# Fingerprints of the quantum space-time in time-dependent quantum mechanics:
An emergent geometric phase

Anwesha Chakraborty , Partha Nandi , Biswajit Chakraborty
Department of Theoretical Sciences
S.N. Bose National Centre for Basic Sciences
JD Block, Sector III, Salt Lake, Kolkata 700106, India
<EMAIL_ADDRESS>parthanandi<EMAIL_ADDRESS>
###### Abstract

We show the emergence of Berry phase in a forced harmonic oscillator system
placed in the quantum space-time of Moyal type, where the time
$\textquoteleft{t}$’ is also an operator. An effective commutative description
of the system gives a time dependent generalised harmonic oscillator system
with perturbation linear in position and momentum. The system is then
diagonalised to get a generalised harmonic oscillator and then its adiabatic
evolution over time-period $\mathcal{T}$ is studied in Heisenberg picture to
compute the expression of geometric phase-shift.

Keywords: Time reparametrization invariant formalism, Non-commutative
geometry, Space-time non-commutativity, Berry phase, Forced Harmonic
Oscillator, Generalized Harmonic Oscillator.

##  1 Introduction

The general consensus is that the quantum gravitational effects are expected
to be observed only near the Planck length scale $l_{p}=\sqrt{\frac{\hbar
G}{c^{3}}}\approx 10^{-33}$ cm. It was first proposed by M. P. Bronstein [1]
and thereafter Doplicher et. al. [2] that, at very low length scale, like in
the vicinity of Planck length scale, the concept of localization of an event
in space-time breaks down in the sense that any attempt towards this
localization will inevitably result in gravitational collapse. One way to
avoid such a scenario is to postulate a non-commutative (NC) algebra between
the space-time coordinates, which are now promoted to the level of operators.
It thus becomes quite fruitful and interesting to build models of quantum
mechanics (QM) and quantum field theory (QFT) in such length scale. However,
the connection between quantum gravity and quantum field theory still remains
quite elusive, primarily due to this localisation problem. So therefore it
becomes quite imperative to study different aspects of quantum space-time
including the case where the time is also an operator and for that one clearly
needs to take the first step towards the formulation of a consistent version
of QM in quantum space-time and eventually the QFT. In this context, we can
point out some earlier works in this direction [3, 4, 5, 6] where some
fascinating results were obtained, like discretization of time [6, 7]
corroborating similar observations made earlier by ’t Hooft in the context of
(2+1) dimensional quantum gravity [8] and correction of spin-statistics
theorem in NC spacetime leading to non-Pauli like transitions [9]. Although,
the effects of non-commutativity in the space-time sector should presumably
become significant at (or before) very high energy scale, for instance, close
to Planck scale energy, it is intriguing to speculate that there should be
some relics of the effects of spacetime non commutativity in the low energy
regime[10], because of the inadequate decoupling mechanism between the high
and low energy sectors. In fact in [11], some of us have shown how to
formulate non-relativistic NCQM in 1+1 dimensional Moyal spacetime, in a user
friendly way, which mandates the formulation of an equivalent commutative
theory, circumventing the famous Pauli’s objection [12] and also facilitates
the study of some physical implications.
Moreover, it is quite interesting to point out here that, the authors in [13]
have shown how one can extend the notion of classical space time to that of a
quantum space-time, such that the well-known Unruh effect and Hawking
radiation, in quantum space-time background can be shown to arise from the
relevant nonzero space time commutator. On the other hand, recently in [14],
the authors have shown a connection between semi-classical quantization of
gravity and geometric phase. Furthermore, it has been shown that, due to its
motion in space-time, the state of a detector coupled to a quantum field ,
acquires a geometric phase which is a functional of the trajectory of the
detector and depends on Unruh temperature. Interestingly, this phase turns out
to be within observable limits for values of acceleration as low as $\sim$10
m/$s^{2}$. So by measuring the geometric phase or namely Berry phase one can
experimentally detect Unruh effect even at this low acceleration. At this
stage, one may recall that it was M. Berry [15, 16] who showed that if the
Hamiltonian of a quantum system depends on a number of parameters, varying
adiabatically in a cyclic fashion, then the state of the system acquires an
extra phase factor in addition to the usual dynamical phase. This additional
factor, known as Berry phase, depends only on the geometrical structure of the
parameter space and not on the time of evolution. This effect was further
generalized to arbitrary cyclic evolution by Aharonov and Anandan [17] and
this has been tested quite extensively for two level systems [18]. Returning
back to our main point, we note that the result in the previously cited paper
[14] shows that the geometric phase encodes the information of the Unruh
temperature indicating the existence of a relation between Berry phase and
Unruh effect. Although the universality of Berry phase can be seen in variety
of contexts like Born-Oppenheimer approximation [19], quantum Hall effect [20,
21, 22] etc, there is still a gap in literature suggesting any connection
between Berry phase and quantum gravity or for that matter in the simplest
set-up of quantum space-time of the type we have mentioned above. So it is
quite appropriate to study the occurence of geometric phase, if any, in such a
quantum space-time. With this motivation in mind, we undertake the study of
the relation between NC space time and emergence of geometric phase in a
dynamical system. For that we have constructed the effective commutative
Hamiltonian of a forced harmonic oscillator placed in a 2 dimensional quantum
(Moyal) space-time, by employing the prescription provided in [11]. This gives
rise to a generalised harmonic oscillator system with perturbations linear in
position and momentum respectively. An adiabatic evolution of the total
Hamiltonian is then considered through the slowly varying time dependent
coefficients. As in [11], here too we have studied the time evolution of the
ladder operators (with which the system is diagonalised) in Heisenberg’s
picture and have shown that the adiabatic evolution of the system over a total
time period $\mathcal{T}$ gives rise to a total phase consisting of the usual
dynamical phase along with an extra geometric phase shift in the operator,
which vanishes in the commutative limit.

The paper is organized as follows: In sec-2 we have provided a review of time-
reparametrization invariant formulation of non-relativistic particle moving in
the commutative space-time and eventual emergence of Schrödinger equation as a
constraint equation at the quantum level paving the way for its generalization
to noncommutative space-time in (1+1) D which we take up in the next sec-3,
where we have also shown how the of non-commutativity of space-time appears
through the Dirac brackets in a classical toy model and then quantized it,
Here we have introduced the Hilbert Schmidt (HS) operatorial formulation to
discuss NC quantum mechanics. In sec-4 we have considered a forced harmonic
oscillator system in NC space-time and then made use of coherent state basis
to obtain an effective commutative Hamiltonian responsible for the time
evolution through the effective commutative Schrödinger equation. By carrying
out an additional time-dependent unitary transformation the system is
transformed to a time-dependent generalised harmonic oscillator (GHO) system.
We then consider an adiabatic time evolution, over the time period
$\mathcal{T}$, of the ladder operators of the GHO system, which is then shown
to yield the geometric i.e. Berry phase along with a dynamical phase in sec-5.
Finally we conclude and try to give some future prospects of the work in
sec-6. Apart from these sections in main text, we have also included three
appendices to complement the meterials presented in the text. And they are
given as followes: In appendix-A.1, we have shown the effect of space-time
noncommutativity on the dynamics of a time independent system. Next in
appendix-A.2, we have carried out constraint analysis and computed the Dirac
brackets for the phase space variables for a particular (1+1) D classical toy
model, which produces non-vanishing Dirac brackets between the configuration
space variables. Then in appendix-A.3, we provide a detailed derivation of the
inner product in the space of square integrable functions
$L^{2}_{\star}(\mathbb{R}^{1})$ in the non-commutative space-time, starting
from that of usual commutative space-time : $L^{2}(\mathbb{R}^{1})$, where the
functions are composed using point-wise multiplication. Finally in
appendix-A.4, we have shown explicit computation of the geometric phase shift
for the system of interest.

##  2 A classical picture of non-commutative space-time from time
reparametrization symmetry

In this section we first briefly sketch the idea of time-reparametrization
invariant form of classical action. We introduce a modified Lagrangian
formalism (non-relativistic), where the usual commutative time $t$ is treated
as an additional configuration space variable and is on an equal footing with
the position coordinates [23].
We begin by considering the Lagrangian $L^{t}(x_{i},\dot{x}_{i},t)$ of a
dynamical system in 1+1 dimension and treat time $t$, as the zero-th component
of generalised coordinate $x^{\mu}$ : $x^{0}=t$. Next we introduce a new
evolution parameter $\tau$ for the trajectory of the sytem and for that all we
want is to just require that, $t$ is a monotonically increasing function of
$\tau:\,\,t=t(\tau)$. So with the identification $dt=\dot{t}d\tau$ and
$\dot{x}_{i}=\left(\frac{dx_{i}}{d\tau}\right)$, with overdots, from here
onwards, representing the derivative with respect to $\tau$, the time-
reparametrization invariant form of action can be written as

$S=\int\,dt\,L^{t}(x_{i},\frac{dx_{i}}{dt},t)=\int\,d\tau\,\,\dot{t}\,L^{t}(x_{i},\dot{x}_{i},t,\dot{t})=\int\,\,d\tau
L^{\tau}(x^{\mu},\dot{x}^{\mu})$ (1)

which remains invariant under the world line reparametrization
$\tau\to\tau^{\prime}=\tau^{\prime}(\tau),\,\,\,\,$ with $x^{\mu}(\tau)$
transforming as world-line scalars:
$x^{\mu}(\tau)\,\to\,x^{{}^{\prime}\mu}(\tau^{\prime})=x^{\mu}(\tau)$ under
this reparametrization. Note that we have identified
$L^{\tau}(x^{\mu},\dot{x}^{\mu})$ the integrand in (1) as our modified
Lagrangian. At this point let us consider the simple example of Lagrangian of
a non-relativistic particle in presence of a potential $V(x,t)$:

$L^{t}(x,\dot{x},t)=\frac{1}{2}m\left(\frac{dx}{dt}\right)^{2}-V(x,t)$ (2)

Considering the evolution parameter to be $\tau$ , the corresponding modified
Lagrangian of the theory is obtained as,

$L^{\tau}=\frac{1}{2}m\frac{\dot{x}^{2}}{\dot{t}}-\dot{t}V(x,t)$ (3)

The canonical momenta corrensponding to the generalised coordinates $x(\tau)$
and $t(\tau)$ are now given by,

$\displaystyle p_{x}$
$\displaystyle=\frac{dL^{\tau}}{d\dot{x}}=m\frac{\dot{x}}{\dot{t}}=m\frac{dx}{dt}$
$\displaystyle p_{t}$
$\displaystyle=\frac{dL^{\tau}}{d\dot{t}}=-\frac{m}{2}\left(\frac{dx}{dt}\right)^{2}-V(x,t)=-\frac{p_{x}^{2}}{2m}-V(x,t)=-H$
(4)

where $H=\frac{p_{x}^{2}}{2m}+V(x,t)$ is the canonical Hamiltonian
corresponding to (2). The above equation (4) indicates the presence of a
primary constraint as

$\phi=p_{t}+H\approx 0$ (5)

where the notation $\approx$ is used in sense of weak equality a la Dirac
[24]. The new Hamiltonian corresponding to the redefined Lagrangian $L^{\tau}$
(3) is proportional to $\phi$ (5) and therefore also vanishes weakly:

$H^{\tau}=p_{t}\dot{t}+p_{x}\dot{x}-L^{\tau}=\dot{t}\phi\approx 0$ (6)

So we see that the system is endowed with a first class constraint $\phi$. The
total Hamiltonian according to Dirac’s prescription for constrained theory is
then given by

$H_{T}^{\tau}=H^{\tau}+\sigma(\tau)\phi=\sigma(\tau)\phi$ (7)

$\sigma(\tau)$ being the Lagrange’s multiplier. We can now write the
Lagrangian in the first order form given by the inverse Legendre
transformation as

$L_{f}^{\tau}=p_{\mu}\dot{x}^{\mu}-\sigma(\tau)(p_{t}+H),\,\,\,\,\,\mu=0,1$
(8)

Note that in the first order formalism, the configuration space coincides with
the phase space i.e the configuration space is enlarged further by taking the
canonical momenta also as generalised coordinates. Now we can easily run
through the algorithm of Dirac’s constraint analysis [24] for the above
Lagrangian (8) to find one primary first class constraint and two pairs of
primary second class constraints. The second class constraints of the system
are then implemented strongly by introducing Dirac’s brackets (DB), which can
be computed easily and finally the DBs among the phase space variables are now
obtained as:

