Some Aspects of Quantum Mechanics and Quantum Field Theory on Quantum Space-
Time

Thesis Submitted For The Degree Of
Doctor of Philosophy (Science)
In

Physics (Theoretical)
By

Partha Nandi

University of Calcutta, India
2021

To my beloved Mother ….

## Acknowledgments

It is a pleasant duty to acknowledge my indebtedness to all those who have
contributed in different ways to the completion of this thesis. I thank them
all for their co-operation, my heartfelt gratitude is to my supervisor
Professor Biswajit Chakraborty (S. N. Bose National Centre for Basic Sciences)
whom I owe my understanding of Physics. He is the person who suggested to me
the ideas that have been developed in the following pages and actively guided
me all along till the completion of this task.

My heartfelt thanks go out to my great teacher Prof. Debashis Chatterjee
(Vijaygarh Joytish Roy College). I sincerely thank him for all those intense
academic discussions. I would like to thank Prof. Frederik G. Scholtz
(Universiteit Stellenbosch) for sharing with me his insightful and innovative
ideas which led to some very fruitful discussions. I thank Prof A. P.
Balachandran and Prof. Michel Berry, Prof. V.P Nair for always being there to
help me in my work. I take this opportunity to convey my sincere gratitude to
Prof. Sibasish Ghosh (IMSc, India ) for organising a short visit and holding
valuable discussions. I am also grateful to Prof. Rabin Banerjee for his
enthusiastic discussions on physics to clarify very elementary but at same
time subtle concepts in physics.

I would also like to express my gratitude Prof. Amitava Lahiri, Prof. Archan
S. Majumdar and Prof. Manu Mathur and Prof. Sunandan Gangopadhyay for helping
me in my academics at this Centre. I am indebted to Prof. Anirban Kundu, Prof.
Indrajit Mitra and Prof. Debnarayana Jana, University of Calcutta, for giving
me crucial advices and guidance, whenever I required.

I am indebted to Prof. S.K Roy, ex. Director of SNBNCBS, for many facilities I
enjoyed in this centre. I thank all the academic and administrative staff of
S.N Bose centre for helping me in many ways. I am thankful to them for their
services. This works been made possible through a fellowship of the S. N. Bose
National Centre for Basic Sciences. I have been fortunate enough to count some
very bright and brilliant persons namely Sayan Pal, Saptarshi Biswas, Anwesha
Chakraborty, Sankarshan Sahu. It has been a rewarding experience working with
them.

My thesis will be incomplete if I don’t acknowledge the role my mother has
played. Without her unflinching support, this journey would have been
impossible. I dedicate this thesis to her.

## List of publications

  1. 1.

“A note on broken dilatation symmetry in planar noncommutative theory”,
Partha Nandi, Sankarshan Sahu, Sayan Kumar Pal
e-Print: arXiv:2101.07076v3 [hep-th] (To appear in Nucl.Phys.B).

  2. 2.

“Emergence of a geometric phase shift in planar noncommutative quantum
mechanics”
Saptarshi Biswas, Partha Nandi, Biswajit Chakraborty
Published in:  Phys.Rev.A 102 (2020) 2, 022231
e-Print: arXiv: 1911.03196 [hep-th].

  3. 3.

“Revisiting quantum mechanics on non-commutative space-time”
Partha Nandi, , Sayan Kumar Pal, Aritra N. Bose, Biswajit Chakraborty
Published in:  Annals Phys. 386 (2017) 305-326.
e-Print: arXiv: arXiv:1708.04769v1 .

  4. 4.

“Particle dynamics and Lie-algebraic type of non-commutativity of space–time”
Partha Nandi, Sayan Kumar Pal, Ravikant Verma
Published in:  Nucl.Phys.B 935 (2018) 183-197.
e-print: 1807.05062 [physics.gen-ph].

  5. 5.

“Connecting dissipation and noncommutativity: A Bateman system case study”
Sayan Kumar Pal,Partha Nandi, Biswajit Chakraborty
Published in:  Phys.Rev.A 97 (2018) 6, 062110
e-print: arXiv: 1803.03334 [quant-ph].

  6. 6.

“Effect of Non-commutativity of space-time on Thermodynamics of Photon gas”
Ravikant Verma, Partha Nandi
Published in:  Gen.Rel.Grav. 51 (2019) 11, 143
e-print: arXiv: 1810.07005 [physics.gen-ph].

  7. 7.

“Effect of dynamical noncommutativity on the limiting mass of white dwarfs”
Sayan Kumar Pal, Partha Nandi
Published in:  Phys.Lett.B 797 (2019) 134859
e-print: arXiv: 1908.11206 [gr-qc].

  8. 8.

“Fuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries”
F.G. Scholtz, Partha Nandi, S.K. Pal, B. Chakraborty
e-print: arXiv: 1905.05018 [hep-th] (Submitted).

  9. 9.

“Emergent entropy of exotic oscillators and squeezing in three-wave mixing
process”
Sayan Kumar Pal, Partha Nandi, Sibasish Ghosh, Frederik G. Scholtz, Biswajit
Chakraborty
Published in: Phys.Lett.A 403 (2021) 127397
e-print: arXiv: 2012.07166 [hep-th].

  10. 10.

“A Conformally Invariant Unified Theory of Maxwell Fields and Linearized
Gravity as Emergent Fields”
Partha Nandi, Partha Ghose
e-print: arXiv: 2106.07028 [gr-qc] (Submitted).

∗This thesis is based on the publications numbered by [1,2,3].

###### Contents

  1. 1 Introduction
    1. 1.1  Organization of the thesis
  2. 2 Planar noncommutativity and broken dilatation symmetry
    1. 2.1 The classical model of non-commutative space
    2. 2.2 Noncommutativity in quantum picture
      1. 2.2.1 Operator-based formulation of planar non-commutative quantum mechanics
    3. 2.3 Coherent state based path-integral approach
      1. 2.3.1 A position and momentum basis for the quantum Hilbert space
      2. 2.3.2  The Path integral action and generating functional
      3. 2.3.3 Equivalent commutative description
    4. 2.4  Broken dilatation (scale) symmetry and Ward-Takahasi identities
      1. 2.4.1 Non-conserved dilatation charge from Noether’s theorem
      2. 2.4.2 Anomalous Ward-Takahashi Identities
    5. 2.5 Computation of the dilatation anomaly: Fujikawa’s method
    6. 2.6 Discussion
  3. 3  Quantum (phase)-space and emergent geometrical phase
    1. 3.1 Quantum parametric oscillator in non-commutative phase-space
    2. 3.2 Evolution of Ladder operators in Heisenberg picture
    3. 3.3 Geometric phases
    4. 3.4 Classical analogue: Hannay angles
    5. 3.5 Discussion
    6. 3.6  Appendix A
    7. 3.7  Appendix B
  4. 4 Quantum theory on (1+1) dimensional quantum space-time
    1. 4.1 Time re-parametrization symmetry and quantum mechanics in (1+1) dimension
      1. 4.1.1 Canonical Formulation: Dirac’s constraint analysis
      2. 4.1.2 Quantum theory: Commutative space-time picture
    2. 4.2 Quantum space-time: Non-commutative picture
      1. 4.2.1 Quantum theory: The quantum Hilbert space
    3. 4.3 Recovery of effective commutative theory
      1. 4.3.1 Schödinger equation and an induced inner product
    4. 4.4 Free particle wave packet in the noncommutative space-time
    5. 4.5 Schrödinger equation and energy spectra of harmonic oscillator
    6. 4.6 Transition probability in presence of space-time noncommutativity
    7. 4.7 Discussion
    8. 4.8 Appendix A
    9. 4.9 Appendix B
  5. 5 Conclusions

##  Chapter 1 Introduction

A reconciliation of a plausible structure of space-time at a short length
scale i.e. in the vicinity of Planck scale and the eventual emergence of
space-time as we perceive it at a longer length scale as a pseudo-Riemannian
differentiable manifold, is arguably one of the most challenging problems in
modern physics: the so-called “ holy grail” of contemporary theoretical
physics. In order to formulate a full quantum theory of gravity, one needs a
more systematic and sophisticated approach to confront the question regarding
the nature of space-time itself at very short length scales or equivalently
very high energy scales, like in the vicinity of Planck scale. As far as we
are aware, it was Bronstein [1] who was first person to advance the argument
suggesting that any attempt to localize an event down to such a length scale
will inevitably result in gravitational collapse, forming a black hole with an
attendant event horizon. This, in turn, will hide all the sought-after
information, thereby rendering these information inaccessible to any outside
observer. This was carried forward further by Sergio Doplicher, Klaus
Fredenhagen, and John Roberts [2, 3] in a more rigorous approach, where they
could demonstrate the inevitability of such a scenario, if the basic tenets of
both general relativity and quantum theory are extrapolated to that scale.
They pointed out the need for a quantum notion of space -time at such short
length scales where the quantum nature of space-time manifested through the
non-commuting nature of the coordinates, which are now promoted to the level
of operators.

On the other hand, one can understand, some what heuristically, the main
discrepancy between quantum mechanics and general relativity from Einstein’s
equations:

$G_{\mu\nu}+\lambda g_{\mu\nu}=8\pi GT_{\mu\nu},$ (1.1)

where the gravitational field, i.e. that of the geometry of space-time, is
captured by the curvature tensor (and hence by the metric tensor) which occurs
in the left hand side of the above equation through a certain contracted form
of the curvature tensor and gets related to the matter content of the system
and is given by the energy-momentum tensor ($T_{\mu\nu}$) that occurs in the
right hand side. But quantum theory implies that the energy-momentum tensor is
necessarily operator valued and will have a significant role to play at a very
high energy scale. This then readily implies that the left-hand side of the
equation (1.1) also must be quantized at this scale. Hence, a quantum
structure of space-time seems quite plausible [4]. The geometry of such
quantum spaces possibly can only be described by the non-commutative geometry,
as formulated by Alain Connes [5].

Although envisaged initially to be an alternative and novel approach to
quantum gravity, Connes and his collaborators, however, applied this framework
of Non-commutative geometry to develop a completely unified description of the
Standard model of particle physics [6, 7, 8, 9, 10, 11], albeit at the
classical level. Among its many successes includes a unified description of
Higgs field and other Yang-Mills gauge fields present in the model, which
eventually facilitated the exact computation of Higgs mass [12, 13]. Despite
all these successes, there are certain shortcomings, which are still present
in the construction. Primary among them is that the model takes the space-time
to be a $4D$ compact Riemannian differentiable manifold $M$ of Euclidean
signature-rather than Lorentzian. However, one should note that modelling the
space-time, thus by a Riemannian manifold $M$, of course serves its purpose
adequately at the scale of Standard model, as one can expect to see any
granular structure, associated with the above mentioned quantum space-time
only in the vicinity of Planck scale. One can recall in this context that the
Standard model scale is many orders below the GUT scale and which, in turn, is
a couple of orders below the Planck scale. And for them the total space is
taken to be the so-called “Almost commutative space” (AC): $M\times F$, where
$F$ is the space which is described by suitable matrix algebra, which is
responsible for the non-commutative structure in the whole construction. Gauge
symmetries appear here through the group of inner automorphism , which is a
normal sub-group of the total automorphism group of $M\times F$, while the
outer automorphism group, obtained by quotienting, gets identified with the
group of diffeomorphism. Despite its beautiful mathematical and conceptual
edifice, however, it is clear that the quantum structure of space-time should
be taken seriously and be incorporated suitably, so that one can go beyond
this framework of AC space, even if one tries to build a tentative and toy
model of quantum gravity.

Before the success of the renormalization procedure had gained acceptance, the
idea of quantized space-time was first proposed by Heisenberg himself in the
late 1930s with the hope of taming the short-distance i.e. ultraviolet
divergences of quantum fields. He shared his idea with Rudolf Peierls, who
eventually made use of it in electronic systems. On the other hand, Rudolf
Peierls informed Pauli about this idea, who in turn told J. Robert Oppenheimer
about it and finally, in 1947, his student Hartland Snyder, using this idea,
published his articles on the subject[14, 15]. In the same year C. N. Yang
[16] generalized it to curved de-Sitter space-time. However, due to the
success of the renormalization scheme, Snyder’s work(s) did not receive much
attention among his contemporaries. But eventually when activities towards the
formulation of quantum gravity was initiated again, it started to become quite
transparent gradually that the usual models of space-time as a differentiable
manifold has to given away and possibly replaced by some form of quantized
space-time for which an intrinsic non-commutative space-time can be a good
example. Furthermore, it was anticipated that the quantized version of gravity
may act as a regulator of quantum field theories. In fact, it has been pointed
out in [17] that the classical self-energy of a charged particle is
automatically regularized in the presence of classical gravity itself. But if
one goes further to incorporate the quantum effects to the self-energy of the
charged particle, this will then introduce additional divergences and to
regulate these divergences, one will most probably require a theory of quantum
gravity and a non-commutative geometry stemming from a non-commutative
structure of space-time may become one of the strong contenders in this case.

Apart from the quantum gravity point of view, the concept of space-time
noncommutativity (with time being an ordinary c-numbered parameter) has found
application in various condensed matter phenomenon [18, 19, 22, 20, 21] as
well. The physics of the anomalous quantum Hall effect [23], spin Hall effect
[24, 25] are other areas in which the non-commutativity among the coordinates
show up. Besides, a Berry curvature in momentum space for a semi-classical
Bloch electron has been used to explain the anomalous/spin/optical Hall
effects. In fact, constant Berry curvature effects in this system arising due
to the breaking of time-reversal symmetry induce a canonical type of non-
commutativity among the spatial coordinates [26]. On the other hand, a
rigorous derivation of the quantization of the Hall conductivity in quantum
Hall effect used methods of non-commutative geometry[27]. In the $2+1$
dimension, the non-commutative geometry of the lowest Landau level also plays
a vital role in the recent work of Haldane [28] on the geometrical description
of the fractional quantum Hall effect. Also, the effects of coordinate
noncommutativity in $2+1$ quantum gravity can be realized by the introduction
of anyons with fractional spin wherein non-commutative parameter is related to
the fractional spin parameter, as has been discussed in [29]. In fact, a close
correspondence between space-time quantization and $2+1$ dimensional gravity
was shown by G.’t Hooft in [30].

Furthermore, in quantum space-time, spatial noncommutativity can be correlated
to a possible curvature of momentum space [31], just as we know the curvature
in configuration space enforces momentum-space operators to satisfy non-
trivial commutation algebra. Indeed the non-commutative coordinates are
realized as translation generators on momentum space. A few years back,
Townsend [32] had shown that Planck length also appears in gravity due to the
noncommutativity between the generators of space-time translations. It has
been naturally suggested that, along with the spatial components of space-time
coordinates, the canonically conjugate momentum components too should satisfy
a non-commutative algebraic structure [33] at the quantum level because
momentum noncommutativity arise naturally as a consequence of coordinate
noncommutativity. The reciprocity theorem proposed by Max Born in 1938 [34]
had earlier hinted at this. In a similar vein, [35] discovered that the
noncommutative structure of the spatial and momentum components can be
connected to the respective curvatures in momentum and coordinate spaces.

String theory also exhibits space-time uncertainty, as first pointed out in
[36], which is another approach to describe the quantum theory of gravity. Not
only that, the method of non-commutative geometry has been used rigorously to
study the duality symmetries of string theory [37, 38]. It has also been shown
that non-commutative geometry appears due to the toroidal compactification of
the matrix model[39]. In fact, these matrix models lead to a non-commutative
version of Yang-Mills theory as an effective quantum field theory. Seiberg and
Witten in their seminal paper [40] showed how this space-time noncommutativity
can also appear as a low energy effect in string theory as a kind of effective
theory by studying the physics of $D$ branes in the background of nonzero
Neveu-Schwarz $B_{\mu\nu}$ field. They further demonstrated, by explicit
construction, the existence of a nontrivial transformation connecting ordinary
gauge fields and non-commutative gauge fields, which is usually called the
Seiberg-Witten map and using which one can construct an equivalent commutative
description of non-commutative gauge theory. Another important feature of this
map is concerned with the notion of the emergent gravity phenomenon. But,
there are some novel ambiguities [41, 42] in the Seiberg-Witten map which are
particularly more significant in the presence of matter fields. In fact, a few
years back, it has been shown by Scholtz et al. in [43] that this map does not
always generate a spectrum preserving transformation for interacting quantum
Hall system. All these studies, nevertheless, indicates that space and time
will become non-commutative operators in a more fundamental formulation of
string/M theory. This triggered renewed interest in quantum (non-commutative)
space-time and formulation on non-commutative quantum mechanics, and quantum
field theories, on such spaces.

To describe field theory on a quantum (non-commutative) space-time, there are
two approaches. In one approach, we need to define fields whose base manifold
is non-commutative and treated as operators in some Hilbert space. In the
other method, fields are considered as some functions of usual commutative
space-time variables, and noncommutativity among these variables is
incorporated through an appropriate “star” ($\star$) product [44, 45], which
corresponds to a deformed multiplication map and is obtained by deforming
point wise multiplication of functions. Perhaps the most commonly studied case
is that of the Moyal space-time:
$[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu}$; i. e. when the non-
commutative parameter ($\theta$) is taken as a constant matrix. Field theories
defined on such a quantum (non-commutative) space-time are based on the Weyl-
Wigner correspondence, in which all products are simply replaced by the Moyal
“star” products ($\star_{M}$) [46], defined by

$(O_{1}\star_{M}O_{2})(x)=e^{\frac{i}{2}\theta^{\mu\nu}\partial_{\mu}\partial^{{}^{\prime}}_{\nu}}O_{1}(x)O_{2}(x^{{}^{\prime}})|_{x^{{}^{\prime}}\rightarrow
x}$ (1.2)

As a result the mass terms in the action are not modified by the
$\star_{M}$-product because it contributes up to an additional surface term in
the action; terms beyond quadratic interaction terms are only modified. Hence
one may arrive at an undeformed free propagator. Historically the first
unexpected behavior of non-commutative quantum field theories was pointed out
by Minwalla, van Raamsdonk and Seiberg [47]. Using Moyal star product
formulation, these authors could show that the perturbative analysis of
quantum field theories on non-commutative space-time are afflicted with some
novel kind of divergences, where the ultraviolet and infrared divergences
(UV/IR mixing) gets mixed up. In other words, the high and low energy scales
of energy get entangled due to presence of noncommutativity among space-time
coordinates which may affect the renormalization procedure. In fact this UV/IR
mixing problem is not only the hallmark of relativistic fields, non-
relativistic scalar field theory over $2+1$ dimensional non-commutative
manifolds also shows this behavior [48]. On the other hand, in a more recent
paper [49], it has been pointed out that the Voros star product is more
appropriate in dealing with quantum mechanics on Moyal plane, as it lends
itself to a probabilistic interpretation, unlike Moyal star product. The
analysis of [49] were, of course, restricted to the level of non-commutative
quantum mechanics, where the position variables where the hermitian
observable. On the other hand, at the level of QFT on non-commutative spaces
also similar results were obtained by [50], although the position variables
are no longer observables there and are merely the labels of continuous
degrees of freedom. This should be pointed out in this context that the Voros
star product naturally arises whenever coherent states are used as basis in
the quantum Hilbert space. However, a slight variant of this coherent state-
based formulation of quantum field theory on non-commutative space-time has
been developed recently in [51]. In this approach, instead of using star
product, any element of algebra generated by the non-commutative space-time
operators are replaced by the coherent state expectation values of the non-
commutative space-time and introduce a modified coherent state representation
of plane wave solution in an ad-hoc way which may not satisfy the
orthonormality condition without Voros start product which leads to a
nontrivial modification at the level of free propagators. The basic advantage
of this new star-product independent formulation is that the whole formalism
of the perturbative field theory is UV finite which indicates that there is no
UV/IR mixing problem [52]. But, a major disadvantage of this formulation is
that it does not provide a straightforward reduction to single-particle non-
commutative quantum mechanics.

