# Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

Vipul Kumar Pandey111e-mail address<EMAIL_ADDRESS>Department of
Physics, Chandigarh University, Mohali \- 140413, INDIA.

###### Abstract

We have previously developed the BRST quantization on the hypersurface
$V_{N-1}$ embedded in N dimensional Euclidean space $R_{N}$ in both
Hamiltonian and Lagrangian formulation. We generalize the formalism in the
case of L dimensional manifold $V_{L}$ embedded in $R_{N}$ with $1\leq L<N$.
The result is essentially the same as the previous one. We have also verified
the results obtained here using a simple example of particle motion on a torus
knot.

##  I Introduction

In the previous work VKP  we have considered the BRST quantization of the
motion on a hypersurface $V_{N}$ embedded in the N dimensional Euclidean space
$R^{N}$ based on Batalin-Fradkin-Fradkina-Tyutin (BFFT) Abelianized Batalin-
Fradkin-Vilkovisky (BFV) and Batalin-Vilkovisky (BV) formalisms. It is a well
known result in differential geometry that any $N$ dimensional Riemann
manifold can be locally embedded in the $\frac{N(N+1)}{2}$ dimensional
Eucliden space but can not be embedded in $(N+1)$ dimension generally YAS . In
this sense, the considerations in the previous manuscript VKP  should be
extended to general Riemannian manifolds. This is the motivation of the
present manuscript VKP2 .

It is well known in the literature that the quantization of the system in
curved space has been extensively studied about the ordering problem using two
different approaches, canonical and path integral HEWCRS ; RS ; TK ; TKRS ;
TORS1 ; TORS2 ; TKTORS ; MOHS ; FSTK ; FKST ; TKW ; HKTK ; JLGAN ; MO ; BSKK .
At the same time, the quantization of dynamical systems constrained to curved
manifolds embedded in the higher-dimensional Euclidean space has been
extensively investigated as one of the quantum theories DWOS ; AFFR ; NDBPD ;
LPDT ; IBMGSL1 ; IBMGSL2 ; IGSLAS . Here, a non-relativistic particle
constrained to move on a curved surface embedded in the higher dimensional
Euclidean space NOKFAK ; NOKFCK  has been taken. These systems and their
various properties they possess, have been investigate by many authors NOG ;
INTT ; STTT ; CFFM ; NO ; FOUC ; CFK ; AS ; LHL ; NOG1 ; NOG2 ; NOMN ; DPO ;
FGK ; AVG ; KFNO ; NOM1 ; NOM2 ; MN ; PM1 ; PM2 ; NCAR1 ; NCAR2 ; NCAR3 ;
DOBSRT . This has motivated us to extend our previous workVKP  to more general
class of systems discussed in NOKFCK .

Becchi-Rout-Stora-Tyutin (BRST) quantization CRS1 ; CRS2 ; CRS3 ; IVT  is on
of the most significant technique to deal with a system with constraints. It
has also been found to be symmetry of general class of constrained systems
PAMD1 ; JLAPGB ; PAMD2 ; KS ; MHCT ; MH . In this quantization method, we
enlarge the total Hilbert space of the gauge system under study and bring back
the gauge symmetry of the gauge fixed action in the extended phase space,
keeping the physical contents of the theory unchanged. BRST symmetry plays
very significant role in renormalization of spontaneously broken gauge
theories like, standard model and hence is of very high significance for
different kind of systems. To the best of our knowledge, there is no
literature available which studied the BRST symmetry for a particle moving on
a hypersurface $V_{L}(1\leq L<N)$ embedded in the Euclidean space $R_{N}$.
This motivates us in the study of BRST symmetry for this system. We will do
the constraints analysis of this sytem using the Dirac’s technique. The system
is shown to contain second-class constraints. The BFFT method will be used to
convert the second class constraints to first class one LFSS ; IABESF1 ;
IABESF2 ; BFFR ; IABIVT ; EERM ; RBBN ; SG ; CBAPMT ; IBVKAR ; ILBAAR ; AAR ;
IBAR ; VPRT ; MPT . Then the BRST charge and symmetries will be constructed
for this BFFT Abelianized system using BFV method ESFGV ; IABGV ; IABESF  with
the help of Faddeev-Senjanovic technique LDFAD ; PSENJ . In the limit
$L\rightarrow(N-1)$ the system will return to sytem in VKP . The results
developed in the manuscript have been verified using a model of particle on
torus knot VS ; PDSG ; VKPBPM ; VPBM ; ASSG . At the end BV-BRST quantization
of the BFFT Abelianized sytem will be investigated IABGAV1 ; IABGAV2 ; JGJPSS
; RART . Recently Lagrangian Abelianization procedure for constrained systems
has also been developed SLL ; VAASLL .

This is the second and final part of the work. In this part we will discuss
BRST quantization of embedding $V_{L}$ in Euclidean space $R_{N}$ where $1\leq
L<N$. The paper has been organized in the following way. In the second
section, we have reviewed motion in curved space and also calculated all the
possible constraints of the theory using Dirac’s constraints analysis method.
In the third section, we have reviewed BFFT formalism. In the section four, we
have constructed first class constraints and Hamiltonians for the general
class of systems under study. In the next section we will construct BRST
symmetry for this system based on BFV Formalism. In the section six, we have
shown consistency of our results in the limit $L\rightarrow(N-1)$. In the
section seven, we have given a simple example of this kind of systems. In the
section eight, we have discussed BV quantization of this system based on BFFT
formalism. In the next section concluding remarks have been made. In the end
we have discussed some important calculations in the appendix.

##  II Classical Mechanics on $V_{L}$ in $R^{N}$

Let us consider an N dimensional Euclidean space ${R_{N}}$, a point in which
is specified by a set of Cartesian coordinates

$\displaystyle X^{A}:\\{x^{1},x^{2},...,x^{N}\\}.$ (1)

Further consider in $R_{N}$ an $L$ dimensional Riemann subspace, $V_{L}(1\leq
L<N)$, a point of which is specified by a set of coordinates $q^{k}$ NOKFCK ,

$\displaystyle q^{k}:\\{q^{1},q^{2},...,q^{L}\\}$ (2)

The metric of this system is defined as $g_{ij}(q^{k})$. We can construct in
$R_{N}$ a set of curvilinear coordinates including $(q^{1},q^{2},...,q^{L})$,

$\displaystyle q^{\mu}:\\{q^{1},q^{2},...,q^{L},q^{L+1},...,q^{N}\\}$ (3)

Let us assume that $\\{q^{L+a}\\}(a:1\sim{N-L})$ are the intrinsic coordinates
normal to $V_{L}$ NOKFCK . We can also use the notation

$\displaystyle Q^{a}\equiv q^{L+a},\quad a:1\sim{N-L}$ (4)

Then the subspace $V_{L}$ can be defined as

$\displaystyle Q^{a}=q^{L+a}=0$ (5)

The metric for the curvilinear coordinates $q^{\mu}$ in $R_{N}$ is defined as

${\tilde{g}}_{\mu\nu}=\begin{pmatrix}{\tilde{g}}_{ij}&N_{ib}\\\
N_{ja}&{\tilde{G}}_{ab}\\\ \end{pmatrix}$ (6)

where $N_{ia}$ and ${\tilde{G}}_{ab}$ are defined as

$\displaystyle
N_{ia}\equiv{\tilde{g}}_{\mu=i,\nu=L+a},\quad{\tilde{G}}_{ab}\equiv{\tilde{g}}_{\mu=L+a,\nu=L+b}$
(7)

It is worth notice that the metric $g_{ij}={\tilde{g}}_{ij}(q^{k},Q^{a}=0)$ is
induced metric on $V_{L}$ and the metric
$G_{ab}={\tilde{G}}_{ab}(q^{k},Q^{a}=0)$ can be defined as some function on
$V_{L}$. Using this assumption, we will get $N=0$ when $Q=0$. So the metric on
$V_{L}$ have form

$g_{\mu\nu}=\begin{pmatrix}{\tilde{g}}_{ij}&0\\\ 0&{\tilde{G}}_{ab}\\\
\end{pmatrix}$ (8)

and the inverse matrix is defined as

$g^{\mu\nu}=\begin{pmatrix}g^{ij}&0\\\ 0&G^{ab}\\\ \end{pmatrix}$ (9)

which implies that

$\displaystyle g^{\mu\zeta}.g_{\zeta\nu}={\delta^{\mu}}_{\nu}$ (10)

which can further be written as,

$\displaystyle g^{ij}\cdot g_{jk}=\delta^{i}_{k},\quad G^{ab}\cdot
G_{bc}={\delta^{a}}_{c}$ (11)

We know that ${\tilde{g}}_{\mu\nu}$ can also be written as

$\displaystyle{\tilde{g}}_{\mu\nu}=(\frac{\partial x}{\partial
q^{\mu}})\cdot(\frac{\partial x}{\partial q^{\nu}}).$ (12)

From here, we can obtain following relations NOKFCK

$\displaystyle 0$ $\displaystyle=$ $\displaystyle(\frac{\partial x}{\partial
q^{k}})\cdot(\frac{\partial x}{\partial Q^{a}})|_{Q=0}=\sum_{A}e_{k}^{A}\cdot
h_{a}^{A}$ $\displaystyle g_{ij}(q^{k})$ $\displaystyle=$
$\displaystyle(\frac{\partial x}{\partial q^{i}})\cdot(\frac{\partial
x}{\partial q^{j}})|_{Q=0}=\sum_{A}e_{i}^{A}\cdot e_{j}^{A}$ $\displaystyle
G_{ab}(q^{k})$ $\displaystyle=$ $\displaystyle(\frac{\partial x}{\partial
Q^{a}})\cdot(\frac{\partial x}{\partial Q^{b}})|_{Q=0}=\sum_{A}h_{a}^{A}\cdot
h_{b}^{A}$ (13)

where $e_{i}^{A}$ and $h_{a}^{A}$ are defined as

$\displaystyle e_{i}^{A}(q^{k})\equiv(\frac{\partial x^{A}}{\partial
q^{i}})|_{Q=0},\quad h_{a}^{A}(q^{k})\equiv(\frac{\partial x^{A}}{\partial
Q^{a}})|_{Q=0}$ (14)

Here, $e_{i}^{A}$ are called the natural frame and gives the induced metric on
$V_{L}$. The inverse metric of ${\tilde{g}}_{\mu\nu}$ is given by

$\displaystyle{\tilde{g}}^{\mu\nu}=\nabla q^{\mu}\cdot\nabla q^{\nu}$ (15)

where $\nabla\equiv\frac{\partial}{\partial x}$. From this, we obtain

$\displaystyle g^{\mu\nu}={\tilde{g}}^{\mu\nu}|_{Q=0}=\nabla
q^{\mu}\cdot\nabla q^{\nu}|_{Q=0}.$ (16)

The Lagrangian for the particle motion on $V_{L}$ is defined as NOKFCK ,

$\displaystyle
L=\frac{1}{2}\cdot{\dot{x}}^{A}{\dot{x}}_{A}-V(x)+\lambda_{a}Q^{a}(x)$ (17)

Here $``A"$ varies between 1 to $N$ and $``a"$ from 1 to $N-L$. The metric for
the coordinate $x^{A}$ is $\delta_{AB}$. $\lambda_{a}^{\prime}S$ are variables
independent of $x^{A}$ and the dot denotes the time derivative. The canonical
momentum conjugate to $x^{A}$ and $\lambda^{a}$ can be written as

$\displaystyle P_{A}$ $\displaystyle\equiv$ $\displaystyle\frac{\partial
L}{\partial{\dot{x}}^{A}}={\dot{x}}^{A}$ $\displaystyle\Pi^{a}$
$\displaystyle\equiv$ $\displaystyle\frac{\partial
L}{\partial{\dot{\lambda}}_{a}}\approx 0$ (18)

Hamiltonian corresponding to Lagrangian in eqn (17) can be written as,

$\displaystyle H_{0}=\frac{1}{2}\cdot P_{A}P^{A}+V(x)-\lambda_{a}Q^{a}(x)$
(19)

###  II.1 Hamiltonian Analysis

The primary constraint for the system under study is defined as,

$\Pi_{a}\approx 0$ (20)

After including the primary constraint, the new Hamiltonian is written as,

$H_{T}=\frac{1}{2}\cdot P_{A}P^{A}+V(x)-\lambda_{a}Q^{a}(x)+u_{a}\Pi^{a}$ (21)

where $u_{a}^{\prime}S$ are a set of Lagrange multipliers for the system. Now,
we will perform the constraint analysis of the given system using the Dirac’s
technique of constraints analysis PAMD1 ; JLAPGB ; PAMD2 ; KS ; MHCT ; MH .
All the constraints of the theory can be calculated in the following manner
NOKFCK ,

$\displaystyle{\dot{\Pi}}^{a}$ $\displaystyle=$
$\displaystyle\\{\Pi^{a},H_{T}\\}_{P}=Q^{a}$ $\displaystyle{\ddot{\Pi}}^{a}$
$\displaystyle=$
$\displaystyle\\{Q^{a},H_{T}\\}_{P}=P^{A}\cdot{\partial_{A}Q^{a}}$
$\displaystyle{\Pi^{a}}^{(3)}$ $\displaystyle=$
$\displaystyle\\{DQ^{a},H_{T}\\}_{P}=D^{2}Q^{a}-\nabla
Q^{a}\cdot\nabla(V-\lambda_{d}Q^{d})$ (22)

The constraint $\Pi^{a(4)}=0$ determines the $u_{a}^{\prime}S$ and the
procedure is over. So, the explicit form of the constraints are,

$\displaystyle\Phi_{1}^{a}$ $\displaystyle=$ $\displaystyle\Pi^{a}\approx 0$
$\displaystyle\Phi_{2}^{a}$ $\displaystyle=$ $\displaystyle Q^{a}\approx 0$
$\displaystyle\Phi_{3}^{a}$ $\displaystyle=$ $\displaystyle DQ^{a}\approx 0$
$\displaystyle\Phi_{4}^{a}$ $\displaystyle=$ $\displaystyle
P^{A}P^{B}\partial_{A}\partial_{B}Q^{a}-\nabla
Q^{a}\cdot{\nabla}(V-\lambda_{d}Q^{d})=D^{2}Q^{a}-\nabla
Q^{a}\cdot{\nabla}\Phi\approx 0$ (23)

Here D and $\Phi$ are defined as, $D=P^{A}\partial_{A}$,
$\Phi=(V-\lambda_{a}Q^{a})$. Also, the product of partial derivatives is
defined as, $\nabla f\cdot\nabla
g\equiv\sum_{A}\partial_{A}f\cdot\partial_{A}g$.

