On symplectic gauge invariant reformulation of a particle on a torus knot

Anjali S and Saurabh Gupta
Department of Physics, National Institute of Technology Calicut,
Kozhikode - 673 601, Kerala, India
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We quantize a particle confined to move on a torus knot satisfying
constraint condition ($p\theta+q\phi)\approx 0$, within the context of a
geometrically motivated approach – Faddeev-Jackiw formalism. We also deduce
the constraint spectrum and discern the basic brackets in a more economical
way vis-à-vis Dirac formalism. We further reformulate the original gauge non-
invariant theory into a physically equivalent gauge theory, which is free from
any additional Wess-Zumino variables, by employing symplectic gauge invariant
formalism. In addition, we analyze the reformulated gauge invariant theory
within the framework of BRST formalism to establish the off-shell nilpotent
and absolutely anti-commuting (anti-)BRST symmetries. Finally, we show the
conserved (anti-)BRST charges, that happen to be generators of corresponding
symmetries, satisfy the physicality criteria which is consistent with the
Dirac quantization procedure.

PACS: 11.10.Ef, 11.15.-q, 11.30.-j

Keywords: Faddeev-Jackiw formalism, Symplectic gauge invariant formalism,
Gauge symmetries, (anti-)BRST symmetries.

##  1 Introduction

Knot theory plays a key role in a wide range of physical scenarios ranging
from string theory, statistical physics, polymer physics to various studies in
molecular biology [1, 2, 3]. It establishes a connection to the exploration of
algebraic structure inherent in renormalizable quantum field theories [4].
Knots are also expected to determine gauge-invariant observables in some
quantum gravity approaches [5]. Consequently, some generalizations of knot
theory have been employed to construct operator algebra which may express
quantum geometry at non-perturbative level [6]. The classical and quantum
mechanics of a free particle constrained to move on a knot that winds a torus
have been explored to deduce the energy spectrum [7] and Hamiltonian analysis
has been carried out to account for dynamics and symmetries [8]. The Becchi-
Rouet-Stora-Tyutin (BRST) and anti-BRST symmetries have been established via
Batalin-Fradkin-Vilkovisky formalism [9]. Moreover, the problem of a charge
moving in a torus knot in the presence of a uniform magnetic field has been
explored in a semi-classical quantization procedure [10].

As long as the conventional canonical quantization process is concerned, it is
not applicable in a straightforward manner to the systems with singular
Lagrangian or constrained systems [11, 12]. To deal with it, Anderson-Bergmann
and Dirac independently developed an approach for a consistent and systematic
treatment of constraints [13, 14, 15]. Dirac’s formalism introduces an
effective procedure for deducing all constraints in a theory and at the same
time classifying them into primary, secondary, tertiary etc. and further into
first-class and second-class. For second-class constrained systems, Dirac
introduced a generalized bracket known as Dirac bracket [15]. However, the
procedure of finding Dirac brackets is found to be convoluted. Alternatively,
Faddeev and Jackiw developed a geometrically motivated methodology in dealing
with constrained systems which makes use of the symplectic nature of phase
space [16]. In Faddeev-Jackiw formalism, the singular nature of symplectic
two-form matrix predicts the existence of constraints and they are procured in
an iterative fashion (cf. for details [17, 18, 19, 20]). This formalism is
further modified to incorporate Dirac-Bergmann algorithm [21, 22] and
implemented in various systems such as four-dimensional BF theory [23], anti
self-dual Yang-Mills equations [24] and Christ-Lee model [25].

It is a well-known fact that the quantization of a gauge non-invariant theory
(system endowed with second-class constraints) confronts the problem of factor
ordering [26, 27]. Consequently, numerous approaches have been developed to
construct an equivalent gauge invariant version of a gauge non-invariant
theory. To mention some approaches, Batalin-Fradkin-Fradkina-Tyutin proposed a
reformulation technique by enlarging the phase space with the introduction of
Wess-Zumino variable [28, 29]. Further, for the system of solely second-class
constraints, Mitra and Rajaraman developed a methodology to construct gauge
theory with a subset of half of the constraints having mutual vanishing
Poisson brackets among them [30, 31]. In addition, another formalism is
introduced which deforms the structure of local symmetries of Lagrangian
formulation in the conversion procedure [32]. Furthermore, a methodology based
on symplectic formalism is also proposed where an arbitrary function depending
upon the phase space variables and Wess-Zumino variable is introduced to
reformulate a gauge non-invariant theory [33, 34, 35].

On the other hand, gauge theories - endowed with the first-class constraints
in the language of Dirac’s prescription for the classification of constraints
- play a crucial role in theoretical physics as they capacitate the
investigations of fundamental interactions. Quantization of such theories
requires special attention due to the presence of superfluous degrees of
freedom. The BRST formalism is one of the most favorite schemes of
quantization for a gauge theory [36, 37, 38]. In BRST formalism, the Hilbert
space of gauge theory is enlarged and the notion of gauge invariance is
replaced by BRST invariance which contains anti-commuting variables - Faddeev-
Popov ghosts (and corresponding anti-ghosts). The resulting theory possesses a
more generalized gauge invariance, known as BRST invariance, where the effect
of gauge fixing is achieved without breaking the BRST symmetry [39].

Keeping the above in mind, our first motive of present investigation is to
explore the constraint structure and subsequently quantize the system of
particle on a torus knot via a geometrically motivated approach - the Faddeev-
Jackiw formalism. The second motive is to reformulate the system as a gauge
theory by employing symplectic gauge invariant formalism and disclose hidden
gauge symmetry. Additionally, we wish to establish the off-shell nilpotent and
absolutely anti-commuting (anti-)BRST symmetries for this system.

The contents of the paper are organized as follows. Section 2 gives a brief
preface about a particle restricted to move on a torus knot. We discuss the
constraint structure and Faddeev-Jackiw quantization in section 3. Our section
4 comprises the reformulation of the system as a gauge theory via symplectic
gauge invariant formalism and gives an account of the gauge symmetries. Our
section 5 contains the derivation of off-shell nilpotent and absolutely anti-
commuting (anti-)BRST transformations and corresponding conserved charges.
Finally, we summarize our results in section 6.

Our Appendix A consists of explicit computation of Dirac brackets and Appendix
B deals with the calculation of equations of motion.

##  2 Preliminaries

We begin with toroidal coordinates $(\eta,\theta,\phi)$ which are related with
Cartesian coordinates $(x,y,z)$ in the following fashion [40]

$x=\frac{a\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},\quad
y=\frac{a\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},\quad
z=\frac{a\sin\theta}{\cosh\eta-\cos\theta},$ (1)

where the coordinates ranges from $0\leq\eta<\infty$, $-\pi<\theta\leq+\pi$
and $0\leq\phi<2\pi$. The parameter $a$ is defined as $a^{2}=R^{2}-r^{2}$,
where $R$ is major radius and $r$ is minor radius of the torus. Moreover, the
toroidal coordinates satisfy following relations

$\displaystyle\tan\phi=\displaystyle\frac{y}{x},\quad
x^{2}+y^{2}+\left(z-a\cot\theta\right)^{2}=\frac{a^{2}}{\sin^{2}\theta},$ (2)
$\displaystyle x^{2}+y^{2}+z^{2}+a^{2}=\displaystyle
2a\left(x^{2}+y^{2}\right)^{\frac{1}{2}}\coth\eta,$

where, $\eta$ taking a constant value represents a toroidal surface, $\theta$
being a constant generates spherical bowls and a constant value of $\phi$
gives half planes [40, 41]. Here we are interested in toroidal surface and
coordinate $\eta$ taking a specific value, say e.g.
$\eta_{c}=\cosh^{-1}(\frac{R}{r})$ gives a toroidal surface [7]. A larger
value of $\eta_{c}$ generates a torus of smaller thickness.

