}^{3}q_{\beta}\right)+\mathcal{O}(\gamma^{2})\right]\end{split}$
(18) $\begin{split}\times\exp\left[\frac{i\Delta
tm}{2\hbar}\left[\left(\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{h_{\beta+1}-h_{\beta}}{4\Delta
t}q_{\beta}\right)^{2}-\varpi^{2}q_{\beta}^{2}\right]\right]\exp\left[-\frac{i\Delta
t}{2m\hbar}\left[p_{\beta}-\left(\frac{m(q_{\beta+1}-q_{\beta})}{\Delta
t}-\frac{m(h_{\beta+1}-h_{\beta})q_{\beta}}{4\Delta
t}\right)\right]^{2}\right]~{}.\end{split}$

To do the momentum integral for each $\beta$ value, we shall do the following
coordinate transformation

$\bar{p}_{\beta}=p_{\beta}-\left(\frac{m(q_{\beta+1}-q_{\beta})}{\Delta
t}-\frac{m(h_{\beta+1}-h_{\beta})q_{\beta}}{4\Delta t}\right)~{}.$ (19)

Using eq.(19) in eq.(18), the propagation kernel upto $\sim\gamma,h$ can be
recast as

$\begin{split}\left<q_{f},t_{f}|q_{i},t_{i}\right>\cong&\lim\limits_{N\rightarrow\infty}\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\prod\limits_{\beta=0}^{N-1}\int_{-\infty}^{+\infty}\frac{d\bar{p}_{\beta}}{2\pi\hbar}\left[1-\frac{i\gamma\Delta
t}{m\hbar}\biggr{[}\biggr{(}\bar{p}_{\beta}+\left(\frac{m(q_{\beta+1}-q_{\beta})}{\Delta
t}-\frac{m(h_{\beta+1}-h_{\beta})q_{\beta}}{4\Delta t}\right)\right)^{4}+\\\
&\frac{h_{\beta+1}-h_{\beta}}{2\Delta
t}\left(\bar{p}_{\beta}+\left(\frac{m(q_{\beta+1}-q_{\beta})}{\Delta
t}-\frac{m(h_{\beta+1}-h_{\beta})q_{\beta}}{4\Delta
t}\right)\right)^{3}q_{\beta}\biggr{]}+\mathcal{O}(\gamma^{2})\biggr{]}\exp\left[-\frac{i\Delta
t}{2m\hbar}{\bar{p}_{\beta}}^{2}\right]\\\ &\times\exp\left[\frac{i\Delta
tm}{2\hbar}\left[\left(\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{h_{\beta+1}-h_{\beta}}{4}q_{\beta}\right)^{2}-\varpi^{2}q_{\beta}^{2}\right]\right]~{}.\end{split}$
(20)

The momentum integral in eq.(20) can be obtained as follows

$\begin{split}\left<q_{\beta+1},t_{\beta+1}|q_{\beta},t_{\beta}\right>\cong&\sqrt{\frac{m}{2\pi
i\hbar\Delta t}}\biggr{\\{}1-6\gamma
m^{2}\left(\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{h_{\beta+1}-h_{\beta}}{4\Delta t}q_{\beta}\right)^{2}-\frac{3\gamma
m^{2}(h_{\beta+1}-h-\beta)}{2\Delta
t}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta t}\\\
&-\frac{(h_{\beta+1}-h_{\beta})}{4\Delta
t}q_{\beta}\biggr{)}q_{\beta}+\frac{3i\gamma m\hbar}{\Delta
t}\biggr{\\}}~{}\exp\biggr{[}\frac{im\Delta
t}{2\hbar}\biggr{[}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{2}-2\gamma m^{2}\\\
&\times\biggr{[}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{4}+\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{2}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{3}\biggr{]}-\varpi^{2}q_{\beta}^{2}\biggr{]}\biggr{]}~{}.\end{split}$
(21)

Using eq.(21) in eq.(20), we obtain the form of the propagation kernel upto
some constant factor as follows