It is worthwhile to mention at this stage that, like in the commutative case
[53], non-commutative quantum mechanics can also be recovered from
relativistic non-commutative quantum field theory through some suitable non-
relativistic limit, as has been extensively studied in [54]. Except for the
new feature that particles of opposite charges should be associated with
opposite non-commutative parameter ($\theta$). On the other hand, a rigorous
formulation of non-commutative quantum mechanics with spatial
noncommutativity, making use of Hilbert-Schmidt (HS) operators has been
studied in great detail [55, 56, 57]. The advantage of this HS operatorial
formulation is that it bypasses the use of any star product and enables one to
confront head-on the operatorial nature of spatial coordinates and evade any
kind of ambiguities that may arise from the use of different kinds of star
product, like that Moyal or Voros [58, 59, 60]. However, there was no follow-
up work to extend this HS operatorial formulation beyond the so-called first
quantization to second quantization or for that matter the full-fledged
relativistic quantum field theory (QFT). Despite this, one can expect to
capture some of the surviving features, if not all, of this completely non-
commutative relativistic QFT even at the level of non-commutative quantum
mechanics. This is simply because of the fact that, like in the commutative
case [61], here too one can definitely expect to interpret non commutative
quantum mechanics as $0+1$ dimensional QFT. In other words, non-commutative
quantum mechanics may be considered as a laboratory where some of the
properties of non-commutative quantum field theories can still be studied in a
simpler setting. In fact, one can go other way around and try to formulate
QFT, as a sort of a unique and inevitable formulation, starting from that of
relativistic quantum mechanics, in a bottom up approach , as was shown by
Steven Weinberg in his famous book on QFT [62]. In other words, the way we
know about QFT today, is the only possibility, where a merger of both the
principles of special relativity and quantum mechanics is envisaged.

###  1.1  Organization of the thesis

In the present thesis, however, which is mainly based on the works [63, 64,
65], no attempts will be made to address any of the issues related to quantum
gravity directly. Rather, this thesis is devoted to studying various aspects
of quantum mechanics on non-commutative space-time and to capture some of the
surviving aspects of symmetries of quantum field theory on such space-time,
illustrated through toy models in $(0+1)$ dimension. This allows one to gain
some insights into this and other related issues in a more transparent manner,
as we have explained above. The outline of this thesis is as follows:

In chapter 2, we discuss the emergence of quantum (non-commutative) space by
considering a quantum Hall system involving a pair of interacting oppositely
charged particles in a region of high magnetic field (uniform and static) in
the low mass limit. We then show how the Lagrangian of the system can be
directly mapped to a harmonic oscillator in Moyal plane, referred to here as
exotic oscillators. Therefore, noncommutativity is an emergent feature of this
physical model here, unlike in many other works where it is conferred with a
fundamental status. Subsequently, a quantum mechanical analysis of this model
is carried out using the Hilbert-Schmidt operator formulation of non-
commutative quantum mechanics. Then a functional integral approach is adopted
to show that Ward-Takahashi (W-T) identities corresponding to the dilation
transformations are anomalous in the sense that additional contributions in
the form of quantum corrections from the Jacobian due to time dilatation
transformation is obtained. This is over and above the classical dilatation
symmetry breaking term. To the best of our knowledge, this has remained
unexplored in the literature from the point of view of $0+1$ dimension
nonlocal field theory. The anomalous term is then computed and regularized
following Fujikawa’s method.

The appearance of the adiabatic geometrical phase in the system of a planar
isotropic time-dependent harmonic oscillator on a non-commutative phase space
has been discussed in chapter 3. Using a non-canonical phase space
transformation we study the effect of noncommutativity analytically. The
emergent nature of geometric phase-related issues is addressed in Heisenberg
picture. It is shown that the additional geometrical phase over and above the
dynamical phase is then found to be given in terms of a product of the non-
commutative parameters. Then we also discuss the correspondence between
Berry’s phase and classical Hannay angle shifts.

In the last chapter (Chapter 4) we develop the concept of quantum space-time
in $1+1$ dimension, taken to be of Moyal type. Then we discuss non-
relativistic quantum mechanics on such a space-time. It is demonstrated that
utilising a coherent state basis, an effective commutative theory can be
written down starting from an abstract form of the Schr‘̀odinger equation
using Hilbert-Schmidt operators. The coherent state basis offers a very
natural description of canonical type non-commutative space-time wherein one
can obtain the commutative limit in an explicit manner, which also provides
some calculation advantages. These are, apart from the fact that in this
coherent state based formulation, the probabilistic interpretation comes out
in a very transparent manner. We, next, concern ourselves with several
problems that are often associated with quantum mechanics. Finally, we end up
with some concluding remarks in chapter 5.

##  Chapter 2 Planar noncommutativity and broken dilatation symmetry

By now there is already a huge body of literature dedicated to the study of
quantum theories on the background of quantum spacetime. Postulating a simple
structure for the non-vanishing commutator algebra between operator-valued
space-time coordinates is expected to shed some insights into the future
theory of quantum gravity [66], through these quantum theories. Normally
people introduce noncommutativity between the spatial coordinates by hand.
However, in certain non-relativistic planar system, non-commutative
coordinates arise naturally. For example, the Landau problem in a strong
magnetic field with the lowest Landau-level projection [67, 68]. Also, the
quantum space (non-commutative space) indicates the existence of a minimal
length scale in the theory. On the other hand, scale or dilatation symmetry
(exact or broken) in the corresponding commutative theory has a long history
[69, 70]. In this chapter, we would like to study the fate of dilatation
symmetry in the planar non-commutative theory both in classical and quantum
level. Note that the breaking of dilatation symmetry is somewhat expected at
the classical level itself, because of the occurrence of a natural scale
coming from the non-commutative parameter. What we are interested in to is
study here is to ascertain whether there is any additional contribution to the
breaking term, arising from the process of quantization i.e over and above the
classical effect, which we can refer to as the anomaly term. And for that we
consider here a non-relativistic interacting quantum Hall system [71, 72] with
the usual constant magnetic field. Here noncommutativity between spatial
coordinates emerges naturally as a consequence of the deformed symplectic
structure among the spatial coordinates at a large magnetic field limit.

The chapter is organized as follows. In sec. 2.1, following the “Peierls
substitution” scheme [73], we demonstrate how deformed symplectic structure
arises naturally at the classical level itself, among the spatial coordinates
of a two-dimensional harmonic oscillator in a planar system of electric dipole
under a very strong magnetic field, normal to the plane, where an additional
harmonic interaction between the charges with an impurity is also considered.
As we shall see that a first-order formulation is quite adequate in this case
i.e. in the limit of large magnetic field. In sec.2.2 we will move on to
discuss the quantum picture of our considered model where we will give a short
review on the formulation of non-commutative quantum mechanics for a two-
dimensional harmonic oscillator in the framework of Hilbert-Schmidt operators.
This is completely an operatorial formulation of non-commutative quantum
mechanics, initiated mainly in [74]. Next, in sec 2.3, by using the coherent
state approach, we construct a path-integral action and functional integral
for the non-commutative dynamical system. This also makes a connection with
Chern-Simons quantum mechanics. Symplectic brackets can then be computed,
rather trivially from this path-integral action and the corresponding non-
commutative structure between the position coordinates is seen to to emerge
naturally here and therefore also serves as a consistency check. In section
2.4, we treat the non-commutative quantum mechanics as a (0+1) dimensional
non-canonical (non-commutative) field theory. Then by exploiting Darboux’s
theorem at the level of functional integral we explicitly compute Ward-
Takahashi identities due to the time dilatation symmetry of an equivalent
commutative effective theory. Thereafter we shall demonstrate that the Ward-
Takahashi identities associated with dilatation symmetry (broken) are
anomalous. Also from these identities we infer that there are three fold
breaking of dilatation symmetry in non-commutative theory: (i) explicit
breaking due to presence of spring constant at the classical level itself,
(ii) deformation in symplectic structure also provides another classical
breaking, and finally (iii) the third is due to the anomaly (a purely quantum
effects) which is induced from noncommutativity. The anomaly term needs to be
regularized and is done in section 2.5 using Fujikawa’s regulator, which is
then found to be proportional to the product of the spring constant of the
harmonic oscillator and the non-commutative parameter. Sec.2.6 is reserved for
the concluding remarks.

###  2.1 The classical model of non-commutative space

We consider a planar system consisting of a pair of interacting oppositely
charged particles having the same mass $m$ and subjected to a constant
magnetic field $B$ (ignoring the other’s effect like Coulomb and radiation
interaction) pointed along the normal to the plane. The system is described by
the following standard Lagrangian in C.G.S units :

$L=\frac{1}{2}m(\dot{x}^{2}_{i}+\dot{y}^{2}_{i})+\frac{eB}{2c}\epsilon_{ij}(x_{j}\dot{x}_{i}-y_{j}\dot{y}_{i})-\frac{k_{0}}{2}(x_{i}-y_{i})^{2}-\frac{k_{1}}{2}x^{2}_{i}$
(2.1)

where $c$ denotes the speed of light in vacuum, and $x_{i}$ and $y_{i}$ are
coordinates of the positive and negative charge respectively. Apart from the
kinetic term and magnetic interaction the third term of the above Lagrangian
(2.1) represent the harmonic interaction between the two charges and the
fourth term describes additional interactions of the negative charge with an
impurity.

Here, we will be interested only in the limit of the strong magnetic field $B$
and small mass $m$, in which the kinetic term is quite redundant and potential
energy playing an important role [75]. Hence the kinetic terms from (2.1) can
be effectively ignored [68, 76] in the limit $\frac{m}{eB}\rightarrow 0$. Thus
the effective Lagrangian describing the high magnetic field limit essentially
reduces to the following form:-

$L_{0}=\frac{eB}{2c}(\epsilon_{ij}x_{j}\dot{x}_{i}-\epsilon_{ij}y_{j}\dot{y}_{i})-V(x_{i},y_{i})~{},$
(2.2)

with
$V(x_{i},y_{i})=\frac{k_{0}}{2}(x_{i}-y_{i})^{2}+\frac{k_{1}}{2}x^{2}_{i}$.
Thus the classical equations of motion associated with the above first-order
Lagrangian (2.2) is given by -

$\dot{x_{i}}=\frac{c}{eB}\epsilon_{ij}\frac{\partial V}{\partial
x_{j}}~{}~{}~{};\dot{y_{i}}=-\frac{c}{eB}\epsilon_{ij}\frac{\partial
V}{\partial y_{j}}.$ (2.3)

The canonical Hamiltonian corresponding to (2.2) is constructed by the usual
Legendre transformation :

$H_{0}=\frac{\partial L_{0}}{\partial\dot{x}_{i}}\dot{x}_{i}+\frac{\partial
L_{0}}{\partial\dot{y}_{i}}\dot{y}_{i}-L_{0}=V(x_{i},y_{i}).$ (2.4)

Now, in order to establish the equivalence between the Hamiltonian and
Lagrangian formalisms [77, 78], we consider the Hamilton’s equations of
motion:

$\displaystyle\dot{x_{i}}$ $\displaystyle=\\{x_{i},H_{0}\\}=\\{x_{i},V\\}$
(2.5) $\displaystyle\dot{y_{i}}$
$\displaystyle=\\{y_{i},H_{0}\\}=\\{y_{i},V\\},$

with the potential $V(x_{i},y_{i})$ playing the role of the Hamiltonian [79].
The symplectic structure can readily be obtained now by comparing the
Lagrangian equations of motion (2.3) with the form of Hamilton’s equations of
motion (3.82) to yield the following brackets :

$\\{x_{i},x_{j}\\}=\frac{c}{eB}\epsilon_{ij};~{}~{}\\{y_{i},y_{j}\\}=-\frac{c}{eB}\epsilon_{ij};~{}\\{x_{i},y_{j}\\}=0.$
(2.6)

Here we can eliminate the degrees of freedom of the negative charge ($y_{i}$)
by defining a new pair of variables as,

$p_{i}=\frac{eB}{c}\epsilon_{ij}(x_{j}-y_{j})~{}~{}and~{}~{}x_{i},$ (2.7)

satisfying the following symplectic structure :

$\\{x_{i},x_{j}\\}=\frac{c}{eB}\epsilon_{ij};~{}~{}\\{x_{i},p_{j}\\}=\delta_{ij};~{}\\{p_{i},p_{j}\\}=0,$
(2.8)

where $p_{i}$ play the role of abelian translation generators of the
cordinates $x_{i}$. Thus at very high magnetic field and low mass limit, the
canonical Hamiltonian (2.4) can be rewritten as -

$H_{0}=\frac{p_{i}^{2}}{2m_{B}}+\frac{1}{2}k_{1}x_{i}^{2},$ (2.9)

where $m_{B}=\frac{e^{2}B^{2}}{k_{0}}$. Therefore at large magnetic field
limit, the dynamical system (2.1) is governed by a bi-dimensional harmonic
oscillator with the non-canonical symplectic structure (2.8). Notice that
noncommutativity of the coordinates has been already established at the
classical level as the symplectic bracket between coordinates $x_{i}$ is
nonzero. In the next section we will discuss the quantum picture of this
theory.

###  2.2 Noncommutativity in quantum picture

We now proceed ahead to describe the quantum theory of the above model at
sufficiently strong magnetic field ($B$)limit in a systematic manner with the
Hamiltonian operator (2.9):

$\hat{H}=\frac{\hat{p}_{i}^{2}}{2m_{B}}+\frac{1}{2}k_{1}\hat{x}_{i}^{2},$
(2.10)

where phase-space variables (operators) ($\hat{x}_{i},\hat{p}_{i}$) satisfy
the following planar non-commutative Heisenberg algebra (NCHA) :

$[\hat{x}_{i},\hat{x}_{j}]=i\theta\epsilon_{ij},~{}[\hat{x}_{i},\hat{p}_{j}]=i\hbar\delta_{ij},~{}[\hat{p}_{i},\hat{p}_{j}]=0;~{}~{}for~{}i,j=1,2$
(2.11)

with $\theta=\frac{\hbar c}{eB}>0$. So, our system (2.10) of interest is
nothing but a two-dimensional isotropic harmonic oscillator living in
effective non-commutative plane [80].

####  2.2.1 Operator-based formulation of planar non-commutative quantum
mechanics

It has been pointed out recently in [55, 74, 81, 82] that non-commutative
quantum mechanics may be formulated in a completely abstract operatorial
level, where quantum systems can be identified with the space of rays on the
Hilbert space of Hilbert-Schmidt operators, which act on an auxiliary Hilbert
space. This latter auxiliary Hilbert space will be referred to as the
configuration space, as its sole purpose is to serve as a space, which
furnishes a representation of just the coordinate sub-algebra. Here, we
present a very short review of the formulation of the Hilbert-Schmidt operator
approach in an appropriate physical setting which also will then pave the way
for a path-integral formulation.

In two dimension, the NC coordinate sub-algebra is given by :

$[\hat{x}_{i},\hat{x}_{j}]=i\theta\epsilon_{ij};~{}for~{}i=j=1,2$ (2.12)

A Hilbert space $\mathcal{H}_{c}$, furnishing a representation of this
coordinate sub-algebra, i.e. the configuration space in our case, can be
easily identified to be the one, which is isomorphic to that of one-
dimensional harmonic oscillator in the usual quantum mechanics. Thus the
construction of this Hilbert space proceeds along the same line. We thus begin
by first introducing the annihilation and creation operators as

$\hat{b}=\frac{1}{\sqrt{2\theta}}(\hat{x}_{1}+i\hat{x}_{2}),~{}~{}\hat{b}^{\dagger}=\frac{1}{\sqrt{2\theta}}(\hat{x}_{1}-i\hat{x}_{2}),$
(2.13)

satisfying the commutation relation
$[\hat{b},\hat{b}^{\dagger}]=\mathbb{I}_{c}$. One then defines non-commutative
configuration space which is isomorphic to boson Fock space as :

$\mathcal{H}_{c}=~{}span\\{\left|n\right\rangle=\frac{1}{\sqrt{n!}}(\hat{b}^{\dagger})^{n}\left|0\right\rangle\\}$
(2.14)

where the span is understood to be over the complex numbers. Here the ground
state $\left|0\right\rangle$ such that $\hat{b}\left|0\right\rangle=0$ so that

$\hat{b}\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle;~{}~{}\hat{b}^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle$
(2.15)

We would like to emphasize here that this $\mathcal{H}_{c}$ furnishes only the
representation of the coordinate algebra (2.12) and not of the entire non-
commutative Heisenberg algebra (2.11) and the reasons for that will become
clear very soon. Now, in order to construct a corresponding Hilbert space,
which furnishes a representation of this entire non-commutative Heisenberg
algebra, we need to consider the Hilbert space formed by the set of all
Hilbert-Schmidt (HS) operators acting on this configuration space
$\mathcal{H}_{c}$. The abstract quantum states of our system can be identified
with these HS operators. And a generic HS operator
$\psi(\hat{x}_{1},\hat{x}_{2})=\psi(\hat{b},\hat{b}^{\dagger})$ is an algebra
element, generated by $(\hat{x}_{1},\hat{x}_{2})$ or equivalently by
$(\hat{b},\hat{b}^{\dagger})$, having a finite HS norm:

$\|\psi\|_{HS}:=\sqrt{tr_{c}(\psi^{\dagger}\psi)}<\infty,$ (2.16)

which in turn is a dense subspace of space bounded operators
($\mathcal{B}(\mathcal{H}_{c})$) acting on $\mathcal{H}_{c}.$

This space of HS operators forms a Hilbert space of its own and is our
ultimate quantum Hilbert space $\mathcal{H}_{q}$ and is defined as:

$\mathcal{H}_{q}=span\\{\
\psi(\hat{x}_{1},\hat{x}_{2})=|\psi):\psi(\hat{x}_{1},\hat{x}_{2})\in\mathcal{B}(\mathcal{H}_{c}),\|\psi\|_{HS}<\infty\\}\,$
(2.17)

This space is equipped with the inner product:

$(\phi|\psi)=tr_{c}(\phi^{\dagger}(\hat{x}_{1},\hat{x}_{2})\psi(\hat{x}_{1},\hat{x}_{2})),$
(2.18)

where the subscript $c$ refers to tracing over $\mathcal{H}_{c}$.