In our present investigation, we consider the case of a particle constrained
to move on a torus knot. A torus knot is defined to be a knot which resides on
an unknotted torus without crossing under or over themselves on the torus
surface. Every torus knot is a $(p,q)$-torus knot for some pair of integers,
where $p$ and $q$ are relatively prime. Here the torus knot wraps $p$ times
along the meridian curve and $q$ times along the longitudinal curve (cf. [42]
for details).

In order to obtain the case of a particle restricted to move on a knot that
winds the torus, we impose the constraint condition $p\theta+q\phi\approx 0$
into the system, where $p$ and $q$ are mutually prime numbers [8, 10]. Thus,
the Lagrangian describing a particle of mass $m$ constrained to move on a
$(p,q)$-torus knot is given as

$L\;=\;\frac{ma^{2}}{2(\cosh\eta-\cos\theta)^{2}}\left[\dot{\eta^{2}}+\dot{\theta^{2}}+\sinh^{2}\eta\;\dot{\phi^{2}}\right]+\lambda(p\theta+q\phi),$
(3)

where $\lambda$ represents the Lagrange multiplier and overdot indicates time
derivative. The canonical momenta corresponding to variables $\eta$, $\theta$,
$\phi$ and $\lambda$ are, respectively, given as

$P_{\eta}=\frac{ma^{2}\dot{\eta}}{(\cosh\eta-\cos\theta)^{2}},\quad
P_{\theta}=\frac{ma^{2}\dot{\theta}}{(\cosh\eta-\cos\theta)^{2}},\quad
P_{\phi}=\frac{ma^{2}\sinh^{2}\eta\dot{\phi}}{(\cosh\eta-\cos\theta)^{2}},\quad
P_{\lambda}=0.$ (4)

The canonical Hamiltonian corresponding to the Lagrangian (3) takes the
following form

$H\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right]-\lambda(p\theta+q\phi).$
(5)

In the context of Dirac formalism, we infer the primary constraint in the
system as

$\varphi_{1}\;\equiv\;P_{\lambda}\;\approx\;0.$ (6)

The consistency condition of $\varphi_{1}$ leads to the secondary constraint
$\varphi_{2}$ as

$\dot{\varphi}_{1}\;=\;\left\\{\varphi_{1},H_{p}\right\\}\approx
0\quad\Longrightarrow\quad\varphi_{2}\;\equiv\;(p\theta+q\phi)\approx 0,$ (7)

where the explicit form of primary Hamiltonian $H_{p}$ is given by

$H_{p}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right]-\lambda(p\theta+q\phi)+\upsilon
P_{\lambda},$ (8)

with $\upsilon$ is a Lagrange multiplier. The consistency condition of
secondary constraint $\varphi_{2}$ leads to the tertiary constraint
$\varphi_{3}$, as

$\dot{\varphi}_{2}\;=\;\left\\{\varphi_{2},H_{p}\right\\}\approx
0\quad\Longrightarrow\quad\varphi_{3}\;\equiv\;\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\approx
0.$ (9)

There are no further constraints present. Among these three constraints,
$\varphi_{1}$ has vanishing Poisson brackets with $\varphi_{2}$ and
$\varphi_{3}$ whereas the Poisson bracket between $\varphi_{2}$ and
$\varphi_{3}$ is given as
$\\{\varphi_{2},\varphi_{3}\\}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)$.
Thus, $\varphi_{1}$ is first-class and $\varphi_{2}$ and $\varphi_{3}$ are
second-class constraints.

##  3 Faddeev-Jackiw formalism: A concise recipe

In this section, we investigate the system of particle on a torus knot in the
context of Faddeev-Jackiw formalism [16, 17]. To begin with, we consider the
first-order Lagrangian where no quadratic terms of generalized velocities are
present. The general form of first-order Lagrangian can be given as

$L^{(0)}=a_{i}^{(0)}\dot{\zeta}^{(0)}_{i}-V^{(0)}(\zeta),$ (10)

where $a_{i}^{(0)}$ and $V^{(0)}(\zeta)$ represent the canonical one-form and
symplectic potential, respectively and the subscript $i$ denotes the dimension
of extended configuration space. In this formalism, the geometric structures
are obtained with the aid of symplectic two-form $f^{(0)}_{ij}$, defined as
the exterior product of canonical one-form [17]. The equations of motion are
derived in terms of symplectic matrix $f_{ij}^{(0)}$ in the following manner

$f_{ij}^{(0)}\dot{\zeta}^{(0)j}\;=\;\frac{\partial
V^{(0)}(\zeta)}{\partial\zeta^{(0)i}},$ (11)

where the symplectic matrix is defined by

$\displaystyle f_{ij}^{(0)}$ $\displaystyle=$ $\displaystyle\frac{\partial
a^{(0)}_{j}(\zeta)}{\partial\zeta^{(0)i}}-\frac{\partial
a^{(0)}_{i}(\zeta)}{\partial\zeta^{(0)j}}.$ (12)

The (non-)singular nature of this symplectic matrix determines whether the
system under investigation is (un)constrained. In Faddeev-Jackiw formalism,
the constraints are procured in an iterative fashion from the zero-modes of
singular symplectic matrix. Successive incorporation of all constraints into
the canonical sector of first-order Lagrangian yields a non-singular matrix
(in case of gauge non-invariant theory). Further, from the inverse of non-
singular symplectic matrix, the basic brackets are deduced (cf. [17, 18] for
details).

###  3.1 A particle on a torus knot

In the case of a particle on a torus knot, taking the canonical momenta as the
auxiliary variables, we construct the first-order Lagrangian as

$L_{f}^{(0)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}-V^{(0)},$
(13)

where the symplectic potential $V^{(0)}$ is given by

$V^{(0)}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right]-\lambda(p\theta+q\phi).$
(14)

In order to deduce the constraint structure we calculate the form of zeroth-
iterated symplectic matrix $f_{ij}^{(0)}$. To start with, we choose the set of
symplectic variables as
$\zeta^{(0)}\;=\;\left\\{\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\lambda\right\\}.$
The corresponding components of canonical one-forms are listed below:

$a^{(0)}_{\eta}\;=\;P_{\eta},\quad a^{(0)}_{\theta}\;=\;P_{\theta},\quad
a^{(0)}_{\phi}\;=\;P_{\phi},\quad
a^{(0)}_{P_{\eta}}\;=\;a^{(0)}_{P_{\theta}}\;=\;a^{(0)}_{P_{\phi}}\;=\;a^{(0)}_{\lambda}\;=\;0.$
(15)

We obtain the following form of symplectic matrix $f_{ij}^{(0)}$ as

$\displaystyle f_{ij}^{(0)}\;=\;\begin{pmatrix}0&-1&0&0&0&0&0\\\
1&0&0&0&0&0&0\\\ 0&0&0&-1&0&0&0\\\ 0&0&1&0&0&0&0\\\ 0&0&0&0&0&-1&0\\\
0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0\end{pmatrix},$ (16)

which is a singular matrix. This indicates the system under study is a
constrained system. The zero-mode of $f_{ij}^{(0)}$ is
$(\nu^{(0)})^{T}\;=\;(0\;0\;0\;0\;0\;0\;\nu_{\lambda})$, where $\nu_{\lambda}$
is an arbitrary constant. To obtain the constraints, we contract this zero-
mode with Euler-Lagrange equations (cf. (11)) as illustrated below

$\displaystyle\Omega^{(0)}\;=\;(\nu^{(0)})^{T}\frac{\partial
V^{(0)}(\zeta)}{\partial\zeta^{(0)}}\;=\;0\quad\Longrightarrow\quad\Omega^{(0)}\;=\;\nu_{\lambda}(p\theta+q\phi)\;=\;0.$
(17)