$\begin{split}\left<q_{f},t_{f}|q_{i},t_{i}\right>&=\int_{-\infty}^{+\infty}\prod\limits_{\alpha=1}^{N-1}dq_{\alpha}\exp\biggr{[}\sum\limits_{\beta=0}^{N-1}\frac{im\Delta
t}{2\hbar}\biggr{\\{}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{2}-2\gamma
m^{2}\biggr{(}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta t}\\\
&-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{4}+\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{2}\biggr{(}\frac{q_{\beta+1}-q_{\beta}}{\Delta
t}-\frac{(h_{\beta+1}-h_{\beta})q_{\beta}}{4}\biggr{)}^{3}\biggr{)}-\varpi^{2}q_{\beta}^{2}\biggr{\\}}\biggr{]}~{}.\end{split}$
(22)

Imposing the $\Delta t\rightarrow 0$ limit in eq.(22), the final form of the
propagation kernel has the usual configuration space path integral structure
as follows

$\left<q_{f},t_{f}|q_{i},t_{i}\right>=\mathcal{N}(T,\gamma,\dot{h})\int\mathcal{D}qe^{\frac{i}{\hbar}S}~{}.$
(23)

In the above equation the configuration space structure of the action $S$ is
given as follows

$\begin{split}S=&\int_{t_{i}}^{t_{f}}dt\left(\frac{m}{2}\left(\dot{q}-\frac{\dot{h}q}{4}\right)^{2}-\frac{1}{2}m\varpi^{2}q^{2}-\gamma
m^{3}\left(\dot{q}-\frac{\dot{h}q}{4}\right)^{4}-\frac{\gamma
m^{3}\dot{h}q}{2}\left(\dot{q}-\frac{\dot{h}q}{4}\right)^{3}\right)\\\
\cong&\int_{t_{i}}^{t_{f}}dt\left(\frac{m}{2}\dot{q}^{2}-\frac{1}{2}m\varpi^{2}q^{2}-\frac{m\dot{h}\dot{q}q}{4}-\gamma
m^{3}\dot{q}^{4}+\frac{1}{2}m^{3}\gamma\dot{h}\dot{q}^{3}q\right)~{}.\end{split}$
(24)

In the last line of the above eq.(24), we have kept terms upto
$\mathcal{O}(h,\gamma)$. The Lagrangian can be easily read off from eq.(24) as
follows

$L=\frac{m}{2}\dot{q}^{2}-\frac{1}{2}m\varpi^{2}q^{2}-\frac{m\dot{h}\dot{q}q}{4}-\gamma
m^{3}\dot{q}^{4}+\frac{1}{2}\gamma m^{3}\dot{h}\dot{q}^{3}q~{}.$ (25)

The equation of motion following from the Lagrangian reads

$\ddot{q}-\frac{\ddot{h}q}{4}+\varpi^{2}q-12m^{2}\gamma\ddot{q}\dot{q}^{2}+3\gamma
m^{2}\dot{h}\ddot{q}\dot{q}q+\frac{3}{2}\gamma
m^{2}\ddot{h}\dot{q}^{2}q+\gamma m^{2}\dot{h}\dot{q}^{3}=0~{}.$ (26)

In the next section we calculate the classical solution for the above equation
of motion.

## 4 Obtaining the classical solution for a periodic circularly polarized
gravitational wave

To obtain the classical solution we shall consider a circularly polarized
gravitational wave in the transverse traceless gauge. Now for a periodic
circularly polarized gravitational wave the perturbation term $h$ containing
the polarization information reads

$h_{kl}(t)=2f_{0}\left(\varepsilon_{\times}(t)\sigma^{1}_{kl}+\varepsilon_{+}(t)\sigma^{3}_{kl}\right)~{};~{}k,l=1,2$
(27)

where $2f_{0}$ is the amplitude of the gravitational wave (here $f_{0}$ is
very small), $\sigma^{1}$ and $\sigma^{3}$ are the Pauli spin matrices. In
eq.(27) $(\varepsilon_{+}(t),\varepsilon_{\times}(t))$ are the two possible
polarization states of the gravitational wave satisfying the condition
$\varepsilon_{+}(t)^{2}+\varepsilon_{\times}(t)^{2}=1$. In this particular
scenario the chosen functional forms of the polarization states can be written
as follows