Note also these HS operators $\Psi(\hat{x}_{1},\hat{x}_{2})$ can also be
expanded as,

$\psi(\hat{x}_{1},\hat{x}_{2})=\psi(\hat{b},\hat{b}^{\dagger})=\sum_{n,m}c_{n,m}|n><m|,$
(2.19)

where $c_{n,m}=<n|\psi(\hat{b},\hat{b}^{\dagger})|m>\in\mathbb{C}$ with
$\sum_{n,m}\mid c_{n,m}\mid^{2}<\infty.$ This shows that $\mathcal{H}_{q}$ can
also be identified as $\mathcal{H}_{c}\otimes\mathcal{H}^{*}_{c}$, with
$\mathcal{H}^{*}_{c}$ being the dual of $\mathcal{H}_{c}$ (2.14) and this is
the space which can furnish the representation of the entire NCHA, as can be
seen easily by defining appropriate actions of phase space operators on it.
But before we discuss about this point, we pause for a while to introduce the
notations we are going to employ to designate the states in $\mathcal{H}_{c}$
and $\mathcal{H}_{q}$: (i) states in $\mathcal{H}_{c}$ are denoted by angular
kets $|.>$, and (ii) states in the $\mathcal{H}_{q}$ are denoted by a “round
ket” $\psi(\hat{x}_{1},\hat{x}_{2})=|\psi)$.

Now like in the commutative case, here too we look for a unitary
representation of the non-commutative Heisenberg algebra (2.11) on quantum
Hilbert space in terms of operators $\hat{X}_{i}$, $\hat{P}_{i}$, whose
actions on $\mathcal{H}_{q}$ are given as follows in the following way -

$\hat{X}_{i}|\psi)=|\hat{x}_{i}\psi),~{}~{}\hat{P}_{i}=\frac{\hbar}{\theta}\epsilon_{ij}|[\hat{x}_{j},\psi])=\frac{\hbar}{\theta}\epsilon_{ij}[\hat{X}_{j}-\hat{X}^{R}_{j}]||\psi).$
(2.20)

Here we have used the capital letters $\hat{X}_{i}$ and $\hat{P}_{i}$ to
denote the representations of the operators $\hat{x}_{i}$ and $\hat{p}_{i}$
acting on $\mathcal{H}_{q}$. Note that here we have taken the action of
$\hat{X}_{i}$ by direct left action on an abstract state vector $|\psi)$, by
default, and the momentum operator is taken to act adjointly, which is clearly
seen to involve the right action on the quantum Hilbert space and is defined
in the following way :

$\hat{X}^{R}_{i}|\psi)=|\psi\hat{x}_{i});~{}[\hat{X}^{R}_{i},\hat{X}^{R}_{j}]|\psi)=|\psi[\hat{x}_{j},\hat{x}_{i}])=-\theta\epsilon_{ij}|\psi)~{}~{}\forall\psi\in\mathcal{H}_{q}$
(2.21)

It can be checked easily that the left and right action commutes:

$[\hat{X}_{i},\hat{X}^{R}_{j}]=0$ (2.22)

We can recognize at this stage this $\hat{X}^{R}_{i}$ can be interpreted as an
additional degree of freedom and can be identified with the quantum
counterpart of $y_{i}$ in (2.7) and (2.20, 2.21) are analogous to (2.6, 2.7)
in the quantum setting. Therefore, the classical picture discussed in the
previous section provides a clear physical realization of the abstract
formulation of non-commutative quantum mechanics in terms of Hilbert-Schmidt
operators. We introduce one further notational convention for any operator
$\hat{O}$ acting on the quantum Hilbert space (2.17), one may define left and
right action (denoted by superscripted L and R) as follows:

$\hat{O}^{L}|\psi)=\hat{O}|\psi),~{}\hat{O}^{R}|\psi)=|\psi)\hat{O};~{}~{}\forall|\psi)=\psi(\hat{x}_{1},\hat{x}_{2})\in\mathcal{H}_{q}$
(2.23)

Finally, it is a matter of straightforward verification to see that the phase
space operators acting on the quantum Hilbert space indeed obey the following
the commutation relations, which are precisely the NCHA:

$[\hat{X}_{i},\hat{X}_{j}]=i\theta\epsilon_{ij},~{}[\hat{X}_{i},\hat{P}_{j}]=i\hbar\delta_{ij},~{}[\hat{P}_{i},\hat{P}_{j}]=0~{}~{}for~{}i,j=1,2;$
(2.24)

It is now useful to define the following non-hermitian operators on the
quantum Hilbert space which we shall require later:

$\hat{B}=\frac{\hat{X}_{1}+i\hat{X}_{2}}{\sqrt{2\theta}},~{}\hat{B}^{\ddagger}=\frac{\hat{X}_{1}-i\hat{X}_{2}}{\sqrt{2\theta}},~{}\hat{P}=\hat{P}_{1}+i\hat{P}_{2},~{}\hat{P}^{\ddagger}=\hat{P}_{1}-i\hat{P}_{2};~{}~{}[\hat{B},\hat{B}^{\ddagger}]=\mathbb{I}_{q}$
(2.25)

with corresponding actions

$\hat{B}\psi(\hat{x}_{1},\hat{x}_{2})=\hat{b}\psi(\hat{x}_{1},\hat{x}_{2}),~{}~{}\hat{B}^{\ddagger}\psi(\hat{x}_{1},\hat{x}_{2})=\hat{b}^{\dagger}\psi(\hat{x}_{1},\hat{x}_{2}),$
(2.26)

where we have used the symbol $\ddagger$ specifically for the operator adjoint
on $\mathcal{H}_{q}$.

Therefore, the representation of our system Hamiltonian (2.10) on
$\mathcal{H}_{q}$ is given by

$\hat{H}=\frac{1}{2m_{B}}(\hat{P}_{1}^{2}+\hat{P}_{2}^{2})+\frac{1}{2}k_{1}(\hat{X}_{1}^{2}+\hat{X}_{2}^{2}),$
(2.27)

where phase-space operators $\hat{X}_{i}$ and $\hat{P}_{i}$ satisfy the same
i.e. isomorphic algebra (2.24) while acting on the quantum Hilbert space
$\mathcal{H}_{q}.$

Further it is noted that -

$\hat{X}_{1}=\sqrt{\frac{\theta}{2}}(\hat{B}+\hat{B}^{\ddagger}),~{}\hat{X}_{2}=-i\sqrt{\frac{\theta}{2}}(\hat{B}-\hat{B}^{\ddagger}),~{}\hat{P}_{1}=\frac{\hbar}{\theta}(\hat{X}_{2}-\hat{X}^{R}_{2}),~{}\hat{P}_{2}=-\frac{\hbar}{\theta}(\hat{X}_{1}-\hat{X}^{R}_{1})$
(2.28)

On the other hand, for instance, the complex momenta $\hat{P}$ in (2.25) can
be written as -

$\hat{P}=i\hbar\sqrt{\frac{2}{\theta}}[\hat{B}^{R}-\hat{B}],~{}~{}\hat{P}^{\ddagger}=i\hbar\sqrt{\frac{2}{\theta}}[\hat{B}^{\ddagger}-\hat{B}^{R\ddagger}],$
(2.29)

where $\hat{B}^{R}=\frac{\hat{X}^{R}_{1}+i\hat{X}^{R}_{2}}{\sqrt{2\theta}}$
and it can be easily checked that -

$[\hat{B}^{R},\hat{B}^{R\ddagger}]=-1;~{}~{}[\hat{B},\hat{B}^{R}]=0$ (2.30)

Accordingly the system Hamiltonian (2.27) can be re-casted in a much simpler
form

$\displaystyle\hat{H}$
$\displaystyle=\frac{1}{2m_{B}}\hat{P}^{\ddagger}\hat{P}+k_{1}\theta(\hat{B}^{\ddagger}\hat{B}+\frac{1}{2})$
(2.31)
$\displaystyle=\alpha_{1}\hat{B}^{\ddagger}\hat{B}+\alpha_{2}\hat{B}^{R\dagger}\hat{B}^{R}-\alpha_{3}(\hat{B}^{\dagger}\hat{B}^{R}+\hat{B}^{R\ddagger}\hat{B})+\alpha_{4},$

where

$\alpha_{1}=k_{1}\theta+\frac{\hbar^{2}}{m_{B}\theta},~{}\alpha_{2}=\frac{\hbar^{2}}{m_{B}\theta},~{}\alpha_{3}=\frac{\hbar^{2}}{2m_{B}},~{}~{}\alpha_{4}=\frac{k_{1}\theta}{2}.$
(2.32)

On recognizing that the system Hamiltonian is obtainable from that for the
standard quadratic Hermitian form by a canonical transformation, namely,

$\displaystyle\hat{B}^{{}^{\prime}}$
$\displaystyle=\sqrt{\frac{m_{B}\omega\theta}{2\hbar\alpha}}\left[(\frac{1}{2}+\frac{\hbar\alpha}{m_{B}\omega\theta})\hat{B}+(\frac{1}{2}-\frac{\alpha\hbar}{m_{B}\omega\theta})\hat{B}^{R}\right]$
(2.33) $\displaystyle\hat{B}^{{}^{\prime}R}$
$\displaystyle=\sqrt{\frac{m_{B}\omega\theta}{2\hbar\alpha}}\left[(\frac{1}{2}-\frac{\hbar\alpha}{m_{B}\omega\theta})\hat{B}+(\frac{1}{2}+\frac{\alpha\hbar}{m_{B}\omega\theta})\hat{B}^{R}\right],$

with $\omega=\sqrt{\frac{k_{1}}{m_{B}}}$ and
$\alpha=\sqrt{1+\frac{m^{2}_{B}\omega^{2}\theta^{2}}{4\hbar^{2}}},$ satisfying
the commutation relation
$[\hat{B}^{{}^{\prime}},\hat{B}^{{}^{\prime}\ddagger}]=\mathbb{I}_{q}$ and
$[\hat{B}^{{}^{\prime}R},\hat{B}^{{}^{\prime}R\ddagger}]=-\mathbb{I}_{q}$ like
their un-primed counterparts, renders the system Hamiltonian to decompose into
two decoupled harmonic oscillators of frequencies $\omega_{+}$ and
$\omega_{-}$:

$\hat{H}=\hbar\omega_{+}\hat{B}^{{}^{\prime}\ddagger}\hat{B}^{{}^{\prime}}+\hbar\omega_{-}\hat{B}^{{}^{\prime}R}\hat{B}^{{}^{\prime}R\ddagger}+\frac{\hbar}{2}(\omega_{+}+\omega_{-})\mathbb{I}_{q}$
(2.34)

where the characteristic frequencies $\omega_{\pm}$ are given by,
$\omega_{\pm}=\alpha\omega\pm\frac{m_{B}\omega^{2}\theta}{2\hbar}$. Here the
quadratic Hermitian form of the Hamiltonian (2.34) act on the quantum Hilbert
space $\mathcal{H}_{q}$ (2.17), and this vector space can be represented by
another complete orthonormal set of basis vector
$|n_{1},n_{2})^{{}^{\prime}}$:

$\mathcal{H}_{q}=span\\{|n_{1},n_{2})^{{}^{\prime}}=\left|n_{1}\right\rangle^{{}^{\prime}}\langle
n_{2}|^{{}^{\prime}}=\frac{(\hat{B}^{{}^{\prime}\ddagger})^{n_{1}}(\hat{B}^{{}^{\prime}R})^{n_{2}}}{\sqrt{n_{1}~{}!n_{2}!}}\left|0\right\rangle^{{}^{\prime}}\langle
0|^{{}^{\prime}}\\}^{\infty}_{n_{1}=n_{2}=0}~{}~{},$ (2.35)

with the inner product

${}^{{}^{\prime}}(\tilde{n}_{1},\tilde{n}_{2}|n_{1},n_{2})^{{}^{\prime}}=tr_{c}[(\left|\tilde{n}_{1}\right\rangle^{{}^{\prime}}\langle\tilde{n}_{2}|^{{}^{\prime}})^{\ddagger}\left|n_{1}\right\rangle^{{}^{\prime}}\langle
n_{2}|^{{}^{\prime}}]=\delta_{\tilde{n}_{1},n_{1}}\delta_{\tilde{n}_{2},n_{2}}.$
(2.36)

The basis vectors of $\mathcal{H}_{q}$ constitute a complete set and
accordingly the completeness relation reads:

$\sum_{n_{1},n_{2}}|n_{1},n_{2})^{{}^{\prime}}(n_{1},n_{2}|^{{}^{\prime}}=\mathbb{I}_{q}$
(2.37)

Here the operators $\hat{B}^{{}^{\prime}\ddagger}$ and
$\hat{B}^{{}^{\prime}R}$ appearing in (2.35) and their respective Hermitian
conjugates act on the left and right sectors of the states of quantum Hilbert
space (2.35) respectively, and $\left|0\right\rangle^{{}^{\prime}}\langle 0|$
standing for the vacuum state of $\mathcal{H}_{q}$:

$\displaystyle\hat{B}^{{}^{\prime}}\left|0\right\rangle^{{}^{\prime}}\langle
0|^{{}^{\prime}}$ $\displaystyle=0$ (2.38)
$\displaystyle\hat{B}^{{}^{\prime}R\ddagger}\left|0\right\rangle^{{}^{\prime}}\langle
0|^{{}^{\prime}}$ $\displaystyle=\left|0\right\rangle^{{}^{\prime}}\langle
0|^{{}^{\prime}}\hat{B}^{\dagger}=0$

The eigenvalue equation of this Hamiltonian is,

$\hat{H}\left|n_{1},n_{2}\right)^{{}^{\prime}}=E_{n_{1}n_{2}}\left|n_{1},n_{2}\right)^{{}^{\prime}},$
(2.39)

whose solution spectrum can virtually be read-off from as,

$E_{n_{1},n_{2}}=(n_{1}\hbar\omega_{+}+n_{2}\hbar\omega_{-})+E_{0,0},$ (2.40)

where the additional constant $E_{0,0}$ is

$E_{0,0}=\frac{1}{2}\hbar(\omega_{+}+\omega_{-}),$ (2.41)

which is nothing but the finite zero point energy of the system Hamiltonian.
This, however, being a constant $c$-number and not an operator will give the
same value for all states and can easily be disposed off by giving a shift to
the zero of energy as,

$\hat{\widetilde{H}}:=\hat{H}-\mathbb{I}_{q}E_{0,0},$ (2.42)

Therefore

$\hat{\tilde{H}}\left|n_{1},n_{2}\right)^{{}^{\prime}}=\tilde{E}_{n_{1}n_{2}}\left|n_{1},n_{2}\right)^{{}^{\prime}},$
(2.43)

with $\hat{\widetilde{H}}|0,0)^{{}^{\prime}}=0$ and
$\tilde{E}_{n_{1}n_{2}}=n_{1}\hbar\omega_{+}+n_{2}\hbar\omega_{-}$. From now
on, we only work with $\hat{\widetilde{H}}.$ We now, as discussed at the
outset, proceed to carry out the path-integral of the system in order to study
its scaling properties.

###  2.3 Coherent state based path-integral approach

Like in commutative case, here too we would like to introduce the notion of
position states, however, in view of the absence of common eigenstates of
$\hat{x}_{1}$ and $\hat{x}_{2}$ operators, the best we can do is to introduce
the minimal uncertainty states i.e. maximally localized states (coherent
states) on $\mathcal{H}_{c}$ [83] as,

$\left|z\right\rangle=e^{-\bar{z}\hat{b}+z\hat{b}^{\dagger}}\left|0\right\rangle=e^{-\frac{\mid
z\mid^{2}}{2}}e^{z\hat{b}^{\dagger}}\left|0\right\rangle~{}\in\mathcal{H}_{c},$
(2.44)

where $z=\frac{x_{1}+ix_{2}}{\sqrt{2\theta}}$ is a dimensionless complex
number. As is well known that these coherent state $\left|z\right\rangle$ is
an eigenstate of the annihilation operator,

$\hat{b}\left|z\right\rangle=z\left|z\right\rangle$ (2.45)

These states provide a complete (albeit over-complete) set on
$\mathcal{H}_{c}$ as,

$\int\frac{dzd\bar{z}}{\pi}\left|z\right\rangle\langle z|=\mathbb{I}_{c},$
(2.46)

where the integration is over the entire complex $z$ plane. And, these states
constitutes a minimum uncertainty states, namely

$\Delta x_{1}\Delta x_{2}=\frac{\theta}{2}$ (2.47)

As these are minimum uncertainty states, the real and imaginary parts of $z$
are the closest we can get to points in space obtained from measurement.

####  2.3.1 A position and momentum basis for the quantum Hilbert space

The quantum Hilbert space was defined in sec 2.1 as the space of HS operators
acting on the configuration space (Fock space) $\mathcal{H}_{c}$. Using the
coherent state states (2.44), we can construct a coherent states (operator) in
quantum Hilbert space $\mathcal{H}_{q}$ as follows :

$|z,\bar{z})=|x_{1},x_{2})=\frac{1}{\sqrt{2\pi\theta}}\left|z\right\rangle\langle
z|~{}\in\mathcal{H}_{q};~{}~{}~{}(w.\bar{w}|z,\bar{z})=\frac{1}{2\pi\theta}e^{-\mid
w-z\mid^{2}},$ (2.48)

satisfying

$\hat{B}|z,\bar{z})=\hat{b}\left|z\right\rangle\langle z|=z|z,\bar{z}).$
(2.49)

Furthermore, the coherent state (”position”) representation of a general state
$|\psi)=\psi(\hat{x}_{1},\hat{x}_{2})$ can be expressed as,

$\psi(x_{1},x_{2})=(z,\bar{z}|\psi)=\frac{1}{\sqrt{2\pi\theta}}tr_{c}(\left|z\right\rangle\langle
z|\psi(\hat{x}_{1},\hat{x}_{2}))=\frac{1}{\sqrt{2\pi\theta}}\langle
z|\psi(\hat{x}_{1},\hat{x}_{2})\left|z\right\rangle.$ (2.50)

This looks exactly like the coordinate representation of the state vectors in
commutative quantum mechanics and the states $|z,\bar{z})$ playing a role
analogous to position basis in commutative quantum mechanics. From now on, we
will speak loosely of the states $|z,\bar{z})=|x_{1},x_{2})$ as position
states, keeping in mind that this is merely an analogy.

Claim:

The resolution of identity of the quantum Hilbert space $\mathcal{H}_{q}$ can
be given in terms of the coherent states (2.48) through the following equation

$\int 2\theta dzd\bar{z}~{}|z,\bar{z})\star_{V}(z,\bar{z}|=\int
dx_{1}dx_{2}~{}|x_{1},x_{2})\star_{V}(x_{1},x_{2}|=\mathbb{I}_{q},$ (2.51)

where
$\star_{V}=e^{\stackrel{{\scriptstyle\leftarrow}}{{\partial_{\bar{z}}}}\stackrel{{\scriptstyle\rightarrow}}{{\partial_{z}}}}$
is called Voros [84] star product.