$\\{x^{\mu},x^{\nu}\\}_{D}=0=\\{p_{\mu},p_{\nu}\\}_{D};\,\,\,\,\,\,\\{x^{\mu},p_{\nu}\\}_{D}=\delta^{\mu}\,_{\nu}$
(9)

Canonical quantization is now carried out by elevating the phase space
variables ($t,x,p_{t},p_{x}$) to the level of operators and elevating the
Dirac brackets to the level of commutator brackets as

$[\hat{t},\hat{x}]=0=[\hat{p}_{t},\hat{p}_{x}],\,\,[\hat{t},\hat{p}_{t}]=i=[\hat{x},\hat{p}_{x}]\,\,\,\,(\hbar=1)$
(10)

We then look for an appropriate Hilbert space on which these operators act and
furnish a suitable representation of this algebra. The quantum counterpart of
the 2D configuration space is now the Hilbert space $L^{2}(\mathbb{R}^{2})$.
We now introduce the simultaneous “spatio-temporal” eigenbasis
$\left|x,t\right\rangle$ of the commutating operators $\hat{t}$ and $\hat{x}$
satisfying

$\hat{t}\left|x,t\right\rangle=t\left|x,t\right\rangle,~{}~{}~{}\hat{x}\left|x,t\right\rangle=x\left|x,t\right\rangle.$
(11)

which fulfills the following completeness and orthonormality relations :

$\int dtdx\left|x,t\right\rangle\langle x,t|=\mathbb{I},~{}~{}~{}\left\langle
x,t|t^{{}^{\prime}},x^{{}^{\prime}}\right\rangle=\delta(t-t^{{}^{\prime}})\delta(x-x^{{}^{\prime}}).$
(12)

The coordinate representations of phase space operators are given by

$\begin{split}\left\langle x,t|\hat{x}|\psi\right\rangle=x\left\langle
x,t|\psi\right\rangle,&\left\langle
x,t|\hat{t}|\psi\right\rangle=t\left\langle x,t|\psi\right\rangle\\\
\left\langle x,t|\hat{p}_{x}|\psi\right\rangle=-i\partial_{x}\left\langle
x,t|\psi\right\rangle,&\left\langle
x,t|\hat{p}_{t}|\psi\right\rangle=-i\partial_{t}\left\langle
x,t|\psi\right\rangle\ \end{split}$ (13)

where $\psi(x,t)=\left\langle x,t|\psi\right\rangle\in L^{2}(\mathbb{R}^{2})$
and can be formally identified with the function satisfying the following
square-integrable norm:

$\left\langle\psi|\psi\right\rangle=\int
dtdx~{}\psi^{\ast}(x,t)\psi(x,t)<\infty.$ (14)

The associated inner product is given by

$<\psi|\phi>_{=}\int dtdx\psi^{\star}(x,t)\phi(x,t)$ (15)

In order to obtain an appropriate probabilistic interpretation of non-
relativistic QM, however, we need to consider a physical Hilbert space which
is the linear span of all possible physical states and can be identified by
projecting out physical states ($\left|\psi_{phy}\right\rangle$) by imposing
the subsidiary condition

$\hat{\phi}\left|\psi_{phy}\right\rangle=(\hat{p_{t}}+\hat{H})\left|\psi_{phy}\right\rangle=0.$
(16)

where $\hat{\phi}$ is the operator form of the first class constraint (5)
[23]. This is tantamount to demanding the gauge invariance of physical states.
The coordinate representation of quantum constraint equation (16) readily
yields the time evolution of the physical states or wave-functions
($<x,t|\psi_{phy}>:=\psi_{phy}(x,t)$) as

$i\frac{\partial}{\partial
t}\psi_{phy}(x,t)=\left(-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}+V(x,t)\right)\psi_{phy}(x,t)$ (17)

which we recognize as the time-dependent Schrödinger equation. Note that it is
independent of the parameter $\tau$, as its $\tau$-evolution, which is nothing
but the unfolding of gauge transformation generated by the first class
constraint $\phi$ (5), is now frozen, as can be easily observed by using
(6,16).
Now to extract the probabilistic interpretation, we recall that the continuity
equation corresponding to this Schrödinger equation, given by

$\frac{\partial\rho}{\partial t}+\frac{\partial J_{x}}{\partial x}=0$ (18)

with $\rho=\psi^{\star}_{phy}(x,t)\psi_{phy}(x,t)$ and
$J_{x}=\frac{1}{2m}Im(\psi^{\star}_{phy}(x,t)\overleftrightarrow{\partial_{x}}\psi_{phy}(x,t))$.
Correspondingly, after performing a spatial integration in both side of (18)
ranging from -$\infty$ to +$\infty$ we can write,

$\partial_{t}\int_{-\infty}^{\infty}\rho
dx=-\int_{-\infty}^{\infty}(\partial_{x}J_{x})dx=0$ (19)

where we have used the fact that our physical wave-function satisfies
$\psi_{phy}(x,t)\to 0$ as $x\to\pm\infty$ unlike in (14) where we also
necessarily require $\psi_{phy}(x,t)\to 0$ as $t\to\pm\infty$, which clearly
will not be suitable for a probabilistic interpretation. Therefore the
nonnegative $\rho$ can indeed be interpreted as a probability density and this
notion of physical state defined on a “space-like surface” indicates the
conservation of the total probability $\int_{-\infty}^{\infty}\rho dx$. From
the expression of the continuity equation (18), the variable $t$ can be
identified with time which now plays the role of the evolution parameter. We
thus see that for $\psi_{phy}(x,t)$ to be the so called “well behaved” it has
to be normalizable and must satisfy the following square integrability
condition at a constant time slice as

$\langle\psi_{phy}|\psi_{phy}\rangle_{t}=\int_{-\infty}^{\infty}\,dx\,\,\psi_{phy}^{*}(x,t)\psi_{phy}(x,t)<\infty$
(20)

In other words, the physical states satisfying (17) cannot be elements of the
Hilbert space $L^{2}(\mathbb{R}^{2})$. We shall, however, restrict our
attention to physical states and we shall henceforth suppress ‘phy” in the
subscript and simply write it as $\psi(x,t)$ for brevity and will insert it
back in the subscript as and when required explicitly. More precisely, the
probabilistic notion in QM is recovered by replacing the inner-product (15) by
the one, which involves only a spatial integration at a constant time slice:

$\left\langle\psi|\phi\right\rangle_{t}:=\int_{t}dx~{}\psi^{\ast}(x,t)\phi(x,t).$
(21)

We shall refer this as “induced inner product”. Clearly, any normalizable
states with $L^{2}(\mathbb{R}^{1})$ inner product (21) may not be so with
respect to that of $L^{2}(\mathbb{R}^{2})$ (15).

As an example, one may consider a stationary state, associated with energy
$E$, of the form $\psi_{E}(x,t)=e^{-iEt}\psi(x)$. Clearly, this
$\psi_{E}(x,t)\in L^{2}(\mathbb{R}^{1})$ but $\psi_{E}(x,t)\notin
L^{2}(\mathbb{R}^{2})$. Further, note that this state can be written as a
superposition of plane waves corresponding to different momentum $p$ as
$\psi_{E}(x,t)=\int\,dp\,e^{-i(Et-px)}\tilde{\psi}(p)$, where the exponential
$\frac{1}{2\pi}e^{-i(Et-px)}:=\langle x,t|E,p\rangle$ is the overlap of
simultaneous eigenstate $|E,p\rangle$ of both $\hat{p}_{t}$ and $\hat{p}_{x}$
: $\hat{p}_{t}|E,p\rangle=-E|E,p\rangle$ and
$\hat{p}_{x}|E,p\rangle=p|E,p\rangle$, with the basis $|x,t\rangle$ (11). Now
if we consider a pair of such stationary states $|E,p\rangle$ and
$|E^{\prime},p^{\prime}\rangle$, then using the original inner product (15),
one gets

$\langle
E^{\prime},p^{\prime}|E,p\rangle=\delta(E^{\prime}-E)\,\delta(p^{\prime}-p)$
(22)

This clearly shows that for coincident energy eigenvalues i.e. $E^{\prime}=E$,
the right hand side in (22) diverges. In contrast, the induced inner product
(21) in this case yields

$\langle E,p^{\prime}|E,p\rangle_{t}=\frac{1}{2\pi}\delta(p^{\prime}-p)$ (23)

which is nothing but the usual orthonormality condition for momentum
eigenstates which is free of such divergence. As we shall demonstrate in the
sequel that an exactly similar situation arises in our kind of NC space-time
as well. In either case, therefore, we shall primarily be working with this
induced inner product or more generally between pair of wave packets like
$|\psi\rangle$ and $|\phi\rangle$ (21), so that we can recover the
probabilistic interpretation using this inner product of
$L^{2}(\mathbb{R}^{1})$ only.

Finally, note that the self-adjoint property of the derivative representation
of $\hat{p}_{t}=-i\partial_{t}$ in (13) does not hold anymore in the Hilbert
space $L^{2}(\mathbb{R}^{1})$ with associated inner product (21), as it is
impossible to demand that $\mid\psi(x,t)\mid\to 0$ as $\mid t\mid\to\infty$ as
mentioned above, which is required in $L^{2}(\mathbb{R}^{2})$, to allow one to
carry out integration by parts with respect to ${}^{\prime}t^{\prime}$ and
drop boundary contribution. We shall thus exclude $\hat{p}_{t}$ from the
dynamical phase-space variables, along with $\hat{t}$. The latter, when
‘demoted’ to a $c$-number parameter, is now identified with the new evolution
parameter with $\left(-i\partial_{t}\right)$ having no association with
$\hat{p}_{t}$ anymore, so that (17) has now the status of a postulate.

##  3 Quantum mechanics on quantum space-time

It has already been shown in [25] that the study of a typical time independent
system on NC space produces new dynamics of the system. The form of the
propagator of the system, gets modified and we also get a correction in the
wave function of the system. On the other hand, when such a system is placed
in a NC space-time (i.e. with time also being an operator) background, the
system dynamics gives equivalent result (see Appendix-A.1) and no NC
correction is obtained in the Hamiltonian and therefore in the spectrum of the
system. This motivates us to examine a time dependent system placed in a NC
space time and look for signatures, if any, of this noncommutativity on the
dynamics of the system. And in this regard, the discussion on QM on
commutative space-time in the preceeding section, will play an important role
as we shall be basically emulating the same template to formulate the QM in
our (1+1) D Moyal plane. And for that we consider a time dependent Hamiltonian
of a forced harmonic oscillator (FHO) which is known to have many applications
in quantum optics. This was primarily motivated by the structure of the
Hamiltonian, proposed by Mehta and Sudarshan [26], where it is shown that
coherent states persists to be coherent after time evolution, if we require
the general form of the Hamiltonian to be that of a FHO. Also in [27] authors
have taken similar Hamiltonian to show that a FHO coupled to a transient
classical force can easily be described in terms of coherent states, which is
very useful in describing light from optical masers. Thus our primary goal is
to look for the above mentioned signature, in the form of geometric phase in
such system and for that we first need to set up the formalism of QM itself of
NC space time with time also being an operator. We intend to deal with this in
this sub-section. For that we will first very briefly demonstrate how the NC
configuration space can appear in the context of a classical dynamical system.

As indicated previously we have an enlarged configuration space in the first
order formulation, which allows more freedom for theoretical interst. In our
purpose, we can add a Chern-Simon (CS) like term in the momentum space to the
existing first order Lagrangian $L^{\tau}_{f}$ (8) [28]. This is motivated
from [29, 30], where it was shown that the phase-space representation of
action, of a particle placed in a NC plane in presence of an arbitrary
potential gives rise to an additional “exotic” term in action, which is a
Chern-Simon (CS) like term in momentum space and the corresponding CS
parameter is essentially of NC origin. Also the presence of this CS term in
the Lagrangian does not affect the Gallelian symmetry of the system.111For a
Gallelian transformation given by $\delta x=vt,\delta t=0$ we get the
transformation in the following momentum as : $\delta p_{x}=mv,\delta
p_{t}=-\delta H_{free}=-p_{x}v$ where $H_{free}$ is the free particle
Hamiltonian. With this transformation we can easily verify that the Chern-
Simon term $\epsilon_{\mu\nu}p_{\mu}\dot{p}_{\nu}$ transforms upto a total
derivative which can be neglected in the action. So the CS action is quasi
invariant under Gallelian transformation.
So we now take the following Lagrangian to obtain a NC generalisation of the
commutative space-time in a classical setting,

$L_{f}^{\tau,\theta}=p_{\mu}\dot{x}^{\mu}+\frac{\theta}{2}\epsilon^{\mu\nu}p_{\mu}\dot{p}_{\nu}-\sigma(\tau)(p_{t}+H),\,\,\,\,\,\mu,\nu=0,1$
(24)

where $p_{\mu}=(p_{t},p_{x})$ . We can again run the Dirac’s algorithm
(Appendix-A.2) to get the following Dirac brackets between the phase space
variables.