$\varepsilon_{+}(t)=\cos(\Omega t)~{},~{}\varepsilon_{\times}(t)=\sin(\Omega
t)$ (28)

with $\Omega$ being the frequency of the gravitational wave. In our case, we
will consider that the only non zero polarization state is
$\varepsilon_{+}(t)=\cos(\Omega t)$. Therefore, in one dimension the
perturbation term can be written as $h=2f_{0}\cos(\Omega t)$. The equation of
motion in eq.(26) upto $\mathcal{O}(f_{0},\gamma)$ takes the form as follows

$\ddot{q}+\omega^{2}q-12m^{2}\gamma\ddot{q}\dot{q}^{2}=0$ (29)

where $\omega^{2}=\varpi^{2}-\frac{\ddot{h}}{4}$. For the equation of motion
in eq.(29), we consider a solution upto $\mathcal{O}(f_{0},\gamma)$ as

$q(t)=q_{0}(t)+f_{0}q_{f_{0}}(t)+\gamma q_{\gamma}(t)~{}.$ (30)

For the form $q(t)$ in the above equation, we obtain the solution of eq.(26)
as a linear combination as $q_{0}(t),q_{f_{0}}(t)\text{ and }q_{\gamma}(t)$.
The analytical forms of $q_{0}(t),q_{f_{0}}(t)\text{ and }q_{\gamma}(t)$ are
given as follows

$\displaystyle q_{0}(t)=$ $\displaystyle\mathcal{A}_{1}\cos(\varpi
t)+\mathcal{A}_{2}\sin(\varpi t)~{},$ (31) $\displaystyle q_{f_{0}}(t)=$
$\displaystyle\mathcal{A}_{3}\cos(\varpi t)+\mathcal{A}_{4}\sin(\varpi
t)-\frac{\Omega}{2(4\varpi^{2}-\Omega^{2})}[\Omega\cos(\Omega
t)\left\\{\mathcal{A}_{1}\cos(\varpi t)+\mathcal{A}_{2}\sin(\varpi
t)\right\\}$ $\displaystyle-2\varpi\sin(\Omega
t)\left\\{\mathcal{A}_{2}\cos(\varpi t)-\mathcal{A}_{1}\sin(\varpi
t)\right\\}]~{},$ (32) $\displaystyle q_{\gamma}(t)=$
$\displaystyle\mathcal{A}_{5}\cos(\varpi t)+\mathcal{A}_{6}\sin(\varpi
t)-\frac{3m^{2}\varpi^{2}}{2}[t\varpi\mathcal{A}_{1}(\mathcal{A}_{1}^{2}+\mathcal{A}_{2}^{2})\sin(\varpi
t)-t\varpi\mathcal{A}_{2}(\mathcal{A}_{1}^{2}+\mathcal{A}_{2}^{2})\cos(\varpi
t)$
$\displaystyle+\frac{\mathcal{A}_{1}}{4}(\mathcal{A}_{1}^{2}-3\mathcal{A}_{2}^{2})\cos(3\varpi
t)-\frac{\mathcal{A}_{2}}{4}(\mathcal{A}_{2}^{2}-3\mathcal{A}_{1}^{2})\sin(3\varpi
t)]$ (33)

where
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3},\mathcal{A}_{4},\mathcal{A}_{5}\text{
and }\mathcal{A}_{6}$ are arbitrary constants which we will calculate for the
$q_{cl}(t)$. To obtain the form of the above constants we will apply the
following set of the initial conditions

$\begin{split}q(t)=\biggr{\\{}\begin{matrix}&q_{0}&\text{for}&t=0\\\
&q_{f}&\text{for}&t=T\end{matrix}~{}.\end{split}$ (34)

Using the initial conditions in eq.(34), the constants can be obtained as
follows