Proof:

In order to derive the resolution of the identity in (2.48), let us consider
the following relations

$\displaystyle|\psi)$ $\displaystyle=$
$\displaystyle\mathbb{I}_{c}\psi(\hat{x}_{1},\hat{x}_{2})\mathbb{I}_{c}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int
dzd\bar{z}dwd\bar{w}|z\rangle\langle w|\langle z|\psi|w\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int
dzd\bar{z}dvd\bar{v}|z\rangle\langle z+v|\langle z|\psi|z+v\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\pi}\int dzd\bar{z}\frac{1}{\pi}\int
d^{2}ve^{-|v|^{2}}|z\rangle\langle
z|e^{\bar{v}\overleftarrow{\partial_{\bar{z}}}+v\overrightarrow{\partial_{z}}}\langle
z|\psi|z\rangle.$

where we have made use of $w=z+v$ with $d^{2}w=d^{2}v$, and
$e^{v\partial_{z}}f(z)=f(z+v)$.

Now, consider the integral over $v,\bar{v}$ first:

$\int d^{2}ve^{-\mid
v\mid^{2}}e^{\bar{v}\overleftarrow{\partial_{\bar{z}}}+v\overrightarrow{\partial_{z}}}$
(2.53)

where $d^{2}v=dvd\bar{v}$ represents an element of area in the complex $v$
plane and the integration is to be carried out over the entire plane.
Resorting to a polar representation of and we have

$\displaystyle\int d^{2}ve^{-\mid
v\mid^{2}}e^{\bar{v}\overleftarrow{\partial_{\bar{z}}}+v\overrightarrow{\partial_{z}}}$
$\displaystyle=$ $\displaystyle\int^{\infty}_{0}rdr\int^{\pi}_{-\pi}d\theta
e^{-r^{2}}\sum_{n,m}\frac{1}{n!m!}(r\overleftarrow{\partial_{\bar{z}}})^{n}(r\overrightarrow{\partial_{z}})^{m}e^{i\theta(n-m)}$
$\displaystyle=$ $\displaystyle
2\pi\sum_{n}\frac{1}{n!n!}(\overleftarrow{\partial_{\bar{z}}}\overrightarrow{\partial_{z}})^{n}\frac{n!}{2}$
$\displaystyle=$ $\displaystyle\pi
e^{\stackrel{{\scriptstyle\leftarrow}}{{\partial_{\bar{z}}}}\stackrel{{\scriptstyle\rightarrow}}{{\partial_{z}}}}.$

Therefore, using (LABEL:+)

$\displaystyle|\psi)$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\int
dzd\bar{z}\frac{1}{\pi}\int d^{2}ve^{-|v|^{2}}|z\rangle\langle
z|e^{\bar{v}\overleftarrow{\partial_{\bar{z}}}+v\overrightarrow{\partial_{z}}}\langle
z|\psi|z\rangle$ (2.55) $\displaystyle=$ $\displaystyle\frac{1}{\pi}\int
2\theta
dzd\bar{z}|z,\bar{z})e^{\overleftarrow{\partial_{\bar{z}}}\overrightarrow{\partial_{z}}}(z,\bar{z}|\psi)=\mathbb{I}_{q}|\psi).$

Since the state $|\psi)$ is arbitrary, we can conclude that the identity
operator on quantum Hilbert space may be represented as

$\int 2\theta
dzd\bar{z}~{}|z,\bar{z})e^{\overleftarrow{\partial_{\bar{z}}}\overrightarrow{\partial_{z}}}(z,\bar{z}|=\mathbb{I}_{q},$
(2.56)

which completes the proof.

Here we recognize that the above exponential operator to be the Voros star-
product operator between two functions $f(z,\bar{z})$ and $g(z,\bar{z})$,
which is defined as-

$f(z,\bar{z})\star_{V}g(z,\bar{z})=f(z,\bar{z})e^{\stackrel{{\scriptstyle\leftarrow}}{{\partial_{\bar{z}}}}\stackrel{{\scriptstyle\rightarrow}}{{\partial_{z}}}}g(z,\bar{z})$
(2.57)

We now introduce a natural definition of a momentum basis [85] in the quantum
Hilbert space such that -

$|p,\bar{p})=\sqrt{\frac{\theta}{2\pi\hbar^{2}}}e^{i\sqrt{\frac{\theta}{2\hbar^{2}}}(\bar{p}\hat{b}+p\hat{b}^{\dagger})};~{}\hat{P}_{i}|p,\bar{p})=p_{i}|p,\bar{p}),~{}p=p_{1}+ip_{2}$
(2.58)

and these states satisfy the usual resolution of identity and orthonormality
condition -

$\int
dp~{}d\bar{p}~{}|p,\bar{p})(p,\bar{p}|=\mathbb{I}_{q},~{}~{}~{}(p^{{}^{\prime}},\bar{p}^{{}^{\prime}}|p,\bar{p})=\delta(p^{{}^{\prime}}-p)$
(2.59)

The overlap of a state $|z,\bar{z})$ with momentum states on the non-
commutative plane is

$(z,\bar{z}|p,\bar{p})=\frac{1}{\sqrt{2\pi\hbar^{2}}}e^{-\frac{\theta}{4\hbar^{2}}p\bar{p}}e^{i\sqrt{\frac{\theta}{2\hbar^{2}}}(p\bar{z}+\bar{p}z)}.$
(2.60)

In this context, we would like to mention that the authors in [86, 87] have
also pointed out that the effective low energy dynamics of relativistic
quantum field theories in the presence of sufficiently strong magnetic field
can be described by the Voros star product.

####  2.3.2  The Path integral action and generating functional

Now, following [85], and using the completeness relations for the momentum
(2.59) and coordinate basis (2.51) in place, we proceed to write down the path
integral representation of the overlap between a pair of coherent states on
the two dimensional Moyal plane as,

$(z_{f},t_{f}|z_{0},t_{0})=\lim_{n\to\infty}\int^{\infty}_{-\infty}(2\theta)^{n}\prod_{j=1}^{n}dz_{j}d\bar{z}_{j}(z_{f},t_{f}|z_{n},t_{n})\star_{V_{n}}(z_{n},t_{n}|...........|z_{1},t_{1})\star_{V_{1}}(z_{1},t_{1}|z_{0},t_{0})$
(2.61)

The propagation over an infinitesimal step $\tau=\frac{t_{f}-t_{0}}{n+1}$ are

$\displaystyle(z_{j+1},t_{j+1}|z_{j},t_{j})$ $\displaystyle=$
$\displaystyle(z_{j+1}|e^{-\frac{i}{\hbar}\tau\hat{\widetilde{H}}}|z_{j})$
(2.62) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}d^{2}p_{j}~{}e^{-\frac{\theta}{2\hbar^{2}}\bar{p}_{j}p_{j}}e^{i\sqrt{\frac{\theta}{2\hbar^{2}}}\left[p_{j}(\bar{z}_{j+1}-\bar{z}_{j})+\bar{p}_{j}(z_{j+1}-z_{j})\right]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
e^{-\frac{i}{\hbar}\epsilon[\frac{\bar{p}_{j}p_{j}}{2m_{B}}+\frac{k_{1}\theta}{2}(2\bar{z}_{j+1}z_{j}+c_{0})]}$

where $c_{0}=1-\frac{2}{\theta k_{1}}E_{0,0}$ . Now substituting the above
expression in (2.61) and computing star products explicitly, we obtain (apart
from a constant factor) -

$\displaystyle(z_{f},t_{f}|z_{0},t_{0})$ (2.63)
$\displaystyle=e^{-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}}\lim_{n\rightarrow\infty}\int\prod_{j=1}^{n}(dz_{j}d\bar{z}_{j})\prod_{j=0}^{n}d^{2}p_{j}\exp\sum_{j=0}^{n}\tau\left[\frac{i}{\hbar}\sqrt{\frac{\theta}{2}}\left[p_{j}\left\\{\frac{\bar{z}_{j+1}-\bar{z}_{j}}{\tau}\right\\}+\bar{p}_{j}\left\\{\frac{z_{j+1}-z_{j}}{\tau}\right\\}-\sqrt{2\theta}k_{1}\bar{z}_{j+1}z_{j}\right]\right]$
$\displaystyle\exp\sum_{j=0}^{n}\tau\left[-\frac{i}{2m\hbar}\bar{p}_{j}p_{j}+\frac{\theta}{2\hbar^{2}}\frac{(p_{j+1}-p_{j})}{\tau}\bar{p}_{j}\right],$

where, $\sigma=-i(\frac{\tau}{2m_{B}\hbar}-i\frac{\theta}{2\hbar^{2}})$ and
$z_{n+1}=z_{f}$. Finally, in the $\tau\rightarrow 0$ or $n\rightarrow\infty$
limit, symbolically, we write the path integral kernel in phase space
representation as,

$(z_{f},t_{f}|z_{0},t_{0})=e^{-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}}\int^{z(t_{f})=z_{f}}_{z(t_{0})=z_{0}}\mathcal{D}\bar{z}(t)\mathcal{D}z(t)\int\mathcal{D}\bar{p}(t)\mathcal{D}p(t)e^{\frac{i}{\hbar}S_{0}[z(t),\bar{z}(t),p(t),\bar{p}(t)]}$
(2.64)

where $S_{0}$ is the phase-space action given by -

$S_{0}=\int^{t_{f}}_{t_{0}}dt\left[\sqrt{\frac{\theta}{2}}(p\dot{\bar{z}}+\bar{p}\dot{z})-\frac{i\theta}{2\hbar}\bar{p}\dot{p}-\frac{1}{2m_{B}}\bar{p}p-\theta
k_{1}\bar{z}z\right].$ (2.65)

It may be noted that the action (2.65) contains an exotic $\bar{p}\dot{p}$
term which is a Chern-Simons like term in momentum space and is responsible
for rendering the configuration space action to be highly non-local [88]. On
recognizing that the first-order action (2.65) can be taken to be the one
written in an extended phase space, where both $z(t)$ and $p(t)$ are
considered as configuration space variables of the system, enables us to
recast the action in the so-called symplectic form [89] as -

$S_{0}=\int^{t_{f}}_{t_{i}}dt\left[\frac{1}{2}\xi^{\alpha}f_{\alpha\beta}\dot{\xi^{\beta}}-V(\xi)\right],~{}~{}~{}~{}\xi^{\alpha}:=\\{z,\bar{z},p,\bar{p}\\};~{}~{}~{}\alpha=1,2..,4$
(2.66)

with

$f=\begin{pmatrix}\huge 0&-\sqrt{\frac{\theta}{2}}\sigma_{1}\\\
\sqrt{\frac{\theta}{2}}\sigma_{1}&-\frac{\theta}{2}\sigma_{2}\end{pmatrix}.$
(2.67)

where $\sigma_{1}$, $\sigma_{2}$ are the usual Pauli matrices, and

$V(\xi)=\frac{1}{2m_{B}}\bar{p}p+\theta k_{1}\bar{z}z$ (2.68)

Therefore one can easily read off the symplectic brackets by using the
Faddeev-Jackiw formalism [89] as:-

$\\{z,\bar{z}\\}=-\frac{i}{\hbar},~{}~{}~{}~{}~{}\\{z,\bar{p}\\}=\\{\bar{z},p\\}=\sqrt{\frac{2}{\theta}},~{}~{}~{}~{}\\{p,\bar{p}\\}=0$
(2.69)

These symplectic brackets 2.69 are consistent with the classical version of
the non-commutative Heisenberg algebra given in (2.8).
Looking at the phase space representation of the path integral kernel, one can
now re-interpret non-commutative quantum mechanics as a $0+1$ dimensions
complex scalar field theory where the coordinates are complex fields and which
depends on time as $z(t)$. Now after providing this brief review of the path-
integral formalism in Moyal plane, our next task will be to write down the
corresponding matrix elements of the time-ordered product of a source
dependent term for all $t$ with $t_{i}<t<t_{f}$ as a path integral -

$\displaystyle(z_{f},t_{f}|z_{0},t_{0})_{J,\bar{J}}$
$\displaystyle=(z_{f},t_{f}|\large{T}exp\left[\frac{i}{\hbar}\int^{t_{f}}_{t_{i}}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|z_{0},t_{0})$
(2.70)
$\displaystyle=e^{-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}}\int^{z(t_{f})=z_{f}}_{z(t_{0})=z_{0}}\mathcal{D}\bar{z}\mathcal{D}z\mathcal{D}\bar{p}\mathcal{D}p~{}~{}exp\left[\frac{i}{\hbar}(S_{0}+\int^{t_{f}}_{t_{i}}\sqrt{\frac{\theta}{2}}[\bar{z}(t)J(t)+z(t)\bar{J}(t)]\right]$
$\displaystyle=e^{-\vec{\partial}_{z_{f}}\vec{\partial}_{\bar{z}_{0}}}\int^{z(t_{f})=z_{f}}_{z(t_{0})=z_{0}}\mathcal{D}\bar{z}\mathcal{D}z\mathcal{D}\bar{p}\mathcal{D}p~{}~{}exp\left(\frac{i}{\hbar}S_{J,\bar{J}}\right),$

with

$S_{J,\bar{J}}=\int^{t_{f}}_{t_{0}}dt\left[\sqrt{\frac{\theta}{2}}(p\dot{\bar{z}}+\bar{p}\dot{z})-\frac{i\theta}{2\hbar}\bar{p}\dot{p}-\frac{1}{2m_{B}}\bar{p}p-\theta
k_{1}\bar{z}z+\sqrt{\frac{\theta}{2}}[\bar{z}(t)J(t)+z(t)\bar{J}(t)\right],$
(2.71)

where $J(t)=J_{1}(t)+iJ_{2}(t)$ is a time dependent source vanishing at
$t\rightarrow\pm\infty$. Now in quantum theories, the prime interest is the
vacuum to vacuum persistence amplitude in the presence of an adiabatically
varying external source. A standard way to obtain this is to go back to the
propagation kernel (2.70) and introduce complete sets of energy states as :-

$\displaystyle(z_{f},t_{f}|z_{0},t_{0})_{J,\bar{J}}$ (2.72)
$\displaystyle=\sum_{n_{1},n_{2}}\sum_{m_{1},m_{2}}(z_{f},t_{f}|n_{1},n_{2})(n_{1},n_{2}|\large{T}\exp\left[\frac{i}{\hbar}\int^{t_{f}}_{t_{0}}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|m_{1},m_{2})\times$
$\displaystyle(m_{1},m_{2}|z_{0},t_{0})$
$\displaystyle=\sum_{n_{1},n_{2}}\sum_{m_{1},m_{2}}(n_{1},n_{2}|\large{T}\exp\left[\frac{i}{\hbar}\int^{t_{f}}_{t_{0}}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|m_{1},m_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times(m_{1},m_{2}|e^{\frac{i}{\hbar}t_{0}\hat{\tilde{H}}}|z_{0})(z_{f}|e^{-\frac{i}{\hbar}t_{f}\hat{\tilde{H}}}|n_{1},n_{2})$
$\displaystyle=(0,0|\large{T}\exp\left[\frac{i}{\hbar}\int^{t_{f}}_{t_{0}}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|0,0)\times(0,0|z_{0})(z_{f}|0,0)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\sum_{n_{1},n_{2}\neq
0}\sum_{m_{1},m_{2}\neq
0}(n_{1},n_{2}|\large{T}\exp\left[\frac{i}{\hbar}\int^{t_{f}}_{t_{i}}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|m_{1},m_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times(m_{1},m_{2}|z_{0})(z_{f}|n_{1},n_{2})e^{-\frac{i}{\hbar}t_{f}\tilde{E}_{n_{1},n_{2}}+\frac{i}{\hbar}t_{0}\tilde{E}_{m_{1},m_{2}}}~{}~{}~{},$

where we have used the fact of (2.43). To accomplish the projection onto the
ground state (vacuum) of the system Hamiltonian 2.10, we now introduce a
parameter $T$ with units of time and replace -

$t_{f}\rightarrow t_{f}=T(1-i\epsilon),~{}~{}~{}~{}~{}~{}~{}t_{0}\rightarrow
t_{0}=-T(1-i\epsilon),$ (2.73)

and take $T\rightarrow\infty$, with $\epsilon$ being a very small positive
constant. In the end, we will take the limit $\epsilon\rightarrow 0$, but only
after the infinite limit of $T$ is carried out. In the limit
$T\rightarrow\infty$, the exponentials in (2.72) oscillate out to zero except
for the ground state. Hence in this asymptotic limit, we obtain

$\displaystyle\lim_{\epsilon\to
0}\lim_{\begin{subarray}{c}t_{f}\rightarrow\infty(1-i\epsilon)\\\
t_{0}\rightarrow-\infty(1-i\epsilon)\end{subarray}}(z_{f},t_{f}|z_{0},t_{0})_{J,\bar{J}}$
$\displaystyle=(0,0|\large{T}exp\left[\frac{i}{\hbar}\int^{\infty}_{-\infty}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|0,0)$
(2.74)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times(0,0|z_{0})(z_{f}|0,0).$

However, one can also observe that -

$\lim_{\epsilon\to
0}\lim_{\begin{subarray}{c}t_{f}\rightarrow\infty(1-i\epsilon)\\\
t_{0}\rightarrow-\infty(1-i\epsilon)\end{subarray}}(z_{f},t_{f}|z_{0},t_{0})_{J=\bar{J}=0}=(0,0|z_{0})(z_{f}|0,0)$
(2.75)

Consequently, we can write -

$(0,0|T\exp\left[\frac{i}{\hbar}\int^{\infty}_{-\infty}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|0,0)=\lim_{\epsilon\to
0}\lim_{\begin{subarray}{c}t_{f}\rightarrow\infty(1-i\epsilon)\\\
t_{0}\rightarrow-\infty(1-i\epsilon)\end{subarray}}\frac{(z_{f},t_{f}|z_{0},t_{0})_{J,\bar{J}}}{(z_{f},t_{f}|z_{0},t_{0})_{J=\bar{J}=0}}$
(2.76)

Since the left hand side of (2.76) is not sensitive to the boundary condition
$z_{0}$ and $z_{f}$ imposed at $t\rightarrow\pm\infty$, the right-hand side
should also be independent of the boundary conditions imposed at
$t\rightarrow\pm\infty$, if one chooses the same boundary conditions in the
numerator and the denominator. Moreover, the right hand side has the structure
of a functional integral and we can write (2.76) as,

$\displaystyle(0,0|\large{T}exp\left[\frac{i}{\hbar}\int^{\infty}_{-\infty}\sqrt{\frac{\theta}{2}}(J(t)\hat{B}^{\dagger}+\bar{J}(t)\hat{B})dt\right]|0,0)$
$\displaystyle=(0,0;\infty|0,0;-\infty)_{J,\bar{J}}$ (2.77)
$\displaystyle=N^{-1}\int\mathcal{D}\bar{z}\mathcal{D}z\mathcal{D}\bar{p}\mathcal{D}p~{}~{}e^{\frac{i}{\hbar}\left[S_{0}+\int^{t_{f}=\infty}_{t_{0}=-\infty}\sqrt{\frac{\theta}{2}}[\bar{z}(t)J(t)+z(t)\bar{J}(t)dt]\right]},$

where the normalization factor $N$ in (3.114) is fixed so as to ensure the
normalized ground state ($(0,0|0,0)=1$).