$\\{x^{\mu},x^{\nu}\\}_{D}=\theta\epsilon^{\mu\nu};\,\,\,\,\,\\{p_{\mu},p_{\nu}\\}_{D}=0;\,\,\,\,\,\,\\{x^{\mu},p_{\nu}\\}_{D}=\delta^{\mu}\,_{\nu}$
(25)

So we can see that the NC nature of configuration space emerges very naturally
from a constrained classical system. Note that this is shown just to motivate
the appearance of non-commutativity between coordinate variables naturally at
the classical level before setting up a quantum mechanical description of the
quantum space-time. In this paper we shall, of course, be dealing with a
different dynamical system in the next section.
Now to provide a description for the non-relativistic QM in quantum space-
time, we will elevate the classical brackets (25) to the level of commutation
brackets as given below,

$[\hat{t},\hat{x}]=i\theta$ (26)

with $\theta$ being the NC parameter, along with

$[\hat{p}_{t},\hat{p}_{x}]=0,\,\,[\hat{t},\hat{p}_{t}]=i=[\hat{x},\hat{p}_{x}].$
(27)

Recall that we are working in the natural unit $\hbar=1$ throughout this
paper. (26) and (27) as a whole represents the NC Heisenberg algebra (NCHA).
A representation of NC coordinate sub-algebra in (26) is furnished by the
Hilbert space [11, 31, 32, 33],

$\mathcal{H}_{c}=Span\left\\{|n\rangle=\frac{(b^{\dagger})^{n}}{\sqrt{n!}}|0\rangle;\,\,b|0\rangle=\frac{\hat{t}+i\hat{x}}{\sqrt{2\theta}}|0\rangle=0\right\\}$
(28)

Let us now introduce the associative NC operator algebra
($\hat{\mathcal{A}}_{\theta}$) generated by ($\hat{t},\hat{x}$) or
equivalently ($\hat{b},\hat{b}^{\dagger}$) acting on this configuration space
$\mathcal{H}_{c}$ (28) as

$\hat{\mathcal{A}}_{\theta}=\\{|\psi)=\psi(\hat{t},\hat{x})=\psi(\hat{b},\hat{b}^{\dagger})=\sum_{m,n}c_{n,m}|m\rangle\langle
n|\\}$ (29)

which is basically the set of all equivalence class of polynomials in
$(\hat{t},\hat{x})$ or ($\hat{b},\hat{b}^{\dagger}$), subject to the
identification $[\hat{b},\hat{b}^{\dagger}]=1$, which also can be thought of
as the universal enveloping algebra corresponding to (26). At this stage one
should note that $\hat{\mathcal{A}}_{\theta}$ has not yet been endowed with
any inner product and norm structure and consequently can’t be identified with
any Hilbert or Banach space at this stage.
We can now introduce a subspace
$\mathcal{H}_{q}\subset\mathcal{B}(\mathcal{H}_{c})\subset\hat{\mathcal{A}}_{\theta}$
of $\hat{\mathcal{A}}_{\theta}$ (29) as the space of ‘Hilbert Schmidt’ (HS)
operators, which are bounded and compact operators with finite HS norm, again
acting on $\mathcal{H}_{c}$ (28), and is given by,

$\mathcal{H}_{q}=\left\\{\psi(\hat{t},\hat{x})\equiv\Big{|}\psi(\hat{t},\hat{x})\Big{)};\,||\psi||_{HS}:=\sqrt{tr_{c}(\psi^{\dagger}\psi)}<\infty\right\\}\subset\hat{\mathcal{A}}_{\theta}$
(30)

where $||\psi||_{HS}$ is the Hilbert-Schmidt norm and $tr_{c}$ denotes trace
over the Hilbert space $\mathcal{H}_{c}$ and
$\mathcal{B}(\mathcal{H}_{c})\subset\hat{\mathcal{A}}_{\theta}$ is a set of
bounded operators on $\mathcal{H}_{c}$. This space is equipped with the inner
product

$\Big{(}\psi(\hat{t},\hat{x}),\phi(\hat{t},\hat{x})\Big{)}:=tr_{c}\Big{(}\psi^{\dagger}(\hat{t},\hat{x})\phi(\hat{t},\hat{x})\Big{)}$
(31)

and so forms a Hilbert space on its own. Note that as a matter of notations
the elements of $\mathcal{H}_{c}$ and $\hat{\mathcal{A}}_{\theta}$ are denoted
by the angular ket $|.\rangle$ and round ket $|.)$ respectively. We now define
the quantum space-time coordinates ($\hat{T},\hat{X}$) and corresponding
conjugate momentum operators ($\hat{P_{t}},\hat{P}_{x}$) by their actions on a
typical element $\Big{|}\psi(\hat{t},\hat{x})\Big{)}\in\mathcal{H}_{q}$ as,

$\displaystyle\hat{T}\Big{|}\psi(\hat{t},\hat{x})\Big{)}=\Big{|}\hat{t}\psi(\hat{t},\hat{x})\Big{)},\,\,\,\,\,\hat{X}\Big{|}\psi(\hat{t},\hat{x})\Big{)}=\Big{|}\hat{x}\psi(\hat{t},\hat{x})\Big{)},$
$\displaystyle\hat{P}_{x}\Big{|}\psi(\hat{t},\hat{x})\Big{)}=-\frac{1}{\theta}\Big{|}[\hat{t},\psi(\hat{t},\hat{x})]\Big{)},\,\,\,\hat{P}_{t}\Big{|}\psi(\hat{t},\hat{x})\Big{)}=\frac{1}{\theta}\Big{|}[\hat{x},\psi(\hat{t},\hat{x})]\Big{)}$
(32)

Note that here upper case letters $\hat{T}$ and $\hat{X}$ are used to
distinguish them from their the lower case counterparts $\hat{t}$ and
$\hat{x}$, as their domain of actions are different; while $(\hat{t},\hat{x})$
act on $\mathcal{H}_{c}$, $(\hat{T},\hat{X})$ act on $\mathcal{H}_{q}$. But
since they satisfy isomorphic commutator algebra:$[\hat{T},\hat{X}]=i\theta$,
$(\hat{T},\hat{X})$ can be regarded as the representation of
($\hat{t},\hat{x}$). It is easily verified using (32) that
($\hat{T},\hat{X},\hat{P}_{t},\hat{P}_{x}$) satisfy the same NC Heisenberg
algebra (26,27). These new-fashioned operators can be thought as “super
operators”, as they operate again on space of operators. In fact, they can act
on the entire $\hat{\mathcal{A}}_{\theta}$.
However, it is clear from the discussion of the previous section that HS norm
(30) has its commutative counterparts in (14). Clearly, the associated inner-
product (31), like the counterpart (15), needs also to be modified to some
appropriate form analogous to (21) if we want to have a proper probabilistic
interpretation. We take it up in the next sub-section.

####  3.0.1 Schrödinger equation and an induced inner product

Now it is clear that in view of $\theta\neq 0$, that we cannot find a
counterpart of the common space-time eigenstate $|x,t\rangle$ (11). However,
we can still recover an effective commutative theory by making use of the
coherent state. For that we choose the Sudarshan-Glauber coherent state made
out of the Fock states $|n\rangle$ belonging to $\mathcal{H}_{c}$ (28) as

$|z\rangle=e^{-\bar{z}\tilde{b}+z\tilde{b}^{\dagger}}|0\rangle\,\in\mathcal{H}_{c}$
(33)

which is an eigen state of the anihilation operator :
$\tilde{b}|z\rangle=z|z\rangle$, where
$\tilde{b}=\frac{\hat{t}+i\hat{x}}{\sqrt{2\theta}}$ and $z$ is dimensionless
complex number and is given by,

$z=\frac{t+ix}{\sqrt{2\theta}};\,\,\,\,\,\,\,\,\,t=\langle
z|\hat{t}|z\rangle,x=\langle z|\hat{x}|z\rangle$ (34)

Here $t$ and $x$ are effective commutative coordinate variables. We can now
construct the counterpart of coherent state basis in $\mathcal{H}_{q}$ (30),
made out of the bases $|z\rangle\equiv|x,t\rangle$ (33), by taking their outer
product as

$|z,\bar{z})\equiv|z)=|z\rangle\langle
z|=\sqrt{2\pi\theta}\,\,|x,t)\,\in\mathcal{H}_{q}\,\,\,\,\,\,\textrm{fulfilling}\,\,\,B|z)=z|z)$
(35)

where the annihilation operator
$\hat{B}=\frac{\hat{T}+i\hat{X}}{\sqrt{2\theta}}$ is a representation of the
operator $\tilde{b}$ in $\mathcal{H}_{q}$ (30). In this context, note that
$|z)\in\mathcal{H}_{q}$ (35) saturates the space-time uncertainty:
$\Delta\hat{T}\,\Delta\hat{X}=\frac{\theta}{2}$, implying that such a state
represents a maximally localized “point”, or rather an event in space-time. In
fact this state being a pure density matrix can be regarded as a pure state of
the algebra $\hat{\mathcal{A}}_{\theta}$ (29) and plays the role of a point,
represented by Dirac’s delta functional, in the corresponding commutative
algebra $C^{\infty}(\mathbb{R}^{2})$ describing (1+1) D commutative plane [34,
35].
It can also be checked that the basis $|z,\bar{z})\equiv|z)$ satisfies the
over-completeness property:

$\int\,\,\frac{d^{2}z}{\pi}|z,\bar{z})\,\star_{V}\,(z,\bar{z}|=\int
dtdx\,|x,t)\star_{V}(x,t|=\textbf{1}_{q},$ (36)

where $*_{V}$ represents the Voros star product and is given by,

$\star_{V}=e^{\overleftarrow{\partial_{z}}\overrightarrow{\partial_{\bar{z}}}}=e^{\frac{i\theta}{2}(-i\delta_{ij}+\epsilon_{ij})\overleftarrow{\partial_{i}}\overrightarrow{\partial_{j}}};\,\,\,\,i,j=0,1;\,\,\,x^{0}=t,x^{1}=x;\,\,\,\,\,\epsilon_{01}=1$
(37)

Then the coherent state representation of an abstract state
$\psi(\hat{t},\hat{x})$ gives the usual coordinate representation of a state
just like formal QM:

$\psi(x,t)=\frac{1}{\sqrt{2\pi\theta}}\Big{(}z,\bar{z}|\psi(\hat{x},\hat{t})\Big{)}=\frac{1}{\sqrt{2\pi\theta}}tr_{c}\Big{[}|z\rangle\langle
z|\psi(\hat{x},\hat{t})\Big{]}=\frac{1}{\sqrt{2\pi\theta}}\langle
z|\psi(\hat{x},\hat{t})|z\rangle$ (38)

and is called the symbol of the HS operator $\psi(\hat{x},\hat{t})$. The
corresponding representation of a composite operator say
$\psi(\hat{x},\hat{t})\phi(\hat{x},\hat{t})$ is given by composing the
corresponding symbols through Voros star product and is given as

$\Big{(}z\Big{|}\psi(\hat{x},\hat{t})\phi(\hat{x},\hat{t})\Big{)}=\Big{(}z\Big{|}\psi(\hat{x},\hat{t})\Big{)}\,\star_{V}\,\Big{(}z\Big{|}\phi(\hat{x},\hat{t})\Big{)}$
(39)

This establishes an isomorphism between the space of HS operators
$\mathcal{H}_{q}$ and the space of their respective symbols, whereas in the
former case, the composition rule is given by the usual product of operators,
in the latter case it is given by the Voros star product. Now, using (36) the
overlap of two arbitrary states ($|\psi),|\phi)$) in the quantum Hilbert space
$\mathcal{H}_{q}$ can be written in the form