Now we incorporate the constraint $\Omega^{(0)}$ into the canonical sector of
first-order Lagrangian (13) with the help of Lagrange multiplier. The form of
first-iterated Lagrangian is given as

$L_{f}^{(1)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\beta}\left(p\theta+q\phi\right)-V^{(1)},$
(18)

where $\beta$ denotes a Lagrange multiplier. Here the form of symplectic
potential $V^{(1)}$ is given by

$V^{(1)}\;=\;V^{(0)}|_{\Omega^{(0)}=0}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right].$
(19)

The set of first-iterated symplectic variables becomes
$\zeta^{(1)}\;=\;\left\\{\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\beta\right\\}$
and the corresponding canonical one-forms are given as

$a^{(1)}_{\eta}\;=\;P_{\eta},\quad a^{(1)}_{\theta}=P_{\theta},\quad
a^{(1)}_{\phi}=P_{\phi},\quad a^{(1)}_{\beta}=p\theta+q\phi,\quad
a^{(1)}_{P_{\eta}}\;=\;a^{(1)}_{P_{\theta}}\;=\;a^{(1)}_{P_{\phi}}\;=\;0.$
(20)

With this the first-iterated symplectic matrix $f_{ij}^{(1)}$ turns out to be

$\displaystyle f_{ij}^{(1)}\;=\;\begin{pmatrix}0&-1&0&0&0&0&0\\\
1&0&0&0&0&0&0\\\ 0&0&0&-1&0&0&p\\\ 0&0&1&0&0&0&0\\\ 0&0&0&0&0&-1&q\\\
0&0&0&0&1&0&0\\\ 0&0&-p&0&-q&0&0\end{pmatrix}.$ (21)

We observe the symplectic matrix $f_{ij}^{(1)}$ is singular and possesses
zero-mode of order $1\times 7$. This zero-mode is given as
$(\nu^{(1)})^{T}\;=\;(0\;0\;0\;p\;0\;q\;1)$. Contracting this zero-mode with
the gradient of first-iterated symplectic potential, we deduce the next
constraint as

$\Omega^{(1)}=(\nu^{(1)})^{T}\frac{\partial
V^{(1)}(\zeta)}{\partial\zeta^{(1)}}\;=\;0\quad\Longrightarrow\quad\Omega^{(1)}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)=0.$
(22)

We now construct the second-iterated Lagrangian by incorporating constraint
$\Omega^{(1)}$ into first-iterated Lagrangian (18) with the help of Lagrange
multiplier, say $\rho$, as

$L_{f}^{(2)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\beta}\left(p\theta+q\phi\right)+\dot{\rho}\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)-V^{(2)},$
(23)

where the symplectic potential $V^{(2)}$ is given by

$V^{(2)}\;=\;V^{(1)}|_{\Omega^{(1)}=0}=\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right].$
(24)

To construct the second-iterated symplectic matrix, we choose the set of
second-iterated symplectic variables as
$\zeta^{(2)}\;=\;\left\\{\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\beta,\rho\right\\}$
and identifying corresponding canonical one-forms as

$\displaystyle a^{(2)}_{\eta}\;=\;P_{\eta},\quad
a^{(2)}_{\theta}\;=\;P_{\theta},\quad a^{(2)}_{\phi}\;=\;P_{\phi},\quad
a^{(2)}_{\beta}\;=\;p\theta+q\phi,$ (25) $\displaystyle\displaystyle
a^{(2)}_{\rho}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right),\quad
a^{(2)}_{P_{\eta}}\;=\;a^{(2)}_{P_{\theta}}\;=\;a^{(2)}_{P_{\phi}}\;=\;0.$

Thus, the second-iterated symplectic matrix $f_{ij}^{(2)}$ looks like

$\displaystyle f_{ij}^{(2)}\;=\;\begin{pmatrix}0&-1&0&0&0&0&0&f_{\eta\rho}\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&-1&0&0&p&f_{\theta\rho}\\\
0&0&1&0&0&0&0&f_{P_{\theta}\rho}\\\ 0&0&0&0&0&-1&q&0\\\
0&0&0&0&1&0&0&f_{P_{\phi}\rho}\\\ 0&0&-p&0&-q&0&0&0\\\
-f_{\eta\rho}&0&-f_{\theta\rho}&-f_{P_{\theta}\rho}&0&-f_{P_{\phi}\rho}&0&0\end{pmatrix},$
(26)

where the explicit form of entries $f_{\eta\rho}$, $f_{\theta\rho}$,
$f_{P_{\theta}\rho}$ and $f_{P_{\phi}\rho}$ are given as

$\displaystyle\displaystyle
f_{\eta\rho}=\frac{2(\cosh\eta-\cos\theta)}{ma^{2}}\left[p\sinh\eta
P_{\theta}+\frac{qP_{\phi}}{\sinh\eta}-\frac{(\cosh\eta-\cos\theta)qP_{\phi}\cosh\eta}{\sinh^{3}\eta}\right],$
(27) $\displaystyle\displaystyle
f_{\theta\rho}=\frac{2(\cosh\eta-\cos\theta)\sin\theta}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right),$
$\displaystyle\displaystyle
f_{P_{\theta}\rho}=\frac{(\cosh\eta-\cos\theta)^{2}p}{ma^{2}},\quad\displaystyle
f_{P_{\phi}\rho}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\frac{q}{\sinh^{2}\eta}.$

It is worth mentioning that the second-iterated symplectic matrix
$f_{ij}^{(2)}$ is found to be non-singular. Thus, it does not have any zero-
mode and we can conclude that we have obtained all the constraints in the
system. This, in turn, indicates that we can stop the iterative process and
procure all the basic brackets in the theory.

At this juncture we would like to point out a key feature of constrained
systems, when explored within the framework of Faddeev-Jackiw formalism, if
the symplectic matrix is singular even after incorporating all the constraints
into the canonical sector then it hints the presence of some gauge symmetry.
Accordingly, a suitable gauge condition has to be chosen in order to get a
non-singular symplectic matrix [18, 25].

###  3.2 Symplectic quantization: Basic brackets

As we have already mentioned that the presence of non-singular symplectic
matrix indicates that our system is gauge non-invariant. Thus, we can compute
the inverse and in turn procure all basic brackets in the theory from its
components. The inverse of the symplectic matrix $f_{ij}^{(2)}$ is calculated
as

$\displaystyle(f_{ij}^{(2)})^{-1}=\begin{pmatrix}0&1&0&0&0&0&0&0\\\
-1&0&0&\frac{pf_{\eta\rho}}{c}&0&\frac{qf_{\eta\rho}}{c}&\frac{f_{\eta\rho}}{c}&0\\\
0&0&0&\frac{qf_{P_{\phi}\rho}}{c}&0&-\frac{qf_{P_{\theta}\rho}}{c}&-\frac{f_{P_{\theta}\rho}}{c}&0\\\
0&-\frac{pf_{\eta\rho}}{c}&-\frac{qf_{P_{\phi}\rho}}{c}&0&\frac{pf_{P_{\phi}\rho}}{c}&\frac{qf_{\theta\rho}}{c}&\frac{f_{\theta\rho}}{c}&-\frac{p}{c}\\\
0&0&0&-\frac{pf_{P_{\phi}\rho}}{c}&0&\frac{pf_{P_{\theta}\rho}}{c}&-\frac{f_{P_{\phi}\rho}}{c}&0\\\
0&-\frac{qf_{\eta\rho}}{c}&\frac{qf_{P_{\theta}\rho}}{c}&-\frac{qf_{\theta\rho}}{c}&-\frac{pf_{P_{\theta}\rho}}{c}&0&0&-\frac{q}{c}\\\
0&-\frac{f_{\eta\rho}}{c}&\frac{f_{P_{\theta}\rho}}{c}&-\frac{f_{\theta\rho}}{c}&\frac{f_{P_{\phi}\rho}}{c}&0&0&-\frac{1}{c}\\\
0&0&0&\frac{p}{c}&0&\frac{q}{c}&\frac{1}{c}&0\end{pmatrix},$ (28)

where
$c=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)$.
So, the basic brackets can be obtained with the help of following relationship