The Poisson brackets between the constraints are defined as NOKFCK ,

$\displaystyle\\{\Phi_{1}^{a},\Phi_{4}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle-\nabla Q^{a}\cdot\nabla Q^{b}\equiv-\alpha^{ab},$
$\displaystyle\\{\Phi_{2}^{a},\Phi_{3}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle\nabla Q^{a}\cdot\nabla Q^{b}\equiv\alpha^{ab},$
$\displaystyle\\{\Phi_{2}^{a},\Phi_{4}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle 2\nabla Q^{a}\cdot(\nabla DQ^{b})\equiv-\beta^{ab},$
$\displaystyle\\{\Phi_{3}^{a},\Phi_{4}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle 2\nabla(DQ^{a})\cdot\nabla(DQ^{b})-\nabla
Q^{a}\cdot\nabla\Phi_{4}^{b}\equiv-\gamma^{ab},$
$\displaystyle\\{\Phi_{3}^{a},\Phi_{3}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle\nabla(DQ^{a})\cdot\nabla Q^{b}-\nabla
Q^{a}\cdot\nabla(DQ^{b})\equiv\rho^{ab}$
$\displaystyle\\{\Phi_{4}^{a},\Phi_{4}^{b}\\}_{P}$ $\displaystyle=$
$\displaystyle
2\left[\nabla\Phi_{4}^{a}\cdot\nabla(DQ^{a})-\nabla\Phi_{4}^{b}\cdot\nabla(DQ^{a})\right]\equiv\epsilon^{ab}$
(24)

Other Poisson brackets vanish. It is worth notice that

$\displaystyle\alpha^{ab}$ $\displaystyle=$
$\displaystyle\alpha^{ba},\mathrm{(symmetric)}$ $\displaystyle\rho^{ab}$
$\displaystyle=$ $\displaystyle-\rho^{ba},\mathrm{(antisymmetric)}$
$\displaystyle\epsilon^{ab}$ $\displaystyle=$
$\displaystyle-\epsilon^{ba},\mathrm{(antisymmetric)}$ (25)

Thus the matrix $\Delta_{ij}^{ab}$ between the constraints has the form

$\Delta_{ij}^{ab}\equiv\\{{\Phi_{i}}^{a},{\Phi_{j}}^{b}\\}_{P}=\begin{bmatrix}0&0&0&-{\alpha}^{ab}\\\
0&0&{\alpha}^{ab}&-{\beta}^{ab}\\\
0&-{\alpha}^{ab}&{\rho}^{ab}&-{\gamma}^{ab}\\\
{\alpha}^{ab}&{\beta}^{ba}&{\gamma}^{ba}&{\epsilon}^{ab}\end{bmatrix}$ (26)

It is worth note here that $a,b=1,....,(N-1)$. Hence each element of the
matrix $\Delta_{ij}^{ab}$ is a $(N-1)\times(N-1)$ matrix. Thus, the matrix
$\Delta_{ij}^{ab}$ is a $4(N-1)\times 4(N-1)$ matrix.

##  III BFFT Formalism

In this section we will discuss the main results of BFFT technique, which is
used to Abelianize the second class constraint systems. The basic idea behind
the scheme is to introduce additional phase space variables $\Theta_{m}^{n}$,
besides the existing physical degrees of freedom $(q,p)$ of the system such
that all the constraints in the extended space of the system are first class.
This means that the original constraints and Hamiltonian have to be modified
accordingly by putting BFFT-extension terms in them. To achieve this, we will
use the results discussed in IABIVT ; RBBN ; SG . Let us consider a set of
constraints $(\Phi_{n}^{m},\Lambda_{j})$ and an Hamiltonian operator $H$. We
know from the Dirac’s constraint analysis that second-class constraints of a
constrained system satisfy an open algebra. These constraints and Hamiltonian
satisfy following algebra

$\displaystyle\\{\Phi_{n}^{m}(q,p),\Phi_{r}^{s}(q,p)\\}\approx\Delta_{nr}^{ms}(q,p)\neq
0,$ $\displaystyle\\{\Phi_{n}^{m}(q,p),\Lambda_{r}^{j}(q,p)\\}\approx 0,$
$\displaystyle\\{\Lambda^{j}(q,p),\Lambda_{t}(q,p)\\}\approx 0,$
$\displaystyle\\{\Lambda^{j}(q,p),H(q,p)\\}\approx 0$ (27)

$``\approx"$ means that the equality holds on the constraint surface. The
additional fields satisfy the symplectic algebra,

$\displaystyle\\{\Theta_{m}^{n},\Theta_{s}^{r}\\}=\omega_{ms}^{nr}$ (28)

where $\omega_{ms}^{nr}$ is a constant quantity and $\det\omega_{ms}^{nr}\neq
0$. The constraints are now defined in terms of auxiliary field
$\Theta_{m}^{n}$ as

$\displaystyle{\tilde{\Phi}}_{n}^{m}={\tilde{\Phi}}_{n}^{m}(q,p;\Theta_{m}^{n}),$
(29)

This modified constraint satisfies the boundary condition

$\displaystyle{\tilde{\Phi}}_{n}^{m}(q,p;0)=\Phi_{n}^{m}(q,p),$ (30)

These modified constraints should satisfy first class constraints algebra. So
the Poisson bracket between the constraints are defined as

$\displaystyle\\{{\tilde{\Phi}}_{n}^{m},{\tilde{\Phi}}_{r}^{s}\\}=0$ (31)

The solution of eqn (31) can be achieved by considering an expansion of
$\Phi_{n}^{m}$, as

$\displaystyle{\tilde{\Phi}}_{n}^{m}=\sum_{k=0}^{\infty}{\tilde{\Phi}}_{n}^{m(k)},$
(32)

where ${\tilde{\Phi}}^{m(k)}\approx O(\Theta^{k})$. The first order correction
in the field is IABIVT ; RBBN ; SG

$\displaystyle{\tilde{\Phi}}_{n}^{m(1)}=X_{nr}^{ms}(q,p)\Theta_{s}^{r}$ (33)

Putting the expression of eqn (33) in eqn (31) and using the boundary
condition given in eqns (30), (27) as well as eqn (28), we get

$\displaystyle\Delta_{nr}^{ms}+X_{nc}^{md}\omega_{df}^{ce}X_{re}^{sf}=0$ (34)

We notice that the eqn (34) does not give a single solution for $X_{ij}^{ab}$,
because there is still unknown matrix $\omega_{ab}^{ij}$. We can make choices
for $\omega_{ab}^{ij}$ in such a way that the newly defined variables are
unconstrained in nature. Using the value of the matrix $\omega_{ab}^{ij}$, we
can calculate the possible of $X_{ij}^{ab}$ from the eqn (34). Using the value
of $X_{ij}^{ab}$ we can obtain ${\tilde{\Theta}}_{n}^{m(1)}$. If
${\Theta_{n}^{m}}+{\tilde{\Theta}}_{n}^{m(1)}$ is strongly involutive in
nature, then series will end otherwise it will continue in the same way till
we don’t get strongly involutive constraints. The explicit expression of
higher order corrections in the field $\Phi$ is

$\displaystyle{\tilde{\Phi}}_{n}^{m(k+1)}=-\frac{1}{k+2}\Theta_{b}^{c}X_{ce}^{bd}\omega_{df}^{eg}B_{gn}^{fm(k)}\quad;\quad
k\geq 1$ (35)

where $B_{mn}^{ba}$ is defined as

$\displaystyle B_{rs}^{ba(k)}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{k}\\{{\tilde{\Phi}}_{r}^{b(k-l)},{\tilde{\Phi}}_{s}^{a(l)}\\}_{(q,p)}+\sum_{l=0}^{k-2}\\{{\tilde{\Phi}}_{r}^{b(k-l)},{\tilde{\Phi}}_{s}^{a(l+2)}\\}_{(\Theta)},\quad
k\geq 2$ $\displaystyle B_{rs}^{ba(1)}$ $\displaystyle=$
$\displaystyle\\{{\tilde{\Phi}}_{r}^{b(0)},{\tilde{\Phi}}_{s}^{a(1)}\\}_{(q,p)}-\\{{\tilde{\Phi}}_{r}^{a(0)},{\tilde{\Phi}}_{s}^{b(1)}\\}_{(q,p)}$
(36)

In the above expressions, we have defined

$\displaystyle
X_{mn}^{ab}X_{bc}^{nr}=\omega_{mn}^{ab}\omega_{bc}^{nr}=\delta_{m}^{r}\delta_{c}^{a}$
(37)

Another important part of the BFFT formalism is that any dynamical variable
$f(q,p)$ has also to be modified in the same way as discussed above in order
to be strongly involutive with the modified constraints
${\tilde{\Phi}}_{n}^{m}$. Denoting the modified quantity by $f(q,p;\Theta)$,
we then have

$\displaystyle\\{{\tilde{\Phi}}_{n}^{m},{\tilde{f}}\\}=0$ (38)

Apart from that, modified variable ${\tilde{f}}$ must also satisfy the
boundary condition given below,

$\displaystyle{\tilde{f}}(q,p;0)=f(q,p)$ (39)

To obtain $\tilde{f}$ as an analogous expansion to eqn (32), we will consider

$\displaystyle{\tilde{f}}=\sum_{k=0}^{\infty}{f}^{(k)}$ (40)

where ${\tilde{f}}^{(k)}$ is also a term which is of the order n in
$\Theta^{{}^{\prime}}s$. The expression in eqn (38) above gives us
${\tilde{f}}^{(1)}$

$\displaystyle{\tilde{f}}^{(1)}=-\Theta_{n}^{a}\omega_{ab}^{no}X_{om}^{bc}(q,p)\\{{\tilde{\Phi}}_{c}^{m},f\\},$
(41)

where $\omega_{ab}^{mn}$ and $X_{mn}^{ab}$ are the inverses of
$\omega_{mn}^{ab}$ and $X_{ab}^{mn}$.

The corrections in the physical variable $f$ can be written in the more
general form as,

$\displaystyle{\tilde{f}}^{(k+1)}=-\frac{1}{k+1}\Theta_{n}^{a}\omega_{ab}^{no}X_{om}^{bc}(q,p){G(f)_{c}^{m}}^{(k)},$
(42)

where

$\displaystyle
G_{a}^{b(k)}=\sum_{l=0}^{k}\\{{\tilde{\Phi}}_{r}^{b(k-l)},f^{(l)}\\}_{(q,p)}+\sum_{l=0}^{(k-2)}\\{{\tilde{\Phi}}_{r}^{b(k-l)},f^{(l+2)}\\}_{(\Theta)}+\\{{\tilde{\Phi}}_{r}^{b(k+1)},f^{(1)}\\}_{(\Theta)}$
(43)

In the similar way, we can find the involutive form of other variables using
the BFFT method described above.

Let us take the initial fields as $q$ and $p$. Then the involutive form of
these fields ($\tilde{q}$ and $\tilde{p}$), will satisfy the following
relations.

$\displaystyle\\{{\tilde{\Phi}},{\tilde{q}}\\}=\\{{\tilde{\Phi}},{\tilde{p}}\\}=0$
(44)

Similarly, any function of the physical variables $\tilde{q}$ and $\tilde{p}$
will also satisfy the strong involution relation, since

$\displaystyle\\{{\tilde{\Theta}},{\tilde{F}}({\tilde{q}},{\tilde{p}})\\}=\\{{\tilde{\Theta}},{\tilde{q}}\\}\frac{\partial{\tilde{F}}}{\partial{\tilde{q}}}+\\{{\tilde{\Theta}},{\tilde{p}}\\}\frac{\partial{\tilde{F}}}{\partial{\tilde{p}}}=0$
(45)

So, if we are taking any dynamical variable in the original phase space, it
can be written in involutive form as

$\displaystyle F(q,p)\rightarrow
F({\tilde{q}},{\tilde{p}})=\tilde{F}({\tilde{q}},{\tilde{p}})$ (46)

It is very much obvious that the initial boundary condition in the BFFT
formalism, namely, the reduction of the involutive physical variables to the
original physical variables, when the new fields are set to zero, remains
preserved.

##  IV Construction of the first class constraint Theory

We can easily observe that all the constraints of the theory in eqn (23) are
of second class in nature. To change them in the first class constraints we
will introduce $4L$ set of possible BFFT fields
$\Theta^{a(1)},\Theta^{a(2)},\Theta^{a(3)},\Theta^{a(4)}$. Here each set of
newly introduced fields will correspond to a set of constraints. We will
define some relation between these BFFT fields which will help us in the
Abelianization of the constraints of the theory. Using the relation between
newly introduced fields we will define $\omega$ which will give us the
possible solution of the eqn (34). Here we will discuss the Abelianiaztion of
constraints and Hamiltonian for the Particle motion on the surface
$V_{L}(1\leq L<N)$ in the Riemann manifold $R_{N}$ (based on VPRT .

Our choice of Poisson Bracket between the fields
$\Theta^{a(1)},\Theta^{a(2)},\Theta^{a(3)},\Theta^{a(4)},\ (a=1,...,N-1)$ are

$\displaystyle\\{\Theta^{a(1)},\Theta^{b(3)}\\}$ $\displaystyle=$
$\displaystyle I^{ab},\quad\\{\Theta^{a(2)},\Theta^{b(4)}\\}=I^{ab}$ (47)

where $I^{ab}$ is an $(N-1)\times(N-1)$ unitary matrix.

From the relation between the fields $\Theta$, we can find matrix
$\omega_{ab}^{ij}$ as,

$\omega^{aibj}=\begin{bmatrix}0&0&I^{ab}&0\\\ 0&0&0&I^{ab}\\\ -I^{ab}&0&0&0\\\
0&-I^{ab}&0&0\end{bmatrix}$ (48)

Using the matrix $\omega^{aibj}$ defined above and the matrix
$\Delta_{ij}^{ab}$ between the constraints in the eqn (26), we can calculate
the possible value of matrix $X_{ij}^{ab}$. Now, using the matrix
$X_{ij}^{ab}$, we can write the modified constraints as,

$\displaystyle{\tilde{\Phi}}_{1}^{a}$ $\displaystyle=$
$\displaystyle\Pi^{a}-\Theta^{a(3)},\quad{\tilde{\Phi}}_{2}^{a}=Q^{a}+\Theta^{a(2)},\quad{\tilde{\Phi}}_{3}^{a}=(P^{A}-\partial^{A}{\bar{Q}}_{b}\Theta^{b(4)})\partial_{A}{\bar{Q}}^{a}$
$\displaystyle{\tilde{\Phi}}_{4}^{a}$ $\displaystyle=$
$\displaystyle(P^{A}-\partial^{A}{\bar{Q}}_{b}\Theta^{b(4)})(P^{B}-\partial^{B}{\bar{Q}}_{c}\Theta^{c(4)})\partial_{A}\partial_{B}{\bar{Q}}^{a}\Theta^{b1}-\partial_{A}{\bar{V}}\cdot\partial^{A}{\bar{Q}}^{a}$
(49) $\displaystyle+$
$\displaystyle\lambda_{d}\partial_{A}{\bar{Q}}^{a}\partial^{A}{\bar{Q}}^{d}+\partial_{A}{\bar{Q}}^{a}\partial^{A}{\bar{Q}}_{e}\Theta^{e(1)}$

It is worth mention here that, all the barred quantities defined in eqn (49)
are function of coordinates $x^{k}$ and fields $\Theta^{a(2)}$ and will take
the form of the original unbarred quantities in the limit
$\Theta^{a(2)}\rightarrow 0$. Here, any field $\bar{f}(x^{k},\Theta^{a(2)})$
will be written as VPRT

$\displaystyle{\bar{f}(x^{k},\Theta^{a(2)})}=\sum_{n=0}^{\infty}\frac{f^{(n)}_{a}}{n!}\Theta^{a(2)}_{(n)}$
(50)

also, the partial differentiation of field $\bar{f}$ wrt. any field $x^{k}$
can be written as VPRT

$\displaystyle{\bar{f}_{,i}}=Q_{ai}\\{\bar{f},\Theta^{a(4)}\\}$ (51)

Then the Poisson bracket between these modified set of constraints is,

$\displaystyle\\{{\tilde{\Phi}}_{i}^{a},{\tilde{\Phi}}_{j}^{a}\\}=0$ (52)

where $i,j=1,2,3,4,$ and $a,b=1,2,...,N-1$. We can conclude from eqn (52) that
the modified constraints are involutive in nature. Hence we have successfully
converted the second class constraints of the theory into first class
constraints.