$(\psi|\phi)=\int dtdx~{}\psi^{\ast}(x,t)\star_{V}\phi(x,t)$ (40)

We shall demonstrate in the sequel that with this Voros star product
probability density is indeed positive definite. Also the basis
$|z,\bar{z})_{V}$ which we can refer to as Voros basis as in [25] is
compatible with POVM (Positive operator valued measure) [36] in contrast to
Moyal basis which is associated to a similar kind of star product named as
Moyal star product [25] in the context of (2+1) D QM in the Moyal plane, where
the time was not an operator. Henceforth, we shall thus be using only this
Voros basis and the associated Voros star product for our purpose, so we shall
omit the subscript V from now onwards.
Now to obtain the effective commutative Schrödinger equation in coordinate
space, we introduce coordinate representation of the phase space operators. To
begin with note that the coherent state representation of the left actions of
space-time operators $\\{\hat{X}_{L},\hat{T}_{L}\\}$, acting on any arbitrary
element $|\psi)\in\mathcal{H}_{q}$, can be written by using (39) as,

$\Big{(}x,t\Big{|}\hat{X}_{L}\,\psi(\hat{x},\hat{t})\Big{)}=\frac{1}{\sqrt{2\pi\theta}}\Big{(}z,\bar{z}\Big{|}\hat{x}\psi\Big{)}=\frac{1}{\sqrt{2\pi\theta}}\,\left\langle
z|\hat{x}|z\right\rangle\star_{V}(z,\bar{z}|\psi(\hat{x},\hat{t}))$ (41)

Finally making use the fact of (38) this readily yields

$\Big{(}x,t|\hat{X}_{L}\,\psi(\hat{x},\hat{t})\Big{)}=X_{\theta}^{L}\,\Big{(}x,t|\psi(\hat{x},\hat{t})\Big{)}=X_{\theta}^{L}\,\psi(x,t)$
(42)

with

$X_{\theta}^{L}=\left[x+\frac{\theta}{2}(\partial_{x}-i\partial_{t})\right]$
(43)

Proceeding exactly in the same way, we obtain the representation of $\hat{T}$
as

$T_{\theta}^{L}=\left[t+\frac{\theta}{2}(\partial_{t}+i\partial_{x})\right],$
(44)

so that $[T_{\theta}^{L},X_{\theta}^{L}]=i\theta$ is trivially satisfied when
their actions on arbitrary test functions are considered. Now, it is trivial
to prove the self-adjointness property of both $X_{\theta}$ and $T_{\theta}$,
w.r.t. the inner product (40) in $\mathcal{H}_{q}$ by considering an arbitrary
pair of different states $|\psi_{1})$, $|\psi_{2})\in\mathcal{H}_{q}$ and
their associated symbols, just by exploiting associativity of Voros star
product. We also note that since this analysis will not involve any
integration by parts i.e. it is not sensitive to the integration measure, this
self-adjointness property of $X_{\theta}$ and $T_{\theta}$ will persist to
hold for the ‘induced’ inner product, to be introduced in the sequel in (56)
(see below) - the counterpart of (21), as well. Like-wise the right action of
$\hat{X}_{R}$ and $\hat{T}_{R}$ on $|\psi)$ defined as

$\hat{X}_{R}|\psi)=\Big{|}\psi(\hat{x},\hat{t})\hat{x}\Big{)};\,\,\,\,\,\,\hat{T}_{R}|\psi)=\Big{|}\psi(\hat{x},\hat{t})\hat{t}\Big{)}$
(45)

where the corresponding representations are obtained as,

$X_{\theta}^{R}=\left[x+\frac{\theta}{2}(\partial_{x}+i\partial_{t})\right],\,\,\,\,\,\,\,\,T_{\theta}^{R}=\left[t+\frac{\theta}{2}(\partial_{t}-i\partial_{x})\right]$
(46)

fulfilling $[T_{\theta}^{R},X_{\theta}^{R}]=-i\theta$, appropriate for the
generators of opposite algebra $\hat{\mathcal{A}}_{\theta}^{o}$.
Finally coming to the coherent state representation of momenta operators in
(32) we observe that since momenta operators act adjointly, their actions can
be written equivalently as the corresponding differences between left and
right actions yielding,

$\Big{(}x,t|\hat{P}_{t}\psi(\hat{x},\hat{t})\Big{)}=-i\partial_{t}\psi(x,t)~{};~{}~{}\Big{(}x,t|\hat{P}_{x}\psi(\hat{x},\hat{t})\Big{)}=-i\partial_{x}\psi(x,t)$
(47)

Now to isolate the physical Hilbert space
$\mathcal{H}_{ph}\subset\hat{\mathcal{A}}_{\theta}$, we note that our NC
theory (24) is endowed with a secondary first-class constraint $\phi\approx 0$
(99) (see Appendix-A.2). Thus, to obtain an effective commutative Schrödinger
equation in NC space-time we must impose the condition that the physical
states $|\psi_{phy})=\psi_{phy}(\hat{x},\hat{t})$ are annihilated by the
operatorial version of this constraint $\phi$:

$(\hat{P}_{t}+\hat{H})|\psi_{phy})=0;\qquad\psi_{phy}(\hat{x},\hat{t})\in\hat{\mathcal{A}}_{\theta}$
(48)

where $\hat{H}=\frac{\hat{P}_{x}^{2}}{2m}+V(\hat{X},\hat{T})$. This is just
the counterpart of (16) in the commutative case. We are now ready to write
down the effective commutative time dependent Schrödinger equation in quantum
space-time by taking the representation of (48) in $|x,t)$ basis. Using
(43,44,47) we finally get,

$i\partial_{t}\psi_{phy}(x,t)=\left[-\frac{1}{2m}\partial_{x}^{2}+V(x,t)\,\star_{V}\right]\psi_{phy}(x,t)$
(49)

Taking complex conjugate of this Schrodinger equation we get

$-i\partial_{t}\psi_{phy}^{*}(x,t)=-\frac{1}{2m}\partial_{x}^{2}\psi_{phy}^{*}(x,t)+\psi_{phy}^{*}(x,t)\
\star_{V}V(x,t).$ (50)

Now using (49) and (50) one therefore obtain the continuity equation

$\partial_{t}\rho_{\theta}+\partial_{x}J_{\theta}^{x}=0$ (51)

where

$\displaystyle\rho_{\theta}$
$\displaystyle=\psi_{phy}^{*}(x,t)\,\star_{V}\,\psi_{phy}(x,t),$
$\displaystyle J_{\theta}^{x}$
$\displaystyle=\frac{1}{2im}[\psi_{phy}^{*}\star_{V}(\partial_{x}\psi_{phy})-(\partial_{x}\psi_{phy}^{*})\star_{V}\psi_{phy}].$
(52)

A spatial integration over $x$ on both side of (51) over the entire range of
$\mathbb{R}^{1}$ now yields,

$\partial_{t}\int_{-\infty}^{\infty}\rho_{\theta}dx=-\int_{-\infty}^{\infty}(\partial_{x}J_{\theta}^{\theta})dx=0$
(53)

where we have imposed the following condition on $\psi_{phy}$:
$\psi_{phy}(x,t),\to 0$ as $x\to\pm\infty$ i.e. $\psi_{phy}(x,t)\in
L^{1}_{*}(\mathbb{R}^{1})$ is square integrable 222It may be noted that
$L_{\star}^{2}(\mathbb{R}^{1})$ involves $\theta$-deformed $\star$
multiplication rule, in contest to $L^{2}(\mathbb{R}^{1})$ which involves just
the usual point-wise multiplication., so that right hand side of the above
equation becomes zero giving the conservation of the total probability
$\int_{-\infty}^{\infty}\rho_{\theta}dx$ whereby we can write
$\rho_{\theta}(x,t)$ in a manifestly positive definite form.

$\rho_{\theta}(x,t)=\psi_{phy}^{*}(x,t)\,\star_{V}\,\psi_{phy}(x,t)=\frac{1}{2\pi\theta}\psi_{phy}^{\ast}(z,\bar{z})\star_{V}\psi_{phy}(z,\bar{z})=\frac{1}{2\pi\theta}\sum_{n=0}^{\infty}\frac{1}{n!}|\partial_{z}^{n}\psi_{phy}(z,\bar{z})|^{2}\geqslant
0.$ (54)

In view of the non-negative condition (54) $\rho_{\theta}(x,t)$ can indeed be
interpreted as probability density at a particular time. It is clear from the
continuity equation (51), the variable $t$ here plays the role of the
evolution parameter and for that we require that $\psi_{phy}(x,t)$ to be “well
behaved” i.e. it fulfills the following square integrability condition at a
constant time slice:

$\langle\psi_{phy}|\psi_{phy}\rangle_{*\,t}=\int_{-\infty}^{\infty}\,dx\,\,\psi_{phy}^{*}(x,t)\star_{V}\psi_{phy}(x,t)<\infty,$
(55)

so that $\psi_{phy}(x,t)\in L_{\star}^{2}(\mathbb{R}^{1})$ which is quite
distinct from $L_{\star}^{2}(\mathbb{R}^{2})$. All of these are
straightforward generalization of the corresponding commutative case,
discussed in sec-2. See, particularly, below (21). Clearly, the corresponding
inner-product for the space should also be defined by a spatial i.e.
$x$-integration for a fixed time slice as,

$\langle\psi_{phy}|\phi_{phy}\rangle_{*,\,t}=\int_{-\infty}^{\infty}\,dx\,\,\psi_{phy}^{*}(x,t)\star_{V}\phi_{phy}(x,t)<\infty.$
(56)

Of course, we need to emphasize again that here $t$ and $x$ should not be
identified as time and space coordinate; they are just the coherent state
expectation values, as given in (34).
Since the physical states satisfying the Schrödinger’s equation are
normalizable with respect to the inner product (56) of
$L_{\star}^{2}(\mathbb{R}^{1})$, it clearly indicates the fact that the
physical states $\psi_{phy}(\hat{x},\hat{t})$ should belong to a suitable
subspace of $\hat{\mathcal{A}}_{\theta}$ (29) which is distinct from
$\mathcal{H}_{q}$, for which the norm is obtained from the inner product
defined for $L_{\star}^{2}(\mathbb{R}^{2})$ (40).

##  4 Forced harmonic oscillator in quantum space-time

Having developed the basic formalism, we are now ready to discuss some
application. Forced harmonic oscillator [26, 27] is a good and simple example
to study the QM of time dependent system as it has various application in
physics specially in quantum optics, as mentioned earlier. To this end, we
place a forced harmonic oscillator in a quantum space-time in 1+1 dimension to
investigate whether any effect of non-commutativity shows up in the geometric
phase (which is not present in the system in classical space-time setting),
when the system is evolved adiabatically in a closed loop in parameter space
for a time period $\mathcal{T}$.
The Hamiltonian of a 1 D forced Harmonic oscillator (FHO) in commutative
space-time is given by,

$H=\frac{\hat{p}_{q}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{q}^{2}+f(t)\hat{q}+g(t)\hat{p}_{q}$
(57)

where $\hat{p}_{q}$ and $\hat{q}$ satisfies the usual Heisenberg algebra :
$[\hat{q},\hat{p}_{q}]=i$ and $(f(t)),g(t))$ are a pair of suitably chosen
periodic functions of time with time period $\mathcal{T}$ to be specified
later.
Now in presence of non-commutativity in space-time we take the Hamiltonian to
be given by the following simple operator ordering prescription, to render it
hermitian:

$\hat{H}=\frac{\hat{P}_{x}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{X}^{2}+\frac{1}{2}[f(\hat{T})\hat{X}+\hat{X}f(\hat{T})]+g(\hat{T})\hat{P}_{x}$
(58)

where the coordinate space variables are replaced with $\hat{T}$ and $\hat{X}$
given by (32). Using this Hamiltonian in the Schrödinger constraint equation
(48), and taking its overlap with the coherent state basis (38), we get by
using (32),

$\displaystyle i\partial_{t}(x,t|\psi_{phy})=(x,t|\hat{H}|\psi_{phy})$
$\displaystyle\Rightarrow$ $\displaystyle
i\partial_{t}\psi_{phy}(x,t)=\left[\frac{P_{x}^{2}}{2m}+\frac{1}{2}m\omega^{2}X_{\theta}^{2}+\frac{1}{2}\\{f(T_{\theta})X_{\theta}+X_{\theta}f(T_{\theta})\\}+g(T_{\theta})P_{x}\right]\psi_{phy}(x,t)$
(59) 333 The hermiticity of the Hamiltonian in the right hand side of (59) can
be established as already discussed in section 2.1.1 after (44).