$\\{\zeta_{i}^{(2)},\zeta_{j}^{(2)}\\}=(f_{ij}^{(2)})^{-1}.$ (29)

The generalized basic brackets in the theory are given as below

$\displaystyle\\{\eta,P_{\eta}\\}=1,\quad\\{\theta,P_{\theta}\\}=\displaystyle\frac{q^{2}}{(p^{2}\sinh^{2}\eta+q^{2})},\quad\\{\theta,P_{\phi}\\}=\displaystyle-\frac{qp\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},$
(30)
$\displaystyle\\{P_{\eta},P_{\theta}\\}=\displaystyle\frac{2p}{(p^{2}\sinh^{2}\eta+q^{2})}\left[\frac{\sinh\eta}{(\cosh\eta-\cos\theta)}\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)-q\coth\eta P_{\phi}\right],$
$\displaystyle\\{P_{\eta},P_{\phi}\\}=\displaystyle\frac{2q}{(p^{2}\sinh^{2}\eta+q^{2})}\left[\frac{\sinh\eta}{(\cosh\eta-\cos\theta)}\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)-q\coth\eta P_{\phi}\right],$
$\displaystyle\\{P_{\eta},\beta\\}=\displaystyle\frac{2}{(p^{2}\sinh^{2}\eta+q^{2})}\left[\frac{\sinh\eta}{(\cosh\eta-\cos\theta)}\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)-q\coth\eta P_{\phi}\right],$
$\displaystyle\\{P_{\theta},P_{\phi}\\}=\displaystyle\frac{2q\sin\theta}{(p^{2}\sinh^{2}\eta+q^{2})}\frac{\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)}{(\cosh\eta-\cos\theta)},\quad\\{P_{\theta},\phi\\}=\displaystyle\frac{qp}{(p^{2}\sinh^{2}\eta+q^{2})},$
$\displaystyle\\{P_{\theta},\beta\\}=\displaystyle\frac{2\sin\theta}{(p^{2}\sinh^{2}\eta+q^{2})}\frac{\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)}{(\cosh\eta-\cos\theta)},\quad\\{\phi,P_{\phi}\\}=\displaystyle\frac{p^{2}\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},$
$\displaystyle\\{P_{\theta},\rho\\}=\displaystyle-\frac{ma^{2}}{(\cosh\eta-\cos\theta)^{2}}\frac{p\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},\quad\\{\theta,\beta\\}=-\displaystyle\frac{p\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},$
$\displaystyle\\{P_{\phi},\rho\\}=-\displaystyle\frac{ma^{2}}{(\cosh\eta-\cos\theta)^{2}}\frac{q\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},\quad\\{\phi,\beta\\}=-\displaystyle\frac{q}{(p^{2}\sinh^{2}\eta+q^{2})},$
$\displaystyle\\{\beta,\rho\\}=-\displaystyle\frac{ma^{2}}{(\cosh\eta-\cos\theta)^{2}}\frac{\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})}.$

The remaining brackets turn out to be zero. The basic brackets among the phase
space variables obtained via Faddeev-Jackiw formalism are found to be
coinciding with the corresponding Dirac brackets (cf. Appendix A for explicit
calculation of Dirac brackets). Thus, through the medium of basic brackets
obtained for the system of a particle on a torus knot, we can quantize the
system with the aid of following relation (cf. [18])

$\\{\zeta_{i}^{(2)},\zeta_{j}^{(2)}\\}=-\frac{i}{\hbar}\left[\hat{\zeta}_{i}^{(2)},\hat{\zeta}_{j}^{(2)}\right],$
(31)

where $\hat{\zeta}_{i}^{(2)}$ and $\hat{\zeta}_{j}^{(2)}$ represent the
operator notation of any general variables in the set of second-iterated
symplectic variables.

##  4 Symplectic gauge invariant formalism

The investigation of a particle on a torus knot via Faddeev-Jackiw formalism
affirmed that it is a gauge non-invariant theory. We now wish to reformulate
the same as a gauge invariant theory through the implementation of symplectic
gauge invariant formalism [33, 43, 44]. In order to accomplish this goal the
key idea is to enlarge the original phase space by introducing Wess-Zumino
variable. To be specific, we introduce a new arbitrary function $G$, which
depends upon original phase space variables and Wess-Zumino variable, say
$\alpha$, as given below

$G(\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\alpha)\;=\;\displaystyle\sum_{n=0}^{\infty}\mathcal{G}^{(n)}(\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\alpha),$
(32)

where $\mathcal{G}^{(n)}$ is a $n^{th}$ order term in Wess-Zumino variable
$\alpha$ and the function $G$ satisfies following boundary condition

$G(\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\alpha=0)\;=\;\mathcal{G}^{(0)}\;=\;0.$
(33)

Thus, working within symplectic gauge invariant formalism and introducing the
new function $G$ into (18), the resulting Lagrangian has following form

$\tilde{L}_{f}^{(1)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\beta}(p\theta+q\phi)-\tilde{V}^{(1)},$
(34)

where the symplectic potential $\tilde{V}^{(1)}$ is given by

$\tilde{V}^{(1)}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right]-G.$
(35)

The extended set of symplectic variables can be identified as
$\tilde{\zeta}^{(1)}=\left\\{\eta,P_{\eta},\theta,P_{\theta},\phi,P_{\phi},\beta,\alpha\right\\}$
and the corresponding canonical one-forms are given as

$\displaystyle\tilde{a}^{(1)}_{\eta}\;=\;P_{\eta},\quad\tilde{a}^{(1)}_{\theta}\;=\;P_{\theta},\quad\tilde{a}^{(1)}_{\phi}\;=\;P_{\phi},$
(36)
$\displaystyle\tilde{a}^{(1)}_{\beta}\;=\;p\theta+q\phi,\quad\tilde{a}^{(1)}_{P_{\eta}}\;=\;\tilde{a}^{(1)}_{P_{\theta}}\;=\;\tilde{a}^{(1)}_{P_{\phi}}\;=\;\tilde{a}^{(1)}_{\alpha}\;=\;0.$

With this input the symplectic matrix $\tilde{f}_{ij}^{(1)}$ takes the
following form

$\displaystyle\tilde{f}_{ij}^{(1)}\;=\;\begin{pmatrix}0&-1&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&-1&0&0&p&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&-1&q&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&-p&0&-q&0&0&0\\\ 0&0&0&0&0&0&0&0\end{pmatrix}.$ (37)

This symplectic matrix $\tilde{f}_{ij}^{(1)}$ is singular and has a zero-mode
of following form

$(\tilde{\nu}^{(1)})^{T}\;=\;\begin{pmatrix}0&0&0&p&0&q&1&1\end{pmatrix}.$
(38)

To reformulate the system as a gauge theory by symplectic gauge invariant
formalism, after incorporating a new arbitrary function $G$ into the first-
iterated Lagrangian, we impose the condition that no further constraints are
generated by the zero-mode of the corresponding singular symplectic matrix. As
a consequence, the contraction of zero-mode $(\tilde{\nu}^{(1)})^{T}$ with the
gradient of new symplectic potential ($\tilde{V}^{(1)}=V^{(1)}-G$), must
vanish identically as given below