Now, we will construct the involutive Hamiltonian for this system.

Corrections in the Hamiltonian due to different fields $\Theta$ can be
calculated as follows. We will start it by calculating the inverse of the
matrices $\omega^{aibj}$ and $X_{ij}^{ab}$. The inverse of the matrix
$\omega^{aibj}$ can be easily written as,

$\omega_{ij}^{ab}=\begin{bmatrix}0&0&-I^{ab}&0\\\ 0&0&0&-I^{ab}\\\
I^{ab}&0&0&0\\\ 0&I^{ab}&0&0\end{bmatrix}$ (53)

The total Hamiltonian with corrections due to BFFT field can be written as,

$\displaystyle{\tilde{H}}=\frac{1}{2}\cdot(P_{A}-\partial_{A}{\bar{Q}}_{b}\Theta^{b(4)})(P^{A}-\partial^{A}{\bar{Q}}_{c}\Theta^{c(4)})+\bar{V}(x)-(\lambda_{a}+\Theta_{a(1)})(Q^{a}(x)+\Theta^{a(2)})$
(54)

Here $\alpha,\beta,\gamma,\rho,\epsilon$ are $(N-1)\times(N-1)$ matrices and
$\Theta^{a(i)}(i=1,2,3,4)$ takes $4(N-1)$ possible values.

Now, by calculating the Poisson bracket between the modified constraints and
the Hamiltonian $\tilde{H}$ it can be easily verified that modified
Hamiltonian is involutive in nature.

$\displaystyle\\{\tilde{H},\tilde{\Phi}_{i}^{a}\\}=0$ (55)

where $i=1,2,3,4$ and $a=1,2,...,N-1$.

##  V Hamiltonian BRST Quantization

###  V.1 Charge and Symmetry

In this section we will construct BRST symmetry for the Particle motion on the
surface $V_{L}(1\leq L<N)$ in the Riemann manifold $R_{N}$. To construct the
we will use the Hamiltonian BRST formalism also called BFV-BRST formalism MHCT
; MH ; ESFGV ; IABGV ; IABESF .

In the BFV-BRST technique associated to a general class of system with first
class constraints, we introduce two canonical set of ghost and anti-ghost
fields $(C,\bar{P})$ with ghost number 1 and -1 respectively and $(P,\bar{C})$
with ghost number -1 and 1 respectively with Lagrange multiplier fields
$(N,B)$ for each set of constraints. As there are $4(N-1)$ set of constraints,
we will introduce two $4(N-1)$ sets of canonical ghost and anti-ghost fields
$(C^{ka},{\bar{P}}_{k}^{a})$, $(P^{ka},{\bar{C}}_{k}^{b})$ and Lagrange
multiplier fields $(N^{ka},B_{k}^{a})$. These fields and corresponding momenta
satisfy following super algebra,

$\displaystyle\\{C^{ka},{\bar{P}}_{l}^{b}\\}=\\{P^{ka},{\bar{C}}_{l}^{b}\\}=\\{N^{ka},B_{l}^{b}\\}=\delta^{k}_{l}I^{ab}$
(56)

Now, using the BFV formalism we can write the general expression for nilpotent
BRST charge, gauge-fixing fermion and BRST invariant Hamiltonian for the
system under study as,

$\displaystyle Q_{BRST}=\int
dx^{N}(C^{k}_{a}{\tilde{\Omega}}_{k}^{a}+P^{k}_{a}B_{k}^{a})$ (57)
$\displaystyle\Psi=\int
dx^{N}({\bar{P}}_{k}^{a}N^{k}_{a}+{\bar{C}}^{k}_{a}\chi_{k}^{a})$ (58)
$\displaystyle H_{U}=H_{P}+H_{BF}-\\{Q_{BRST},\Psi\\}$ (59)

In the BFV-BRST formulation the generating functional is doesn’t depend on
gauge fixing fermion ESFGV ; IABESF ; IABGV ; LDFAD ; PSENJ , hence one has
freedom to choose it in the convenient form. It is also worth notice here,
that gauge-fixing fermion $\chi_{k}^{a}$ is Hermitian in nature and of the
similar Grassmann parity as of $\tilde{\Phi}^{a}$. They also satisfy

$\displaystyle\det|\\{\chi_{k}^{a},\tilde{\Phi}^{l}_{b}\\}|\neq 0$ (60)

We will apply the general results obtained above to the particle on surface
$V^{L}$ embedded in $R^{N}$.

$\displaystyle S_{eff}=\int dt\big{[}P_{A}\dot{x}^{A}+\Pi_{\Theta
a}^{k}\dot{\Theta}^{a}_{k}+B_{k}^{a}{\dot{N}}^{k}_{a}+{\dot{P}}^{k}_{a}{\bar{C}}_{k}^{a}+{\dot{C}}_{a}^{k}{\bar{P}}_{k}^{a}-H_{P}-H_{BF}+\left[Q_{BRST},\Psi\right]\big{]}$
(61)

For the system under study, the modified constraints ${\tilde{\Phi}}_{k}^{a}$
have been calculated in eqn (49).

Our choice of gauge condition $\chi_{k}^{a}$ for the motion on the surface
$V_{L}$ in the Riemann manifold $R_{N}$ is $\Phi_{k}^{a}$ in eqn (23).

Here $\tilde{H}=H_{P}+H_{BF}$ is taken as the BFFT modified Hamiltonian for
this system obtained in eqn (54).

We will define the canonical brackets for all dynamical variables as

$\displaystyle[x^{A},P_{B}]={\delta^{A}}_{B};\quad[\Theta^{a}_{i},\Pi^{jb}_{\Theta}]=[\lambda^{a}_{i},\Pi^{bj}]=\delta^{j}_{i}I^{ab};\quad\\{{\bar{C}}_{k}^{a},{\dot{C}}^{bl}\\}=i\delta^{l}_{k}I^{ab};\quad\\{C_{k}^{a},{\dot{\bar{C}}}^{bl}\\}=-i\delta^{l}_{k}I^{ab}$
(62)

Nilpotent BRST transformations corresponding the action in eqn (61) can be
easily constructed using the relation
$S_{\mathrm{BRST}}\Gamma=-[Q_{\mathrm{BRST}},\Gamma]_{\pm}$. It’s relation
with infinitesimal BRST transformation $S_{\mathrm{BRST}}$ can be established
as $\delta_{\mathrm{BRST}}\ =S_{\mathrm{BRST}}\Gamma\delta\Lambda$.
$\delta\Lambda$ defined here is an infinitesimal BRST parameter. Here ‘$-$’
and ‘$+$’ sign are defined for bosonic and fermionic variables respectively.
The BRST transformation for the particle motion on general class of Riemann
surfaces embedded in higher dimensional Euclidean spaces is written as,

$\displaystyle S_{\mathrm{BRST}}N^{ak}=P^{ak},\quad
S_{\mathrm{BRST}}{\bar{P}}^{ak}={\tilde{\Phi}}^{ak},\quad
S_{\mathrm{BRST}}{\bar{C}}^{ak}=-B^{ak}$ $\displaystyle
S_{\mathrm{BRST}}P^{A}=S_{\mathrm{BRST}}C^{ak}=S_{\mathrm{BRST}}\Pi^{ak}=S_{\mathrm{BRST}}{P}^{ak}=0$
(63)

It can be easily verified that these transformations are nilpotent in nature.

Using the expressions for BRST charge $Q_{BRST}$ and gauge-fixing fermion
$\Psi$, effective action in eqn (61) can be written as

$\displaystyle S_{eff}=\int dt\big{[}P_{A}{\dot{x}}^{A}+\Pi_{\Theta
a}^{k}\dot{\Theta}_{k}^{a}+B_{k}^{a}{\dot{N}}^{k}_{a}+{\dot{P}}^{k}_{a}{\bar{C}}_{k}^{a}+{\dot{C}}^{k}_{a}{\bar{P}}_{k}^{a}-\tilde{H}-P^{k}_{a}{\bar{P}}_{k}^{a}+N_{k}^{a}\tilde{\Phi}^{k}_{a}+B_{k}^{a}\chi^{k}_{a}+{\bar{C}_{k}^{a}}C^{k}_{a}\big{]}$
(64)

The generating functional for this effective action can be written as

$\displaystyle Z_{\Psi}$ $\displaystyle=$
$\displaystyle\int[D\varphi]e^{(iS_{eff})}$ (65)

The Liouville measure $D\varphi$ for the generating functional is defined as,

$\displaystyle D\varphi=\prod_{i}d\Xi_{i}$ (66)

Here $\Xi_{i}$ are all dynamical variables $(P_{A},x^{A},\Pi_{\Theta
a}^{k},\Theta^{a}_{k},N^{k}_{a},B_{k}^{a},{\bar{C}}_{a}^{k},P^{a}_{k},C_{a}^{a},{\bar{P}}^{a}_{k})$
of the theory. Now performing the integration over ghost and antighost momenta
$P^{a}_{k}$ and ${\bar{P}}^{a}_{k}$, we will get

$\displaystyle{Z_{\psi}}=\int D\varphi^{\prime}\exp\big{[}i\int
dt\big{[}P_{A}{\dot{x}}^{A}+\Pi_{\Theta
a}^{k}\dot{\Theta}^{a}_{k}+B_{k}^{a}\dot{N}^{k}_{a}+\dot{C}_{k}^{a}{\dot{\bar{C}}^{k}_{a}}-\tilde{H}+N_{k}^{a}\tilde{\Phi}^{k}_{a}-C^{k}_{a}\bar{C}_{k}^{a}-B_{k}^{a}\chi^{k}_{a}\big{]}\big{]}$
(67)

Here $D\varphi^{\prime}$ is the new path integral measure for the effective
action after the performance of integral over the fields $P$ and $\bar{P}$.
Further performance of integral over fields $B^{k}_{a}$ will give us the new
effective generating functional as

$\displaystyle{Z_{\psi}}=\int D\phi^{\prime\prime}\exp\big{[}i\int
dt\big{[}P_{A}{\dot{x}}^{A}+\Pi_{\Theta
a}^{k}\dot{\Theta}^{a}_{k}+\dot{C}_{k}^{a}{\dot{\bar{C}}^{k}_{a}}-\tilde{H}+N_{k}^{a}\tilde{\Phi}^{k}_{a}-C^{k}_{a}\bar{C}_{k}^{a}-\frac{\\{\dot{N_{k}^{a}}-\chi_{k}^{a}\\}^{2}}{2}\big{]}\big{]}$
(68)

where $D\varphi^{\prime\prime}$ is the new path integral measure which
corresponds to all the dynamical variables left out after integration. Now,
the new BRST transformation for the modified action in eqn (68) is written as

$\displaystyle S_{\mathrm{BRST}}N^{ak}={\dot{C}}^{ak},\quad
S_{\mathrm{BRST}}{\bar{C}}^{ak}=-{\dot{N}}^{ak}-\chi^{ak}$ $\displaystyle
S_{\mathrm{BRST}}P^{A}=S_{\mathrm{BRST}}C^{ak}=S_{\mathrm{BRST}}B^{ak}=S_{\mathrm{BRST}}{P}^{ak}=0$
(69)

It is well known in the literature that BRST charges are nilpotent in nature.
We also know that the operation of these charges on the states of total
Hilbert space gives us the physical subspace of the system.

$\displaystyle{Q_{\mathrm{BRST}}}|\mathrm{phys}\rangle=0,\quad|\mathrm{phys}\rangle\neq{Q_{\mathrm{BRST}}}|....\rangle$
(70)

which can be further written in more explicit form for this system as,

$\displaystyle iC_{k}^{a}\tilde{\Phi}^{k}_{a}|\mathrm{phys}\rangle=0,\quad
i\dot{\bar{C}}_{k}^{a}N^{k}_{a}|\mathrm{phys}\rangle=0$ (71)

The result of eqn (71) implies that the first class constraints of the system
under study will annihilate the physical subspace of the total Hilbert space
of the system.

##  VI Consistency with Previous result

We can check the consistency of the results of this system in the limit of
$N-L=1$ VKP . In this limit, elements of the matrix will transform as,

$\alpha^{ab}\rightarrow\alpha,\quad\beta^{ab}\rightarrow\beta,\quad\gamma^{ab}\rightarrow\gamma$
(72)

From the antisymmetry (25)

$\rho^{ab}\rightarrow 0,\quad\epsilon^{ab}\rightarrow 0$ (73)

The inverse matrix elements will transform under this limit as

$A\rightarrow-\frac{\gamma}{\alpha^{2}},\quad
B\rightarrow\frac{\beta}{\alpha^{2}},\quad
C\rightarrow\frac{\gamma}{\alpha^{2}},\quad
D\rightarrow-\frac{\beta}{\alpha^{2}},\quad{E,F}\rightarrow 0$ (74)

Here $E,F$ vanishes due to asymmetry.