where we have made use of the coherent state representation of the phase space
variables, which are given by (43,44,47) [11],

$P_{t}=-i\partial_{t};\,\,\,P_{x}=-i\partial_{x};\,\,\,X_{\theta}=x+\frac{\theta}{2}(\partial_{x}-i\partial_{t});\,\,\,T_{\theta}=t+\frac{\theta}{2}(\partial_{t}+i\partial_{x})$

Interestingly, this $X_{\theta}$ and $T_{\theta}$ can be related to
commutative $x$ and $t$, defined in (34), by making use of similarity
transformations as,

$X_{\theta}=SxS^{-1};\,\,\,\,\,T_{\theta}=S^{\dagger}t(S^{\dagger})^{-1}$ (60)

where

$S=e^{\frac{\theta}{4}(\partial_{t}^{2}+\partial_{x}^{2})}e^{-\frac{i\theta}{2}\partial_{t}\partial_{x}}$
(61)

is a non-unitary operator and this operator can be used to define a map from
$L^{2}_{*}(\mathbb{R}^{1})$ to $L^{2}(\mathbb{R}^{1})$ as,

$\displaystyle S^{-1}\,:\,\,L_{*}^{2}(\mathbb{R}^{1})\,\,$
$\displaystyle\to\,\,L^{2}(\mathbb{R}^{1})$ $\displaystyle
S^{-1}\big{(}\psi_{phy}(x,t)\big{)}\,$
$\displaystyle:=\,\psi_{c}(x,t)\,\,\in\,\,L^{2}(\mathbb{R}^{1})$ (62)

where we have used the notation $\psi_{c}$ to denote the elements of
$L^{2}(\mathbb{R}^{1})$ space. Now one can easily verify that the inner
product (56) in $L^{2}_{*}(\mathbb{R}^{1})$ becomes identical to its
commutative counterpart in $L^{2}(\mathbb{R}^{1})$ (see Appendix-A.3)

$\Big{\langle}\psi_{phy}\,,\,\phi_{phy}\Big{\rangle}_{*,\,t}=\langle\psi_{c}\,\,,\phi_{c}\rangle_{t}\,\,\,\,\forall\,\psi_{phy},\phi_{phy}\,\in\,L^{2}_{*}(\mathbb{R}^{1})$
(63)

where we have made use of integration by parts and dropped some boundary
terms. This equality shows that once in $L^{2}(\mathbb{R}^{1})$, we can indeed
replace non-local Voros star product, only within the integral, with the local
point-wise multiplication as in the usual commutative QM. It should be
emphasised here that, although the results of the integration as a whole are
equal in both sides of (63), the integrands, by themselves are not :
$\rho_{\theta}(x,t)=\psi_{phy}^{*}(x,t)\star\psi_{phy}(x,t)\,\neq\,\psi_{c}^{*}(x,t)\psi_{c}(x,t)=|\psi_{c}(x,t)|^{2}$,
as certain boundary terms had to be dropped in deriving (63), as we have
mentioned already. Physically, this means that both $\rho_{\theta}(x,t)$ and
$|\psi_{c}(x,t)|^{2}$ can’t be interpreted as the probability density
simultaneously for observing the prticle at the point $x$. And, as we have
shown that $\rho_{\theta}$ (54) represents the probability density in NC
spacetime, while $|\psi_{c}(x,t)|^{2}$ can be interpreted as probability
density only in the commutative limit $\theta\to 0$. Further the integrands
themselves are elements of two different algebras: non-commutative and
commutative, which are not clearly $\star$-isomorphic [37] to each other and
the effect of non-commutativity is manifested in different fashion [38]
through the considered dynamical model when we go from
$L_{*}^{2}(\mathbb{R}^{1})$ to $L^{2}(\mathbb{R}^{1})$ with help of the non-
unitary transformation $S^{-1}$.
Using (59,60,62) ,can now be recast in the following form,

$i\partial_{t}\psi_{c}(x,t)=\left[\frac{p_{x}^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2}+\frac{1}{2}\left\\{f(S^{-1}S^{\dagger}t(S^{\dagger})^{-1})x+xf(S^{-1}S^{\dagger}t(S^{\dagger})^{-1}S)\right\\}+g(S^{-1}S^{\dagger}t(S^{\dagger})^{-1})p_{x}\right]\psi_{c}(x,t)$
(64)

Further using the identity
$S^{-1}S^{\dagger}=e^{i\theta\partial_{t}\partial_{x}}:=U$, with $U$ being
unitary, (64) can be simplified as,

$i\partial_{t}\psi_{c}(x,t)=\left[\frac{p_{x}^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2}+\frac{1}{2}\\{f(UtU^{-1})x+xf(UtU^{-1})\\}+g(UtU^{-1})p_{x}\right]\psi_{c}(x,t)$
(65)

Now using the relation $UtU^{-1}=t-\theta p_{x}$, as follows by using Hadamard
identity and the Taylor expansion upto first order in $\theta$, by taking
$\theta$ to be very small, we can write,

$F(t-\theta p_{x})\approxeq F(t)-\theta p_{x}\dot{F}(t)$ (66)

where $F(t)$ stands collectively for $f(t)$ and $g(t)$ and $\dot{F}(t)$ is
their first order time derivatives. Note that $\dot{f}(t)$ and $\dot{g}(t)$
are also periodic functions of ‘$t$’ like $f(t)$ and $g(t)$. We now impose the
condition that $\dot{f}(t)$ and $\dot{g}(t)$ are slowly varying function of
time. This will facilitate the use of adiabatic approximation in the system
under consideration. Here we want to stress upon the fact that, had we taken
$f(t)$ and $g(t)$ themselves to be adiabatic in nature, then second and higher
order derivative of $f$ and $g$ would have to be dropped in our calculation
because of adiabatic approximation and the geometric phase would vanish in
that case. That is why we will choose such periodic functions $f(t)$ and
$g(t)$ very judiciously, so that only their first order derivatives are slowly
varying periodic functions of time.
Finally using (66) in (65) we get the effective commutative time dependent
Schrödinger equation for the theory as,

$i\partial_{t}\psi_{c}(x,t)=H_{c}\psi_{c}(x,t)$ (67)

where $H_{c}$ is the corresponding effective commutative Hamiltonian and is
given as

$H_{c}=\alpha(t)p_{x}^{2}+\beta
x^{2}+\gamma(t)(xp_{x}+p_{x}x)+f(t)x+g(t)p_{x}=H_{GHO}+f(t)x+g(t)p_{x}$ (68)

where $H_{GHO}$ stands for Hamiltonian of a generalised time dependent
harmonic oscillator and the time dependent coefficients occuring there are
given by,

$\alpha(t)=\frac{1}{2m}-\theta\dot{g}(t);\,\,\,\,\beta=\frac{1}{2}m\omega^{2};\,\,\,\,\gamma(t)=-\frac{1}{2}\theta\dot{f}(t)$
(69)

whereas the other two terms represent perturbations linear in position and
momentum in coordinate basis. In order to diagonalize the whole Hamiltonian we
try to diagonalize $H_{GHO}$ [39] first. And for that we introduce
annihilation and creation operators as follows:

$a(t)=A(t)[x+(B(t)+iC(t))p_{x}]$ (70)

where,

$\displaystyle
A(t)=\sqrt{\frac{\beta}{\Omega(t)}}\,\,\,\,\,\,\textrm{where}\,\,\,\,\,\Omega(t)=2\sqrt{\alpha\beta-\gamma^{2}};$
$\displaystyle
B(t)=\frac{\gamma(t)}{\beta};\,\,\,\,C(t)=\frac{\Omega(t)}{2\beta}$ (71)

are time dependent real functions of time. Correspondingly,

$a^{\dagger}(t)=A(t)[x+(B(t)-iC(t))p_{x}]$ (72)

fulfilling $[a(t),a^{\dagger}(t)]=\textbf{1}$.
Note that the coefficients $A,B,C$ and $\Omega$ are time dependent, as they
depend upon $\alpha,\gamma$ and thus $\dot{f}(t)$ and $\dot{g}(t)$. So we can
say that $A,B,C$ are slowly varying function of time and we shall in future
neglect the second and higher order time derivative of these variables in our
adiabatic approximation.
With this, the Hamiltonian (68) can be written in terms of the ladder operator
as ,

$H_{c}=\Omega(t)\Big{(}a^{\dagger}(t)a(t)+\frac{1}{2}\Big{)}+P(t)a(t)+\bar{P}(t)a^{\dagger}(t)$
(73)

where,

$P(t)=A(t)[C(t)f(t)+i(B(t)f(t)-g(t))]=\sqrt{\frac{\beta}{\Omega(t)}}\left[\frac{\Omega}{2\beta}f(t)+i\left(\frac{\gamma(t)f(t)}{\beta}-g(t)\right)\right]$

and $\Omega(t)$ can be identified with the “instantaneous frequency”. And
although, this $\Omega(t)$ can be taken to be a slowly varying function of
time, i.e. compatible with the adiabatic nature, the other time dependent
functions $P(t)$ and $\bar{P}(t)$ are a bit ambiguous as they consist of terms
like the product of a fast varying and a slow varying functions. As a result,
we cannot say that under time evolution the full Hamitonian varies slowly.
In order to circumvent this problem let us now look for a suitable time
dependent unitary transformation $\mathcal{U}(t)$, taking values in a suitable
Lie group and transforming the wave function $\psi_{c}(x,t)$ in (67) as

$\psi_{c}(x,t)\,\to\,\tilde{\psi}_{c}(x,t)=\mathcal{U}(t)\psi_{c}(x,t);\,\,\,\,\,\mathcal{U}^{\dagger}(t)\mathcal{U}(t)=1$
(74)

With this, the corresponding Hamiltonian will transform under this time-
dependent unitary transformation as

$H_{c}\,\to\,\tilde{H}_{c}=\mathcal{U}(t)H_{c}\mathcal{U}^{\dagger}(t)-i\mathcal{U}(t)\partial_{t}\mathcal{U}^{\dagger}(t)$
(75)

This shows that the time evolution of the transformed states
$\tilde{\psi}_{c}(x,t)$ is not governed by just
$\mathcal{U}H_{c}\mathcal{U}^{\dagger}$ anymore, rather by an effective
Hamiltonian $\tilde{\mathcal{H}}_{c}$, which is obtained by augmenting
$\mathcal{U}H_{c}\mathcal{U}^{\dagger}$ by a suitable “connection” like term:

$i\partial_{t}\tilde{\psi}_{c}(x,t)=\tilde{H}_{c}\tilde{\psi}_{c}(x,t)$ (76)

The instantaneous energy eigenstates of the Hamilotonian $H_{c}$ are not
anymore the eigenstates of those of $\tilde{H}_{c}$. Our purpose is to get rid
of the terms involving fast variables like $P(t)$ and $\bar{P}(t)$ in
$\tilde{\mathcal{H}}_{c}$ so that we are left with just the first term in (73)
by a suitable choice of $\mathcal{U}(t)$. Indeed, it is quite gratifying to
see that such a $\mathcal{U}(t)$ exists and is given by

$\mathcal{U}(t)=e^{-(wa-\bar{w}a^{\dagger}+il)},\,\,\,\,\,\,\,\,w\,\in\mathbb{C},\,\,l\,\in\,\mathbb{R}$
(77)

where $w$ and $l$ are given by,

$\displaystyle w(t)$
$\displaystyle=-ie^{i\int_{0}^{t}\,dt^{\prime}\,\Omega(t^{\prime})}\int_{0}^{t}\,dt^{\prime\prime}\,P(t^{\prime\prime})e^{-i\int_{0}^{t^{\prime\prime}}\,d\tilde{t}\,\Omega(\tilde{t})}$
$\displaystyle l(t)$
$\displaystyle=\int_{0}^{t}\,dt^{\prime}\,\left[\Omega(t^{\prime})|w(t^{\prime})|^{2}-\frac{i}{2}\Big{(}w(t^{\prime})\dot{\bar{w}}(t^{\prime})-\dot{w}(t^{\prime})\bar{w}(t^{\prime})\Big{)}\right]$
(78)

This structure of $\mathcal{U}(t)$ in (77) can be identified with the non-
abelian Heisenberg group, obtained by exponentiating the elements of
Heisenberg Lie algebra $\textswab{h}$, generated by
$(a,a^{\dagger},\mathbb{I})$, where $\mathbb{I}$ gives the central extension.
With this we can show that (75) becomes,

$\tilde{H}_{c}=\Omega(t)(a^{\dagger}a+\frac{1}{2})=H_{GHO}$ (79)

So by giving a time dependent unitary transformation (77) we are able to show
that the total Hamiltonian $H_{c}$ (68) transforms to a generalised hamonic
oscillator Hamiltonian (79). Having diagonalized $\tilde{H}_{c}$, the
computation of geometric phase acquired by $\psi_{c}(x,t)$ is reduced to that
of obtaining the corresponding phase acquired by $\tilde{\psi}_{c}(x,t)$,
which we take up in the next section.