$(\tilde{\nu}^{(1)})^{T}\frac{\partial\tilde{V}^{(1)}}{\partial\tilde{\zeta}^{(1)}}\;=\;(\tilde{\nu}^{(1)})^{T}\Big{(}\frac{\partial
V^{(1)}}{\partial\tilde{\zeta}^{(1)}}-\displaystyle\sum_{n=0}^{\infty}\frac{\partial\mathcal{G}^{(n)}}{\partial\tilde{\zeta}^{(1)}}\Big{)}\;=\;0.$
(39)

Thus, from the above expression, we deduce the explicit forms of various order
correction terms of arbitrary function $G$. For this, we take the aid of
condition that the above expression (cf. (39)) must vanish identically null
for each power of Wess-Zumino variable. In order to determine the first-order
correction in $\alpha$ (i.e. $\mathcal{G}^{(1)}$), considering the terms
independent of Wess-Zumino variable and then, by integrating the resultant
equation, we compute $\mathcal{G}^{(1)}$ as

$\mathcal{G}^{(1)}\;=\;\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\alpha.$
(40)

Now this correction term $\mathcal{G}^{(1)}$ is included into the Lagrangian
(34) in the following fashion

$\tilde{L}_{f}^{\prime(1)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\beta}(p\theta+q\phi)-\tilde{V}^{\prime(1)},$
(41)

where the form of symplectic potential $\tilde{V}^{\prime(1)}$ is given by

$\displaystyle\tilde{V}^{\prime(1)}=\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}-2\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\alpha\right].$
(42)

When the zero-mode $(\tilde{\nu}^{(1)})^{T}$ is being contracted with the
gradient of new symplectic potential $\tilde{V}^{\prime(1)}$, it leads to a
non-zero term as follows

$(\tilde{\nu}^{(1)})^{T}\frac{\partial\tilde{V}^{\prime(1)}}{\partial\tilde{\zeta}^{(1)}}\;=\;-\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left[p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right]\alpha,$
(43)

which contradicts the condition that zero-mode should not lead to any new
constraints. Thus, we have to calculate the second-order correction in
$\alpha$ (i.e. $\mathcal{G}^{(2)}$) which can be carried out by looking for
the linear order terms in Wess-Zumino variable in (39). The expression for
$\mathcal{G}^{(2)}$ can be obtained by integrating the aftermath expression
and given as

$\mathcal{G}^{(2)}\;=\;-\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right]\alpha^{2}.$
(44)

The second-order correction $\mathcal{G}^{(2)}$ is incorporated into the
Lagrangian (41) in the following manner

$\tilde{L}_{f}^{\prime\prime(1)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\beta}(p\theta+q\phi)-\tilde{V}^{\prime\prime(1)},$
(45)

where the form of symplectic potential $\tilde{V^{\prime\prime}}^{(1)}$ is
given by

$\displaystyle\tilde{V}^{\prime\prime(1)}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\Big{[}P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}-2\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\alpha$
$\displaystyle+$
$\displaystyle\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\alpha^{2}\Big{]}.$

Now it is straightforward to check that the zero-mode
$(\tilde{\nu}^{(1)})^{T}$ contracting on the gradient of potential
$\tilde{V}^{\prime\prime(1)}$ generates identically null. In other words, the
zero-mode $(\tilde{\nu}^{(1)})^{T}$ does not produce any new constraints and,
thus, we can stop the iterative process of determining higher-order correction
terms in $G$ with $n\geq 3$. Consequently, this zero-mode acts as the
generator of infinitesimal gauge transformations.

Therefore, the zeroth-iterated symplectic potential $(\tilde{V}^{(0)})$, which
can also be identified as the canonical Hamiltonian of the system, is given as

$\displaystyle\tilde{V}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\Big{(}P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}-2\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\alpha$
$\displaystyle+$
$\displaystyle\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\alpha^{2}\Big{)}-\lambda(p\theta+q\phi)\quad\equiv\quad\tilde{H}.$

Correspondingly, the first-order Lagrangian can be expressed as

$\tilde{L}_{f}^{(0)}\;=\;\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}-\tilde{V}^{(0)}.$
(48)

As we have already mentioned, the infinitesimal gauge transformations can be
generated from the zero-mode (cf. (38)) in the following fashion

$\delta\tilde{\zeta}^{(1)}=(\tilde{\nu}^{(1)})^{T}\epsilon,$ (49)

where $\epsilon$ is the infinitesimal parameter of gauge transformations and
$\tilde{\zeta}^{(1)}$ represents the extended set of first-iterated symplectic
variables. The full set of gauge transformations ($\delta$) are listed below

$\delta P_{\theta}=p\epsilon,\quad\delta
P_{\phi}=q\epsilon,\quad\delta\lambda=\dot{\epsilon},\quad\delta\alpha=\epsilon,\quad\delta\eta=\delta
P_{\eta}=\delta\theta=\delta\phi=0.$ (50)

Under the gauge transformations (50), the zeroth-iterated Lagrangian (48)
transforms to a total time derivative as follows

$\delta\tilde{L}_{f}^{(0)}\;=\;\frac{d}{dt}\big{[}\epsilon(p\theta+q\phi)\big{]}.$
(51)

Thus, we have obtained the gauge invariant Lagrangian (48) and corresponding
Hamiltonian (4) within the framework of symplectic gauge invariant formalism.

###  4.1 Constraint structure and gauge symmetries

We now wish to deduce the constraint structure by employing Dirac’s formalism.
For this, we identify the canonical conjugate momenta corresponding to
$\lambda$ and Wess-Zumino variable $\alpha$ as primary constraints. These
primary constraints $(\psi_{1},\bar{\psi}_{1})$ generate two constraint
chains. The first chain is as given below

$\psi_{1}\;=\;P_{\lambda}\approx 0,\quad\psi_{2}\;=\;(p\theta+q\phi)\approx
0,$ (52)

and the second constraint chain is listed as

$\displaystyle\bar{\psi}_{1}=P_{\alpha}\approx
0,\quad\bar{\psi}_{2}=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}-\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\alpha\right)\approx
0.$ (53)

Here, $P_{\alpha}$ is canonical conjugate momentum to Wess-Zumino variable
$\alpha$. At this juncture, we would like to mention that in the first
constraint chain the consistency condition of secondary constraint $\psi_{2}$
leads tertiary constraint
$\psi_{3}=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}-\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\alpha\right)\approx
0,$ which is equal to $\bar{\psi_{2}}$ in the second constraint chain. Since
$\psi_{3}$ is not unique, so it is possible to put this constraint in one of
the chains and simultaneously close the other one [30]. Thus, we have put this
constraint in the second chain and closed the first one.

It is worthwhile to point out that $\psi_{1}$ has vanishing Poisson brackets
with rest of the constraints. Furthermore, there exist non-vanishing Poisson
brackets among other constraints such as
$\left\\{\psi_{2},\bar{\psi}_{2}\right\\}=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)=\left\\{\bar{\psi}_{1},\bar{\psi}_{2}\right\\}$,
which indicate that there is a mixture of both first-class and second-class
constraints. So, we split the constraint chains into a set of first-class and
second-class constraints through constraint combination [15, 33]. As a result,
the set of first-class constraints is

$\chi_{1}=P_{\lambda},\quad\chi_{2}=p\theta+q\phi-P_{\alpha},$ (54)

whereas the set of second-class constraints turns out to be

$\displaystyle\bar{\psi}_{1}=P_{\alpha},\quad\bar{\psi}_{2}=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}-\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\alpha\right).$
(55)

Now, the second-class constraints are assumed in a strong way, with this, the
Hamiltonian in (4) is reduced to the following form [12]

$\displaystyle\tilde{H^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left(P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\right)-\lambda(p\theta+q\phi).$
(56)

At this juncture, we would like to point out that in the symplectic gauge
invariant formalism even though we begin with the inclusion of Wess-Zumino
variable by enlarging the phase space, we obtained the form of the Hamiltonian
without the presence of any additional variables. Thus, we have reformulated
the original gauge non-invariant system as a gauge theory which is free from
Wess-Zumino variable.