Also under the transformation $Q^{a}\rightarrow f(x)$ the modified constraints
in eqn (49) takes the form of modified constraints of $L=N-1$ case,

$\displaystyle\tilde{\Phi}_{1}$ $\displaystyle=$
$\displaystyle\Pi-\Theta^{(3)}$ $\displaystyle\tilde{\Phi}_{2}$
$\displaystyle=$ $\displaystyle f(x)+\Theta^{(2)}$
$\displaystyle\tilde{\Phi}_{3}$ $\displaystyle=$
$\displaystyle(\bar{P}^{k}-{\partial^{k}{\bar{f}(x)}}\Theta^{(4)})\bar{\partial_{k}f(x)}$
$\displaystyle\tilde{\Phi}_{4}$ $\displaystyle=$
$\displaystyle(\bar{P}^{k}-\partial^{k}\bar{f}(x)\Theta^{(4)})(\bar{P}^{l}-\partial^{l}\bar{f}(x)\Theta^{(4)})\partial_{k}\partial_{l}\bar{f}(x)-\partial_{k}\bar{V}\cdot\partial^{k}\bar{f}(x)$
(75)
$\displaystyle+\lambda\partial_{k}\bar{f}(x)\partial^{k}\bar{f}(x)+\partial_{k}\bar{f}(x)\partial^{k}\bar{f}(x)\Theta^{1}$

The form of modified Hamiltonian in eqn (54) under these transformations is

$\displaystyle\tilde{H}=\frac{1}{2}\cdot(\bar{P}_{k}-\partial_{k}\bar{f}(x)\Theta^{(4)})(\bar{P}^{k}-\partial^{k}\bar{f}(x)\Theta^{(4)})+\bar{V}(x)-(\lambda+\Theta^{(1)})(f(x)+\Theta^{(2)})$
(76)

Similary, BRST charge and BRST symmetry can be written in this limit as

$\displaystyle Q_{\mathrm{BRST}}=C^{k}{\tilde{\Phi}}_{k}+P^{k}B_{k}$
$\displaystyle S_{\mathrm{BRST}}N^{k}={\dot{C}}^{k},\quad
S_{\mathrm{BRST}}{\bar{C}}^{k}=-{\dot{N}}^{k}-\chi^{k}$ $\displaystyle
S_{\mathrm{BRST}}P^{a}=S_{\mathrm{BRST}}C^{k}=S_{\mathrm{BRST}}B^{k}=S_{\mathrm{BRST}}{P}^{k}=0$
(77)

These are the same constraints, Hamiltonian, BRST charge and symmetry which we
have obtained in our previous work VKP . This shows that all the results
obtained here (49,54,63,69) using BFFT formalism are consistent with the
previous results VKP  in the limit $L\rightarrow(N-1)$.

##  VII  Examples of $L(1\leq L<N)$ Dimensional Embedding in $R^{N}$

As an example of $L$ dimensional embedding in $R^{N}$ we will discuss particle
on torus knot VS ; PDSG ; VKPBPM ; VPBM ; ASSG . We will discuss all the
important results developed for general system in this case.

###  VII.1 Particle on Torus Knot

Particle on torus knot is a one dimensional surface embedded in three
dimensional space. It is a special kind of knot that lies on the surface of
un-knotted torus in $R^{3}$. It is specified by a set of co-prime integers $p$
and $q$. A torus knot of type $(p,q)$ winds $p$ times around the rotational
symmetry axis of the torus and $q$ times around a circle in the interior of
the torus. The toroidal co-ordinate system is a suitable choice to study this
system. Toroidal co-ordinates are related to Cartesian co-ordinates
($x_{1},x_{2},x_{3}$) in following ways

$\displaystyle x_{1}=\frac{a\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},\quad
x_{2}=\frac{a\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},\quad
x_{3}=\frac{a\sin\theta}{\cosh\eta-\cos\theta}$ (78)

where, $0\leq\eta\leq\infty$, $-\pi\leq\theta\leq\pi$ and $0\leq\phi\leq
2\pi$. A toroidal surface is represnted by some specific value of $\eta$ (say
$\eta_{0}$). Parameters $a$ and $\eta_{0}$ are written as $a^{2}=R^{2}-d^{2}$
and $\cosh\eta_{0}=\frac{R}{D}$ where $R$ and $D$ are major and minor radius
of torus respectively.

Lagrangian for a particle constrained to move on the surface of torus knot is

$\displaystyle
L=\frac{1}{2}ma^{2}\frac{{\dot{\eta}}^{2}+{\dot{\theta}}^{2}+{\sinh^{2}\eta}{\dot{\phi}}^{2}}{(\cosh\eta-\cos\theta)^{2}}+\lambda(p\theta+q\phi)$
(79)

where $(r,\theta,\phi)$ are toroidal co-ordinates for toric geometry and
$\lambda$ is the Lagrange multiplier. The canonical Hamiltonian corresponding
to the Lagrangian in eqn (79)is then written as,

$\displaystyle
H=\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[{p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{{p^{2}_{\phi}}}{{\sinh^{2}\eta}}\right]-\lambda(p\theta+q\phi)$
(80)

where $p_{\eta}$, $p_{\theta},p_{\phi}$ and ${p}_{\lambda}$ are the canonical
momenta conjugate to the coordinate $\eta$, $\theta$, $\phi$ and $\lambda$
respectively, defined as

$\displaystyle
p_{\eta}=\frac{ma^{2}{\dot{\eta}}}{(\cosh\eta-\cos\theta)^{2}},\quad
p_{\theta}=\frac{ma^{2}\dot{\theta}}{(\cosh\eta-\cos\theta)^{2}},\quad
p_{\phi}=\frac{ma^{2}\sinh^{2}\eta\dot{\phi}}{(\cosh\eta-\cos\theta)^{2}},\quad{p}_{\lambda}\approx
0$ (81)

The ${p}_{\lambda}$ is the primary constraint of the theory.

After inclusion of primary constraint our new Hamiltonian has the form

$H_{T}=\frac{(\cosh\eta-\cos\theta)^{2}}{2ma^{2}}\left[{p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{{p^{2}_{\phi}}}{{\sinh^{2}\eta}}\right]-\lambda(p\theta+q\phi)+up_{\lambda}$
(82)

Now, using Dirac’s method of Hamiltonian analysis, we will calculate all the
possible constraints of the theory as,

$\displaystyle{\dot{p}}_{\lambda}=\\{{p}_{\lambda},H_{T}\\}_{P}=(p\theta+q\phi)\approx
0$ (83)
$\displaystyle{\ddot{p}}_{\lambda}=\\{(p\theta+q\phi),H_{T}\\}_{P}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left[pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta}\right]\approx
0$ (84)
$\displaystyle{p}_{\lambda}^{(3)}=\\{\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left[pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta}\right],H_{T}\\}_{P}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}[\\{2pp_{\theta}\sinh{\eta}+\frac{2qp_{\phi}}{\sinh{\eta}}$
$\displaystyle-\frac{2qp_{\phi}\cosh{\eta}(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}\\}p_{\eta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}+2\sin{\theta}(pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta})p_{\theta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}$
$\displaystyle-p\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\\{\sin{\theta}({p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{p^{2}_{\phi}}{\sinh^{2}\eta})-\lambda
p\\}+\lambda\frac{q^{2}}{{\sinh^{2}\eta}}]\approx 0$ (85)

${p}_{\lambda}^{(4)}$ will vanish and the value of $u$ will be determined from
it. All the constraints can be written as,

$\displaystyle\Phi_{1}^{a}=p_{\lambda}$
$\displaystyle\Phi_{2}^{a}=Q^{a}=(p\theta+q\phi)$
$\displaystyle\Phi_{3}^{a}=DQ^{a}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\left[pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta}\right]$
$\displaystyle\Phi_{4}^{a}=P^{A}P^{B}\partial_{A}\partial_{B}Q^{a}-\nabla
Q^{a}\cdot{\nabla}(V-\lambda_{d}Q^{d})=D^{2}Q^{a}-\nabla
Q^{a}\cdot{\nabla}\Phi$
$\displaystyle=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}[\\{2pp_{\theta}\sinh{\eta}+\frac{2qp_{\phi}}{\sinh\eta}-\frac{2qp_{\phi}\cosh{\eta}(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}\\}p_{\eta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}$
$\displaystyle+2\sin{\theta}(pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta})p_{\theta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}-p\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\\{\sin{\theta}({p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{p^{2}_{\phi}}{\sinh^{2}\eta})-\lambda
p\\}$ $\displaystyle+\lambda\frac{q^{2}}{\sinh^{2}\eta}]$ (86)

Now, the Poisson brackets between the constraints have following values,

$\displaystyle\\{\Phi_{1}^{a},\Phi_{4}^{b}\\}_{P}=-\nabla Q^{a}\cdot\nabla
Q^{b}=-\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}[p^{2}+\frac{q^{2}}{\sinh^{2}\eta}]\equiv-\alpha^{ab}$
(87) $\displaystyle\\{\Phi_{2}^{a},\Phi_{3}^{b}\\}_{P}=\nabla Q^{a}\cdot\nabla
Q^{b}=\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}[p^{2}+\frac{q^{2}}{\sinh^{2}\eta}]\equiv\alpha^{ab}$
(88) $\displaystyle\\{\Phi_{2}^{a},\Phi_{4}^{b}\\}_{P}=2\nabla
Q^{a}\cdot(\nabla
DQ^{b})=2\frac{(\cosh\eta-\cos\theta)^{3}}{m^{2}a^{4}}[p\\{pp_{\eta}\sinh\eta+\sin{\theta}(pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta})\\}$
$\displaystyle+q\\{\frac{qp_{\eta}}{\sinh{\eta}}(1-\frac{\cosh{\eta}(\cosh\eta-\cos\theta)}{\sinh^{2}{\eta}})+\frac{\sin{\theta}}{\sinh^{2}\eta}(qp_{\theta}-pp_{\phi})\\}]\equiv-\beta^{ab}$
(89)

Similar the Poisson bracket between other constraints
$(\Phi_{3}^{a},\Phi_{4}^{b})$ (107), $(\Phi_{3}^{a},\Phi_{3}^{b})$ (108)and
$(\Phi_{4}^{a},\Phi_{4}^{b})$ (109) has been explicitly calculated in the
appendix. All these brackets will be nonzero and will be equal to
$\gamma^{ab},\rho^{ab}$ and $\epsilon^{ab}$. Thus the matrix between the
constraints will take exactly the same form of matrix $\Delta_{ij}^{ab}$ in
eqn (26).

As all the constraints of the theory (86) are second class, we will follow the
method of section IV and introduce four possible fields
$\Theta^{a(1)},\Theta^{a(2)},\Theta^{a(3)},\Theta^{a(4)}$ corresponding to
each constraint. Relation between these fields will provide us possible value
of $\omega^{abij}$. Our choice of Poisson bracket between the fields will be
same as one taken for the general case. Hence the matrix $\omega^{abij}$ will
have the form of eqn (48).

Using the matrix $\omega^{abij}$ and the matrix $\Delta_{ab}^{ij}$, in the eqn
(33), one can find many possible value of matrix $X_{ab}^{ij}$.

Now, applying the results developed in section IV we can calculate the
modified constraints as

$\displaystyle\tilde{\Phi}_{1}^{a}$ $\displaystyle=$ $\displaystyle
p_{\lambda}-\Theta^{a(3)}$ $\displaystyle\tilde{\Phi}_{2}^{a}$
$\displaystyle=$ $\displaystyle(p\theta+q\phi)+\Theta^{a(2)}$
$\displaystyle\tilde{\Phi}_{3}^{a}$ $\displaystyle=$
$\displaystyle\frac{\big{(}\cosh\eta-\cos{(\theta-\frac{\Theta^{a(2)}}{2p})}\big{)}^{2}}{ma^{2}}\big{[}(p_{\theta}-p\Theta^{(4a)})p+\frac{(p_{\phi}-q\Theta^{(4a)})q}{\sinh^{2}\eta}\big{]}$
$\displaystyle\tilde{\Phi}_{4}^{a}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos{(\theta-\frac{\Theta^{a(2)}}{2p})})^{2}}{ma^{2}}[\big{\\{}2p(p_{\theta}-p\Theta^{(4a)})\sinh{\eta}+\frac{2q(p_{\phi}-q\Theta^{(4a)})}{\sinh\eta}$
(90) $\displaystyle-$
$\displaystyle\frac{2q(p_{\phi}-q\Theta^{(4a)})\cosh{\eta}\big{(}\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p})\big{)}}{\sinh^{3}{\eta}}\big{\\}}p_{\eta}\frac{(\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p}))}{ma^{2}}$
$\displaystyle+$ $\displaystyle
2\sin{(\theta-\frac{\Theta^{a(2)}}{2p})}(p(p_{\theta}-p\Theta^{(4a)})+\frac{q(p_{\phi}-q\Theta^{(4a)})}{\sinh^{2}\eta})(p_{\theta}-p\Theta^{(4a)})\frac{(\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p}))}{ma^{2}}$
$\displaystyle-$ $\displaystyle
p\frac{(\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p}))}{ma^{2}}\\{\sin{(\theta-\frac{\Theta^{a(2)}}{2p})}({p^{2}_{\eta}}+{(p_{\theta}-p\Theta^{(4a)})^{2}}+\frac{(p_{\phi}-q\Theta^{(4a)})^{2}}{\sinh^{2}\eta})-\lambda
p\\}$ $\displaystyle+$
$\displaystyle\lambda\frac{q^{2}}{\sinh^{2}\eta}+\big{(}\frac{(\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p}))}{ma^{2}}p^{2}+\frac{q^{2}}{\sinh^{2}\eta}\big{)}\Theta^{a(1)}]$

The Poisson bracket between these modified constraints vanishes which shows
that modified constraints are involutive. Hence we have converted the second
class constraints of the theory into first class.

Now, we will construct first class Hamiltonian for this system using the
results in section V.

The total involutive Hamiltonian for this system will take the form as VPRT ,

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle\frac{(\cosh\eta-\cos(\theta-\frac{\Theta^{a(2)}}{2p}))^{2}}{2ma^{2}}\left[{p^{2}_{\eta}}+{(p_{\theta}-p\Theta^{(4a)})^{2}}+\frac{(p_{\phi}-q\Theta^{(4a)})^{2}}{\sinh^{2}\eta}\right]$
(91) $\displaystyle-$
$\displaystyle(\lambda+\Theta^{a(1)})(p\theta+q\phi+\Theta^{a(2)})$

It can be easily verified that the Hamiltonian $\tilde{H}$ is involutive by
computing it’s Poisson bracket with modified constraints of the theory.

$\displaystyle\\{\tilde{H},\tilde{\Phi}_{i}^{a}\\}=0$ (92)

where $i=1,2,3,4$.

BRST charge for this first class system can be written using above expression,
as

$\displaystyle Q_{BRST}=iC^{i}_{a}\tilde{\Phi}_{i}^{a}+iP^{i}_{a}B_{i}^{a}$
(93)

and corresponding BRST symmetry transformation can be written as

$\displaystyle S_{\mathrm{BRST}}N^{ak}=P^{ak},\quad
S_{\mathrm{BRST}}{\bar{P}}^{ak}={\tilde{\Phi}}^{ak},\quad
S_{\mathrm{BRST}}{\bar{C}}^{ak}=-B^{ak}$ $\displaystyle
S_{\mathrm{BRST}}P^{A}=S_{\mathrm{BRST}}C^{ak}=S_{\mathrm{BRST}}\Pi^{ak}=S_{\mathrm{BRST}}{P}^{ak}=0$
(94)

This shows that result obtained in section IV is true for any $L(1\leq L<N)$
dimensional surface embedded in $R^{N}$.