##  5 Evolution of the ladder operators and appearance of geometric phase:

It is quite clear that the time evolved form of $\tilde{\psi}_{c}(x,t)$ in
(76) is considerably simpler to obtain than its precursor wave function
$\psi_{c}(x,t)$ occuring in (67) and evolving with Hamiltonian $H_{c}$ (68),
using the diagonalised form of the corresponding Hamiltonian $\tilde{H}_{c}$
(79) in our adiabatic approximation, particularly because of the absence of
terms involving the fast variables $P(t)$ and $\bar{P}(t)$. In fact, it
becomes even more simpler to isolate the geometric phase in the Heisenberg
picture, as was done in [11]. We undertake this task in this section.
The Heisenberg equation of motion using (79) for the operators $a$ and
$a^{\dagger}$ are given by,

$\frac{da^{\dagger}}{dt}=\xi(t)a^{\dagger}+X(t)a\hskip
20.0pt\textrm{and}\hskip
20.0pt\frac{da}{dt}=\bar{\xi}(t)a+\bar{X}(t)a^{\dagger}$ (80)

where

$X(t)=-A^{2}(\dot{C}+i\dot{B});\,\,\,\,\xi(t)=Y(t)+i\Omega(t);\,\,\,\,Y(t)=A[(2\dot{A}C+A\dot{C})+iA\dot{B}]$
(81)

We now make use of the adiabaticity of the parameters $\alpha$ and $\gamma$
and thus those of $A,B,C$ by dropping their second and higher order time
derivatives. The final form of the creation operator at the end of a cycle of
time period $t=\mathcal{T}$ is given by (the details of the calculation is
provided in the Appendix-A.4),

$a^{\dagger}(\mathcal{T})=a^{\dagger}(0)exp\left[i\int_{0}^{\mathcal{T}}\Omega\,\,d\tau+i\int_{0}^{\mathcal{T}}\left(\frac{1}{\Omega}\right)\frac{d\gamma}{d\tau}d\tau\right]$
(82)

The second term in the exponential of (82) represents the additional phase
over and above the dynamical phase $e^{i\int\Omega(t)dt}$, which is due the
adiabatic evolution of the Hamiltonian around a closed loop $\Gamma$ in the
parameter space in time $\mathcal{T}$. This geometric phase commonly known as
Berry phase (in Heisenberg picture) can be written in a more familiar form,
given as a functional of $\Gamma$,

$\Phi_{G}[\Gamma]=\int_{\Gamma=\partial
S}\frac{1}{\Omega}\nabla_{\textbf{R}}\gamma.d\textbf{R}=-\frac{\theta}{2}\int\int_{S}\nabla_{\textbf{R}}\left(\frac{1}{\Omega}\right)\times\nabla_{\textbf{R}}\Big{(}\dot{f}(t)\Big{)}\,.\,d\textbf{S}$
(83)

where we have used
$\frac{d}{d\tau}=\frac{d\textbf{R}}{d\tau}\,.\,\nabla_{\textbf{R}}$,
$\textbf{R}\equiv(\alpha,\beta,\gamma)$ being a vector in the parameter space
whose components are time dependent and in the second equality, we have made
use of Stoke’s theorem to convert the line integral into the surface integral.
The expression of the above equation (83), explicitly depends on the NC
parameter $\theta$ and vanishes in the commutative limit $\theta\to 0$. Note
that, we can relate this geometric phase shift obtained in the Heisenberg
picture, to a more familiar form of Berry phase obtained by the state vectors,
by going over to Schrödinger picture.
Finally observe that, had we worked in the Schrödinger’s picture, we would
have obtained the same geometric phase $\Phi_{G}[\Gamma]$ acquired by the wave
function $\tilde{\psi}_{c}(x,t)$ (74) in time $\mathcal{T}$. Eventually since
the original wave function $\psi_{c}(x,t)$ is simply
$\mathcal{U}^{\dagger}(t)\tilde{\psi}_{c}(x,t)$ with $\mathcal{U}(t)$ given in
(77,78) is a linear and unitary operator, it is clear that $\psi_{c}(x,t)$ too
will acquire the same Berry phase in time $\mathcal{T}$, as the Berry phase,
being a simple number (83) will not be affected by action of the Heisenberg
group. We would like to make some pertinent observations before we conclude
this section.

  * •

It can be observed easily from (83) that the geometric phase depends upon the
parameter $\gamma$ given in (69) which vanishes for $\theta=0$. So the non-
commutativity of the space-time plays here a crucial role for the appearance
of the geometric phase. Note in this context that the authors in [39] have
shown that the occurence of the dialatation term in the Hamiltonian is a
neccessary element for a system to produce the geometric phase. And although
our original system i.e. the forced harmonic oscillator (57) did not contain
such term, placing the system in a NC space-time has enabled us to generate
such a term in the effective commutative Hamiltonian (68).

  * •

The Hamiltonian of the F.H.O in (68) can be expanded in terms of the
generators of the double photon algebra [40]. For that let us define

$K_{+}=\frac{ix^{2}}{2},\,\,\,K_{-}=\frac{ip^{2}}{2},\,\,\,K_{0}=\frac{i(xp+px)}{4},\,\,\,A_{+}=x,A_{-}=p$
(84)

Taking $\mathbb{I}$ as a central extension we can show that the above
generators follow the following Lie algebra:

$\displaystyle[K_{0},K_{\pm}]=\pm K_{\pm},\,\,\,\,[K_{+},K_{-}]=-2K_{0}$ (85)
$\displaystyle[K_{+},A_{-}]=-A_{+},\,\,\,[K_{-},A_{+}]=A_{-},\,\,\,[K_{0},A_{+}]=\frac{A_{+}}{2},\,\,\,[K_{0},A_{-}]=-\frac{A_{-}}{2},\,\,\,[A_{+},A_{-}]=i\mathbb{I}$
(86)

where the sub-algebra (85) is the su(1,1) Lie-algebra and (85),(86) as a whole
represents the double photon algebra [40].
Now the Hamiltonian (68) can therefore be written in terms of the generators
(84) as

$H_{c}=B^{\mu}(t)K_{\mu}+f(t)A_{+}+g(t)A_{-};\,\,\,\,\mu=+,-,0;\,\,\,\,B^{+}=-2i\alpha(t),B^{-}=-2i\beta(t),B^{0}=-2i\gamma(t)$
(87)

The corresponding transformed Hamiltonian $\tilde{H}_{c}$, on the other hand,
takes its value only in the su(1,1) sub-algebra.

$H_{c}\,\to\tilde{H}_{c}\,=\mathcal{U}H_{c}\mathcal{U}^{\dagger}-i\mathcal{U}\frac{\partial\mathcal{U}^{\dagger}}{\partial
t}=B^{\mu}(t)K_{\mu}$ (88)

At this stage, we may point out that the Hamiltonians in [39] were also, for
example, su(1,1) and su(2) Lie algebra valued, in analogy with ours. So like
in their case, we too can anticipate the occurance of geometric phase in our
case.

##  6 Conclusion and future direction

Experimental detection of Planck scale phenomena involving quantum structure
of space-time, in a non-relativistic (NR) system, is a challanging and
difficult task. However, in some previous works in the literature [41, 42],
authors have tried to find plausible scenarios, which can possibly help to
find out the effects caused by quantum structure of space-time, in low energy
regime. That motivated us to study the fingerprinters of Planck scale physics
in NR quantum system, which is the main objective of our present work. Here we
will briefly summerize our findings.
We have shown that a system of a forced harmonic oscillator, when placed in a
non-commutative space-time background, acquires a geometric phase shift. The
effective commutative system Hamiltonian is shown to be as that of a time
dependent generalised harmonic oscillator with perturbation linear in position
and momentum. The system is then subjected to a unitary transformation to get
rid of these perturbations and then diagonalised with the help of the time
dependent unitary transformation. We then derived the equation of motion of
the ladder operators in Heisenberg picture, which shows that an extra phase
over and above the dynamical phase is produced, depending on the geometry of
the parameter space, when the system is transported adiabatically around a
closed loop. This phase directly depends on the non-commutative parameter
$\theta$ in our expression, computed upto first order in $\theta$ and vanishes
in commutative limit $\theta\to 0$. It should be noted in this context that
our computations, carried out in Heisenberg picture, as in [32], is rather a
bit of an unconventional approach to calculate the geometric phase. Its
primary advantage is that, it provides a natural framework for semi-classical
correspondence. The geometric phase calculated in this procedure can shown to
be equal to the Hannay’s angle, which is generated by the corresponding
classical adiabatic evolution.

Finally, we would like to comment on some future prospects of our work. In
[43], the authors had shown an interesting relationship between the Fubini-
Study metric defined in the projective Hilbert space of a time dependent
quantum system and the energy-time uncertainty relation. Later, authors in
[44] has pointed out that, the coherent states provide a natural framework for
the realization of the above mentioned relationship and demonstrated this
explicitly in their computation involving a spin system in a magnetic field
and a generalised harmonic oscillator system respectively. In light of the
present paper, it seems that we can also extend our study to compute the
energy-time uncertainty relation in our system as well, by making use of the
existing Fubini-Study metric of the projective Hilbert space. Presumably, the
space-time noncommutativity, which is solely responsible for giving rise to
the geometric phase and inducing a geometry in the parameter space, should
have some tangible effect in the energy-time uncertainty relation and it
should be quite interesting to investigate this point in our future work.
On the other hand, in [26], it was shown that a coherent state persists to be
coherent at all times if the system Hamiltonian, with which the operators are
evolved (in Heisenberg picture), is that of a time dependent forced harmonic
oscillator. We can extend this computation for our system, which is basically
FHO, in the presence of noncommutativity in space-time. The effective
commutative system, which is a system of generalised forced harmonic
oscillator (GFHO), will produce states which are squeezed coherent states of
an usual FHO [45]. This encourages us to study the fate of these squeezed
coherent states under time evolution and its application in quantum optics and
quantum information theory, if any.

Furthermore, note that the idea of noncommutativity in the space-time sector
may also be introduced through the non-relativistic, second quantized
formalism. This would indeed be a good starting point for considering NC
space-time because, unlike in QM, here, space and time can naturally be
treated on an equal footing, in the sense that the spatial coordinates do not
correspond to any observables anymore and to begin with, they are just
c-numbered parameters (lebelling the continuous spatial degrees of freedom of
the system) like the evolution parameter time. Importantly, the second
quantized formalism allows us to investigate the impact of space-time
noncommutativity on the one-particle sector of quantum field theory for a
generic external potential [46], which is similar to NC quantum mechanical
analysis in the first quantized formalism. We believe that the nontrivial
changes in the Schrödinger equation is indeed originating from the space-time
noncommutativity which has not hitherto been explored in the literature, to
the best of our knowledge. We hope to return with some of these issues in our
future works.

## Acknowledgement

P.N. would like to thank Prof. A. Deriglazov for valuable and constructive
suggestions during the development of the work and Prof. A.P. Balachandran for
carefully going through the manuscript and pointing out certain shortcomings
in the earlier version of the draft. A.C. would like to express sincere
gratitude to Mr. Sayan Kumar Pal for many helpful discussions for this paper.
A.C. would like to thank DST-India for providing financial support in the form
of fellowship during the course of this work and P.N. would like to thank S N
Bose National Centre for Basic Sciences, Kolkata, for providing financial
support during the project tenure.

## Appendices

### A.1 On the effect of space-time noncommutativity on a time-independent
system

Let us take a time-independent Hamiltonian placed in a NC space-time.