Therefore, the generating functional or the vacuum persistence amplitude in
presence of external sources $J(t)$ and $\bar{J}(t)$ is defined as,

$Z[J,\bar{J}]\equiv
Z[J_{1},J_{2}]=N^{-1}\int\mathcal{D}\bar{z}\mathcal{D}z\mathcal{D}\bar{p}\mathcal{D}p~{}~{}e^{\frac{i}{\hbar}\left[S_{0}+\int^{t_{f}=\infty}_{t_{0}=-\infty}\sqrt{\frac{\theta}{2}}[\bar{z}(t)J(t)+z(t)\bar{J}(t)]dt\right]}.$
(2.78)

####  2.3.3 Equivalent commutative description

Here we notice that the non-canonical brackets (2.69) can be realized in terms
of canonical variables $\alpha,p$ as,

$\displaystyle z(t)$
$\displaystyle=\alpha(t)+\frac{i}{2\hbar}\sqrt{\frac{\theta}{2}}p(t)$ (2.79)
$\displaystyle\bar{z}(t)$
$\displaystyle=\bar{\alpha}(t)-\frac{i}{2\hbar}\sqrt{\frac{\theta}{2}}\bar{p}(t),$

satisfying

$\\{\alpha,\bar{\alpha}\\}=0,~{}~{}~{}~{}~{}\\{\alpha,\bar{p}\\}=\\{\bar{\alpha},p\\}=\sqrt{\frac{2}{\theta}},~{}~{}~{}~{}\\{p,\bar{p}\\}=0$
(2.80)

However, in order to avoid the non-locality [90] in the first-order action
(2.65), we may apply the change of variables (2.79) (change of variables,
which are dummy in nature is quite harmless in the path-integral framework and
this leaves the integration measure invariant) in the generating functional
(2.78) to re-write it as

$Z[J,\bar{J}]=N^{-1}\int\mathcal{D}\bar{\alpha}\mathcal{D}\alpha\mathcal{D}\bar{p}\mathcal{D}p~{}\exp(\frac{i}{\hbar}S_{J,\bar{J}}[\alpha,\bar{\alpha},p,\bar{p}]),$
(2.81)

where

$\displaystyle
S_{J,\bar{J}}[\alpha,\bar{\alpha},p,\bar{p}]=\int_{-\infty}^{\infty}dt$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\
\left[\sqrt{\frac{\theta}{2}}(\dot{\bar{\alpha}}-\frac{i\theta}{2\hbar}k_{1}\bar{\alpha}+\frac{i}{\hbar}\sqrt{\frac{\theta}{2}}\bar{J})p+\sqrt{\frac{\theta}{2}}(\dot{\alpha}+\frac{i\theta}{2\hbar}k_{1}\alpha-\frac{i}{\hbar}\sqrt{\frac{\theta}{2}}J)\bar{p}\right.$
(2.82)
$\displaystyle\left.-\frac{1}{2m_{B}}(1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}})\bar{p}p-\theta
k_{1}\bar{\alpha}\alpha+\sqrt{\frac{\theta}{2}}(\alpha\bar{J}+\bar{\alpha}J)\right]$

In this context, it is important to mention that the canonical phase space
variables ($\alpha,\bar{\alpha}$ and $p$, $\bar{p}$) can be identified with
the center of mass coordinates and their corresponding canonical conjugate
momenta of our two particles system (LABEL:sp). This has been pointed out in
[108, 109, 110]. However, the equivalence between two actions (2.82 and 2.65)
in absence of source has already been established [111] from Batalin-Tyutin
point of view. But of course this change of variables follows from the
Faddeev-Jackiw reduction scheme [77, 91] to carry out the standard canonical
quantization scheme of a theory initially described in terms of non-canonical
(constrained) variables. This method relies on Darboux’s theorem [92] which
asserts that any non-standard symplectic two-form can always be brought back
to the standard canonical form through a redefinition of the phase-space
variables, see [93, 94] for related applications in this direction. This
procedure yields a canonical (equivalent commutative) description of the given
constrained (noncommutative) theory. Here, one can shift the burden of
deformation from the symplectic structure to the Hamiltonian which then
acquires a new interaction term induced from the deformation parameter
($\theta$) which will be made explicit as we move ahead. Now for simplicity,
under rescaling the dynamical variables as -

$\displaystyle\alpha(t)\rightarrow Z(t)$
$\displaystyle=\frac{1}{\sqrt{1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}}}}\alpha(t)$
(2.83) $\displaystyle p(t)\rightarrow P(t)$
$\displaystyle=\sqrt{1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}}}p(t)~{},$

the functional integral (2.81) yields :-

$Z[J,\bar{J}]=N^{-1}\int\mathcal{D}\bar{Z}\mathcal{D}Z\mathcal{D}\bar{P}\mathcal{D}P~{}e^{\frac{i}{\hbar}S_{J,\bar{J}}[Z,\bar{Z},P,\bar{P}]}$
(2.84)

with

$\displaystyle S_{J,\bar{J}}[Z,\bar{Z},P,\bar{P}]$
$\displaystyle=\int_{-\infty}^{\infty}dt\left[\sqrt{\frac{\theta}{2}}(\dot{\bar{Z}}-\frac{i\theta}{2\hbar}k_{1}\bar{Z}+\frac{i\lambda}{2\hbar}\sqrt{\frac{\theta}{2}}\bar{J})P+\sqrt{\frac{\theta}{2}}(\dot{Z}+\frac{i\theta}{2\hbar}k_{1}Z-\frac{i\lambda}{2\hbar}\sqrt{\frac{\theta}{2}}J)\bar{P}\right.$
(2.85) $\displaystyle\left.-\frac{1}{2m_{B}}\bar{P}P-\theta
k_{1}(1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}})\bar{Z}Z+\lambda(1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}})\sqrt{\frac{\theta}{2}}(Z\bar{J}+\bar{Z}J)\right]$

where $\lambda=(1+\frac{k_{1}\theta^{2}m_{B}}{4\hbar^{2}})^{-\frac{1}{2}}.$
The generating functional in configuration space is now easily derived by
integrating over the momenta variables. Indeed, the dependence on momenta in
the exponent of (2.84) is at most quadratic in nature and one may perform the
gaussian integrations over the momenta to obtain -

$Z[J,\bar{J}]=\tilde{N}^{-1}\int\mathcal{D}\bar{Z}\mathcal{D}Z~{}e^{\frac{i}{\hbar}S_{eff}[Z,\bar{Z}]}e^{\frac{i}{\hbar}S_{I}[Z,\bar{Z},J,\bar{J}]},$
(2.86)

where $\tilde{N}$ is a normalization constant which arises from the
integration of momenta, and

$\displaystyle S_{eff}[Z,\bar{Z}]=\int_{-\infty}^{\infty}dt$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\
\left[m_{B}\theta\dot{\bar{Z}}\dot{Z}-\frac{im_{B}\theta^{2}k_{1}}{2\hbar}(\bar{Z}\dot{Z}-\dot{\bar{Z}}Z)-k_{1}\theta\bar{Z}Z\right];$
(2.87)

$\displaystyle
S_{I}[Z,\bar{Z},J,\bar{J}]=\int_{-\infty}^{\infty}dt\lambda\sqrt{\frac{\theta}{2}}\left[\bar{Z}(J+\frac{im_{B}\theta}{2\hbar}\dot{J})+Z(\bar{J}-\frac{im_{B}\theta}{2\hbar}\dot{\bar{J}})+\frac{m_{B}\lambda\theta}{2\hbar^{2}}\sqrt{\frac{\theta}{2}}\bar{J}J\right]~{}~{}~{}$
(2.88)

Now, writing $J(t)=J_{1}(t)+iJ_{2}(t)$,
$Z(t)=\frac{q_{1}(t)+iq_{2}(t)}{\sqrt{2\theta}}$, the above interacting part
of the action (2.88) can be rewritten as -

$\displaystyle
S_{I}=\int_{-\infty}^{\infty}dt{\lambda}(J_{i}q_{i}-\epsilon_{ij}\frac{m_{B}\theta}{2\hbar}\dot{J}_{j}q_{i})+\int_{-\infty}^{\infty}dt\left(\frac{m_{B}\lambda^{2}{\theta}^{2}}{4\hbar^{2}}J_{i}^{2}\right)$
(2.89)

Here we note that the interaction part of the action (2.89) contains three
terms: (i) standard source terms ($J_{i}q_{i}$), (ii) a new quadratic source
term and, (iii) a time derivative of source term coupled with coordinates
($\epsilon_{ij}\dot{J}_{j}q_{i}$).

Thus the functional integral can be written in terms of an effective theory
comprising the usual variables as,

$Z[J_{1},J_{2}]=\tilde{N}^{-1}e^{\frac{im_{B}}{\hbar}(\frac{\lambda\theta}{2\hbar})^{2}\int_{-\infty}^{\infty}dtJ_{i}^{2}}\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)~{}~{}~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[L_{eff}+\lambda(J_{i}q_{i}-\epsilon_{ij}\frac{m_{B}\theta}{2\hbar}\dot{J}_{j}q_{i})\right]}$
(2.90)

where,

$L_{eff}=\frac{1}{2}m_{B}\dot{q}^{2}_{i}-\frac{m_{B}\theta
k_{1}}{2\hbar}\epsilon_{ij}q_{j}\dot{q}_{i}-\frac{1}{2}k_{1}q^{2}_{i}$ (2.91)

Defining the new column vectors:-

$J=\begin{pmatrix}J_{1}&&J_{2}\end{pmatrix}^{T}~{}~{};~{}~{}X=\begin{pmatrix}q_{1}&&q_{2}\end{pmatrix}^{T}$
(2.92)

we can recast the form of the action (2.90) in the following way:

$\displaystyle Z[J_{1},J_{2}]$ $\displaystyle=$
$\displaystyle\tilde{N}^{-1}e^{\frac{im_{B}}{\hbar}(\frac{\lambda\theta}{2\hbar})^{2}\int_{-\infty}^{\infty}dt{J}^{T}{J}}\int\mathcal{D}X(t)~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[X^{\dagger}\mathcal{R}X+\lambda
J^{T}X-\lambda\frac{m_{B}\theta}{2\hbar}\tilde{J}^{T}X\right]}$ (2.93)
$\displaystyle=$
$\displaystyle\tilde{N}^{-1}e^{\frac{i}{\hbar}\int_{-\infty}^{\infty}dt\bigg{(}\frac{m_{B}{\lambda}^{2}{\theta}^{2}}{4\hbar^{2}}{J}^{T}{J}-\frac{{\lambda^{\prime}}^{2}}{{4}}\tilde{J}^{T}\mathcal{R}^{-1}\tilde{J}+\frac{\lambda\lambda^{\prime}}{2}J^{T}\mathcal{R}^{-1}\tilde{J}\bigg{)}}$
$\displaystyle\times\int\mathcal{D}X(t)e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[(X-\frac{\lambda^{\prime}}{2}\mathcal{R}^{-1}\tilde{J})^{\dagger}\mathcal{R}(X-\frac{\lambda^{\prime}}{2}\mathcal{R}^{-1}\tilde{J})+\lambda
J^{T}(X-\frac{\lambda^{\prime}}{2}\mathcal{R}^{-1}\tilde{J})\right]}$
$\displaystyle=$
$\displaystyle\tilde{N}^{-1}e^{\frac{im_{B}}{\hbar}(\frac{\lambda\theta}{2\hbar})^{2}\int
dt{J}^{T}{J}}~{}e^{\frac{i}{\hbar}\int
dt(\frac{\lambda\lambda^{\prime}}{2}J^{T}\mathcal{R}^{-1}\tilde{J}-\frac{\lambda^{\prime
2}}{4}\tilde{J}^{T}\mathcal{R}^{-1}\tilde{J})}\int\mathcal{D}X(t)e^{\frac{i}{\hbar}\int_{-\infty}^{\infty}dt[X^{\dagger}\mathcal{R}X+\lambda
J^{T}X]}~{}~{}~{}~{}~{}~{}~{}$

Here we have redefined the variable (X) in the last line by giving it a
suitable shift as

$X\rightarrow X-\frac{\lambda^{\prime}}{2}\mathcal{R}^{-1}\tilde{J}$ (2.94)

with $\tilde{J}=\begin{pmatrix}\dot{J}_{2}&&-\dot{J}_{1}\end{pmatrix}^{T}$;
$\mathcal{R}=-(m_{B}\frac{d^{2}}{dt^{2}}+k_{1})I+im_{B}\sigma_{y}k_{1}\frac{\theta}{\hbar}\frac{d}{dt}$;
and $\lambda^{\prime}=\frac{m_{B}\theta}{2\hbar}\lambda$.

Finally, we can identify the effective commutative generating functional in
terms of $q_{i}$’s as,

$Z[J_{1},J_{2}]=\tilde{N}^{-1}\mathcal{K}[J_{1},J_{2}]Z_{eff}[J_{1},J_{2};\lambda]~{}~{}~{};~{}~{}~{}~{}~{}~{}~{}~{}Z_{eff}[J_{1},J_{2};\lambda]=\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)~{}e^{\frac{i}{\hbar}\int_{-\infty}^{\infty}dt[L_{eff}+\lambda
J_{i}q_{i}]}$ (2.95)

where the quadratic source term and its higher order time derivatives have
been denoted by $\mathcal{K}[J_{1},J_{2}]$ and $L_{eff}$ is the same as in
(2.91).

Thus at strong magnetic field limit, our non-commutative system (2.10) can be
described through an effective Lagrangian (2.91), which can be interpreted as
the Lagrangian of a charged particle moving in the commutative $q_{1}$-$q_{2}$
plane in presence of a constant effective magnetic field with an additional
quadratic potential where the second term in the right hand side of (2.91)
describes the interaction of a charged particle (of charge e) with a constant
magnetic field $(\vec{B}_{eff})$ pointing along the normal to the plane. The
components of the corresponding vector potential $\vec{A}_{eff}$ in symmetric
gauge can be read off from (2.91),

$(A_{eff})_{i}=-\frac{m_{B}\theta k_{1}c}{2\hbar e}\epsilon_{ij}q_{j},$ (2.96)

with $\vec{B}_{eff}=\vec{\nabla}\times\vec{A}_{eff}$. Thus, we have an exact
mapping between the planar non-commutative system and “generalized” Landau
problem [95], where the magnetic field is a manifestation of the presence of
non-commutativity. Therefore, the classical dynamics of the planar non-
commutative system may be described by the following action :

$S_{eff}=\int
dt\left[\frac{1}{2}m_{B}\dot{q}^{2}_{i}+\frac{e}{c}(A_{eff})_{i}\dot{q}_{i}-\frac{1}{2}k_{1}q^{2}_{i}\right].$
(2.97)

This form of the effective (commutative equivalent) action reminds us of that
of a charged particle interacting with planar magnetic point vertex, written
in terms of some appropriate variables for facilitating an alternate
consistent quantization procedure [96]. Also, the corresponding action was
shown there to be scale or dilatation invariant. We now move on to study the
scale symmetry of the action (2.97).

###  2.4  Broken dilatation (scale) symmetry and Ward-Takahasi identities

In relativistic case, space and time both transform uniformly under scale
transformation, while in non-relativistic situation scale transformation can
act “anisotropically” [98], in the sense that there is no compulsion for the
space and time coordinates to have the same scaling factor and the case in
which these scaling factors are really different, is known as ‘Lifshitz
scaling’. This type of scaling plays a vital role in condensed matter
systems[99]. In this section, we study the time dilatation symmetry (broken)
on noncommutative space by using the effective commutative description.
Further we will be deriving the anomalous Ward-Takahashi (W-T) identities.

####  2.4.1 Non-conserved dilatation charge from Noether’s theorem

The well-known paradigms of planar quantum mechanics [100] of charged
particles in a background magnetic field are the magnetic vortex and Landau
problems. Few years back, in a series of papers, Jackiw [96, 101] has pointed
out that the existence of dilatation symmetry for a charged particle
interacting with magnetic point vortex and magnetic monopole at the classical
level. On the other hand, recently in [102] the time dilatation symmetry of
the non-relativistic Landau problem has been investigated. However, it has
been found that there is no exact dilatation symmetry associated with the non-
relativistic Landau problem, thus giving rise to a non-conserved dilatation
charge at the classical level and further a dilatation anomaly due to
quantization. It may be noted that all these problems has a similar form for
the Lagrangian but their scale transformation properties are different. These
observations naturally encourage us to investigate whether the generalized
Landau problem respects this symmetry and the dilatation anomaly, if any, pops
up.

Now, by inspection we see that only the kinetic term of the effective action
(2.97) is invariant under global time dilatation transformation:

$t\rightarrow t^{{}^{\prime}}=e^{-\gamma}t,$ (2.98)

when the coordinates must concomitantly undergo a change:

$q_{i}(t)\rightarrow
q^{{}^{\prime}}_{i}(t^{{}^{\prime}})=e^{-\frac{\gamma}{2}}q_{i}(t),$ (2.99)

where $\gamma$ is a real parameter. Following Jackiw [70, 101], the Lie
differential of the field $q_{i}(t)$ due to infinitesimal dilatation
transformation is given by -

$\delta_{D}{q}_{i}=\gamma(t\dot{{q}}_{i}-\frac{1}{2}{q}_{i}),$ (2.100)

where $\delta_{D}q_{i}(t)=q^{{}^{\prime}}_{i}(t)-q_{i}(t)$. It is now possible
to write the non-conserved dilatation charge [69] from Noether’s analysis of
(2.97). To compute the generator of the dilatation transformation, it will be
convenient to allow the parameter $\gamma$ in (2.100) to depend on time:
$\gamma=\gamma(t)$ i.e., infinitesimal local variation viz.