##  VIII Batalin - Vilkovisky Quantization

In the current section of the manuscript, we are going to discuss the
quantization of the Particle motion on the surface $V_{L}(1\leq L<N)$ in the
Riemann manifold $R_{N}$ using the field-antifield formalism IABGAV1 ; IABGAV2
; JGJPSS  developed for BFFT systems in RART . We will start by introducing
$4(N-1)$ set of antifields
$\varpi^{k\star}_{\mu}=(x^{\star}_{A},\Theta^{k\star}_{a},\lambda^{k\star}_{a},C^{k\star}_{a})$
corresponding to the fields
$\varpi^{\mu}_{k}=(x^{A},\Theta^{a}_{k},\lambda^{a}_{k},C^{a}_{k})$. Here,
fields $x^{A},\Theta^{ak}$ and $\lambda^{ak}$ are bosonic in nature and have
ghost number zero whereas the ghost fields $C^{ak}$ are fermionic in nature
and have ghost number one. Antifields corresponding to these fields have
opposite Grassmann parity and their ghost numbers are given by minus the ghost
number of the corresponding fields minus one.

BV-action for this system in terms of fields and antifields is written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{0}+\int
dt\big{[}x^{\star}_{A}\\{x^{A},{\tilde{\Phi}}_{a}^{k}\\}C^{a}_{k}+{\Theta}^{\star}_{bk}\\{\Theta^{bk},{\tilde{\Phi}}_{a}^{l}\\}C^{a}_{l}+\lambda^{k\star}_{a}{\dot{C}}^{a}_{k}\big{]}$
(95)

where action $S_{0}$ is defined as

$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle\int
dt\big{[}P_{A}{\dot{x}}^{A}+\Pi_{a}^{k}{\dot{\Theta}}^{a}_{k}-\lambda^{a}_{k}{\tilde{\Phi}}_{a}^{k}-{\tilde{H}}\big{]}$
(96)

Here ${\tilde{\Phi}^{a}_{i}}$ and ${\tilde{H}}$ are modified constraints and
modified Hamiltonian in eqn (49) and eqn (54) respectively. The BV-action
defined in eqn (95) satisfies the classical master equation

$\displaystyle\frac{1}{2}(S,S)=0$ (97)

where the antibracket between any two dynamical variables
$X[\varpi,\varpi^{\star}]$ and $Y[\varpi,\varpi^{\star}]$ is defined as

$\displaystyle(X,Y)=\frac{\delta_{r}X}{\delta\varpi^{\mu}}\frac{\delta_{l}Y}{\delta\varpi^{\star}_{\mu}}-\frac{\delta_{r}X}{\delta\varpi^{\star}_{\mu}}\frac{\delta_{l}Y}{\delta\varpi^{\mu}}$
(98)

Here, de Witt’s notation of sum and integration over intermediary variables
will be assumed, whenever necessary. The BRST differential in is the BV
formalism can be introduced using the relation $sX=(X,S)$ for any local
functional $X[\varpi,\varpi^{\star}]$ of fields. Nilpotentcy of the BRST
operator $s$ can be proved using classical master equation and Jacobi
identity. So, the BV action satisfying the master equation is equivalent to
it’s BRST invariance.

To fix gauges, we will extend the Hilbert space to introduce $4(N-1)$ pairs of
ghost-antighost fields and corresponding momenta
$({\bar{C}}_{a}^{k},P_{a}^{k})$, $(\bar{C^{k\star}_{a}},P^{k\star}_{a})$, as
well as gauge-fixing fermions $\Psi$. These antifields can be easily
eliminated by choosing
$\varpi^{\star}_{\mu}=\frac{\partial\Psi}{\partial\varpi^{\mu}}$. One of the
possible forms of $\Psi$, we can choose for the given system is,

$\displaystyle\Psi={\bar{C}}_{a}^{k}\Theta^{a}_{k}$ (99)

We have liberty to make other possible choices also. Now, we will extend the
BV action defined above to a nonminimal action,

$\displaystyle S\rightarrow S_{nm}=S+\int dtP_{a}^{k}{\bar{C}}^{\star a}_{k}$
(100)

in order to implement the many set of gauge fixing conditions introduced by
$\Psi$. Now, the generating functional for gauge-fixed action is defined as

$\displaystyle{Z_{\psi}}$ $\displaystyle=$
$\displaystyle\int[d\varpi^{\mu}][d\omega]^{-\frac{1}{2}}[df]^{-\frac{1}{2}}\exp{\frac{i}{\hbar}S_{nm}[\varpi^{\mu},\varpi^{\star}_{\mu}=\frac{\partial\Psi}{\partial{\varpi^{\mu}}}]}$
(101)

Now, we will replace the original classical field–antifield action $S$ by some
quantum action $\Sigma$ which is expressed as a local functional of fields and
antifields and also satisfy a new equation called quantum master equation
defined as

$\displaystyle\frac{1}{2}(\Sigma,\Sigma)-i{\hbar}\Delta\Sigma=0$ (102)

then the gauge symmetries of the extended action are not obstructed at quantum
level. Here $\Delta$ act as an operator and is defined as

$\displaystyle\Delta\equiv(\frac{\delta_{r}}{\delta\varpi^{\mu}})(\frac{\delta_{l}}{\delta\varpi^{\star}_{\mu}})$
(103)

It is also assumed here that the quantum action $\Sigma$ can be expanded in
powers of $\bar{h}$ in the following manner,

$\displaystyle\Sigma[\varpi^{\mu},\varpi^{\star}_{\mu}]=S[\varpi^{\mu},\varpi^{\star}_{\mu}]+\sum_{p=1}^{\infty}{\hbar}^{p}M_{p}[\varpi^{\mu},\varpi^{\star}_{\mu}]$
(104)

The first two term of the quantum master eqn (102) reads as,

$\displaystyle(S,S)=0$ $\displaystyle(M_{1},S)=i\Delta S$ (105)

It can be easily observed that if $\Delta S$ is non-zero and gives a
nontrivial result, then there exists some $M_{1}$ which can be expressed in
terms of local fields such that eqn (105) is satisfied. Using the
cohomological arguments, it can be easily shown that the quantum master
equation for the first order systems with pure second class constraints
converted to first class by the use of the BFFT procedure, can always be
solved. BRST transformations for the fields and antifields for the BFFT
Abelianized system can be written as follows,

$\displaystyle S_{\mathrm{BRST}}N^{k\nu}={\dot{C}}^{k\nu},\quad
S_{\mathrm{BRST}}C^{k\nu}=0,\quad
S_{\mathrm{BRST}}{\bar{C}}^{k\nu}=P^{k\nu},\quad S_{\mathrm{BRST}}P^{k\nu}=0$
$\displaystyle S_{\mathrm{BRST}}x^{\star}_{A}=-\frac{\partial S}{\partial
x^{A}},\quad
S_{\mathrm{BRST}}\Theta^{k\star}_{\nu}=-\frac{\partial^{k}S}{\partial{\Theta}^{\nu}},\quad
S_{\mathrm{BRST}}N^{k\star}_{\nu}={\tilde{\Phi}^{k}}_{\nu},\quad
S_{\mathrm{BRST}}{\bar{P}}^{{k\star}\nu}={\bar{C}}^{{k\star}\nu}$
$\displaystyle
S_{\mathrm{BRST}}C^{k\star}_{\nu}=-x^{\star}_{A}\\{x^{A},{\tilde{\Phi}}_{\nu}^{k}\\}-{\Theta}^{i\star}_{l}\\{\Theta^{l}_{i},{\tilde{\Phi}}_{\nu}^{k}\\}-{\dot{N}}^{k\star},\quad
S_{\mathrm{BRST}}\bar{C^{k\star}_{\nu}}=0$ (106)

The symmetry transformations obtained in the eqn (106) are identical to the
one obtained in eqn (63) using BFV formalism. Also, we can easily show on the
basis of argument given in RART  that the enlarged symmetries due to the
compensating fields (BFFT variables) are non-anomalous in nature. These BFFT
fields also plays very significant role at the quantum level because of the
existence of a counterterm, by modifying the expectation values of the
relevant physical quantities.

##  IX Result and Discussion

The BRST symmetry for a particle moving in a curved space $V_{L}(1\leq L<N)$
embedded in a Euclidean space $R_{N}$ is investigated in both Hamiltonian and
Lagrangian formalism. All the constraints of the system have been calculated
using Dirac’s Hamiltonian analysis. Using the algebra of constraints, we have
found that all the constraints of the system are second class. To construct a
gauge invariant theory, we have used the BFFT technique. Using this technique,
all the second class constraints of the system are converted into first class
constraints and corresponding Hamiltonian is also constructed explicitly.
Using the involution of Hamiltonian with first class constraints, this
Hamiltonian is shown to be first class. In the limit of $\Theta\rightarrow 0$
the constraints and Hamiltonian returns to original second class constraints
and Hamiltonian. Now, using this gauge invariant system, we have constructed
BRST charge, symmetries and the BRST invariant action. For constructing BRST
symmetry from the first class constraint system, BFV formalism is used. These
BRST charges acting on the stetes of the total Hilbert space, will annihilate
the physical subspace of it. From it, we have deduced that first class
constraints operating on total Hilbert space of the system will annihilate its
physical subspace which can be used as a physicality criteria for the BRST
invariant system. We have shown that the general results derived here for any
surface embedded in $R^{N}$ is consistent with the results of previous work
VKP . We have also discussed particle motion on the torus knot surface as an
example of this kind of system. In this example, we have explicitly calculated
all the constraints of this system and converted them to first class
constraints. It has been found that diagonal elements $\rho$ and $\epsilon$ of
matrix (of the Poisson’s bracket between the constraints) are nonzero if we
take two different torus knot systems. In the limit of $a\rightarrow b$, the
$\rho,\epsilon$ will vanish and we will achieve the commutative case of VKP .
We have also constructed the first class Hamiltonian, BRST charge and symmetry
for this system. It has been shown that all the results deduced for the
general system are consistent with this system. At the end we have discussed
Batalin - Vilkovisky quantization of this system based on BFFT formalism. Here
also we have explicitly calculated the BRST symmetry for the general system
which is consistent with the symmetry derived from Hamiltonian formalism. This
again proves the equivalence between the Hamiltonian and Lagrangian formalism.
Recently a more general technique of Lagrangian Abelianization has been
developed SLL ; VAASLL . It will be interesting to apply this technique to the
motion in Riemann manifold.

##  X Appendix

The calculation of Poisson bracket between some of the constraints for
particle on the torus knot model is straightforward but cumbersome process. We
have calculated these brackets explicitly here.

The Poisson bracket between constraints $(\Phi_{3}^{a},\Phi_{4}^{b})$ is
written as,

$\displaystyle\\{\Phi_{3}^{a},\Phi_{4}^{b}\\}_{P}=2\nabla(DQ^{a})\cdot\nabla(DQ^{b})-\nabla
Q^{a}\cdot\nabla\Phi_{4}^{b}=2\frac{(\cosh\eta-\cos\theta)^{4}}{m^{3}a^{6}}\\{\sinh\eta(pp_{\theta}+\frac{q\phi}{\sinh^{2}\eta})$
$\displaystyle-\frac{qp_{\phi}\cosh\eta(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}\\}\cdot\\{pp_{\theta}\sinh\eta+\frac{qp_{\phi}}{\sinh\eta}-\frac{qp_{\phi}\cosh\eta(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}-pp_{\eta}\sin\theta\\}$
$\displaystyle+4\frac{(\cosh\eta-\cos\theta)^{4}}{m^{3}a^{6}}\sin\theta(pp_{\theta}+\frac{q\phi}{\sinh^{2}\eta})\\{pp_{\eta}\sinh\eta+\sin\theta(pp_{\theta}+\frac{q\phi}{\sinh^{2}\eta})\\}-p\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}$
$\displaystyle\big{[}2\sin\theta\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\cdot\\{\\{2pp_{\theta}\sinh{\eta}+\frac{2qp_{\phi}}{\sinh{\eta}}-\frac{2qp_{\phi}\cosh{\eta}(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}\\}p_{\eta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}$
$\displaystyle+2\sin{\theta}(pp_{\theta}+\frac{q\phi}{\sinh^{2}\eta})p_{\theta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}-p\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\\{\sin{\theta}({p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{p^{2}_{\phi}}{\sinh^{2}\eta})-\lambda
p\\}$
$\displaystyle+\lambda\frac{q^{2}}{{\sinh^{2}\eta}}\\}+\frac{(\cosh\eta-\cos\theta)^{2}}{ma^{2}}\\{-2q\sin\theta
p_{\phi}\frac{\cosh\eta}{\sinh^{3}\eta}\cdot
p_{\eta}\frac{(\cosh\eta-\cos\theta)}{ma^{2}}\\}+\\{2pp_{\theta}\sinh{\eta}$
$\displaystyle+\frac{2qp_{\phi}}{\sinh{\eta}}-\frac{2qp_{\phi}\cosh{\eta}(\cosh\eta-\cos\theta)}{\sinh^{3}{\eta}}\\}\frac{p_{\eta}\sin\theta}{ma^{2}}+2p_{\theta}\cos\theta(pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta})\frac{(\cosh\eta-\cos\theta)}{ma^{2}}$
$\displaystyle+2\sin\theta(pp_{\theta}+\frac{qp_{\phi}}{\sinh^{2}\eta})\frac{p_{\theta}\sin\theta}{ma^{2}}-\frac{p_{\theta}\sin\theta}{ma^{2}}\\{\sin\theta({p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{p^{2}_{\phi}}{\sinh^{2}\eta})-\lambda
p\\}-p\cos\theta$
$\displaystyle\frac{(\cosh\eta-\cos\theta)}{ma^{2}}({p^{2}_{\eta}}+{p^{2}_{\theta}}+\frac{p^{2}_{\phi}}{\sinh^{2}\eta})\big{]}\equiv-\gamma^{ab}$
(107)

Poisson bracket $(\Phi_{3}^{a},\Phi_{3}^{b})$ is written as,

$\displaystyle\\{\Phi_{3}^{a},\Phi_{3}^{b}\\}_{P}=\nabla(DQ^{a})\cdot\nabla
Q^{b}-\nabla Q^{a}\cdot\nabla(DQ^{b})$
$\displaystyle=\\{2\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}(p{p_{\theta^{a}}}+\frac{q\phi^{a}}{\sinh^{2}\eta^{a}})\cdot
p^{\prime}\frac{(\cosh\eta^{b}-\cos\theta^{b})^{2}}{ma^{2}}$
$\displaystyle-p\frac{(\cosh\eta^{a}-\cos\theta^{a})^{2}}{ma^{2}}\cdot
2\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}(p^{\prime}p_{\theta^{b}}+\frac{q^{\prime}\phi^{b}}{\sinh^{2}\eta^{b}})\\}\equiv\rho^{ab}$
(108)