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=q_{0}~{},~{}\mathcal{A}_{2}=\frac{q_{f}-q_{0}\cos(\varpi
T)}{\sin(\varpi T)}~{},$ (35) $\displaystyle\mathcal{A}_{3}$
$\displaystyle=\frac{\mathcal{A}_{1}\Omega^{2}}{2(4\varpi^{2}-\Omega^{2})}~{},$
(36) $\displaystyle\mathcal{A}_{4}$
$\displaystyle=\frac{\Omega\left\\{\cos(\varpi
T)\left[\Omega\mathcal{A}_{1}\cos(\Omega T)-2\varpi\mathcal{A}_{2}\sin(\Omega
T)\right]+\sin(\varpi T)\left[\Omega\mathcal{A}_{2}\cos(\Omega
T)+2\varpi\mathcal{A}_{1}\sin(\omega
T)\right]\right\\}}{2(4\varpi^{2}-\Omega^{2})\sin(\varpi
T)}-\mathcal{A}_{3}\cot(\varpi T)~{},$ (37) $\displaystyle\mathcal{A}_{5}$
$\displaystyle=\frac{3}{8}m^{2}\varpi^{2}\mathcal{A}_{1}\left(\mathcal{A}_{1}^{2}-3\mathcal{A}_{2}^{2}\right)~{},$
(38) $\displaystyle\mathcal{A}_{6}$
$\displaystyle=\frac{3m^{2}\varpi^{2}\left[\varpi
T\left(\mathcal{A}_{1}\sin(\varpi T)-\mathcal{A}_{2}\cos(\varpi
T)\right)(\mathcal{A}_{1}^{2}+\mathcal{A}_{2}^{2})+\frac{\mathcal{A}_{1}(\mathcal{A}_{1}^{2}-3\mathcal{A}_{2}^{2})\cos(3\varpi
T)}{4}-\frac{\mathcal{A}_{2}(\mathcal{A}_{2}^{2}-3\mathcal{A}_{1}^{2})\sin(3\varpi
T)}{4}\right]}{2\sin(\varpi T)}-\mathcal{A}_{5}\cot(\varpi T).$ (39)

Using eq.(30) along with eq.(s)(35-39) in eq.(24) (with $h$ being replaced by
$2f_{0}\cos(\Omega t)$), we obtain the form of the classical action upto
$\mathcal{O}(\gamma,f)$ as follows

$S_{\mathcal{C}}=S_{\mathcal{C}}^{(0)}+S_{\mathcal{C}}^{(\gamma)}+S_{\mathcal{C}}^{(f_{0})}$
(40)

where $S_{\mathcal{C}}^{(0)},~{}S_{\mathcal{C}}^{(\gamma)}$, and
$S_{\mathcal{C}}^{(f_{0})}$ are given by the following equations

$\displaystyle S_{\mathcal{C}}^{(0)}$
$\displaystyle=\frac{m\varpi}{2\sin(\varpi
T)}\left((q_{0}^{2}+q_{f}^{2})\cos(\varpi T)-2q_{0}q_{f}\right)~{},$ (41)
$\displaystyle S_{\mathcal{C}}^{(\gamma)}$ $\displaystyle=-\frac{\gamma
m^{3}\varpi^{3}}{32\sin^{4}(\varpi T)}\biggr{[}12\varpi
T\left(q_{f}^{4}+4q_{f}^{2}q_{0}^{2}+q_{0}^{4}\right)-48q_{0}q_{f}\varpi
T\cos(\varpi T)(q_{f}^{2}+q_{0}^{2})+24q_{0}^{2}q_{f}^{2}\varpi T\cos(2\varpi
T)$ $\displaystyle-44q_{0}q_{f}\sin(\varpi
T)(q_{0}^{2}+q_{f}^{2})+4\sin(2\varpi
T)\left(2q_{0}^{4}+15q_{0}^{2}q_{f}^{2}+2q_{f}^{4}\right)-12q_{0}q_{f}\sin(3\varpi
T)(q_{0}^{2}+q_{f}^{2})+\sin(4\varpi T)(q_{0}^{4}+q_{f}^{4})\biggr{]},$ (42)
$\displaystyle S_{\mathcal{C}}^{(f_{0})}$
$\displaystyle=-\frac{f_{0}m\varpi\Omega}{2\sin(\varpi
T)(4\varpi^{2}-\Omega^{2})}\biggr{[}\frac{\varpi\sin(\Omega T)}{\sin(\varpi
T)}\left(q_{0}^{2}-2q_{0}q_{f}\cos(\varpi T)+q_{f}^{2}\cos(2\varpi
T)\right)+2q_{0}q_{f}\Omega\cos^{2}\left(\frac{\Omega T}{2}\right)$
$\displaystyle-\Omega\cos(\varpi T)\left(q_{0}^{2}+q_{f}^{2}\cos(\Omega
T)\right)\biggr{]}~{}.$ (43)