$\delta^{L}_{D}{q}_{i}(t)=\gamma(t)\mathcal{Q}q_{i}(t)),$ (2.101)

where $\mathcal{Q}q_{i}(t)=(t\frac{\partial}{\partial
t}-\frac{1}{2})q_{i}(t).$ The change in the action due to these
transformations (2.100) is

$\displaystyle\delta^{L}_{D}{S}_{eff}$ $\displaystyle=\int
dt\bigg{[}\frac{\partial
L_{eff}}{\partial\dot{q}_{i}}\dot{\gamma}\mathcal{Q}q_{i}(t)+\gamma(t)\bigg{(}\frac{\partial
L_{eff}}{\partial\dot{q}_{i}}\mathcal{Q}\dot{q}_{i}(t)+\frac{\partial
L_{eff}}{\partial q_{i}}\mathcal{Q}q_{i}(t)\bigg{)}\bigg{]}$ (2.102)
$\displaystyle=\int
dt\bigg{[}p_{i}\dot{\gamma}\mathcal{Q}q_{i}+\gamma(t)\bigg{(}\frac{d}{dt}(tL_{eff})+k_{1}{q}_{i}^{2}+\frac{m_{B}\theta
k_{1}}{2\hbar}\epsilon_{ij}q_{j}\dot{q}_{i}\bigg{)}\bigg{]},$ (2.103)

where $p_{i}=\frac{\partial L_{eff}}{\partial\dot{q}_{i}}$ is the canonical
momentum. Here, we choose $\gamma(t)$ to vanish asymptotically so that one can
safely discard the boundary contribution and the action $S_{eff}$ then changes
as :

$\displaystyle\delta^{L}_{D}{S}_{eff}$ $\displaystyle=\int
dt\gamma(t)\bigg{(}k_{1}{q}_{i}^{2}+m_{B}k_{1}{\frac{\theta}{2\hbar}}\epsilon_{ij}q_{j}\dot{{q}}_{i}-\frac{dD(t)}{dt}\bigg{)},$
(2.104)

where $D(t)=tH_{eff}-\frac{1}{2}(q_{i}p_{i})$, and
$H_{eff}=\dot{q}_{i}p_{i}-L_{eff}.$
The above expression (2.104) holds for any arbitrary field configuration
$q_{i}(t)$ with the specific change $\delta^{L}_{D}q_{i}(t)$. However, when
$q_{i}(t)$ obeys the classical equations of motion then $\delta S_{eff}=0$ for
any $\delta q_{i}$ including the symmetry transformation (2.101) with
$\gamma(t)$ a function of time. This indicate that at the on-shell level, we
have the non-conserved dilatation charge ($D$) corresponding to the (broken)
dilatation symmetry as -

$\frac{dD(t)}{dt}=\Delta_{0}(t)$ (2.105)

where
$\Delta_{0}(t):=\Delta_{0}(q_{i},\dot{q}_{i};t)=k_{1}({q}_{i}^{2}+m_{B}\frac{\theta}{2\hbar}\epsilon_{ij}q_{j}\dot{q}_{i}).$

We can really identify the non conserved dilatation charge $D$ as the
generator of the infinitesimal global scale transformations (2.100). The
generator of the transformation, consistent with
$\delta_{D}q_{i}(t)=\gamma\\{D(t),q_{i}(t)\\}$. At this stage, we observe that
the scale invariance is broken explicitly by the presence of parameter $k_{1}$
in the action (2.97) and the dilatation charge $D$ acquires a non zero time
derivative (2.105). If $k_{1}$ vanishes, the scale transformation (2.98) has
no effect on the dynamics and therefore corresponds to a symmetry at the
classical level. In presence of $k_{1}$, the second term proportional to
$\theta$ arises primarily because of the non-zero scale dimension of $\theta$
and also for fact that the symplectic bracket between the non-commuting
coordinates (2.69) is not invariant under dilatations (2.100). Our observation
therefore corroborates with the findings of [103, 104].

####  2.4.2 Anomalous Ward-Takahashi Identities

We now proceed ahead to discuss the path integral formulation of Ward-
Takahashi (W-T) identities associated with the action described in (2.97). At
zeroth-order, it represents the quantum-mechanical counterpart of Noether’s
current conservation theorem. In this section, we will explicitly calculate
the W-T identities upto 2nd order and we will see that the W-T identities
associated with broken dilatation symmetry is anomalous, indicating an
existence of a non-vanishing dilation anomaly. Here we use the word anomaly to
emphasise its genuine quantum mechanical origin and its contribution is over
and above the terms arising from the breaking of the symmetry at the classical
level itself. During the course of the derivation of these W-T identities, we
adopt the method due to Fujikawa wherein the anomalous terms are identified
with the Jacobian factor arising from the path-integral measure under the
scale transformation. The anomalous term arising here is eventually
regularized using Fujikawa’s prescription [105, 106].

In order to present our analysis we consider the generating functional in
$0+1$ dimensional QFT defined in effective commutative description
$\eqref{parteff}$ :-

$\displaystyle Z_{eff}\left[J_{1},J_{2};\lambda\right]$
$\displaystyle=\int\mathcal{D}X(t)~{}~{}~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left(X^{T}(t)\mathcal{R}X(t)+\lambda
J^{T}.X\right)}$ (2.106)

For our purpose we confine our attention to $Z_{eff}$ instead of $Z$ as the
presence/absence of solely source dependent factors
($\mathcal{K}[J_{1},J_{2}]$) does not change the Ward identities. Here
$\mathcal{R}$ is self-adjoint and its eigenvalues are being considered as
discrete real here -

$\mathcal{R}\phi_{k}(t)=\lambda_{k}\phi_{k}(t),$ (2.107)

where the eigenfunctions $\\{\phi_{k}(t)\\}$ are taken to satisfy the usual
orthogonality and completeness conditions:

$\displaystyle\int dt\phi_{k}^{\dagger}(t)\phi_{j}(t)$
$\displaystyle=\delta_{kj}~{};$ (2.108)
$\displaystyle\sum_{k=1}^{\infty}\phi_{k}(t)\phi^{\dagger}_{k}(t^{{}^{\prime}})$
$\displaystyle=\delta(t-t^{{}^{\prime}})\mathbb{I}_{2}$

In order to specify the functional measure, we expand

$X(t)=\sum_{k=1}^{\infty}a_{k}\phi_{k}(t)$ (2.109)

Now, the functional-integral measure is then defined as,

$\mathcal{D}X=\prod_{k=1}^{k=\infty}da_{k}=\mathcal{D}a$ (2.110)

To obtain the W-T identities, we study the behaviour of the action (2.97) and
the measure defined in (2.110) under infinitesimal local dilatation
transformations (2.101) of the field $q_{i}$ as

$X(t)\rightarrow X^{\prime}(t)=X(t)+\delta^{L}_{D}X(t),$ (2.111)

with $\delta^{L}_{D}X(t)=\gamma(t)(t\frac{\partial}{\partial
t}-\frac{1}{2})X(t),$ the expansion coefficients of (2.109) change to-

$\displaystyle a_{j}\rightarrow a_{j}^{\prime}=a_{j}+\sum_{k}a_{k}\int
dt\gamma(t)\phi^{\dagger}_{j}(t)(t\frac{\partial}{\partial
t}-\frac{1}{2})\phi_{k}(t)).$ (2.112)

Thus, the measure changes as [107] -

$\mathcal{D}X\rightarrow\mathcal{D}X^{{}^{\prime}}=\prod_{j=1}^{j=\infty}da^{{}^{\prime}}_{j}=(\det
c_{jk})\prod_{k=1}^{k=\infty}da_{k}=e^{Tr(ln\;c_{jk})}\mathcal{D}X$ (2.113)

where $det(c_{jk})$ can be read as the Jacobian of the transformation (2.112),
whereas, the transformation matrix elements $c_{jk}$ can be obtained by,

$c_{jk}=\frac{\partial a^{\prime}_{j}}{\partial a_{k}}=\delta_{jk}+\int
dt\gamma(t)\phi_{j}^{\dagger}(t)(t\frac{\partial}{\partial
t}-\frac{1}{2})\phi_{k}(t)$ (2.114)

Now, the above effective generating functional of connected Green’s functions
(2.106) can be re-written under the dilatation transformations (2.111) as,

$\displaystyle Z_{eff}\left[J_{1},J_{2};\lambda\right]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}X^{\prime}(t)~{}~{}~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[X^{{}^{\prime}T}\mathcal{R}X^{{}^{\prime}}+\lambda
J^{T}.X^{{}^{\prime}}\right]}$ (2.115) $\displaystyle=$ $\displaystyle\int
e^{Tr(ln\;c_{jk})}\mathcal{D}X~{}~{}~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[X^{T}\mathcal{R}X+\lambda
J^{T}.X\right]+\delta^{L}_{0}S_{eff}+\left[\lambda\int_{-\infty}^{\infty}dtJ^{T}(t)\delta^{L}_{0}X(t)\right]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

In the first equality, we have simply re-labeled $X(t)$ by
$X^{{}^{\prime}}(t)$ as a dummy variable in the functional integral (2.106).
The second equality is nontrivial and uses the assumed coordinate
transformations (2.111) and (2.113). For infinitesimal local scale parameter
($\gamma(t)$) the generating functional changes to become,

$\displaystyle
Z_{eff}\left[J_{1},J_{2};\lambda\right]=\int\mathcal{D}X(t)~{}e^{\frac{i}{\hbar}\int^{\infty}_{-\infty}dt\left[X^{T}\mathcal{R}X+\lambda
J^{T}.X\right]}e^{\frac{i}{\hbar}[\delta^{L}_{0}S_{eff}+\int_{-\infty}^{\infty}dt(\lambda
J^{T}\delta_{0}^{L}X-i\hbar\gamma(t)A(t))]}$

where,

$A(t)=\sum_{k}\phi_{k}^{\dagger}(t)(t\frac{\partial}{\partial
t}-\frac{1}{2})\phi_{k}(t).$ (2.117)

This $A(t)$ is known as anomalous term. Now reverting back to component
notation and performing the Taylor series expansion of the exponential factor
in (LABEL:dbc) upto terms of $\mathcal{O}(\gamma)$, we obtain :-

$\displaystyle
Z_{eff}\left[J_{1},J_{2};\lambda\right]=\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)e^{\frac{i}{\hbar}\left[S_{eff}+\int^{\infty}_{-\infty}dt~{}\lambda
J_{i}q_{i}\right]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(1+\frac{i}{\hbar}\int
dt\gamma(t)(\Delta_{0}-\frac{dD}{dt}+\lambda
J_{i}(t)\mathcal{Q}{q}_{i}(t)-i\hbar A(t))+\mathcal{O}(\gamma^{2})\right),$
(2.118)

where in the last line we have made use of eq.(2.104). The invariance of the
functional integral under the infinitesimal transformations (2.111) (the
change of variables in integration does not change the integral itself)
represents the W-T identities. Specifically, the variation with respect to
$\gamma(t)$ must vanish since it holds for any value of $\gamma$, i.e.

$\frac{\delta
Z_{eff}[J_{1},J_{2};\lambda]}{\delta\gamma(t^{\prime})}|_{\gamma=0}=0$ (2.119)

This yields

$\displaystyle
0=\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)e^{\frac{i}{\hbar}S_{eff}}$
$\displaystyle\times(1+\frac{i\lambda}{\hbar}\int
dt_{1}J_{i}(t_{1})q_{i}(t_{1})-\frac{{\lambda}^{2}}{2{\hbar}^{2}}\int\int
dt_{1}dt_{2}J_{i}(t_{1})J_{k}(t_{2})q_{i}(t_{1})q_{k}(t_{2})+........)$
$\displaystyle\times\frac{i}{\hbar}\left(\Delta_{0}-\frac{dD}{dt^{{}^{\prime}}}+{\lambda}J_{i}(t^{\prime})\mathcal{Q}{q}_{i}(t^{\prime})-i\hbar
A(t^{\prime})\right),$ (2.120)

Rearrangement of the previous expression yields:

$\displaystyle
0=\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)e^{\frac{i}{\hbar}S_{eff}}(\frac{i}{\hbar}(\Delta_{0}(t^{\prime})-\frac{dD}{dt^{\prime}}-i\hbar
A(t^{\prime}))$ $\displaystyle+\frac{i\lambda}{\hbar}[\frac{i}{\hbar}\int
dt_{1}J_{i}(t_{1})q_{i}(t_{1})(\Delta_{0}(t^{\prime})-\frac{dD}{dt^{\prime}}-i\hbar
A(t^{\prime}))$ $\displaystyle+\int
dt_{1}J_{i}(t_{1})\mathcal{Q}q_{i}(t_{1})\delta(t_{1}-t^{\prime})]$
$\displaystyle-\frac{{\lambda}^{2}}{2{\hbar}^{2}}[\int\int
dt_{1}dt_{2}J_{i}(t_{1})J_{k}(t_{2})q_{i}(t_{1})\mathcal{Q}q_{k}(t_{2})\delta(t_{2}-t^{\prime})$
$\displaystyle+\int\int
dt_{1}dt_{2}J_{i}(t_{1})J_{k}(t_{2})\mathcal{Q}q_{i}(t_{1})q_{k}(t_{2})\delta(t_{1}-t^{\prime})$
$\displaystyle+\frac{i}{\hbar}\int\int
dt_{1}dt_{2}J_{i}(t_{1})J_{k}(t_{2})q_{i}(t_{1})q_{k}(t_{2})({\Delta}_{0}(t^{\prime})-\frac{dD}{dt^{\prime}}-i\hbar
A(t^{\prime}))$
$\displaystyle\times\frac{i}{\hbar}({\Delta}_{0}(t^{\prime})-\frac{dD}{dt^{\prime}}-i\hbar
A(t^{\prime}))]+...............)$ (2.121)

Since W-T identity is actually an infinite number of identities satisfied by
the Green functions, the different orders of Ward-Takahasi identities are
obtained in terms of mean values of dynamical quantities 111where the mean
values are defined as: $\left\langle
T^{*}[............]\right\rangle=\int\mathcal{D}q_{1}(t)\mathcal{D}q_{2}(t)[............]e^{\frac{i}{\hbar}S_{e}ff}$
apart from the aforementioned normalization factor. by taking the functional
derivative of right hand side of (2.4.2) with respect to the source term i.e.
$\frac{\delta}{\delta J_{i}(t^{\prime}_{1})}\frac{\delta}{\delta
J_{m}(t^{\prime}_{2})}.....$ and then setting $J_{l}(t)$’s to $0$.

Thus the zeroth-order W-T identity:-

$\displaystyle\frac{d}{dt^{\prime}}\left\langle
D(t^{\prime})\right\rangle=\left\langle\Delta_{0}(t^{\prime})\right\rangle-i\hbar\left\langle
A(t^{\prime})\right\rangle$ (2.122)

First-order W-T identity:-

$\displaystyle\frac{d}{dt^{\prime}}\left\langle
T^{*}[D(t^{\prime})q_{i}(t^{\prime}_{1})]\right\rangle=$
$\displaystyle\left\langle
T^{*}[\Delta_{0}(t^{\prime})q_{i}(t^{\prime}_{1})]\right\rangle-i\hbar\left\langle
T^{*}[A(t^{\prime})q_{i}(t^{\prime}_{1})]\right\rangle$
$\displaystyle-i\hbar\delta(t^{\prime}_{1}-t^{\prime})\left\langle\mathcal{Q}{q}_{i}(t_{1})\right\rangle$
(2.123)

Second-order W-T identity:-

$\displaystyle\frac{d}{dt^{\prime}}\left\langle
T^{*}[D(t^{\prime}){q}_{k}(t^{\prime}_{2}){q}_{i}(t^{\prime}_{1}))]\right\rangle$
$\displaystyle=\left\langle
T^{*}[\Delta_{0}(t^{\prime}){q}_{k}(t^{\prime}_{2}){q}_{i}(t^{\prime}_{1})]\right\rangle$
$\displaystyle-i\hbar\delta(t^{\prime}_{1}-t^{\prime})\left\langle
T^{*}[{q}_{k}(t^{\prime}_{2})\mathcal{Q}{q}_{i}(t^{\prime}_{1})\right\rangle$
$\displaystyle-i\hbar\delta(t^{\prime}_{2}-t^{\prime})\left\langle
T^{*}[\mathcal{Q}{q}_{k}(t^{\prime}_{2}){q}_{i}(t^{\prime}_{1})]\right\rangle$
$\displaystyle-i\hbar\left\langle
T^{*}[A(t^{\prime}){q}_{k}(t^{\prime}_{2}){q}_{i}(t^{\prime}_{1})]\right\rangle$
(2.124)

Similarly one can also derive the higher-orders Ward-Takahashi identities.
Particularly, the zeroth-order W-T identity tells us that the dilatation
symmetry is explicitly broken due to presence of the spring-constant $k_{1}$
in the Lagrangian (2.91), which is consistent with that from Noether’s
analysis (2.105). But at the quantum level, there is also an additional
correction term on the right which is the contribution from the existence of
anomaly due to quantization and it is the anomaly term $A(t)$ which is solely
responsible for modifying the rate of change of dilatation charge $D(t).$ The
higher-order identities also show up this anomalous behavior up to ”contact
terms”.

Furthermore, it is worthwhile to observe that in this derivation of W-T
identities, the ill-defined expression of the anomaly term in (2.117)
apparently looks divergent. This is because, at each time point $t$ we are
summing over an infinite number of eigenmodes $\phi_{k}(t)$. However, it is
possible to give a physical meaning to this expression by the method of
regularization and subsequently extract out a finite result as we will see in
the next section.