Similarly for $(\Phi_{4}^{a},\Phi_{4}^{b})$ can be written as,

$\displaystyle\\{\Phi_{4}^{a},\Phi_{4}^{b}\\}_{P}=2\left[\nabla\Phi_{4}^{a}\cdot\nabla(DQ^{a})-\nabla\Phi_{4}^{b}\cdot\nabla(DQ^{a})\right]=2\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}\sinh\eta^{a}[\\{2pp_{\theta^{a}}\sinh{\eta^{a}}$
$\displaystyle+\frac{2qp_{\phi^{a}}}{\sinh\eta^{a}}-\frac{2qp_{\phi^{a}}\cosh{\eta^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{3}{\eta^{a}}}\\}p_{\eta^{a}}\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}+2\sin{\theta^{a}}(pp_{\theta^{a}}+\frac{q\phi^{a}}{\sinh^{2}{\eta^{a}}})p_{\theta}$
$\displaystyle\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}-p\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}\\{\sin{\theta^{a}}({p^{2}_{\eta^{a}}}+{p^{2}_{\theta^{a}}}+\frac{p^{2}_{\phi^{a}}}{\sinh^{2}{\eta^{a}}})-\lambda^{a}p\\}+\lambda^{a}\frac{q^{2}}{\sinh^{2}{\eta^{a}}}]$
$\displaystyle+\frac{(\cosh\eta^{a}-\cos\theta^{a})^{2}}{ma^{2}}[\\{2pp_{\theta^{a}}\cosh{\eta^{a}}-\frac{4qp_{\phi^{a}}\cosh{\eta^{a}}}{\sinh^{2}{\eta^{a}}}-\frac{2qp_{\phi^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{2}{\eta^{a}}}$
$\displaystyle+6\frac{qp_{\phi^{a}}\cosh^{2}{\eta^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{4}{\eta^{a}}}\\}p_{\eta^{a}}\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}+\frac{p_{\eta^{a}}\sinh\eta^{a}}{ma^{2}}\\{2pp_{\theta^{a}}\sinh{\eta^{a}}+\frac{2qp_{\phi^{a}}}{\sinh{\eta^{a}}}$
$\displaystyle-\frac{2qp_{\phi^{a}}\cosh{\eta^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{3}{\eta^{a}}}\\}-4\sin\theta^{a}(\frac{2qp_{\phi^{a}}\cosh\eta^{a}}{\sinh^{3}{\eta^{a}}})\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}p_{\theta^{a}}+2\sin\theta^{a}(pp_{\theta^{a}}$
$\displaystyle+\frac{qp_{\phi^{a}}}{\sinh^{2}\eta^{a}})\frac{\sinh\eta^{a}}{ma^{2}}p_{\theta}-p\frac{\sinh\eta^{a}}{ma^{2}}\\{\sin{\theta^{a}}({p^{2}_{\eta^{a}}}+{p^{2}_{\theta^{a}}}+\frac{p^{2}_{\phi^{a}}}{\sinh^{2}{\eta^{a}}})-\lambda^{a}p\\}+2p\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}$
$\displaystyle\\{\sin\theta^{a}\frac{p^{2}_{\phi^{a}}\cosh\eta^{a}}{\sinh^{3}{\eta^{a}}}\\}-2\lambda^{a}\frac{q^{2}\cosh\eta^{a}}{\sinh^{3}{\eta^{a}}}]\cdot
2\frac{(\cosh\eta^{b}-\cos\theta^{b})^{3}}{m^{2}a^{4}}\\{pp_{\theta^{b}}\sinh\eta^{b}+\frac{qp_{\phi^{b}}}{\sinh\eta^{b}}$
$\displaystyle-\frac{qp_{\phi^{b}}\cosh\eta^{b}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{3}{\eta}}-pp_{\eta^{b}}\sin\theta^{b}\\}-2\frac{(\cosh\eta^{a}-\cos\theta^{a})^{3}}{m^{2}a^{4}}\\{pp_{\theta^{a}}\sinh\eta^{a}+\frac{qp_{\phi^{a}}}{\sinh\eta^{a}}$
$\displaystyle-\frac{qp_{\phi^{a}}\cosh\eta^{a}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{3}{\eta^{a}}}-pp_{\eta^{a}}\sin\theta^{a}\\}\cdot
2\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}\sinh\eta^{b}[\\{2pp_{\theta^{b}}\sinh{\eta^{b}}$
$\displaystyle+\frac{2qp_{\phi^{b}}}{\sinh\eta^{b}}-\frac{2qp_{\phi^{b}}\cosh{\eta^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{3}{\eta^{b}}}\\}p_{\eta^{b}}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}+2\sin{\theta^{b}}(pp_{\theta^{b}}+\frac{q\phi^{b}}{\sinh^{2}{\eta^{b}}})p_{\theta^{b}}$
$\displaystyle\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}-p\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}\\{\sin{\theta^{b}}({p^{2}_{\eta^{b}}}+{p^{2}_{\theta^{b}}}+\frac{p^{2}_{\phi^{b}}}{\sinh^{2}{\eta^{b}}})-\lambda^{b}p\\}+\lambda^{b}\frac{q^{2}}{\sinh^{2}{\eta^{b}}}]$
$\displaystyle+\frac{(\cosh\eta^{b}-\cos\theta^{b})^{2}}{ma^{2}}[\\{2pp_{\theta^{b}}\cosh{\eta^{b}}-\frac{4qp_{\phi^{b}}\cosh{\eta^{b}}}{\sinh^{2}{\eta^{b}}}-\frac{2qp_{\phi^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{2}{\eta^{b}}}$
$\displaystyle+6\frac{qp_{\phi^{b}}\cosh^{2}{\eta^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{4}{\eta^{b}}}\\}p_{\eta^{b}}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}+\frac{p_{\eta^{b}}\sinh\eta^{b}}{ma^{2}}\\{2pp_{\theta^{b}}\sinh{\eta^{b}}+\frac{2qp_{\phi^{b}}}{\sinh{\eta^{b}}}$
$\displaystyle-\frac{2qp_{\phi^{b}}\cosh{\eta^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{3}{\eta^{b}}}\\}-4\sin\theta^{b}(\frac{2qp_{\phi^{b}}\cosh\eta^{b}}{\sinh^{3}{\eta^{b}}})\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}p_{\theta^{b}}+2\sin\theta^{b}(pp_{\theta^{b}}$
$\displaystyle+\frac{qp_{\phi^{b}}}{\sinh^{2}\eta^{b}})\frac{\sinh\eta^{b}}{ma^{2}}p_{\theta^{b}}-p\frac{\sinh\eta^{b}}{ma^{2}}\\{\sin{\theta^{b}}({p^{2}_{\eta^{b}}}+{p^{2}_{\theta^{b}}}+\frac{p^{2}_{\phi^{b}}}{\sinh^{2}{\eta^{b}}})-\lambda^{b}p\\}+2p\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}$
$\displaystyle\\{\sin\theta\frac{p^{2}_{\phi^{b}}\cosh\eta^{b}}{\sinh^{3}{\eta^{b}}}\\}-2\lambda^{b}\frac{q^{2}\cosh\eta^{b}}{\sinh^{3}{\eta^{b}}}]+[2\sin\theta^{a}\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}\cdot\\{\\{2pp_{\theta^{a}}\sinh{\eta^{a}}+\frac{2qp_{\phi^{a}}}{\sinh{\eta^{a}}}$
$\displaystyle-\frac{2qp_{\phi^{a}}\cosh{\eta^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{3}{\eta^{a}}}\\}p_{\eta^{a}}\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}+2\sin{\theta^{a}}(pp_{\theta^{a}}+\frac{q\phi^{a}}{\sinh^{2}\eta^{a}})p_{\theta^{a}}$
$\displaystyle\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}-p\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}\\{\sin{\theta^{a}}(p^{2}_{\eta^{a}}+p^{2}_{\theta^{a}}+\frac{p^{2}_{\phi^{a}}}{\sinh^{2}\eta^{a}})-\lambda^{a}p\\}+\lambda^{a}\frac{q^{2}}{{\sinh^{2}\eta^{a}}}\\}$
$\displaystyle+\frac{(\cosh\eta^{a}-\cos\theta^{a})^{2}}{ma^{2}}\\{-2q\sin\theta^{a}p_{\phi^{a}}\frac{\cosh\eta^{a}}{\sinh^{3}\eta^{a}}\cdot
p_{\eta^{a}}\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}\\}+\\{2pp_{\theta^{a}}\sinh{\eta^{a}}+\frac{2qp_{\phi^{a}}}{\sinh{\eta^{a}}}$
$\displaystyle-\frac{2qp_{\phi^{a}}\cosh{\eta^{a}}(\cosh\eta^{a}-\cos\theta^{a})}{\sinh^{3}{\eta^{a}}}\\}\frac{p_{\eta^{a}}\sin\theta^{a}}{ma^{2}}+2p_{\theta^{a}}\cos\theta^{a}(pp_{\theta^{a}}+\frac{qp_{\phi^{a}}}{\sinh^{2}\eta^{a}})\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}$
$\displaystyle+2\sin\theta^{a}(pp_{\theta^{a}}+\frac{qp_{\phi^{a}}}{\sinh^{2}\eta^{a}})\frac{p_{\theta^{a}}\sin\theta^{a}}{ma^{2}}-\frac{p_{\theta^{a}}\sin\theta^{a}}{ma^{2}}\\{\sin\theta^{a}(p^{2}_{\eta^{a}}+p^{2}_{\theta^{a}}+\frac{p^{2}_{\phi^{a}}}{\sinh^{2}\eta^{a}})-\lambda^{a}p\\}-p\cos\theta^{a}$
$\displaystyle\frac{(\cosh\eta^{a}-\cos\theta^{a})}{ma^{2}}(p^{2}_{\eta^{a}}+p^{2}_{\theta^{a}}+\frac{p^{2}_{\phi^{a}}}{\sinh^{2}\eta^{a}})\big{]}\cdot
2\frac{(\cosh\eta^{b}-\cos\theta^{b})^{3}}{m^{2}a^{4}}\\{pp_{\eta^{b}}\sinh\eta^{b}+pp_{\theta^{b}}+\frac{qp_{\phi^{b}}\sin\theta^{b}}{\sinh^{2}\eta^{b}}\\}$
$\displaystyle-2\frac{(\cosh\eta^{a}-\cos\theta^{a})^{3}}{m^{2}a^{4}}\\{pp_{\eta^{a}}\sinh\eta^{a}+pp_{\theta^{a}}+\frac{qp_{\phi^{a}}\sin\theta^{a}}{\sinh^{2}\eta^{a}}\\}\cdot\big{[}2\sin\theta^{b}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}\cdot\\{\\{2pp_{\theta^{b}}$
$\displaystyle\sinh{\eta^{b}}+\frac{2qp_{\phi}^{b}}{\sinh\eta^{b}}-\frac{2qp_{\phi^{b}}\cosh{\eta^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{3}{\eta^{b}}}\\}p_{\eta}^{b}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}+2\sin{\theta^{b}}(pp_{\theta^{b}}+\frac{q\phi^{b}}{\sinh^{2}\eta^{b}})$
$\displaystyle
p_{\theta}^{b}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}-p\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}\\{\sin{\theta^{b}}(p^{2}_{\eta^{b}}+p^{2}_{\theta^{b}}+\frac{p^{2}_{\phi^{b}}}{\sinh^{2}\eta^{b}})-\lambda^{b}p\\}+\lambda^{b}\frac{q^{2}}{{\sinh^{2}\eta^{b}}}\\}$
$\displaystyle+\frac{(\cosh\eta^{b}-\cos\theta^{b})^{2}}{ma^{2}}\\{-2q\sin\theta^{b}p_{\phi^{b}}\frac{\cosh\eta^{b}}{\sinh^{3}\eta^{b}}\cdot
p_{\eta^{b}}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}\\}+\\{2pp_{\theta^{b}}\sinh{\eta^{b}}+\frac{2qp_{\phi^{b}}}{\sinh{\eta^{b}}}$
$\displaystyle-\frac{2qp_{\phi^{b}}\cosh{\eta^{b}}(\cosh\eta^{b}-\cos\theta^{b})}{\sinh^{3}{\eta^{b}}}\\}\frac{p_{\eta^{b}}\sin\theta^{b}}{ma^{2}}+2p_{\theta^{b}}\cos\theta^{b}(pp_{\theta^{b}}+\frac{qp_{\phi^{b}}}{\sinh^{2}\eta^{b}})\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}$
$\displaystyle+2\sin\theta^{b}(pp_{\theta^{b}}+\frac{qp_{\phi^{b}}}{\sinh^{2}\eta^{b}})\frac{p_{\theta^{b}}\sin\theta^{b}}{ma^{2}}-\frac{p_{\theta^{b}}\sin\theta^{b}}{ma^{2}}\\{\sin\theta^{b}(p^{2}_{\eta^{b}}+p^{2}_{\theta^{b}}+\frac{p^{2}_{\phi^{b}}}{\sinh^{2}\eta^{b}})-\lambda^{b}p\\}$
$\displaystyle-p\cos\theta^{b}\frac{(\cosh\eta^{b}-\cos\theta^{b})}{ma^{2}}(p^{2}_{\eta^{b}}+p^{2}_{\theta^{b}}+\frac{p^{2}_{\phi^{b}}}{\sinh^{2}\eta^{b}})\big{]}\equiv\epsilon^{ab}$
(109)