We now wish to disclose the gauge symmetries of the newly reformulated gauge
invariant theory of a particle on a torus knot. For this purpose, we identify
the first-order Lagrangian corresponding to Hamiltonian (56) as

$\displaystyle\tilde{L}^{\prime}_{f}=\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}-\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left(P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\right)+\lambda(p\theta+q\phi).$
(57)

From the above Lagrangian, we derive set of constraints via Dirac formalism as

$\tilde{\chi}_{1}=P_{\lambda}\approx
0,\quad\tilde{\chi}_{2}=(p\theta+q\phi)\approx 0,$ (58)

where the constraints $\tilde{\chi}_{1}$ and $\tilde{\chi}_{2}$ are first-
class in nature. Consequently, the set of first-class constraints acts as the
generator of gauge symmetry and we derive the explicit form of infinitesimal
gauge symmetry transformations as

$\delta P_{\theta}=p\kappa,\quad\delta
P_{\phi}=q\kappa,\quad\delta\lambda=\dot{\kappa},\quad\delta[\eta,P_{\eta},\theta,\phi]=0,$
(59)

where $\kappa$ represents the infinitesimal gauge parameter. Under the gauge
transformations in (59), the first-order Lagrangian in (57) transforms

$\delta\tilde{L}^{\prime}_{f}\;=\;\frac{d}{dt}\Big{[}\kappa(p\theta+q\phi)\Big{]},$
(60)

into a total time derivative. Thus, by employing symplectic gauge invariant
formalism, we have reformulated the system of a particle constrained to move
on a torus knot as a gauge theory.

Before ending this section, a few remarkable features of the newly constructed
gauge theory are in order; first, the expression of first-class constraints
preserve the same form as of the original gauge non-invariant theory. Second,
the gauge invariant theory is found to be independent of any additional
variables even though the symplectic gauge invariant formalism employs a
methodology of enlarging the phase space with Wess-Zumino variable. Finally,
it is worthwhile to mention that the reformulated gauge theory is physically
equivalent to the original gauge non-invariant one as far as the dynamics and
physical degrees of freedom are concerned. To be precise, the equations of
motion determined from the newly reformulated gauge theory coincide with
Euler-Lagrange equations obtained from initial gauge non-invariant system (see
below, Appendix B for details). Moreover, in the calculation of physical
degrees of freedom, taking the case of original gauge non-invariant system,
one first-class and two second-class constraints produce one degree of freedom
[12]. Likewise, the system of gauge invariant theory with two first-class
constraints also provide one degree of freedom. Thus, symplectic gauge
invariant formalism produces same dynamics and preserve the physical degrees
of freedom for the reformulated gauge theory.

##  5 Off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries

We now investigate the reformulated gauge theory of a particle on a torus knot
by means of BRST formalism which benefits in achieving the gauge fixing
without breaking underlying symmetry. For this purpose, we replace the time
dependent gauge parameter by anti-commuting (anti-)ghost variables
$(\bar{c})c$, satisfying $c^{2}=0=\bar{c}^{2},\;c\bar{c}+\bar{c}c=0$. Thus,
the BRST symmetry transformations $(s_{b})$ can be written as

$\displaystyle s_{b}\lambda\;=\;\dot{c},\quad s_{b}P_{\theta}\;=\;cp,\quad
s_{b}P_{\phi}\;=\;cq,\quad s_{b}\bar{c}\;=\;ib,$ (61) $\displaystyle
s_{b}c\;=\;0,\quad
s_{b}b\;=\;s_{b}\eta\;=\;s_{b}P_{\eta}\;=\;s_{b}\theta\;=\;s_{b}\phi\;=\;0,$

where $b$ is Nakanishi-Lautrup auxiliary variable. The BRST invariant first-
order Lagrangian $(L_{b})$ can be constructed by adding a BRST invariant
function to (57) as

$\displaystyle
L_{b}=\tilde{L}^{\prime}_{f}+s_{b}\left[i\bar{c}\left(\dot{\lambda}+P_{\theta}+P_{\phi}-\frac{1}{2}b\right)\right].$
(62)

It can be rewritten in its full blaze of glory as

$\displaystyle L_{b}$ $\displaystyle=$
$\displaystyle\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}-\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\Big{(}P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\Big{)}$
$\displaystyle+$
$\displaystyle\lambda(p\theta+q\phi)-b(\dot{\lambda}+P_{\theta}+P_{\phi})+\frac{1}{2}b^{2}+i\dot{\bar{c}}\dot{c}-i(p+q)\bar{c}c.$

Here the Nakanishi-Lautrup auxiliary variable is used to linearize the gauge
fixing term
$\displaystyle-\frac{1}{2}\left[\dot{\lambda}+P_{\theta}+P_{\phi}\right]^{2}$.
Further, the BRST invariant first-order Lagrangian (5) is also found to be
invariant under another set of symmetries, known as anti-BRST symmetries
($s_{ab}$), of the following form

$\displaystyle s_{ab}\lambda\;=\;\dot{\bar{c}},\quad
s_{ab}P_{\theta}\;=\;{\bar{c}}p,\quad s_{ab}P_{\phi}\;=\;{\bar{c}}q,\quad
s_{ab}c\;=\;-ib,$ (64) $\displaystyle s_{ab}\bar{c}\;=\;0,\quad
s_{ab}b\;=\;s_{ab}\eta\;=\;s_{ab}P_{\eta}\;=\;s_{ab}\theta\;=\;s_{ab}\phi\;=\;0.$

These (anti-)BRST symmetry transformations are off-shell nilpotent (i.e.
$s_{b}^{2}=0=s_{ab}^{2}$) and absolutely anti-commuting (i.e.
$\left\\{s_{b},s_{ab}\right\\}=s_{b}s_{ab}+s_{ab}s_{b}=0$) in nature. The
Hamiltonian corresponding to $L_{b}$ can be expressed as

$H_{b}=\dot{\eta}P_{\eta}+\dot{\theta}P_{\theta}+\dot{\phi}P_{\phi}+\dot{\lambda}P_{\lambda}+\dot{c}\Pi_{c}+\dot{\bar{c}}\Pi_{\bar{c}}-L_{b},$
(65)

with $P_{\lambda}=-b$, $\Pi_{c}=-i\dot{\bar{c}}$ and $\Pi_{\bar{c}}=i\dot{c}$,
we obtain

$\displaystyle H_{b}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\Big{(}P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\Big{)}-i\Pi_{\bar{c}}\Pi_{c}$
$\displaystyle-$
$\displaystyle\lambda(p\theta+q\phi)-\frac{1}{2}P_{\lambda}^{2}-P_{\lambda}(P_{\theta}+P_{\phi})+i(p+q){\bar{c}}c.$

The conserved (anti-)BRST charges $Q_{(a)b}$ can be derived as

$Q_{b}=\dot{c}P_{\lambda}-c(p\theta+q\phi),\quad
Q_{ab}=\dot{\bar{c}}P_{\lambda}-\bar{c}(p\theta+q\phi).$ (67)

It is worth mentioning that the (anti-)BRST charges are nilpotent of order two
(i.e. $Q_{b}^{2}=0=Q_{ab}^{2}$) and absolutely anti-commuting
($\\{Q_{b},Q_{ab}\\}=Q_{b}Q_{ab}+Q_{ab}Q_{b}=0$) in nature. In addition, the
conservation of charges $Q_{(a)b}$ can be verified with the aid of Euler-
Lagrange equations of motion corresponding to the first-order Lagrangian (cf.
(5)) as given below