###  2.5 Computation of the dilatation anomaly: Fujikawa’s method

Now we are in good position to compute the anomalous term in W-T identities.
Following Fujikawa’s prescription [106], the dilatation anomaly can be
regularized by correcting each contribution from $\phi_{k}(t)$ by a factor
$e^{-\lambda_{k}^{2}/M^{2}}$ as,

$A(t)\rightarrow
A_{Reg.}(t):=\lim_{M\to\infty}\sum_{k}\phi_{k}^{\dagger}(t)(t\frac{\partial}{\partial
t}-\frac{1}{2})\phi_{k}(t)e^{-\lambda_{k}^{2}/M^{2}}$ (2.125)

Here $\phi_{k}(t)\phi^{\dagger}_{k}(t)$ is an ill-defined object and can be
interpreted consistently by first separating time points and then taking the
coincident time limit at a appropriate step. A simple manipulation therefore
yields:

$\displaystyle A_{Reg.}(t)$ $\displaystyle=\lim_{M\to\infty}\lim_{t\to
t^{\prime}}~{}tr\sum_{k}(t\frac{\partial}{\partial
t}-\frac{1}{2})e^{-\mathcal{R}^{2}/M^{2}}\phi_{k}(t)\phi_{k}^{\dagger}(t^{\prime})$
$\displaystyle=\lim_{M\to\infty}\lim_{t\to
t^{\prime}}~{}tr\bigg{[}(t\frac{\partial}{\partial
t}-\frac{1}{2})e^{-\mathcal{R}^{2}/M^{2}}\delta(t-t^{\prime})\mathbb{I}_{2}\bigg{]}$
$\displaystyle=\lim_{M\to\infty}\lim_{t\to
t^{\prime}}~{}tr\int_{-\infty}^{\infty}\frac{dz}{2\pi}(t\frac{\partial}{\partial
t}-\frac{1}{2})e^{-\mathcal{R}^{2}/M^{2}}e^{iz(t-t^{\prime})}\mathbb{I}_{2}$
(2.126)

Note that in above, tr refers to only the $2\times 2$ matrix indices whereas
the Tr appearing earlier in (2.113) refers to both functional and matrix
indices. Here we have made use of the completeness relation of $\phi_{k}(t)$
(2.108) and in the last line, the integral representation of Dirac delta
($\delta$)-function has been used. Now we define a classical parameter
${\theta}^{\prime}=\frac{\theta}{2\hbar}$ to be used from now onwards in order
to remove any $\hbar$ dependency of $\theta$. The expression of the anomalous
term (2.5) after computing $\lim_{t\to
t^{\prime}}e^{-\frac{\mathcal{R}^{2}}{M^{2}}}e^{iz(t-t^{\prime})}$ and then
taking the limit $t\to t^{\prime}$ becomes -

$A_{Reg.}(t)=\lim_{M\to\infty}~{}tr~{}\int_{-\infty}^{\infty}\frac{dz}{2\pi}(izt-\frac{1}{2})e^{-a\bf{I}+b\bf{\sigma_{y}}}\mathbb{I}_{2},$
(2.127)

where

$\displaystyle a$
$\displaystyle=\frac{1}{M^{2}}(m_{B}^{2}{z^{4}}+4m_{B}^{2}k_{1}^{2}{\theta}^{\prime
2}{z^{2}}-2m_{B}k_{1}z^{2}+k_{1}^{2})$ $\displaystyle b$
$\displaystyle=\frac{4m_{B}k_{1}{\theta}^{\prime}}{M^{2}}(m_{B}z^{3}-k_{1}z)$

Using $e^{-a\bf{I}}=e^{-a}\bf{I}$,
$e^{b\bf{\sigma_{y}}}=\bf{I}\cosh{b}+\bf{\sigma_{y}}\sinh{b}$ and completing
the trace operation, we have :-

$A_{Reg.}(t)=2\lim_{M\to\infty}\int_{-\infty}^{\infty}\frac{dz}{2\pi}(izt-\frac{1}{2})e^{-a}\cosh{b}$
(2.128)

Since $e^{-a}\cosh{b}$ is an even function in $z$, the first term in the
integral doesn’t contribute. Therefore, the regularized version of the anomaly
term (2.128) can be re-written in a much simpler form as,

$A_{Reg.}(t)=-\lim_{M\to\infty}\frac{1}{2\pi}(I_{1}+I_{2}),$ (2.129)

where the integrals $I_{1}$ and $I_{2}$ are given by -

$\displaystyle I_{1}=\int_{0}^{\infty}dze^{-a+b}$ $\displaystyle
I_{2}=\int_{0}^{\infty}dze^{-a-b}$

Let $z^{2}-2k_{1}{\theta}^{\prime}z=s$, $\frac{m_{B}^{2}}{M^{2}}=p$,
$p{\omega^{\prime}}^{2}=q$ (where
${\omega^{\prime}}^{2}=k_{1}^{2}{\theta}^{\prime 2}+\frac{k_{1}}{m_{B}}$) and
$r=\sqrt{s+k_{1}^{2}{\theta}^{\prime 2}}$. The integral $I_{1}$ becomes :

$\displaystyle I_{1}$
$\displaystyle=\int_{0}^{\infty}\frac{ds}{2\sqrt{s+k_{1}^{2}{\theta}^{\prime
2}}}\exp{-p(s-{\omega}^{2})^{2}}$
$\displaystyle=\int_{k_{1}^{2}{\theta}^{\prime
2}}^{\infty}\frac{dr}{2\sqrt{r}}\exp{-p(r-\omega^{\prime 2})^{2}}$
$\displaystyle=e^{-p{\omega^{\prime}}^{4}}\int_{k_{1}^{2}{\theta}^{\prime
2}}^{\infty}\frac{dr}{2\sqrt{r}}e^{-pr^{2}+2qr}$ (2.130)

Similarly one can show :-

$I_{2}=e^{-p{\omega^{\prime}}^{4}}\int_{k_{1}^{2}{\theta}^{\prime
2}}^{\infty}\frac{dr}{2\sqrt{r}}e^{-pr^{2}+2qr}$ (2.131)

Thus the regularised anomaly factor $A_{Reg.}(t)$ simplifies to -

$\displaystyle A_{Reg.}(t)$
$\displaystyle=-\lim_{M\to\infty}\frac{1}{2\pi}e^{-p{\omega^{\prime}}^{4}}\int_{k_{1}^{2}{\theta}^{\prime
2}}^{\infty}dr\;{r}^{-1/2}e^{-pr^{2}+2qr}$
$\displaystyle=-\frac{1}{2\pi}\lim_{M\to\infty}e^{-p{\omega^{\prime}}^{4}}\left[\int_{0}^{\infty}dr\;{r}^{-1/2}e^{-pr^{2}+2qr}-\int_{0}^{k_{1}^{2}{\theta}^{\prime
2}}dr\;{r}^{-1/2}e^{-pr^{2}+2qr}\right]$ (2.132)

Now, taking the limit $M\to\infty$, the 2nd integral in (2.132) is convergent
and gives a finite result of $2k_{1}\theta^{\prime}$. The first integral can
be evaluated as in [112]:

$\lim_{M\to\infty}e^{-p{\omega^{\prime}}^{4}}\int_{0}^{\infty}\frac{dr}{\sqrt{r}}e^{-pr^{2}+2qr}=\lim_{M\to\infty}(2p)^{-\frac{1}{4}}\Gamma\left(\frac{1}{2}\right)D_{-\frac{1}{2}}(y)e^{\frac{y^{2}}{2}}e^{-p{\omega^{\prime}}^{4}}$
(2.133)

where $y=\frac{p{\omega^{\prime}}^{2}}{\sqrt{2p}}$. We can also write
$D_{-1/2}(y)$ in terms of $K_{1/4}(y)$, where $D_{-1/2}(y)$ and $K_{1/4}(y)$
represent the parabolic cylindrical functions and modified Bessel function
respectively.

$\lim_{M\to\infty}(2p)^{-\frac{1}{4}}\Gamma\left(\frac{1}{2}\right)D_{-\frac{1}{2}}(y)e^{\frac{y^{2}}{2}}=\lim_{M\to\infty}(2p)^{-\frac{1}{4}}e^{\frac{y^{2}}{2}}(\frac{1}{4}y^{2})^{\frac{1}{4}}K_{1/4}(\frac{1}{4}y^{2})$
(2.134)

Using
$K_{1/4}(x)\approx\frac{1}{2}\Gamma(\frac{1}{4})(\frac{x}{2})^{-\frac{1}{4}}$
as $x\rightarrow 0$, we have :-

$\lim_{M\to\infty}e^{-p{\omega^{\prime}}^{4}}\int_{0}^{\infty}\frac{dr}{\sqrt{r}}e^{-pr^{2}+2qr}=\frac{1}{2}\lim_{M\to\infty}\Gamma\left(\frac{1}{4}\right)p^{-\frac{1}{4}}=\frac{1}{2}\lim_{M\to\infty}\Gamma\left(\frac{1}{4}\right)(M/m_{B})^{1/2}$
(2.135)
$A_{Reg.}(t)=-\frac{1}{2\pi}\left[\frac{1}{2}\lim_{M\to\infty}\Gamma\left(\frac{1}{4}\right)(M/m_{B})^{1/2}-2k_{1}{\theta^{\prime}}\right]$
(2.136)

Note that the first term in the expression is independent of the coupling
$k_{1}$ and is the same as in the free particle case of our effective
commutative theory, which is usually considered to be non-anomalous and only
the second term is independent of the cutoff parameter $M$. Therefore,
following conventional wisdom [113], we now concentrate on the nontrivial
finite contribution coming from (2.136). To obtain a non-divergent anomaly
part, we need to renormalize the relation (2.136) in free particle theory
limit $k_{1}\rightarrow 0$ by simply adding a term $i\hbar\gamma(t)A_{f}$ in
the Lagrangian $L_{eff}$ (LABEL:dbc), thus cancelling the divergence in
(2.136). Here, $A_{f}$ is equal to the Fujikawa factor for the free particle
theory,

$A_{f}=-\frac{1}{4\pi}\lim_{M\to\infty}\Gamma\left(\frac{1}{4}\right)(M/m_{B})^{1/2}$
(2.137)

Therefore, after renormalization the correct anomaly is given by :

$A_{renormalized}=A_{Reg.}-A_{f}=\frac{1}{\pi}k_{1}{\theta^{\prime}}$ (2.138)

Taking into account the above scheme of renormalization, only the finite piece
of the anomaly term contributes to obtain renormalized version of the set of
bare W-T identities given in (2.122-2.4.2). Furthermore, expression of the
anomaly term (2.138) is completely independent of field configuration, so, the
form of the anomalous term is the same even if we were to work directly with
noncommutative space variables, except that the dilatation transformation
would have been implemented in a different way. On the other hand, since the
anomalous correction to the W-T identities is a first order in $\theta$, where
$\theta=\frac{\hbar c}{eB}$, which suggests that for commutative limit i.e.
$\theta\rightarrow 0$, one must have $\hbar\rightarrow 0$, indicating thereby
the commutative limit coincides with the classical limit. This indeed attests
to the fact that in this setting, the correct quantum corrections are fully
taken into account only when we switch to the non-commutative quantum
framework.

###  2.6 Discussion

In this chapter, we’ve studied a system of interacting two opposite charged
particles in two dimensions, subject to a strong perpendicular magnetic field
and low mass limit. We have demonstrated clearly how the first-order
Lagrangian of the system gets mapped directly to a deformed harmonic
oscillator. Then we reviewed the quantum picture for the above using Hilbert-
Schmidt operators-based formulation. Here the noncommutativity is given as an
effective description, unlike in many other works where it’s postulated as a
fundamental parameter. We then construct a path-integral formulation of the
functional integral using coherent states, as a theory in 0+1 dimensions. The
commutative equivalent action is then derived through path-integral which then
describes a charged particle moving in a constant background magnetic field
and under the influence of additional harmonic oscillator potential - the
generalized Landau problem. Moreover, the non-commutative harmonic oscillator
and generalized Landau problem are found to be related very closely. The
important point is that the form of this action has the same form as that of a
charged particle interacting with some point magnetic vortex which is scale or
dilatation invariant unlike our present case, thus suggesting the need for a
radical analysis for each individual case. We compute the W-T identities
associated with the broken dilation symmetry which appear to be anomalous. To
the extent, we are aware this has been remained unexplored in the literature
from the point of view of non-commutative physics under scaling
transformations. The anomalous term has been computed and eventually
regularized following Fujikawa’s method.

##  Chapter 3  Quantum (phase)-space and emergent geometrical phase

In the previous chapter, we have discussed the broken dilatation symmetry at
the classical and quantum level for an interacting dipolar system (a two-body
problem) subjected to a large magnetic field at which our system effectively
described as a harmonic oscillator placed in the ambient quantum (non-
commutative) plane. Also, we have established a mapping between the non-
commutative harmonic oscillator and a generalized Landau problem.

Indeed, apart from this spatial noncommutativity in the well known Landau
problem, it is well known that the non-relativistic theory in a $2+1$
dimension [114], planar Galilean group can be endowed with two fold centrally
extended Galilean algebra: (i) one involving the commutator of the Galilean
boost generators $\hat{K}_{i}$ between themselves, which is a non-zero
constant $[\hat{K}_{i},\hat{K}_{j}]=i\hbar\kappa\epsilon_{ij}$, and (ii) the
other involves the commutation relation between the Galilean boost and linear
momentum, which is related to the mass ($m$) of the particle:
$[\hat{K}_{i},\hat{P}_{i}]=i\hbar m\delta_{ij}$, with other commutators taking
their usual structures. Here the central charge $\kappa$ can be identified
with the fractional spin of the anyons, as has been shown in [115, 116], on
using certain nonrelativistic reduction of (2+1)-dimensional Poincare algebra
$iso(2,1)$. The price to be paid for a realization of two fold centrally
extended Galilean algebra is that the Galilean covariant coordinates
necessarily satisfy non-commutative algebra
$[\hat{x}_{i},\hat{x}_{j}]=i\theta\epsilon_{ij}$, with
$\hbar\kappa=m^{2}\theta.$ Now, we can further consider a modified commutator
between the two components of the momentum operators
$[\hat{P}_{i},\hat{P}_{j}]=iB\epsilon_{ij}$ with constant $B$. Here this $B$
can arise in presence of a uniform and constant magnetic field normal to the
plane. This can be seen to correspond to the case of the coupling of non-
relativistic anyons with this background magnetic field. Various interesting
points of this simultaneous noncommutativity in the coordinates and momenta
has been studied in the context of fractional quantum Hall-effect(FQHE)
[117][118],[119]. Besides this, the effects of phase-space noncommutativity
can have nontrivial consequences in gravitational wave detection and such a
situation has been encountered in [120]. On the other-hand, non-commutative
phase-spaces also provide a physical background for exact quantization of
interacting scalar field theory [121, 122].

In more recent times, Xiao et.al. have shown that in the semi-classical
analysis of Bloch electrons in presence of magnetic field, a nonzero Berry
curvature of the band structure leads to a noncommutativity between phase
space variables[26] and the effect of Berry Phase in deformed phase-space
geometry has also been discussed in [123]. As a result, it’s important to look
into whether the Berry phase exists in a quantum-mechanical system where both
the position and momentum operators fulfil non-commutative algebra. We are
also aware of the universal appeal of the Berry phase [124, 125] from both
theoretically such as fractional statistics [126, 127], anomalies in gauge
field theories [128, 129, 130] and also in several other situations
encountered in experiments [131, 132], wherein the system Hamiltonian depends
on some set of parameters and the Berry phase or geometrical phase obtained
through the adiabatic transport of a quantum system around a closed path in
this parameter space. An important feature of this geometric phase is that it
does not depend at all on the rate of traversal of the circuit provided of
course the change is adiabatic. Furthermore, a classical counterpart of this
quantual geometric phase shift was found by Hannay as well as Berry for a
classical integrable system, which is known as Hanny’s angle. A more detailed
discussion is given in [134, 146].

In this chapter, we study the appearance of adiabatic phase shift in the
context of planar quantum mechanics for a quantum two-dimensional simple
harmonic oscillator model with constant noncommutativity in both coordinates
and momenta. However, in this context, we would like to mention a similar work
carried out in [133], though in a different system involving a gravitational
quantum well. In that paper, their original model was defined on the
usual(commutative) phase-space and introduced noncommutativity just rewriting
the commutative variables by their NC counterparts by exploiting inverse of
the Bopp shift (or the Seiberg-Witten map in the parlance of [133]), and found
no geometric contribution in the total phase. Here we follow a different
approach to obtain a non-vanishing Berry phase which does not agree with the
conclusions of [133] for reasons that we explain in the sequel.

The different sections of the chapter are organized as follows. In section 3.1
we introduced a parametric harmonic oscillator placed in the ambient non-
commutative phase-space (both positions and momenta are non-commutative) where
mass and frequency parameter are slowly varying periodic function of time.
Here the energy spectrum is computed by using a generalised non-canonical
phase space transformation, which we refer to as generalised Bopp shift (see
Appendix-A), which connect the non-commutative phase space variables with
their commutative counterparts. In this context, we would like to mention that
in the literature one can find claims by [136] that this mapping from non-
commutative phase space to usual (commutative) phase space is not unique.
However, as we show in the appendix- A that for a particular case, they are
unitary equivalent. In the next section (3.2) we find out, the geometric phase
factor in Heisenberg picture which is acquired by the annihilation (and
corresponding creation) operators under an adiabatic transport in parameter
space of the system. In section 3.3 we discuss the usual expression of
geometric phase in terms of the phase gathered by the state vector by going
over from the Heisenberg picture to Schödinger picture and it is our present
contention that we provide a connection between the quantal description of
geometric phase shift and classical Hannay angle in a rather straightforward
manner in sec 3.4. Concluding remarks are made in sec-3.5. There are two
appendices; in the Appendix-A we discuss different realizations of non-
commutative algebra and their equivalence while in Appendix-B, apart from
reviewing some of the necessary group theoretical aspects related to our
model, we discuss the apparent removability of geometric phase by a time-
dependent unitary transformation, which nevertheless is shown to reappear in
”disguise” in the dynamical phase but retaining its geometrical nature.

###  3.1 Quantum parametric oscillator in non-commutative phase-space

In this section, we basically consider a two-dimensional non-commutative plane
where a harmonic oscillator with time-dependent parameters $P(t)$, $Q(t)$
varying adiabatically with period $T$ is subjected also to momentum
noncommutativity. This system is governed by the Hamiltonian

$\mathcal{H}(t)=P(t)(\hat{p}_{1}^{2}+\hat{p}_{2}^{2})+Q(t)(\hat{x}_{1}^{2}+\hat{x}_{2}^{2}),$
(3.1)

with $P(t),Q(t)>0$ and these time-dependent slowly varying parameters are
assumed to absorb all other parameters like mass, spring constant as mentioned
in the previous section. Our model is constructed in the spirit of [137, 138,
139, 140, 141] where the motivation of having time dependency of the
parameters $P(t)$ and $Q(t)$ in the real physical model was demonstrated. The
entire Heisenberg algebra of Non-commutative phase space is characterized by:

$\begin{array}[]{c}[\hat{x}_{i},\hat{x}_{j}]=i\theta\epsilon_{ij};[\hat{p}_{i},\hat{p}_{j}]=i\eta\epsilon_{ij};[\hat{x}_{i},\hat{p}_{j}]=i\hslash\delta_{ij}\end{array};\theta\eta<0$
(3.2)

where $\theta$,$\eta$ are taken to be constants and we shall restrict ourself
to negative value of $\theta\eta$ for consistent quantization. See for a
detail discussion [142, 143] and references therein. However, this deformed
phase-space algebra 3.2 enforces a symmetry between the coordinates
$\hat{x}_{i}$ and momentum $\hat{p}_{i}$ under the following transformation

$\displaystyle\begin{array}[]{rcl}&&\hat{x}_{i}\rightarrow\hat{p}_{i}\\\
&&\hat{p}_{i}\rightarrow\hat{x}_{i}\\\ &&\theta\rightarrow-\eta\\\
&&\eta\rightarrow-\theta\\\ &&i\rightarrow-i.\end{array}$ (3.8)

Of course, this symmetry between $\hat{x}_{i}$ and $\hat{p}_{i}$ will be lost
in absence of momentum non-commutativity.