##  XI References

## References

  * (1) V. K. Pandey, Advances in High Energy Physics, vol. 2022, Article ID 2158485, 14 pages, 2022.
  * (2) Y. Asanaga, Introduction to Riemann Geometry, Kyoritsu Zennsho No. 182, Tokyo (1978).
  * (3) H. E. Lin, W. C. Lin and R. Sugano, Nucl. Phys. B 16, 43l (1970).
  * (4) R. Sugano, Prog. Theor. Phys. 46, 297 (1971).
  * (5) T. Kimura, Prog. Theor. Phys. 46, 1261 (1971).
  * (6) T. Kimura and R. Sugano, Prog. Theor. Phys. 47, 1004 (1972).
  * (7) T. Ohtani and R. Sugano, Prog. Theor. Phys. 47, 1704 (1972).
  * (8) T. Ohtani and R. Sugano, Prog. Theor. Phys. 50, 1715 (1973).
  * (9) T. Kimura, T. Ohtani and R. Sugano, Prog. Theor. Phys. 48, 1395 (1972).
  * (10) M. Omote and H. Sato, Prog. Theor. Phys. 47, 1367 (1972).
  * (11) K. Fujii, K-I. Sato, N. Toyota and A. P. Kobushukin, Phys. Rev. Lett. 58, 651 (1987).
  * (12) K. Fujii, A. Kobushukin, K-I. Sato and N. Toyota, Phys. Rev. D 37, 3663 (1988).
  * (13) T. Kawai, Foundation of Phys. 143 (1975).
  * (14) H. Kamo and T. Kawai, Nucl. Phys. B 81, 349 (1974).
  * (15) J. L. Garvais and A. Neveu, Phys. Rep. 23, 237 (1976).
  * (16) M. Omote, Nucl. Phys. B 120, 325 (1977).
  * (17) B. Sakita and K. Kikkawa, Quantum Mechanics of Many Degrees of Freedom Basing on Path Integral (Iwanami, Tokyo, 1986).
  * (18) B. S. Dewitt, Phys. Rev. 85, 653 (1952).
  * (19) V. De Alearo, S. Fubini, G. Furlan and M. Roncadelli, Nucl. Phys. B 296, 402 (1988).
  * (20) N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge (1982).
  * (21) L. Parker and D. Toms. Quantum Field Theory in Curved Spacetime, (2009).
  * (22) I. A. Batalin, M. A. Grigoriev and S. L. Lyakhovich, Theoretical and Mathematical Physics, 128, 1109 (2001).
  * (23) I. A. Batalin, M. A. Grigoriev and S. L. Lyakhovich, J. Math. Phys. 46, 072301 (2005).
  * (24) I.V. Gorbunov, S.L. Lyakhovich, A.A. Sharapov, J. Geom. and Phys. 53, 98 (2005).
  * (25) N. Ogawa, K. Fujii, A. Kobushukin, Prog. Theo. Phys. 83, 894 (1990).
  * (26) N. Ogawa, K. Fujii, N. Chepilko and A. Kobushkin, Prog. Theo. Phys. 85, 1189 (1991).
  * (27) N. Ogawa, Phys. Rev. D 62, 085023 (2000).
  * (28) M. Ikegami, Y. Nagaoka, S. Takagi and T. Tanzawa, Prog. of Theor. Phys., 88(2), 229 (1992).
  * (29) S. Takagi and T. Tanzawa, Prog. of Theor. Phys. 87(3), 561 (1992).
  * (30) C. Filgueiras and F.Moraes, Annals Phys. 323, 3150 (2008).
  * (31) N. Ogawa, Phys. Rev. E 81, 061113 (2010).
  * (32) K. Fujii, N. Ogawa, S. Uchiyama and N. Chepilko, Int. J. of Mod. Phys. A 12, 5235 (1997).
  * (33) N. Chepilko, K. Fujii, and A. Kobushkin, Phys. Rev. D 44, 3249 (1991).
  * (34) A. Saa, Class and Quan. Grav. 14, 385 (1997).
  * (35) D. Lian, L. Hu and Q. Liu, Ann. Phys. 530, 1700415 (2018).
  * (36) N. Ogawa, Mod. Phys. Lett. A 12, 1583 (1997) 1583.
  * (37) N. Ogawa, arxiv:hep-th/9703181.
  * (38) N. Okamoto and M. Nakamura, Prog. Theo Phys. 96, 235 (1996).
  * (39) C. Destri, P. Maraner and E. Onofri, II Nuovo Cimento A (1965-1970) 107, 237 (1994) 237.
  * (40) A. Foerster, H. O. Girotti and P. S. Kuhn, Phys. Lett. A 195, 301 (1994).
  * (41) A. V. Golovnev, Rept. Math. Phys. 6, 459 (2009).
  * (42) K. Fujii, N. Ogawa, Prog. Theo. Phys. 89, 575 (1993).
  * (43) M. Nakamura, N. Okamoto and H. Minowa, arXiv:hep-th/9710232.
  * (44) M. Nakamura, N. Okamoto and H. Minowa, II Nuovo Cimento B (1971-1996) 111, 521 (1996).
  * (45) M. Nakamura, arXiv:1503.06541.
  * (46) P. Maraner, J. Phys. A 28, 2939 (1995).
  * (47) P. Maraner, Annals Phys., 325 246 (1996).
  * (48) N. Chepilko and A. Romanenko, Eur. Phys. J. C 21, 369 (2001).
  * (49) N. Chepilko and A. Romanenko, Eur. Phys. J. C 21, 587 (2001).
  * (50) N. Chepilko and A. Romanenko, Eur. Phys. J. C 21, 757 (2001).
  * (51) S. L. de Oliveira, C. M. B. Santos and R. Thibes, Braz. J. Phys. B, 50 (2020) 480.
  * (52) C. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52, 344 (1974).
  * (53) C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42, 127 (1975).
  * (54) C. Becchi, A. Rouet and R. Stora, Ann. Phys. (N. Y.) 98, 287 (1976).
  * (55) I. V. Tyutin, Lebedev Report N FIAN, 39 (1975), arXiv:0812.0580 (2008).
  * (56) P. A. M. Dirac, Can. J. Math. 2, 129 (1950).
  * (57) J. L. Anderson and P. G. Bergmann, Phys. Rev. 83, 1018 (1951).
  * (58) P. A. M. Dirac, Lectures on Quantum Mechanics ,Yeshiva University, New York (1964).
  * (59) K. Sundermeyer, Constrained Dynamics, Lecture notes in Physics, vol. 169 (Springer,Berlin, 1982).
  * (60) M. Henneaux and C. Teitelboim, Quantization of Gauge System, Princeton University Press, Princeton (1992).
  * (61) M. Henneaux, Phys. Rep. 126, 1 (1985).
  * (62) L. D. Faddeev and S. L. Shatashvili, Phys. Lett. B 167, 225 (1986).
  * (63) I. A. Batalin and E. S. Fradkin, Phys. Lett. B 180, 157 (1986).
  * (64) I. A. Batalin and E. S. Fradkin, Nucl. Phys. B 279, 514 (1987).
  * (65) I. A. Batalin, E. S. Fradkin, and T. E. Fradkina, Nucl. Phys. B 314, 158 (1989)(1989)[Erratum: Nucl. Phys. B 323, 734 (1989)].
  * (66) I. A. Batalin and I. V. Tyutin, Int. J. Mod. Phys. A 6, 3255 (1991).
  * (67) E. S. Egorian and R. P. Manvelyan, Theor. Math. Phys. 94, 173 (1993).
  * (68) R. Banerjee and J. Barcelos-Neto, Nucl. Phys. B 499, 453 (1997).
  * (69) S. Ghosh, J. Math. Phys. 42, 5202 (2001).
  * (70) C. Burdik, A. Pashnev and M. Tsulaia, Mod. Phys. Lett. A 16, 731 (2001).
  * (71) I. L. Buchbinder, V. K. Krykhtin and A. A. Reshetnyak, Nucl. Phys. B 787, 211 (2007).
  * (72) I. L. Buchbinder and A. A. Reshetnyak, Nucl. Phys. B 862, 270 (2012).
  * (73) A. A. Reshetnyak, JHEP 1809, 104 (2018).
  * (74) I. L. Buchbinder and A. A. Reshetnyak, Phys. Lett. B 820, 136470 (2021).
  * (75) V. K. Pandey and R. Thibes, arxiv: 2103.05626[hep-th].
  * (76) B. P. Mandal, V. K. Pandey and R. Thibes, arXiv:2202.00715.
  * (77) E. S. Fradkin and G. Vilkovisky, Phys. Lett. B 55, 224 (1975).
  * (78) I. A. Batalin and G. Vilkovisky, Phys. Lett. B 69, 309 (1977).
  * (79) I. A. Batalin and E. S. Fradkin, Phys. Lett. B 122, 157 (1983).
  * (80) L. D. Faddeev, Theor. Math. Phys. 1, (1970) 1.
  * (81) P. Senjanovic, Ann. Phys. 100, 277 (1976); Ann. Phys. 209, 248 (1991) (erratum).
  * (82) V. V. Sreedhar, Ann. Phys., 359 (2015) 20.
  * (83) P. Das and S. Ghosh, Found. of Phys. 46, 1649 (2016).
  * (84) V. K. Pandey and B. P. Mandal, Euro Phys. Lett., 119, 31003 (2017).
  * (85) V. K. Pandey and B. P. Mandal, Springer Proc. in Phys. 203 (2018) 513.
  * (86) Anjali S and Saurabh Gupta, arXiv:2108.08788.
  * (87) I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B. 102, 27 (1981).
  * (88) I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D 28, 2567 (1983); Erratum-ibid: Phys. Rev. D 30, 508 (1984).
  * (89) J. Gomis, J. Paris and S. Samuel, Phys. Rep. 259, 1 (1995).
  * (90) R. Amorim and R. Thibes, J. Math. Phys. 40, 5306 (1999).
  * (91) S.L. Lyakhovich, Eur.Phys.J.C 81, 472 (2021).
  * (92) V.A. Abakumova and S.L. Lyakhovich, Phys.Lett.B 820, 136552 (2021).
  * (93) V. K. Pandey, arxiv: 2008.02112 [hep-th].


0.8pt

11institutetext: CDT Biomedical AI, School of Informatics, University of
Edinburgh
11email<EMAIL_ADDRESS>22institutetext: Centre for Medical
Informatics, University of Edinburgh 33institutetext: School of Informatics,
University of Edinburgh

# Robust and efficient computation of retinal fractal dimension through deep
approximation

Justin Engelmann 11 Ana Villaplana-Velasco 22 Amos Storkey Equal
supervision.33
Miguel O. Bernabeu* 22

###### Abstract

A retinal trait, or phenotype, summarises a specific aspect of a retinal image
in a single number. This can then be used for further analyses, e.g. with
statistical methods. However, reducing an aspect of a complex image to a
single, meaningful number is challenging. Thus, methods for calculating
retinal traits tend to be complex, multi-step pipelines that can only be
applied to high quality images. This means that researchers often have to
discard substantial portions of the available data. We hypothesise that such
pipelines can be approximated with a single, simpler step that can be made
robust to common quality issues. We propose Deep Approximation of Retinal
Traits (DART) where a deep neural network is used predict the output of an
existing pipeline on high quality images from synthetically degraded versions
of these images. We demonstrate DART on retinal Fractal Dimension (FD)
calculated by VAMPIRE, using retinal images from UK Biobank that previous work
identified as high quality. Our method shows very high agreement with
FDVAMPIRE on unseen test images (Pearson $r=0.9572$). Even when those images
are severely degraded, DART can still recover an FD estimate that shows good
agreement with FDVAMPIRE obtained from the original images (Pearson
$r=0.8817$). This suggests that our method could enable researchers to discard
fewer images in the future. Our method can compute FD for over 1,000img/s
using a single GPU. We consider these to be very encouraging initial results
and hope to develop this approach into a useful tool for retinal analysis.

###### Keywords:

Retinal fractal dimension Deep approximation of retinal traits Robust retinal
image analysis.

##  1 Introduction

Retinal fundus images are non-invasive and low-cost. They are important for
ophthalmology and also capture a detailed picture of the retinal vasculature.
Thus, they can be used for studying and potentially predicting diseases such
as diabetes, stroke, hypertension and neurovascular disease [10]. To analyse
the relationships between aspects of the retina and other quantities of
interest, retinal traits (also called features, parameters or phenotypes) are
used as a quantitative description of a specific aspect of the retinal image.
Reducing a complex image to a single, meaningful number is necessary to use
standard statistical methods yet a challenging task. It is challenging to
identify a potentially salient aspect of the retina in the first place and to
then design a method that can reliably quantify this aspect. This is further
complicated by the large variability in retinal images stemming from
idiosyncrasies of the imaged retinas (e.g. due to retinal diseases or rare
phenotypes) and image quality (e.g. due to operator inexperience or time
pressures in large scale cohort studies). Thus, pipelines for extracting such
retinal traits tend to be complex and comprise of multiple steps, and can only
be applied to images of sufficient quality.

Poor image quality is a key problem in retinal image analysis. Particularly
for large scale studies such as UK Biobank, many images are of poor quality
being blurred, obscured, or hazy [9]. Imaging artefacts such as noise, non-
uniform illumination or blur can also lead to poor vessel segmentations [12].
Previous work analysing 2,690 UK Biobank participants found that only 60% had
an image that could be adequately analysed by VAMPIRE [9]. Two recent large-
scale studies using retinal Fractal Dimension (FD) for predicting
cardiovascular disease risk discarded 26% [21] and 43% [16] of the images in
UK Biobank. Although necessary, this is unfortunate as it leads to lower
sample sizes and makes it hard to study rare diseases in particular.

Figure 1: Overview of our proposed framework. a) A typical pipeline for
computing FD: an encoder-decoder neural network for segmentation, potentially
some refinement steps like optic disc segmentation and removal, and a method
to calculate FD of the segmentation (e.g. box counting or multifractal). b)
DART, our proposed approach outputs a deep approximation of FD in a single
step using an encoder-only neural network, with drastically reduced
complexity. c) We can train our model to be robust to image quality issues by
synthetically degrading input images and training our model to minimise the
loss between its output and the FD obtained with the original high quality
image.

We hypothesise that it is possible to approximate pipelines for calculating
retinal traits with a single, simpler step and propose Deep Approximation of
Retinal Traits (DART). Fig. 1 gives a high-level overview of our approach.
DART trains a deep neural network (DNN) to predict the output of an original
method (OM) for calculating a retinal trait. We can then train the model to be
robust to image quality issues by synthetically degrading the input images
during training and asking the DNN model to predict the output of the OM on
the original high quality image. The intuition behind this approach is that
obtaining a high quality segmentation of the entire retina is a much harder
task than describing an aspect of the vasculature like vascular complexity
directly. DART offers a segmentation-free way of computing retinal traits
related to the vasculature, but can also be applied to any other retinal image
analysis method like feature extraction for disease grading or pathology
segmentation.

In the present work, we focus on retinal Fractal Dimension (FD), a key retinal
trait that has been used to predict cardiovascular disease risk [16, 21] and
is associated with neurodegeneration and stroke [6]. We use FD as calculated
by VAMPIRE [15] with the multifractal [14] method as the OM we apply DART to.
At minimum, FDDART should have very high agreement with FDVAMPIRE on high
quality images so that it can be interpreted in the same way. To be a useful
method, it should further be robust to image quality issues and efficient.
Robustness would enable researchers to discard fewer images than currently
necessary while efficiency allows to conduct analyses at large scale without
requiring large compute resources.

##  2 Deep Approximation of Retinal Traits (DART)

###  2.1 Motivation and theory

We hypothesise that it is possible to approximate the entire pipeline of an
original method (OM) for calculating a retinal trait in a single, simpler
step. We denote the distribution of high quality retinal fundus images as
$X^{HQ}$, where each image $x_{i}$ has dimensions height H, width W, and
channels C. The OM can be interpreted as a function $f$ that maps from the
image space to one-dimensional retinal trait space (in our case, FD)
$f:\mathbb{R}^{HxWxC}\to\mathbb{R}^{1}$, i.e. given an image $x_{i}\in X^{HQ}$
the FD computed by the OM is $FD^{OM}=f(x_{i})$. Our goal is to find an
alternative function $g:\mathbb{R}^{HxWxC}\to\mathbb{R}^{1}$ that is both
simpler than $f$ and has high agreement with $f$ for all images of sufficient
quality that the OM can be used, i.e. for all $x_{i}\in X^{HQ}$
$f(x_{i})\approx g(x_{i})$.

Designing such a simpler function by hand would be very challenging. Thus, we
use a deep neural network (DNN). DNNs are universal function approximators in
theory and very effective for image analysis in practice. We can then find a
good approximation of $f$ by simply updating the model parameters $\theta$
(weights, biases, normalisation layer parameters) to minimise some
differentiable measure of divergence between $f(x_{i})$ and $g(x_{i})$, e.g.
mean squared error.

####  2.1.1 Accuracy

The output of the OM is fully determined by the given image, so we would
expect that very high accuracy can be achieved. This contrasts with other
problems, e.g. clincians take into account additional information like
symptoms and family history, and might disagree with each other or even
themselves if shown the same image multiple times.