$\displaystyle\dot{\eta}=\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}P_{\eta},\qquad\dot{\theta}=b+\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\frac{q\left(qP_{\theta}-pP_{\phi}\right)}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)},$
(68)
$\displaystyle\dot{\phi}=b-\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\frac{p\left(qP_{\theta}-pP_{\phi}\right)}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)},\qquad\dot{b}=-(p\theta+q\phi),$
$\displaystyle
b=\dot{\lambda}+P_{\theta}+P_{\phi},\qquad\ddot{c}+(p+q)c=0,\qquad\ddot{\bar{c}}+(p+q)\bar{c}=0,$
$\displaystyle\dot{P_{\phi}}=\lambda
q,\qquad\dot{P_{\theta}}=\displaystyle\lambda
p-\frac{(\cosh\eta-\cos\theta)\sin\theta}{ma^{2}}\Big{[}P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\Big{]},$
$\displaystyle\dot{P}_{\eta}=-\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\sinh\eta\Big{[}P_{\eta}^{2}+\frac{\left(qP_{\theta}-pP_{\phi}\right)^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}$
$\displaystyle-\frac{(\cosh\eta-\cos\theta)\cosh\eta\left(qP_{\theta}-pP_{\phi}\right)^{2}p^{2}}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)^{2}}\Big{]}.$

Further, we can verify that the conserved nilpotent charges $Q_{(a)b}$ are the
generators of (anti-)BRST transformations (cf. (61), (64)) from the following
relation

$s_{(a)b}\psi=i\left[Q_{(a)b},\psi\right]_{\pm},$ (69)

by making use of (anti-)commutation relation
$\left[\eta,P_{\eta}\right]=\left[\theta,P_{\theta}\right]=\left[\phi,P_{\phi}\right]=\left[\lambda,P_{\lambda}\right]=i,\;\left\\{\dot{\bar{c}},c\right\\}=1,$
and $\left\\{\dot{c},\bar{c}\right\\}=-1$. In the above, $\psi$ is any generic
variable of the system and $\pm$ represent the (anti-)commutation relation.
The physicality criteria implies that these (anti-)BRST charges annihilate the
physical subspace $\left|\varPsi\right\rangle_{phys}$, which leads to

$Q_{(a)b}\left|\varPsi\right\rangle_{phys}=0\quad\Longrightarrow\quad(p\theta+q\phi)\left|\varPsi\right\rangle_{phys}=0,\quad
P_{\lambda}\left|\varPsi\right\rangle_{phys}=0.$ (70)

This shows the consistency with the Dirac quantization scheme as the physical
states in the theory are annihilated by the first-class constraints.

##  6 Conclusions

In the present endeavor, we have quantized the system of particle constrained
to move on a torus knot by the means of Faddeev-Jackiw formalism. We have also
derived the constraint structure and further found that the two-form
symplectic matrix remains non-singular even after incorporating all the
constraints into the canonical sector of the first-order Lagrangian. This
shows that the underlying theory is gauge non-invariant and from the inverse
of the non-singular symplectic matrix, we have obtained all the basic
brackets.

We have further employed symplectic gauge invariant formalism to reformulate
this gauge non-invariant theory into a gauge invariant one. To implement this
formalism, we have enlarged the original phase space by introducing a new
variable - Wess-Zumino variable. In the procedure, we have included an
arbitrary function depending upon the original phase space variables and Wess-
Zumino variable into the first-iterated Lagrangian. Then by imposing the
condition that the zero-mode of corresponding singular symplectic matrix does
not lead to further constraints, we have found the explicit form of the
arbitrary function. However, the analysis of constraint structure via Dirac
formalism for the newly constructed theory have revealed the presence of a
mixture of both the first-class and second-class constraints. Later, through
constraint combination, the first-class constraints have been separated out
and second-class constraints are assumed in a strong way. This, in turn,
produced a Hamiltonian (and corresponding Lagrangian) which is free from any
Wess-Zumino variables. Furthermore, we have demonstrated that the thus
obtained, first-order Lagrangian remains invariant under the transformations
generated by the first-class constraints.

Finally, we have deduced BRST invariant first-order Lagrangian and established
the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries
corresponding to the newly reformulated gauge theory. We have also derived the
(anti-)BRST charges which are shown to be nilpotent of order two and
absolutely anti-commuting in nature. Further, these conserved (anti-)BRST
charges have been depicted as the generators of the corresponding (anti-)BRST
symmetries. The physicality condition satisfied by these charges shows that
the first-class constraints annihilate the physical states of the system and
it is on par with the Dirac quantization procedure. For future endeavor, we
wish to explore the possibility of a particle on torus knot as a model for
Hodge theory and aim to establish the physical realizations of cohomological
operators within the BRST formalism [45].

Acknowledgments: The support from FRG scheme of National Institute of
Technology Calicut is thankfully acknowledged.

Appendix A: On equivalence with Dirac brackets

As we have already seen in Section 2, that for the system of particle on a
torus knot described by the Lagrangian (3), there exist one first-class and
two second-class constraints. Therefore, from the second-class constraints
$\varphi_{2}=(p\theta+q\phi)\approx 0$ and
$\varphi_{3}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\approx
0$, we can construct a $2\times 2$ anti-symmetric matrix
$C_{ij}=\left\\{\varphi_{i},\varphi_{j}\right\\}$, with $i,j=2,3$, as

$\displaystyle
C_{ij}\;=\;\begin{pmatrix}0&\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)\\\
\displaystyle-\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)&0\end{pmatrix}.$
(71)

This is a non-singular matrix and from the inverse of $C_{ij}$ we can compute
Dirac brackets. For any dynamical variables, say $A$ and $B$, of a theory the
Dirac brackets $(\left[A,B\right]_{D})$ can be given as

$\left[A,B\right]_{D}=\left[A,B\right]-\left[A,\varphi_{i}\right]C_{ij}^{-1}\left[\varphi_{j},B\right],$
(72)

where $\left[A,B\right]$ defines the Poisson bracket between the dynamical
variables and $C_{ij}^{-1}$ represents the inverse of the anti-symmetric
matrix $C_{ij}$, which, in this case, turns out to be

$\displaystyle
C_{ij}^{-1}\;=\;\begin{pmatrix}0&\displaystyle-\frac{ma^{2}}{(\cosh\eta-\cos\theta)^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)^{-1}\\\
\displaystyle\frac{ma^{2}}{(\cosh\eta-\cos\theta)^{2}}\left(p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\right)^{-1}&0\end{pmatrix}.$
(73)

Thus, Dirac brackets can be worked out and the non-zero ones are listed below

$\displaystyle\left[\eta,P_{\eta}\right]_{D}=1,\quad\left[\theta,P_{\theta}\right]_{D}=\displaystyle\frac{q^{2}}{(p^{2}\sinh^{2}\eta+q^{2})},\quad\left[\phi,P_{\phi}\right]_{D}=\displaystyle\frac{p^{2}\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},$
(74)
$\displaystyle\left[\theta,P_{\phi}\right]_{D}=\displaystyle-\frac{qp\sinh^{2}\eta}{(p^{2}\sinh^{2}\eta+q^{2})},\quad\left[P_{\theta},\phi\right]_{D}=\displaystyle\frac{qp}{(p^{2}\sinh^{2}\eta+q^{2})},$
$\displaystyle\left[P_{\eta},P_{\theta}\right]_{D}=\displaystyle\frac{2p}{(p^{2}\sinh^{2}\eta+q^{2})}\left[\frac{\sinh\eta}{(\cosh\eta-\cos\theta)}\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)-q\coth\eta P_{\phi}\right],$
$\displaystyle\left[P_{\eta},P_{\phi}\right]_{D}=\displaystyle\frac{2q}{(p^{2}\sinh^{2}\eta+q^{2})}\left[\frac{\sinh\eta}{(\cosh\eta-\cos\theta)}\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)-q\coth\eta P_{\phi}\right],$
$\displaystyle\left[P_{\theta},P_{\phi}\right]_{D}=\displaystyle\frac{2q\sin\theta}{(p^{2}\sinh^{2}\eta+q^{2})}\frac{\left(p\sinh^{2}\eta
P_{\theta}+qP_{\phi}\right)}{(\cosh\eta-\cos\theta)}.$

We infer that the above procured Dirac brackets, among the dynamical variables
of the theory, coincide with the basic brackets calculated by employing
Faddeev-Jackiw formalism (cf. (30)).