$H=\frac{\hat{p}_{x}^{2}}{2m}+V(\hat{X})$ (89)

The corresponding Schrödinger equation is obtained by taking its
representation in coherent state basis (35) to get,

$i\partial_{t}\psi_{phy}(x,t)=(x,t|\hat{H}|\psi_{phy})=\Big{[}\frac{p_{x}^{2}}{2m}+V(X_{\theta})\Big{]}\psi_{phy}(x,t);\qquad\psi_{phy}(x,t)\in
L^{2}_{*}(\mathbb{R}^{1})$ (90)

Now, using (60), we get

$i\partial_{t}\psi_{phy}(x,t)=\Big{[}\frac{p_{x}^{2}}{2m}+SV(x)S^{-1}\Big{]}\psi_{phy}(x,t)$

which, when re-written in terms of $\psi_{c}(x,t)=S^{-1}\psi_{phy}(x,t)\in
L^{2}(\mathbb{R}^{1})$ (62), readily yields the usual commutative Schrödinger
equation:

$i\partial_{t}\psi_{c}(x,t)=\big{[}\frac{p_{x}^{2}}{2m}+V(x)\Big{]}\psi_{c}(x,t)$
(91)

thereby showing that one effectively recovers the commutative Schrödinger
equation and leaves the spectrum and therefore the dynamics of the system
unaffected. Rather, it is the wave function of the system that only gets
modified by the transformation : $\psi_{c}=S^{-1}\psi_{phy}$.

### A.2 Dirac’s constraint analysis for the Lagrangian (24)

Starting with the first order form of the Lagrangian (24), here we are going
to carry out Dirac’s constraint analysis and compute the Dirac’s bracket among
the configuration space variables where space-time and their corresponding
canonical momenta are considered to be configuration space variable,

$L_{f}^{\tau,\theta}=p_{\mu}\dot{x}^{\mu}-\sigma(\tau)(p_{t}+H)+\frac{\theta}{2}\epsilon^{\mu\nu}p_{\mu}\dot{p}_{\nu},\,\,\,\,\,\,\mu,\nu=0,1$
(92)

$\sigma(\tau)$ is an arbitrary Lagrange multiplier. Now we can derive the
canonical momenta corresponding to $x^{\mu},p_{\mu}$ and $\sigma(\tau)$ as
following :

$\pi^{x}_{\mu}=\frac{\partial
L^{\tau,\theta}}{\partial\dot{x}^{\mu}}=p_{\mu};\qquad\pi_{p}^{\mu}=\frac{\partial
L^{\tau,\theta}}{\partial\dot{p}_{\mu}}=-\frac{\theta}{2}\epsilon^{\mu\nu}p_{\nu};\qquad\pi_{\sigma}=\frac{\partial
L^{\tau,\theta}}{\partial\dot{\sigma}}=0$ (93)

We can see that the canonical momenta are not related to the generalized
velocities and so that the above equations (93) are interpreted as primary
constraints of the theory and can be written as

$\Phi_{1,{\mu}}=\pi_{\mu}^{x}-p_{\mu}\approx
0,\,\,\,\,\,\,\,\Phi_{2}^{\mu}=\pi^{\mu}_{p}+\frac{\theta}{2}\epsilon^{\mu\nu}p_{\nu}\approx
0,\,\,\,\,\,\,\,\Phi_{3}=\pi_{\sigma}\approx 0$ (94)

Using the following Poission brackets between the canonical pairs,

$\\{x^{\mu},\pi_{\nu}^{x}\\}=\delta^{\mu}\,_{\nu},\,\,\,\,\,\,\\{p_{\mu},\pi^{\nu}_{p}\\}=\delta_{\mu}\,^{\nu}$
(95)

we can derive the Poission brackets between the constraints as (94) as

$\displaystyle\\{\Phi_{3},\Phi_{3}\\}=\\{\Phi_{3},\Phi_{1,\mu}\\}=\\{\Phi_{3},\Phi_{2}^{\mu}\\}=0$
$\displaystyle\\{\Phi_{1,{\mu}},\Phi_{1,{\nu}}\\}=0;\qquad\\{\Phi_{1,{\mu}},\Phi_{2}^{\nu}\\}=-\delta_{\mu}\,^{\nu}\qquad\\{\Phi_{2}^{\mu},\Phi_{2}^{\nu}\\}=\theta\epsilon^{\mu\nu}$
(96)

We can therefore write the canonical Hamiltonian as

$H_{c}=\sigma(\tau)(p_{t}+H)$ (97)

We can see that $\Phi_{3}$ gives zero brackets with all other constraints. So
it can be classified as a first class constraint. The rest of the constraints
are classified as second class constraints as the Poission bracket between
$\Phi_{1,{\mu}}$ and $\Phi_{2,\mu}$ does not vanish. We can therefore
implement the second class constraints strongly through the Dirac brackets.
But, before that note that, the total Hamiltonian can be written by adding all
the constraints to the canonical Hamiltonian $H_{c}$ (97) as,

$H_{T}=\sigma(\tau)(p_{t}+H)+\lambda_{1,\mu}\Phi_{1,\mu}+\lambda_{2,\mu}\Phi_{2}^{\mu}+\lambda_{3}\Phi_{3}$
(98)

where $\lambda_{1,\mu},\lambda_{2,\mu}$ and $\lambda_{3}$ are suitable
multipliers. Now the preservation of $\Phi_{3}$ in time yields the secondary
constraint:

$\Sigma=\dot{\Phi_{3}}=\\{H_{T},\pi_{\sigma}\\}=p_{t}+H\approx 0$ (99)

From the time preservation conditions of the other constraints we just get the
relation between $\sigma(\tau)$ and $\lambda_{1,\mu},\lambda_{2,\mu}$, which
are not of our concern in this context. Rather, we shall implement the other
second class constraints, as mentioned above. And for that we first need to
write down the constraint matrix $\Lambda_{ab}$ and its inverse
$(\Lambda^{-1})_{ab}$ which are now obtained as,

$\Lambda_{ab}=\begin{pmatrix}\\{\Phi_{1,{\mu}},\Phi_{1,{\nu}}\\}&\\{\Phi_{1,{\mu}},\Phi_{2}^{\nu}\\}\\\
\\{\Phi_{2}^{\nu},\Phi_{1,{\mu}}\\}&\\{\Phi_{2}^{\mu},\Phi_{2}^{\nu}\\}\end{pmatrix}=\begin{pmatrix}0&-\delta_{\mu}\,^{\nu}\\\
\delta^{\nu}\,_{\mu}&\theta\epsilon^{\mu\nu}\end{pmatrix};\,\,\,\,\,\,\,\,\,(\Lambda^{-1})_{ab}=\begin{pmatrix}\theta\epsilon_{\mu\nu}&\delta^{\mu}\,_{\nu}\\\
-\delta_{\nu}\,^{\mu}&0\end{pmatrix}$ (100)

fulfilling, $\Lambda_{ab}(\Lambda^{-1})_{bc}=\delta_{ac}$.
Using the following definition for Dirac brackets [24],

$\\{f,g\\}_{D}=\\{f,g\\}-\\{f,\Phi_{a}\\}(\Lambda^{-1})_{ab}\\{\Phi_{b},g\\}$
(101)

we now compute the Dirac brackets between the phase space variables to get the
following structure (25):

$\\{x^{\mu},x^{\nu}\\}_{D}=\theta\epsilon^{\mu\nu},\,\,\,\,\,\,\\{p_{\mu},p_{\nu}\\}_{D}=0,\,\,\,\,\,\,\\{x^{\mu},p_{\nu}\\}_{D}=\delta^{\mu}\,_{\nu}$

These Dirac brackets, when elevated to the quantum level, readily yields the
NCHA (26,27), which includes the space time noncommutative structures.
Finally, we would like to make certain observations regarding the roles of
primary first class constraint $\Phi_{3}$ and the secondary constraint
$\Sigma$. The primary first class constraint $\Phi_{3}$ (94) involves the
canonical momentum corresponding to the Lagrange’s multiplier $\sigma(\tau)$,
which is not physical in nature. So we can ignore this constraint. On the
other hand by using the Dirac brackets (A.2 Dirac’s constraint analysis for
the Lagrangian (24)) we can easily show that the secondary constraint $\Sigma$
(99) is a first class constraint as it has vanishing Dirac brackets with all
other primary constraints. And eventually $\Sigma$ can be shown to give rise
to the $\tau$ evolution of the system in the form of gauge transformation of
the theory.

### A.3 On the mapping $S^{-1}$: $L_{*}^{2}(\mathbb{R}^{1})$ to
$L^{2}(\mathbb{R}^{1})$

Here we would like to prove that, the map (62) i.e.
$S^{-1}:\,L^{2}_{*}(\mathbb{R}^{1})\,\to\,L^{2}(\mathbb{R}^{1})$, with
$S^{-1}=e^{-\frac{\theta}{4}(\partial_{t}^{2}+\partial_{x}^{2})}e^{i\frac{\theta}{2}\partial_{t}\partial_{x}}$,
a non-unitary operator, is an inner product preserving one i.e. the inner
product between a pair of arbitrary states in $L_{*}^{2}(\mathbb{R}^{1})$
space with respect to star multiplication, is equal to the inner product of
the transformed states belonging to $L^{2}(\mathbb{R}^{1})$ with respect to
ordinary point-wise multiplication.
For that, let $\psi_{phy}(x,t;\theta)\,\in\,L^{2}_{*}(\mathbb{R}^{1})$ and
$\psi_{c}(x,t)\,\in\,L^{2}(\mathbb{R}^{1})$ and related by (62) as
$\psi_{phy}(x,t,\theta)=S\psi_{c}(x,t)$, satisfy the respective Schrödinger
equations (59) and (67). Now, if we consider a stationary wave-function in
$L^{2}(\mathbb{R}^{1})$ as

$\psi_{c}(x,t)=e^{-iEt}\psi_{c}(x),$ (102)

then the corresponding physical state in $L^{2}_{\star}(R^{1})$ can be easily
determined by using (62) as

$\psi_{phy}(x,t;\theta)=e^{-iEt}e^{\frac{\theta}{4}(-E^{2}+\partial^{2}_{x})}e^{\frac{\theta}{2}E\partial_{x}}\psi_{c}(x)=e^{-iEt}\psi_{phy}(x;\theta)$
(103)

So here too we see that, the time part gets factored out here in the form of
$e^{-iEt}$ just like (102), which is indicative of the fact that this
$\psi_{phy}(x,t,\theta)$ is also a stationary state with the same energy
eigenvalue $E$. Now the norm of the state $\psi_{c}(x,t)\in
L^{2}(\mathbb{R}^{1})$ is defined as

$\displaystyle\int\,\,dx\,\,\psi_{c}^{*}(x,t)\psi_{c}(x,t)$ $\displaystyle=$
$\displaystyle\int\,\,dx\,\,(S^{-1}\psi_{phy}(x,t;\theta))^{*}\quad(S^{-1}\psi_{phy}(x,t;\theta))$
$\displaystyle=$
$\displaystyle\int\,\,dx\,\,\Big{[}\psi^{*}_{phy}(x,t;\theta)e^{-\frac{\theta}{4}(-E^{2}+\overleftarrow{\partial_{x}}^{2})}e^{-\frac{i\theta}{2}\overleftarrow{\partial_{t}}\overleftarrow{\partial_{x}}}\Big{]}\Big{[}e^{-\frac{\theta}{4}(-E^{2}+\overrightarrow{\partial_{x}}^{2})}e^{i\frac{\theta}{2}\overrightarrow{\partial_{t}}\overrightarrow{\partial_{x}}}\psi_{phy}(x,t;\theta)\Big{]}$
$\displaystyle=$
$\displaystyle\int\,\,dx\,\,\Big{[}\psi^{*}_{phy}(x,t;\theta)e^{-\frac{\theta}{4}(-\overleftarrow{\partial}_{t}\overrightarrow{\partial}_{t}-\overleftarrow{\partial_{x}}\overrightarrow{\partial}_{x})}e^{\frac{i\theta}{2}\overleftarrow{\partial_{t}}\overrightarrow{\partial_{x}}}\Big{]}\Big{[}e^{-\frac{\theta}{4}(-\overleftarrow{\partial}_{t}\overrightarrow{\partial}_{t}-\overleftarrow{\partial_{x}}\overrightarrow{\partial_{x}})}e^{-i\frac{\theta}{2}\overleftarrow{\partial_{x}}\overrightarrow{\partial_{t}}}\psi_{phy}(x,t;\theta)\Big{]}$
(Here we have used integration by parts and dropped the boundary terms.)
$\displaystyle=$
$\displaystyle\int\,\,dx\,\,\Big{[}\psi^{*}_{phy}(x,t;\theta)e^{\frac{\theta}{2}(\overleftarrow{\partial}_{t}\overrightarrow{\partial}_{t}+\overleftarrow{\partial}_{x}\overrightarrow{\partial}_{x})}e^{\frac{i\theta}{2}(\overleftarrow{\partial}_{t}\overrightarrow{\partial}_{x}-\overleftarrow{\partial}_{x}\overrightarrow{\partial}_{t})}\psi_{phy}(x,t;\theta)\Big{]}=\int\,\,dx\,\,\psi^{*}_{phy}(x,t;\theta)\star_{V}\psi_{phy}(x,t;\theta)$
(104)

With this we are able to prove that, the norm of the stationary wave function
$\psi_{phy}(x,t;\theta)$ associated with energy eigen value $E$, in
$L^{2}_{*}(\mathbb{R}^{1})$ space is equal to the norm of the transformed
stationary wave-function $\psi_{c}(x,t)$ in $L^{2}(\mathbb{R}^{1})$ space with
the same energy eigenvalue $E$. The products defining the norms in Hilbert
spaces $L^{2}(\mathbb{R}^{1})$ and $L^{2}_{*}(\mathbb{R}^{1})$ are the usual
point-wise multiplication and Voros star multiplication respectively.