Therefore, we now have the final form of the propagator for the resonant bar
detector interacting with a gravitational wave as follows

$\left<q_{f},T|q_{0},0\right>=\sqrt{\frac{m\varpi}{2\pi i\hbar\sin(\varpi
T)}}\tilde{\mathcal{N}}(T,\gamma,f_{0})e^{\frac{i}{\hbar}S_{cl}}.$ (44)

To obtain an overall structure of the fluctuation parameter in the above
equation we consider the free particle structure involving gravitational wave
(GW) interaction only. In this case, the infinitesimal propagator considering
the particle GW interaction from eq.(15) can be extracted as follows (in the
$\varpi\rightarrow 0$ limit)

$\begin{split}\left<q_{1},\Delta
t|q_{0},0\right>=&\int_{-\infty}^{\infty}\frac{dp_{0}}{2\pi\hbar}\exp\biggr{[}\frac{i\Delta
t}{\hbar}\left(p_{0}\frac{(q_{1}-q_{0})}{\Delta
t}-\left(\frac{p_{0}^{2}}{2m}+\frac{\gamma
p_{0}^{4}}{m}+\frac{p_{0}q_{0}f_{0}}{\Delta t}\left(\cos(\Omega\Delta
t)-1\right)\right)\right)\biggr{]}\\\ \simeq&\sqrt{\frac{m}{2\pi i\hbar\Delta
t}}e^{\frac{im}{2\hbar\Delta
t}(q_{1}-q_{0})^{2}}\biggr{[}1+\frac{3im\gamma\hbar}{\Delta t}-6\gamma
m^{2}\left(\frac{q_{1}-q_{0}}{\Delta t}\right)^{2}-\frac{i\gamma
m^{3}(q_{1}-q_{0})^{4}}{\hbar\Delta t^{3}}\\\
&-\frac{if_{0}q_{0}}{\hbar}\left(\frac{m(q_{1}-q_{0})}{\Delta
t}\right)(\cos(\Omega\Delta t)-1)\biggr{]}~{}.\end{split}$ (45)

Now the total propagator can be written using the set of infinitesimal
propagators as follows

$\begin{split}\left<q_{f},T|q_{0},0\right>&\simeq\left(\frac{m}{2\pi
i\hbar\Delta t}\right)^{\frac{N}{2}}\int dq_{1}dq_{2}\cdots
dq_{N-1}e^{\frac{im}{2\hbar\Delta
t}\left[(q_{1}-q_{0})^{2}+(q_{2}-q_{1})^{2}+\cdots+(q_{f}-q_{N-1})^{2}\right]}\biggr{[}1+\frac{3i\gamma
m\hbar N}{\Delta t}-\frac{6\gamma m^{2}}{\Delta t^{2}}((q_{1}-q_{0})^{2}\\\
&+(q_{2}-q_{1})^{2}+\cdots+(q_{f}-q_{N-1})^{2})-\frac{i\gamma
m^{3}}{\hbar\Delta
t^{3}}((q_{1}-q_{0})^{4}+(q_{2}-q_{1})^{4}+\cdots+(q_{f}-q_{N-1})^{4})\\\
&-\frac{if_{0}m}{\hbar\Delta t^{2}}\left[q_{0}(q_{1}-q_{0})(\cos(\Omega\Delta
t)-1)+\cdots+q_{N-1}(q_{f}-q_{N-1})(\cos(N\Omega\Delta
t)-\cos((N-1)\Omega\Delta t))\right]\biggr{]}~{}.\end{split}$ (46)

In the absence of the gravitational wave[57], the form of the propagator in
eq.(46) reads

$\left<q_{f},T|q_{0},0\right>=\sqrt{\frac{m}{2\pi i\hbar
T}}e^{\frac{im}{2\hbar T}(q_{f}-q_{0})^{2}}\left(1+\frac{3i\gamma
m\hbar}{T}-6\gamma m^{2}\left(\frac{q_{f}-q_{0}}{T}\right)^{2}-\frac{i\gamma
m^{3}}{\hbar T^{3}}(q_{f}-q_{0})^{4}\right)~{}.$ (47)