Appendix B: On Lagrange multipliers

In Faddeev-Jackiw formalism, Lagrange multipliers appear as auxiliary
coordinates with the purpose of incorporating constraints into the canonical
sector of the first-order Lagrangian. Whereas in Dirac formalism these
Lagrange multipliers remain absent. So we provide a new interpretation for
these Lagrange multipliers in terms of dynamical coordinates in the system
[22]. For this purpose we make use of symplectic equations of motion of
second-iterated Lagrangian (23)

$\displaystyle\dot{\zeta}^{(2)j}=(f_{ij}^{(2)})^{-1}\frac{\partial
V^{(2)}(\zeta)}{\partial\zeta^{(2)i}}.$ (75)

The explicit form of the above relationship is given as

$\displaystyle\begin{pmatrix}\dot{\eta}\\\ \dot{P_{\eta}}\\\ \dot{\theta}\\\
\dot{P_{\theta}}\\\ \dot{\phi}\\\ \dot{P_{\phi}}\\\ \dot{\beta}\\\
\dot{\rho}\end{pmatrix}=\begin{pmatrix}0&1&0&0&0&0&0&0\\\
-1&0&0&\frac{pf_{\eta\rho}}{c}&0&\frac{qf_{\eta\rho}}{c}&\frac{f_{\eta\rho}}{c}&0\\\
0&0&0&\frac{qf_{P_{\phi}\rho}}{c}&0&-\frac{qf_{P_{\theta}\rho}}{c}&-\frac{f_{P_{\theta}\rho}}{c}&0\\\
0&-\frac{pf_{\eta\rho}}{c}&-\frac{qf_{P_{\phi}\rho}}{c}&0&\frac{pf_{P_{\phi}\rho}}{c}&\frac{qf_{\theta\rho}}{c}&\frac{f_{\theta\rho}}{c}&-\frac{p}{c}\\\
0&0&0&-\frac{pf_{P_{\phi}\rho}}{c}&0&\frac{pf_{P_{\theta}\rho}}{c}&-\frac{f_{P_{\phi}\rho}}{c}&0\\\
0&-\frac{qf_{\eta\rho}}{c}&\frac{qf_{P_{\theta}\rho}}{c}&-\frac{qf_{\theta\rho}}{c}&-\frac{pf_{P_{\theta}\rho}}{c}&0&0&-\frac{q}{c}\\\
0&-\frac{f_{\eta\rho}}{c}&\frac{f_{P_{\theta}\rho}}{c}&-\frac{f_{\theta\rho}}{c}&\frac{f_{P_{\phi}\rho}}{c}&0&0&-\frac{1}{c}\\\
0&0&0&\frac{p}{c}&0&\frac{q}{c}&\frac{1}{c}&0\end{pmatrix}$ (76)
$\displaystyle\begin{pmatrix}\displaystyle\frac{(\cosh\eta-\cos\theta)\sinh\eta}{ma^{2}}\left(P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right)-\frac{(\cosh\eta-\cos\theta)^{2}\cosh\eta
P_{\phi}^{2}}{ma^{2}\sinh^{3}\eta}\\\
\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}P_{\eta}\\\
\displaystyle\frac{(\cosh\eta-\cos\theta)\sin\theta}{ma^{2}}\left(P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right)\\\
\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}P_{\theta}\\\ 0\\\
\displaystyle\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\frac{P_{\phi}}{\sinh^{2}\eta}\\\
0\\\ 0\end{pmatrix}.$

From the above relationship (76), it is straightforward to identify the
expression for Lagrange multipliers as

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\sinh^{2}\eta}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right),$
(77) $\displaystyle\dot{\beta}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\sinh^{2}\eta}{ma^{2}}\frac{\left(\cosh\eta-\cos\theta\right)}{\left(p^{2}\sinh^{2}\eta+q^{2}\right)}\Big{[}p\sin\theta\left(P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right)$
$\displaystyle+$
$\displaystyle\frac{2q\cosh\eta}{\sinh^{3}\eta}\left(\cosh\eta-\cos\theta\right)P_{\eta}P_{\phi}-2\left(pP_{\theta}+\frac{qP_{\phi}}{\sinh^{2}\eta}\right)\left[\sinh\eta
P_{\eta}+\sin\theta P_{\theta}\right]\Big{]}.$

Thus, from the above expressions it is evident that the Lagrange multipliers,
which appear as auxiliary variables in Faddeev-Jackiw formalism, can be
expressed in terms of generalized coordinates and momenta. These expressions
provide a new interpretation for the Lagrange multipliers. In our
investigation, we have procured all constraints via Faddeev-Jackiw formalism
and subsequently quantized the system. Finally, we wish to write down the
equations of motion and from the second-iterated symplectic matrix these
equations of motion take the following form

$f_{ij}^{(2)}\dot{\zeta}^{(2)j}=\frac{\partial
V^{(2)}(\zeta)}{\partial\zeta^{(2)i}},$ (78)

where the explicit expressions are given as

$\displaystyle\displaystyle\dot{P_{\phi}}-q\dot{\beta}=0,\qquad
p\dot{\theta}+q\dot{\phi}=0,\qquad\dot{\theta}+\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\Big{(}p\dot{\rho}-P_{\theta}\Big{)}=0,$
(79)
$\displaystyle\dot{\eta}-\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}P_{\eta}=0,\qquad\displaystyle\dot{\phi}+\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}\sinh^{2}\eta}\Big{(}q\dot{\rho}-P_{\phi}\Big{)}=0,$
$\displaystyle\displaystyle\dot{P}_{\theta}-p\dot{\beta}-f_{\theta\rho}\dot{\rho}=-\frac{(\cosh\eta-\cos\theta)\sin\theta}{ma^{2}}\left(P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right),$
$\displaystyle\displaystyle\dot{P}_{\eta}-f_{\eta\rho}\dot{\rho}=\frac{(\cosh\eta-\cos\theta)^{2}\cosh\eta}{ma^{2}\sinh^{3}\eta}P_{\phi}^{2}-\frac{(\cosh\eta-\cos\theta)\sinh\eta}{ma^{2}}\left(P_{\eta}^{2}+P_{\theta}^{2}+\frac{P_{\phi}^{2}}{\sinh^{2}\eta}\right),$
$\displaystyle\dot{\eta}f_{\eta\rho}+\dot{\theta}f_{\theta\rho}+\dot{P}_{\theta}f_{P_{\theta}\rho}+\dot{P}_{\phi}f_{P_{\phi}\rho}=0.$

The precise forms of $f_{\theta\rho}$, $f_{\eta\rho}$, $f_{P_{\theta}\rho}$
and $f_{P_{\phi}\rho}$ are given in (27). Thus, we have obtained the Euler-
Lagrange equations corresponding to the second-iterated Lagrangian.

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