In order to obtain the spectra of the model, we introduce a linear realization
of the above non-commutative algebra (3.2) in terms of ordinary phase space
variables ($q_{i},p_{i}$), which we refer to as generalized Bopp’s shift (See
Appendix-A for other kinds of Bopp shift). These are given by,

$\displaystyle\hat{x}_{i}$
$\displaystyle=q_{i}-\frac{\theta}{2\hslash}\epsilon_{ij}p_{j}+\frac{\sqrt{-\theta\eta}}{2\hslash}\epsilon_{ij}q_{j}$
(3.9) $\displaystyle\hat{p}_{i}$
$\displaystyle=p_{i}+\frac{\eta}{2\hslash}\epsilon_{ij}q_{j}+\frac{\sqrt{-\theta\eta}}{2\hslash}\epsilon_{ij}p_{j},$

where the canonical pair $q_{i}$ and $p_{i}$ are commuting coordinates and
momenta respectively obeying the usual Heisenberg algebra:

$[q_{i},q_{j}]=0=[p_{i},p_{j}];[q_{i},p_{j}]=i\hslash\delta_{ij}$ (3.10)

and are distinguished by the absence of overhead hats. Obviously, this
transformation (3.9) is not an unitary one as it changes the basic commutation
relations, but it nevertheless enables us to convert the Hamiltonian in the
non-commutative phase-space into a modified Hamiltonian in the usual canonical
phase-space, where NC effects will manifest through an explicit dependency of
the deformation parameters $\theta$ and $\eta$. Substituting (3.9) in (3.1) we
can obtain the following form of the Hamiltonian:

$\mathcal{H}(t)=\alpha(t)(p_{1}^{2}+p_{2}^{2})+\beta(t)(q_{1}^{2}+q_{2}^{2})+\delta(t)(p_{i}q_{i}+q_{i}p_{i})-\gamma(t)(q_{1}p_{2}-q_{2}p_{1});$
(3.11)

where the time-dependent effective coefficients $\alpha,\beta,\gamma,\delta$
are given by,

$\displaystyle\alpha(t)$
$\displaystyle=P(t)\left\\{1-\frac{\theta\eta}{4\hslash^{2}}\right\\}+Q(t)\left(\frac{\theta}{2\hslash}\right)^{2}$
(3.12) $\displaystyle\beta(t)$
$\displaystyle=Q(t)\left\\{1-\frac{\theta\eta}{4\hslash^{2}}\right\\}+P(t)\left(\frac{\eta}{2\hslash}\right)^{2}$
$\displaystyle\gamma(t)$ $\displaystyle=\frac{1}{\hslash}\left(\eta
P(t)+\theta Q(t)\right)$ $\displaystyle\delta(t)$
$\displaystyle=\left(\frac{\sqrt{-\theta\eta}}{4\hslash^{2}}\right)\left(\eta
P(t)-\theta Q(t)\right)$

At this stage, we can recognize that our system Hamiltonian living effectively
an enlarged parameter space and can be written as a combination of three
terms,

$\mathcal{H}(t)=\mathcal{H}_{gho,1}(t)+\mathcal{H}_{gho,2}(t)+\mathcal{H}_{\boldsymbol{L}}(t)$
(3.13)

where $\mathcal{H}_{gho,i}(t)^{\prime}s$ (i=1 or 2) are like a generalised
time dependent harmonic oscillator Hamiltonian along $i^{th}-direction$,

$\mathcal{H}_{gho,i}(t)=\alpha(t)(p_{i}^{2})+\beta(t)(q_{i}^{2})+\delta(t)(p_{i}q_{i}+q_{i}p_{i})\text{
}\text{ }\text{ }\text{ }\text{(no sum on $i$)}$ (3.14)

and

$\mathcal{H}_{\boldsymbol{L}}(t)=-\gamma(t)(q_{1}p_{2}-q_{2}p_{1})$ (3.15)

which is effectively a Zeeman term in Landau like problem, where a magnetic
field is present perpendicular to the $q_{1}-q_{2}$ plane. In order to
diagonalize the system Hamiltonian, at first we introduce the annihilation
(and corresponding creation) operators

$\boldsymbol{a}_{j}=\left(\frac{\beta}{2\hslash\sqrt{\alpha\beta-\delta^{2}}}\right)^{1/2}\left[q_{j}+\left(\frac{\delta}{\beta}+i\frac{\sqrt{\alpha\beta-\delta^{2}}}{\beta}\right)p_{j}\right];\text{
}j=1,2$ (3.16)

satisfying the commutation relations

$[\boldsymbol{a}_{i},\boldsymbol{a}_{j}^{\dagger}]=\delta_{ij},$ (3.17)

with $\beta>0$ and
$\alpha\beta-\delta^{2}=\left(\frac{P\eta}{2\hslash}-\frac{Q\theta}{2\hslash}\right)^{2}+PQ>0$,
as follows from (3.12) and from the fact that $PQ>0$. The Hamiltonian then
reduces to the form

$\mathcal{H}(t)=\hslash\omega\left(\sum_{j=1,2}\boldsymbol{a_{j}^{\dagger}a_{j}}+1\right)+i\hslash\gamma\left(\boldsymbol{a_{1}^{\dagger}a_{2}-a_{2}^{\dagger}a_{1}}\right);\omega=2\sqrt{\alpha\beta-\delta^{2}}.$
(3.18)

Observe that the non-diagonal second term, is like the Schwinger
representation of $J_{2}$ angular momentum operator
($\overrightarrow{J}=\boldsymbol{a_{i}}^{\dagger}(\overrightarrow{\sigma})_{ij}\boldsymbol{a_{j}}$).
If we now perform yet another unitary transformation,

$\left[\begin{array}[]{c}\boldsymbol{a_{1}}\\\
\boldsymbol{a_{2}}\end{array}\right]\text{ }\rightarrow\text{
}\left[\begin{array}[]{c}\boldsymbol{a_{+}}\\\
\boldsymbol{a_{-}}\end{array}\right]=M\left[\begin{array}[]{c}\boldsymbol{a_{1}}\\\
\boldsymbol{a_{2}}\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&-i\\\
i&-1\end{array}\right]\left[\begin{array}[]{c}\boldsymbol{a_{1}}\\\
\boldsymbol{a_{2}}\end{array}\right]$ (3.19)
$\begin{array}[]{cc}[\boldsymbol{a_{i},a_{j}^{\dagger}}]=\delta_{ij};[\boldsymbol{a_{i},a_{j}}]=0&\text{
}(i,j\in\\{+,-\\})\end{array}$ (3.20)

with $M$ a unitary matrix, which make the term 3.15 into the exact diagonal
form of $J_{3}$, while retaining the diagonal form of the first term.
Accordingly the system Hamiltonian may be rewritten as

$\begin{aligned}
\mathcal{H}(t)&=\hslash\sum_{j=+,-}\omega_{j}\boldsymbol{a_{j}}^{\dagger}\boldsymbol{a_{j}}+\hslash\omega;\text{
}\omega_{\pm}=\omega\mp\gamma\end{aligned}.$ (3.21)

It is useful to define instantaneous eigenvalue equation of this Hamiltonian,
viz.,

$\mathcal{H}(t)\left|n_{1},n_{2};t\right\rangle=E_{n_{1}n_{2}}(t)\left|n_{1},n_{2};t\right\rangle,$
(3.22)

and hence instantaneous eigenvalues and eigenstates can virtually be read-off
from (3.21) as,

$\begin{array}[]{c}E_{n_{1}n_{2}}(t)=\hslash\omega\left(n_{1}+n_{2}+1\right)-\hslash\gamma\left(n_{1}-n_{2}\right);\\\
\left|n_{1},n_{2};t\right\rangle=\frac{\left(\boldsymbol{a_{+}^{\dagger}}\right)^{n_{1}}\left(\boldsymbol{a_{-}^{\dagger}}\right)^{n_{2}}}{\sqrt{n_{1}!}\sqrt{n_{2}!}}\left|0,0;t\right\rangle,~{}~{}\boldsymbol{a}_{\boldsymbol{\pm}}(t)\left|0,0;t\right\rangle=0,\end{array}$
(3.23)

where $n_{1},n_{2}$ are nonnegative integers. This reproduces the spectra
which was also obtained in [147] as well. This spectrum is clearly non-
degenerate and this ensures that there is no crossing of energy levels during
the adiabatic process.

In this light, it’s worth noting that [148] looked at essentially the same
system, but with Bartelomi’s realisation. By carrying out a simple scaling in
(3.76) and (3.77).

$\theta\rightarrow\xi^{-1}\theta;~{}\eta\rightarrow\xi^{-1}\eta;~{}\hbar\rightarrow\hbar_{eff}=\hbar\xi^{-1},$
(3.24)

with $\xi=\xi_{c}$ (3.78), one can readily verify that the phase-space algebra
and spectrum in [148] agree with (3.2) and (3.23), respectively.

Finally, we rewrite our primitive operators
$\boldsymbol{a}_{1},\boldsymbol{a}_{2}$ of (3.16) as follows:

$\displaystyle\boldsymbol{a}_{i}=A(t)[q_{i}+\left(B(t)+iC(t)\right)p_{i}]\text{
( {i} $\in$\\{1,2\\})}$ (3.25) where  $\displaystyle
A(t)=\left(\frac{\beta}{\hslash\omega}\right)^{1/2};~{}B(t)=\frac{\delta}{\beta};~{}C(t)=\frac{\omega}{2\beta}.$

Through the time dependence of $A,B,C,$ we can show that
$\boldsymbol{a_{\pm},a_{\pm}^{\dagger}}$ also have explicit time dependence.

###  3.2 Evolution of Ladder operators in Heisenberg picture

In this section we will study the adiabatic evolution of the system in
Heisenberg picture to obtain the geometric phase shift. The Heisenberg
equation of motion for a generic operator $\hat{O}$ are given by
$\frac{d\hat{O}}{dt}=\frac{1}{i\hslash}[\hat{O},\mathcal{H}]+\frac{\partial\hat{O}}{\partial
t}$. These equations of motion are formally identical to Hamilton’s equations
of classical dynamics so that the connection between quantum adiabatic phase
and the classical anholonomy associated with the classical Hannay angle shifts
will also be quite transparent in this framework.

Working in the Heisenberg representation for the ladder operators which are
explicitly time dependent, relevant Heisenberg equations of motion take the
following forms:

$\begin{array}[]{c}\dfrac{d}{dt}\left[\begin{array}[]{c}a_{+}\\\ a_{-}\\\
a_{+}^{\dagger}\\\
a_{-}^{\dagger}\end{array}\right]=\left[\begin{array}[]{cccc}X_{+}&0&0&Y\\\
0&X_{-}&Y&0\\\ 0&Y^{*}&X_{+}^{*}&0\\\
Y^{*}&0&0&X^{*}\end{array}\right]\left[\begin{array}[]{c}a_{+}\\\ a_{-}\\\
a_{+}^{\dagger}\\\ a_{-}^{\dagger}\end{array}\right]\\\
X_{\pm}=\frac{\dot{A}}{A}\pm i\left(\gamma\mp
2C\beta\mp\frac{(\dot{B}+i\dot{C})}{2C}\right),Y=-\frac{(\dot{B}+i\dot{C})}{2C}.\end{array}$
(3.26)

Until now, we’ve found all of the expressions without any approximation.
However, we shall begin to investigate the adiabatic variation of $P(t)$ and
$Q(t)$ from now on. Note that, due to their explicit dependency,
$A,B,C,\alpha,\beta,\gamma,\delta$ follow the same order of adiabaticity as P
and Q, which is a measure of the parameter’s slow time evolution. If
$\dot{P},\dot{Q}\approx\epsilon$ ,
$\ddot{P},\ddot{Q}\approx\epsilon^{2}\ldots$, then $\dot{F}\approx\epsilon$ ,
$\ddot{F}\approx\epsilon^{2}\ldots$ , where $F$ collectively stands for
$A,B,C,\alpha,\beta,\gamma,\delta$. We won’t overlook any terms that are now
within the adiabatic limit, but will merely keep track of adiabaticity order.
It will eventually become evident that the geometric phase [149, 150] can be
calculated by ignoring the second or higher order term in adiabaticity order.

We are now able to disentangle the four interrelated differential equations
caused by the evolution of annihilation and creation operators in (3.26) by
first differentiating w.r.t. time and rearranging them appropriately, yielding
$\boldsymbol{a}_{+}$ and $\boldsymbol{a}_{-}$ second-order differential
equations:

$\begin{array}[]{c}\frac{d^{2}\boldsymbol{a}_{+}}{dt^{2}}=\frac{d\boldsymbol{a}_{+}}{dt}\left(X_{+}+\frac{\dot{Y}}{Y}+X_{-}^{*}\right)+\boldsymbol{a}_{+}\left(\dot{X}_{+}-\frac{\dot{Y}}{Y}X_{+}+YY^{*}-X_{+}X_{-}^{*}\right)\\\
\frac{d^{2}\boldsymbol{a}_{-}}{dt^{2}}=\frac{d\boldsymbol{a}_{-}}{dt}\left(X_{-}+\frac{\dot{Y}}{Y}+X_{+}^{*}\right)+\boldsymbol{a}_{-}\left(\dot{X}_{-}-\frac{\dot{Y}}{Y}X_{-}+YY^{*}-X_{-}X_{+}^{*}\right)\end{array}$
(3.27)

Now we observe that if $Y\approx\epsilon$, then $\dot{Y}\approx\epsilon^{2}$ ,
and so $\frac{\dot{Y}}{Y}\approx\epsilon$.

Hence substituting $X_{+},X_{-},Y$ from (3.26) and only retaining terms
involving first order of adiabaticity $\frac{\dot{Y}}{Y}$, we deduce

$\frac{d^{2}\boldsymbol{a}_{+}}{dt^{2}}=\frac{d\boldsymbol{a}_{+}}{dt}\left(\mathfrak{P}+\frac{\dot{Y}}{Y}\right)+\boldsymbol{a}_{+}\left(\mathfrak{Q}-\frac{\dot{Y}}{Y}X_{+}\right),$
(3.28)

where

$\displaystyle\mathfrak{P}=$
$\displaystyle\left(2\frac{\dot{A}}{A}+2i\gamma+\frac{\dot{C}}{C}\right)$
(3.29) $\displaystyle\mathfrak{Q}=$ $\displaystyle
i\left(\dot{\gamma}-2\frac{d}{dt}\left(C\beta\right)-\gamma\frac{\dot{C}}{C}-2\frac{\dot{A}}{A}\gamma\right)+\left\\{\gamma^{2}-4C^{2}\beta^{2}-2\dot{B}\beta\right\\}+\mathscr{O}\left(\epsilon^{2}\right)$

As can be seen, the differential equation fulfilled by $\boldsymbol{a}_{-}$
has a similar structure.

To continue, we must write (3.28) in its normal form. To do so, we’ll need a
new time-dependent operator $\boldsymbol{b}(t)$ defined as,

$\boldsymbol{a_{+}}(t)=\boldsymbol{b}(t)e^{\frac{1}{2}\int_{t}\left(\mathfrak{P}+\frac{\dot{Y}}{Y}\right)d\tau}.$
(3.30)

Replacing (3.30) in (3.28) we obtain

$\boldsymbol{\ddot{b}}+\boldsymbol{b}\left(\frac{\dot{\mathfrak{P}}}{2}-\frac{\mathfrak{P}^{2}}{4}-\mathfrak{Q}\right)+\boldsymbol{b}\left(\frac{\dot{Y}}{Y}X_{+}-\frac{\mathfrak{P}}{2}\frac{\dot{Y}}{Y}+\mathscr{O}\left(\epsilon^{2}\right)\right)=0,$
(3.31)

where

$\frac{\dot{Y}}{Y}=\frac{\ddot{B}+i\ddot{C}}{\dot{B}+i\dot{C}}-\frac{\dot{C}}{C}=Z+i\tilde{Z}-\frac{\dot{C}}{C},$
(3.32)

with both $Z$ and $\tilde{Z}$ corresponds respectively to the real and
imaginary part of $\frac{\ddot{B}+i\ddot{C}}{\dot{B}+i\dot{C}}$.

On using the expressions of $\mathfrak{P}$ and $\mathfrak{Q}$ from (3.29) we
can be recast 3.31 in the following form:

$\boldsymbol{\ddot{b}}+\boldsymbol{b}(U+iV)=0$ (3.33)

where,

$\displaystyle U=$ $\displaystyle
4C^{2}\beta^{2}+2\dot{B}\beta+2\tilde{Z}C\beta+\mathscr{O}\left(\epsilon^{2}\right)\approx\mathscr{O}\left(\epsilon^{0}\right)$
(3.34) $\displaystyle V=$ $\displaystyle
2\frac{d}{dt}\left(C\beta\right)-2C\beta\left(Z-\frac{\dot{C}}{C}\right)+\mathscr{O}\left(\epsilon^{2}\right)\approx\mathscr{O}\left(\epsilon\right)$

Note that, since we are working in adiabatic limit, the functions $U$ and $V$
are very slowly varying function with time. Hence, we can apply the formula
for WKB approximation for complex potential [151] given by

$\boldsymbol{b}(t)=\boldsymbol{b}(0)\left[\frac{C_{1}}{\sqrt{\left|\xi(t)\right|}}\exp\left(\int_{0}^{t}\left(i\xi(\tau)-\phi(\tau)\right)d\tau\right)+\frac{C_{2}}{\sqrt{\left|\xi(t)\right|}}\exp\left(\int_{0}^{t}\left(-i\xi(\tau)+\phi(\tau)\right)d\tau\right)\right]$
(3.35)

where, $\sqrt{U+iV}=\xi+i\phi$ and ($C_{1}$, $C_{2}$) are arbitrary
coefficients that can be used to find the general solution of the differential
equation 3.35 in our case:

$\displaystyle\xi=\sqrt{\frac{\sqrt{U^{2}+V^{2}}+U}{2}}$
$\displaystyle\approx\sqrt{U+\frac{V^{2}}{4U}}\approx\sqrt{U}\approx
2C\beta+\frac{\dot{B}\beta+C\beta\tilde{Z}}{2C\beta}$ (3.36)
$\displaystyle\phi=\sqrt{\frac{\sqrt{U^{2}+V^{2}}-U}{2}}$
$\displaystyle\approx\sqrt{\frac{V^{2}}{4U}}\approx\frac{2\frac{d}{dt}\left(C\beta\right)-2C\beta\left(Z-\frac{\dot{C}}{C}\right)}{4C\beta}$

In our adiabatic approximation, we’ve left out terms in the second order (or
beyond) of adiabaticity. We can now see that the solution must adhere to the
following boundary condition: $\boldsymbol{b}(t=0)=\boldsymbol{b}(0)$. The
cyclicity of the parameters also implies that
$\sqrt{\left|\xi(0)\right|}=\sqrt{\left|\xi(T)\right|}$. Finally, only the
phase factor of the second component in the solution (3.35) produces the
dynamical phase of $\boldsymbol{a}_{+}$ with the appropriate sign. This will
become clear as we compute $\boldsymbol{a}_{+}(T)$. As a result, we set
$C_{1}=0$ in (3.35). By combining all of these expressions, the specific
solution of (3.31) is obtained as follows:

$\displaystyle\boldsymbol{b}(T)$
$\displaystyle\approx\boldsymbol{b}(0)\exp\left(\int_{0}^{T}\left\\{-i\left(2C\beta+\frac{\dot{B}\beta+C\beta\tilde{Z}}{2C\beta}\right)+\phi\right\\}d\tau\right)$
(3.37)

It may also be noted that $\frac{\dot{Y}}{Y}=Z+i\tilde{Z}-\frac{\dot{C}}{C}$
is an exact differential. Hence, we can write,

$\int_{0}^{T}\frac{\dot{Y}}{Y}d\tau=\int_{0}^{T}\left(Z+i\tilde{Z}-\frac{\dot{C}}{C}\right)d\tau=\int_{0}^{T}Zd\tau+i\int_{0}^{T}\tilde{Z}d\tau=0$
(3.38)