In presence of the gravitational wave, the propagator has the form given as

$\begin{split}\left<q_{f},T|q_{0},0\right>&\simeq\sqrt{\frac{m}{2\pi i\hbar
T}}e^{\frac{im}{2\hbar T}(q_{f}-q_{0})^{2}}\biggr{(}1+\frac{3i\gamma
m\hbar}{T}-6\gamma m^{2}\left(\frac{q_{f}-q_{0}}{T}\right)^{2}-\frac{i\gamma
m^{3}}{\hbar T^{3}}(q_{f}-q_{0})^{4}\\\
&+\frac{if_{0}mT}{\hbar}\left(\frac{(q_{f}-q_{0})}{T}\right)^{2}[\cos(\Omega
T)-1]-\frac{if_{0}q_{f}}{\hbar}\left(\frac{m(q_{f}-q_{0})}{T}\right)[\cos(\Omega
T)-1]\biggr{)}\\\ &\simeq\sqrt{\frac{m}{2\pi i\hbar
T}}\tilde{\mathcal{N}}(T,\gamma,f_{0})e^{\frac{i}{\hbar}S_{cl}^{(f)}}\end{split}$
(48)

where $S_{cl}^{(f)}$ is the classical action involving free particles and
gravitational waves given by

$\begin{split}S_{cl}^{(f)}=\frac{m}{2T}(q_{f}-q_{0})^{2}-\frac{\gamma
m^{3}}{T^{3}}(q_{f}-q_{0})^{4}-\frac{mf_{0}}{2T}(q_{f}-q_{0})\left[(q_{f}\cos[\Omega
T]-q_{0})-(q_{f}-q_{0})\frac{\sin[\Omega T]}{\Omega T}\right]\end{split}$ (49)

and the form of the fluctuation term is given as follows

$\begin{split}\tilde{\mathcal{N}}(T,\gamma,f_{0})&\simeq 1+\frac{3i\gamma
m\hbar}{T}-6\gamma
m^{2}\left[\frac{q_{f}-q_{0}}{T}\right]^{2}+\frac{if_{0}mT}{\hbar}\left[\frac{q_{f}-q_{0}}{T}\right]^{2}[\cos(\Omega
T)-1]-\frac{if_{0}mq_{f}}{\hbar}\left[\frac{q_{f}-q_{0}}{T}\right][\cos(\Omega
T)-1]\\\ &-\frac{imf_{0}(q_{f}-q_{0})}{2\hbar
T}\biggr{[}\frac{(q_{f}-q_{0})\sin(\Omega T)}{\Omega T}-(q_{f}\cos(\Omega
T)-q_{0})\biggr{]}~{}.\end{split}$ (50)

## 5 Summary

In this work, we have constructed the path integral formalism of the
propagation kernel for a resonant bar detector in the presence of a
gravitational wave in the generalized uncertainty principle framework. In this
framework, we have considered only quadratic order correction in the momentum.
We have obtained the configuration space action for this system using the path
integral formalism. With the action in hand, we have then obtained the
equation of motion of the system. From the equation of motion, we observe that
the overall frequency of the resonant detector shifts due to interaction with
the gravitational wave. Next, we have used the form of the perturbation term
for a circularly polarized gravitational wave to calculate the classical
solution of the detector coordinate $q(t)$. Using this form of $q(t)$ we have
finally obtained the classical action for a resonant bar detector interacting
with a gravitational wave in the generalized uncertainty principle framework.
We have then investigated the quantum fluctuation parameter of the bar
detector in presence of a circularly polarized gravitational wave. In order to
obtain the final form of the fluctuation, we have considered a free particle
interacting with the gravitational wave. The final form of the fluctuation
picks up correction terms due to both GUP correction and gravitational wave
interaction. In this process we have neglected cross terms considering both
GUP and GW interactions as it would result in a much smaller correction to the
fluctuation factor than the other corrections present in the analytical form
of the quantum fluctuation. It would also be important to carry out the above
analysis in a linear GUP framework. However, we would like to report this in
future. From the observational point of view, the importance of our work lies
in the fact that resonant bar detectors have the potential of detecting
gravitational waves with their present sensitivity at distances of the order
of $10^{2}kpc$ from the earth. The propagator captures the quantum effects
also. Hence, detectability of such quantum effects in resonant bar detectors
is also a possibility in the near future. A knowhow of the propagator of the
detector coordinates is therefore necessary if not absolutely essential.

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