Now let us construct a pair of non-stationary physical states by taking linear
combination of stationary state wave functions with suitable coefficients such
as,

$\Psi_{phy}(x,t;\theta)=\sum_{n}A_{n}\psi^{(n)}_{phy}(x,t;\theta),$ (105)
$\Phi_{phy}(x,t;\theta)=\sum_{n}B_{n}\psi^{(n)}_{phy}(x,t;\theta)$ (106)

where $\psi^{(n)}_{phy}(x,t;\theta)=e^{-iE_{n}t}\psi^{(n)}_{phy}(x;\theta)$ is
a stationary state for a particular $n^{th}$ energy eigen value $E_{n}$. Note
that in 1D spatial dimension all bound stationary states are non-degenerate
and here $\psi^{(n)}_{phy}(x,t;\theta)$ forms a complete set of basis and also
therefore can be chosen to orthonormal as,

$\int_{t}~{}dx~{}\psi^{*(n)}_{phy}(x,t;\theta)\star_{V}\psi^{(m)}_{phy}(x,t;\theta)=\delta_{mn}$
(107)

Now let us consider the inner products between this pair of non stationary
states (105) and (106) as

$\int
dx~{}\Phi^{\star}_{phy}(x,t;\theta)\star_{V}\Psi_{phy}(x,t;\theta)=\sum_{n,m}A_{m}B^{\star}_{n}\int
dx~{}\psi^{(n)\star}_{phy}(x,t;\theta)\star_{V}\psi^{(m)}_{phy}(x,t;\theta)$
(108)

Using the orthonormality (107) we arrive at

$\displaystyle\int\,dx\,\Phi^{\star}_{phy}(x,t;\theta)\star_{V}\Psi_{phy}(x,t;\theta)$
$\displaystyle=\sum_{n}A_{n}B^{\star}_{n}\int
dx~{}\psi^{(n)\star}_{phy}(x,t;\theta)\star_{V}\psi^{(n)}_{phy}(x,t;\theta)$
$\displaystyle(\textrm{Now using (\ref{3})})\qquad$
$\displaystyle=\sum_{n}A_{n}B_{n}^{*}\,\int\,\,dx\,\,\psi_{c}^{(n)*}(x,t)\,\psi_{c}^{(n)}(x,t),$
(109)

where we have used $\psi_{c}^{(n)}(x,t)=S^{-1}\psi^{(n)}_{phy}(x,t;\theta)$.
It can be shown by simple argument that, if the non-stationary state
$\Psi_{phy}(x,t,\theta)\,\in\,L^{2}_{*}(\mathbb{R}^{1})$ is given by (105),
then the corresponding non-stationary state in $L^{2}(\mathbb{R}^{1})$ is
given by,

$\Psi_{c}(x,t)=S^{-1}\Psi_{phy}(x,t,\theta)=S^{-1}\sum_{n}A_{n}\psi_{phy}^{(n)}(x,t,\theta)=\sum_{n}A_{n}S^{-1}\psi_{phy}^{(n)}(x,t,\theta)=\sum_{n}A_{n}\psi_{c}^{(n)}(x,t)$
(110)

Using this we can finally write (109) as,

$\int\,\,dx\,\,\Phi^{*}_{phy}(x,t,\theta)\,\star_{V}\,\Psi_{phy}(x,t,\theta)=\int\,\,dx\,\,\Phi_{c}^{*}(x,t)\,.\,\Psi_{c}(x,t)$
(111)

This equality suggests that, the inner product between two non-stationary
states in $L^{2}_{*}(\mathbb{R}^{1})$ with respect to star multiplication is
indeed equal to the inner product between the corresponding pair of non-
stationary states of $L^{2}(\mathbb{R}^{1})$ space with respect to usual
point-wise product which we use for usual commutative QM. This is also
expected because of the following reason: the map, $S^{-1}$ is a non-unitary
transformation, which naturally changes the inner-product of the Hilbert space
and the Voros star product in $L^{2}_{*}(\mathbb{R}^{1})$ gets replaced by
usual point-wise product of $L^{2}(\mathbb{R}^{1})$ space.

### A.4 Calculation of the geometric phase

To solve the first order differential equation of the ladder operators given
by (80), we can decouple the equations in (80) , by taking their time
derivatives and combining them suitably to get,

$\frac{d^{2}a^{\dagger}}{dt^{2}}+Z_{1}(t)\frac{da^{\dagger}}{dt}+Z_{2}(t)a^{\dagger}=0;\,\,\,\,\,\,\,\,\,\frac{d^{2}a}{dt^{2}}+\bar{Z}_{1}(t)\frac{da}{dt}+\bar{Z}_{2}(t)a=0$
(112)

where

$Z_{1}(t)=-\left(\frac{\dot{X}}{X}+\xi+\bar{\xi}\right);\qquad
Z_{2}(t)=-(\dot{\xi}-\frac{\dot{X}}{X}\xi+X\bar{X}-\xi\bar{\xi})$ (113)

where $X(t),\xi(t)$ are time dependent and defined in (81). All the
expressions found here, so far, are exact. We shall now consider the
adiabaticity of $\alpha(t)$ and $\gamma(t)$ in (69). Since, we have taken
$\dot{f}$ and $\dot{g}$ to be slowly varying periodic function of time, it
follows that, $\alpha(t)$ and $\gamma(t)$ (69) too vary adiabatically with
time. Further, since $A(t),B(t)$ and $C(t)$ (71) depends linearly on
$\alpha(t),\beta,\gamma(t)$, they too follow the same order of adiabaticity.
So if $\dot{\alpha},\dot{\gamma}\sim\epsilon$ and
$\ddot{\alpha},\ddot{\gamma}\sim\epsilon^{2}$, then,
$\dot{F}\sim\epsilon\,\,\textrm{and}\,\,\ddot{F}\sim\epsilon^{2}$,where $F$
collectively stands for $A,B,C$. Eventually we shall ignore second and higher
order time derivatives ($\sim\epsilon^{2}$) for considering adiabatic time
evolution. To proceed further we write (112) in its normal form. To do so, we
introduce another time dependent operator, $b(t)$ as,

$a^{\dagger}(t)=b^{\dagger}(t)e^{-\frac{1}{2}\int_{0}^{t}Z_{1}(t^{\prime})dt^{\prime}}$
(114)

With this the first equation of (112) can be recast in terms of $b^{\dagger}$

$\frac{d^{2}b^{\dagger}(t)}{dt^{2}}+\tilde{Z}_{2}(t)b^{\dagger}(t)=0$ (115)

where,

$\displaystyle\tilde{Z}_{2}=Z_{2}-\frac{1}{2}\dot{Z}_{1}-\frac{Z_{1}^{2}}{4}\approx\Omega^{2}-\Omega\tilde{W}+i(2\Omega\frac{\dot{A}}{A}+\Omega)\sim\mathcal{O}(\epsilon)=U+iV(say)$
(116)

where we have taken
$\frac{\ddot{C}+i\ddot{B}}{\dot{C}+i\dot{B}}=W+i\tilde{W}\sim\mathcal{O}(\epsilon)$.
Then (115) can be re-written in the following form:

$\frac{d^{2}b^{\dagger}(t)}{dt^{2}}+(U+iV)b^{\dagger}(t)=0$ (117)

As we are working in the adiabatic regime, both $U$ and $V$ vary slowly with
time as is clear from (116). We can then apply the formula for WKB
approximation for complex potential [47] given by,

$b^{\dagger}(t)=b^{\dagger}(0)\left[\frac{C_{1}}{\sqrt{|\chi(t)|}}exp\left(\int_{0}^{t}(i\chi(\tau)-\phi(\tau))d\tau\right)+\frac{C_{2}}{\sqrt{|\chi(t)|}}exp\left(\int_{0}^{t}(-i\chi(\tau)+\phi(\tau))d\tau\right)\right]$
(118)

where $\sqrt{U+iV}=\chi+i\phi$ so that (118) can be applied to find the
general solution of the differential equation (117). In our case, $\chi$ and
$\phi$ are given as,

$\chi=\sqrt{\frac{\sqrt{U^{2}+V^{2}}+U}{2}}\approx\sqrt{U+\frac{V^{2}}{4U}}\approx\sqrt{U}\approx\Omega-\frac{\tilde{W}}{2}$
$\phi=\sqrt{\frac{\sqrt{U^{2}+V^{2}}-U}{2}}\approx\sqrt{\frac{V^{2}}{4U}}\approx\frac{\dot{A}}{A}+\frac{W}{2}$

where second and higher order derivative terms are ignored everywhere. We also
have implied the parameters to be periodic. Further note that only the first
term in the exponential of (118) yields the correct sign of the dynamical
phase for $a^{\dagger}$. So we shall put $C_{2}=0$. Now combining all the
expressions we get at $t=\mathcal{T}$,

$b^{\dagger}(\mathcal{T})\approx
b^{\dagger}(0)\frac{C_{1}}{\sqrt{|\chi(t)|}}exp\left(\int_{0}^{\mathcal{T}}[i\Omega-\frac{d}{d\tau}(lnA)-\frac{1}{2}(W+i\tilde{W})]d\tau\right)$
(119)

As the integrands are exact differential and as we have the periodic boundary
condition for the functions, i.e. $A(T)=A(0)$, we can drop some terms in the
exponent as following:

$\int_{0}^{\mathcal{T}}\Big{(}\frac{\dot{A}}{A}+\frac{W+i\tilde{W}}{2}\Big{)}d\tau=\int_{0}^{\mathcal{T}}\frac{1}{2}\frac{d}{d\tau}\Big{(}lnX)d\tau=0$

$X$ is given in (81). Substituting (119) in (114), we get

$a^{\dagger}(\mathcal{T})=b^{\dagger}(\mathcal{T})\,\,exp\,\,\Big{(}{\int_{0}^{\mathcal{T}}\Big{[}\frac{d}{dt^{\prime}}(ln\,X+A^{2}C)+iA^{2}B\Big{]}dt^{\prime}}\Big{)}=b^{\dagger}(\mathcal{T})e^{i\int_{0}^{\mathcal{T}}A^{2}\dot{B}dt^{\prime}}$
(120)

The expression of $A,B,C$ are given in (71). Finally making use of (119) in
(120) we can ultimately reproduce (82) as,

$\displaystyle a^{\dagger}(\mathcal{T})$
$\displaystyle=a^{\dagger}(0)exp\left(i\int_{0}^{\mathcal{T}}\Omega\,\,dt^{\prime}\right)exp\left(i\int_{0}^{\mathcal{T}}A^{2}\dot{B}\,\,d\tau\right)$
$\displaystyle=a^{\dagger}(0)exp\left[i\int_{0}^{\mathcal{T}}\Omega\,\,d\tau+i\int_{0}^{\mathcal{T}}\left(\frac{1}{\Omega}\right)d\gamma\,\,\,\right]$
(121)

The two terms in the exponents clearly represents the dynamical and geometric
phase respectively.

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