####  2.1.2 Simplicity & Efficiency

Some readers might not perceive DNNs as simple or efficient. However, modern
pipelines for retinal image analysis tend to use DNNs for vessel segmentation,
so not requiring additional steps implies strictly lower complexity both
computationally and in terms of required code. Furthermore, segmentation
models tend to have an encoder-decoder structure (e.g. UNet) whereas models
for classification/regression only need an encoder and small prediction head,
making them more parameter-, memory-, and compute-efficient. Finally, given
the widespread adoption of deep learning, the frameworks are very mature and
can be very efficiently GPU-accelerated.

####  2.1.3 Robustness

We hypothesise that there images of lower quality that are such that a)
current pipelines would not produce a useful FD number, but b) there is still
sufficient information to give an accurate estimate of the FD number we would
have obtained on a counterfactual high quality image. For example, in an image
with an obstruction, only part of the retina might be visible. Thus, the
resulting vessel segmentation map would be poor and the FD of this map would
be very different from that of the counterfactual high quality image, yet the
visible parts of the retina might contain sufficient information about the
vascular complexity of the retina as a whole to recover an accurate estimate
of the FD.

As we do not observe counterfactual high quality images or objective ground
truth FD values, we artificially degrade high quality images with a
degradation function $\mathrm{degrade}(x_{i})=x_{i}^{\mathrm{degarded}}$ and
train our model to minimise the difference between the predicted FD for the
degraded image and the OM’s FD for the high quality image
$g_{\theta}(x_{i}^{\mathrm{degarded}})\approx f(x_{i})$. If there indeed is
sufficient information in the degraded images, then our model should be able
to predict the OM’s FD from the high quality image reasonably well. However,
this is a much harder task than matching the OM on high quality images, as the
degradations lose information and for a given degraded image there are
multiple possible counterfactual high quality images.

###  2.2 Implementation

####  2.2.1 Model & Training

Our model consists of a pretrained ResNet18 [4] backbone that extracts a
feature map from the images, followed by spatial average pool and a small
multi-layer perceptron with a two hidden layers with 128 and 32 units, and a
single output. Each hidden layer is followed by a layernorm [1] and GELU [5]
activation. No activation is applied to the final output. ResNet is a well-
established architecture that has been shown to perform competitively with
more recent architectures when using modern training techniques [2, 19]. We
use Resnet18 as it is the most light-weight member of the Resnet family. We
initialise the backbones with pre-trained weights on natural images from
Instagram [20]. Those images are very different from retinal images, thus this
is merely a minor refinement on random initialisation. We resize images to
224x224 pixels for computational efficiency and lower memory requirements.
Apart from standard normalisation using channel-wise ImageNet mean and
standard deviations, no further preprocessing is done and all 3 colour
channels are kept.

We train our model using a batchsize of 256 to minimise the mean squared error
between prediction and target after normalizing the target to zero mean and
unit variance, using mean and standard deviation from the training data to
avoid data leakage. The model output can then be mapped back to FD range by
applying the inverse transformation. We use the AdamW optimiser [8]
($\beta_{1}=0.9,\beta_{2}=0.999$, weight decay of $10^{-6}$) and a cosine
learning rate schedule [7]. We train for 35 epochs with a linear learning rate
warmup from $\eta_{min}=10^{-5}$ to $\eta_{max}=10^{-3}$ for 5 epochs,
followed by 3 cycles of 10 epochs each. During each cycle, the current epoch
learning rate is set according to a cosine schedule, and after each cycle
$\eta_{max}$ is decayed by taking the square root. We apply generic data
augmentations (horizontal ($p=0.5$) and vertical flip ($p=0.1$), mild affine
transformations ($p=0.15$, rotation by up to ±10°, shear of up to ±5°, and
scaling by ±5%)) as well as the image degradations described in the next
section with $p=0.75$ (sampling all 5 levels uniformly) to the images during
training. We implemented our code in Python 3.9 using PyTorch and timm [18]
and plan to make it publicly available upon publication.

####  2.2.2 Synthetic degradations

Table 1: Severity levels for the degradations. Brightness, contrast and gamma
changes are independently sampled from the given interval. Dimensions in
pixels.

Severity 1 2 3 4 5 Brightness/Contrast/Gamma ±5% ±10% ±15% ±20% ±25% Mini
Artifacts (holes, height, width) 2-20/1-3/5-8 2-24/1-5/5-12 2-28/1-5/5-16
2-32/1-3/5-20 2-40/1-3/5-24 Square Artifacts (side length) 25 50 75 100 125
Chop Artifacts (% of image removed) 10-15 10-25 10-35 10-45 10-50 Advanced
Blur (kernel size, sigma) 3-5/0.2-0.5 3-7/0.2-0.7 3-9/0.2-0.8 3-11/0.2-0.9
3-13/0.2-1.0 Gaussian Noise (variance) 1-10 5-10 5-20 5-25 5-30

We focus on three types of quality issues in retinal images [12, 9]: Lighting
issues, artifacts/obstructions, and imaging issues. To simulate general
lighting issues, we independently change brightness, contrast and gamma of the
image. To simulate artifacts/obstructions and severely inconsistent lighting,
we introduce one of three artifacts: 1) many smaller rectangular holes placed
across the retina, b) a single large square hole, or c) we “chop” off the
bottom or top part of the image. The latter is inspired by the observation
that in UK Biobank some images only have the top or bottom part properly
illuminated. To simulate general imaging issues, we add pixel-wise Gaussian
noise and blur the image. Standard isotropic Gaussian blur kernels do not
mimic realistic image blur, so we use an advanced anisotropic blurring
technique developed for image super-resolution [17] where the standard
deviations for both dimensions of the kernel are sampled independently, and
the kernel is then rotated and has some noise added before being applied to
the image.

We specify degradation parameters for five levels of severity, shown in Table
1. For a given level, we sample parameters for each image independently from
the given ranges. Degradations are applied after images have already been
downsized to 224x224. We apply an artifact with $p=0.2*s$ where $s$ is the
severity. If an image was chosen to have an artifact applied to it, we then
choose Mini Artifacts with $p=0.85$, Square Artifact with $p=0.10$, and Chop
Artifact with $p=0.05$. Degradations are implemented using the albumentations
package [3].

Figure 2: Random examples of synthetically degraded versions of the same
fundus image. Best viewed zoomed in, especially for the advanced blur. UK
Biobank asks to only reproduce imaging data where necessary, so we demonstrate
the degradations on an image taken from DRIVE [13] which is similar in
appearance to those in UK Biobank.

##  3 Experiments

###  3.1 Data

We apply our DART framework multi-fractal FD [14] calculated with VAMPIRE
[15]. We use only images that had been identified as high quality in a
previous study [16] as for those images FDVAMPIRE should be reliable and can
be considered as a reasonable “ground-truth”. We randomly split the data into
train, validation, and test sets containing 70, 10, and 20% of the
participants in UK Biobank, resulting in 52,242 / 7,478 / 14,907 images
belonging to 32,300 / 4,614 / 9,229 participants in each set. We split at the
participant level such that no images of the same participant occur in
different sets. Images are cropped to square to remove black non-retinal
regions and processed at 224x224 as described above.

###  3.2 Results

####  3.2.1 Agreement & Robustness

Table 2: Agreement between FDVAMPIRE obtained on high quality images, and
FDDART for different levels of degradation measured on 14,907 held-out test
set images.

Degradations $R^{2}$ Pearson $r$ (p-value) Spearman $r$ (p-value) OLS
Regression fit None 0.9160 0.9572 (0.0000) 0.9561 (0.0000) y=0.01 + 1.00x
Severity 1 0.8957 0.9467 (0.0000) 0.9446 (0.0000) y=0.01 + 0.99x Severity 2
0.8859 0.9414 (0.0000) 0.9396 (0.0000) y=0.01 + 0.99x Severity 3 0.8623 0.9287
(0.0000) 0.9282 (0.0000) y=0.00 + 1.00x Severity 4 0.8309 0.9116 (0.0000)
0.9103 (0.0000) y=0.01 + 0.99x Severity 5 0.7773 0.8817 (0.0000) 0.8840
(0.0000) y=0.02 + 0.99x

We find very high agreement between FDVAMPIRE and FDDART on the original
images with Pearson $r=0.9572$ and $r^{2}=0.9160$. Table 2 shows results for
different levels of degradations. When degrading the images and asking our
model to predict the FDVAMPIRE obtained from the high quality image,
agreements goes down as the images become more degraded, which is what we
would expect as these degradations remove substantial information about the
retinal vasculature. However, despite this, we still observe good agreement
with the FDVAMPIRE obtained on the original image even at severity level 5
where extreme degradations are applied (Pearson $r=0.8817$ and
$R^{2}=0.7773$). This suggests that DART can recover good estimates of the
retinal trait that would have been obtained from a counterfactual high quality
image even if the available image has very poor quality. Thus, this might
allow for discarding much fewer images than currently necessary.

(a) Scatterplots of FDDART against FDVAMPIRE obtained
from original images for different levels of degradation.

(b) Boxplots of the
residuals.

Figure 3: Agreement results for 14,907 held-out test set images. Best viewed
zoomed in. a) Red line: best linear fit; dashed black line: $y=x$. b) Faint
red line: $x=0$; vertical black lines: ± one interquartile range (IQR) of
FDVAMPIRE for reference.

For comparison, a previous study comparing FD for arteries and veins
separately between VAMPIRE and SIVA [11] found very poor agreement between the
measures of the two tools ($R^{2}=0.139$ and $R^{2}=0.168$ for arteries and
veins, respectively). Another study comparing vessel caliber-related retinal
traits obtained with VAMPIRE, SIVA, and IVAN found that they agreed with
Pearson $r$s of 0.29 to 0.86. Thus, the observed agreement between FDVAMPIRE
and FDDART with a Pearson $r=0.9572$ and $r^{2}=0.9160$ is very high, and even
when DART is applied the most degraded images the agreement (Pearson
$r=0.8817$ and $R^{2}=0.7773$) is higher than what could be expected when
using two different tools on the same high quality images.

Finally, our method shows very low bias even as degradation severity is
increased (Fig. 3). The best OLS fit is very close to the identity line for
all levels of severity, or equivalently, the optimal linear translation
function from FDDART to FDVAMPIRE is almost simply the identity function. This
also implies that no post-hoc adjustment for image quality is needed and
FDDART values obtained for images of varying quality are on the same scale
out-of-the-box. As degradation severity increases, the variance of the
residuals also increases but most residuals are still less than one
interquartile range (IQR), a robust equivalent of the standard deviation, even
when applying the strongest degradation.

####  3.2.2 Speed

Images were loaded into RAM so that hard disk speed is not a factor. We then
measured the time it took to process all 52,242 training images, including
normalisation, moving them from RAM to GPU VRAM, as well as the time to move
the results back to RAM. We used a modern workstation (Intel i9-9920X 24 core
CPU, single Nvidia RTX A6000 24GB GPU, 126GB of RAM) and a batchsize of 440.
With ResNet18 as backbone, our model processed all 52,242 images in 48.5s ±
93.6 ms (mean±std over 5 runs), yielding a rate of 1077 img/s.

##  4 Conclusion

We have shown that we can use DART to approximate the multi-step pipeline for
obtaining FDVAMPIRE with very high agreement. Our resulting model can compute
FDDART for over 1,000img/s using a GPU. Furthermore, our model can compute
FDDART values from severely degraded images that still match the FDVAMPIRE
values obtained on the high quality images well. This could allow researchers
interested in studying retinal traits to discard fewer images than currently
necessary and thus have higher sample sizes. We consider these to be very
encouraging initial results.

There are a number of directions for future work. First, the proposed
framework can be easily applied to other retinal traits like vessel tortuosity
or width, or FD as calculated by other pipelines. We would expect that this
would be similarly successful. Second, the robustness of the resulting DART
model should be evaluated in more depth and the cases with extreme residuals
should be manually examined. We expect that robustness can be further
improved, especially if we identify common failure cases and use those as data
augmentations. Third, many straight-forward, incremental technical
improvements should be possible such as improved training procedures to
further increase performance, trying different architectures and resolutions,
and speeding up inference speed further through common tricks like fusing
batch norm layers into the convolutional layers. Finally, we hope that our
approach will eventually enable other researchers to conduct better analyses,
e.g. by not having to discard as many images and thus having a larger sample
size available.

## Acknowledgements

We thank our colleagues for their help and support.

This research has been conducted using the UK Biobank Resource under project
72144. This work was supported by the United Kingdom Research and Innovation
(grant EP/S02431X/1), UKRI Centre for Doctoral Training in Biomedical AI at
the University of Edinburgh, School of Informatics. For the purpose of open
access, the author has applied a creative commons attribution (CC BY) licence
to any author accepted manuscript version arising.

## References

  * [1] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016)
  * [2] Bello, I., Fedus, W., Du, X., Cubuk, E.D., Srinivas, A., Lin, T.Y., Shlens, J., Zoph, B.: Revisiting resnets: Improved training and scaling strategies. arXiv preprint arXiv:2103.07579 (2021)
  * [3] Buslaev, A., Iglovikov, V.I., Khvedchenya, E., Parinov, A., Druzhinin, M., Kalinin, A.A.: Albumentations: Fast and flexible image augmentations. Information 11(2) (2020). https://doi.org/10.3390/info11020125, https://www.mdpi.com/2078-2489/11/2/125
  * [4] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 770–778 (2016)
  * [5] Hendrycks, D., Gimpel, K.: Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415 (2016)
  * [6] Lemmens, S., Devulder, A., Van Keer, K., Bierkens, J., De Boever, P., Stalmans, I.: Systematic review on fractal dimension of the retinal vasculature in neurodegeneration and stroke: assessment of a potential biomarker. Frontiers in neuroscience 14, 16 (2020)
  * [7] Loshchilov, I., Hutter, F.: SGDR: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983 (2016)
  * [8] Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101 (2017)
  * [9] MacGillivray, T.J., Cameron, J.R., Zhang, Q., El-Medany, A., Mulholland, C., Sheng, Z., Dhillon, B., Doubal, F.N., Foster, P.J., Trucco, E., et al.: Suitability of uk biobank retinal images for automatic analysis of morphometric properties of the vasculature. PLoS One 10(5), e0127914 (2015)
  * [10] MacGillivray, T., Trucco, E., Cameron, J., Dhillon, B., Houston, J., Van Beek, E.: Retinal imaging as a source of biomarkers for diagnosis, characterization and prognosis of chronic illness or long-term conditions. The British journal of radiology 87(1040), 20130832 (2014)
  * [11] McGrory, S., Taylor, A.M., Pellegrini, E., Ballerini, L., Kirin, M., Doubal, F.N., Wardlaw, J.M., Doney, A.S., Dhillon, B., Starr, J.M., et al.: Towards standardization of quantitative retinal vascular parameters: comparison of siva and vampire measurements in the lothian birth cohort 1936. Translational vision science & technology 7(2), 12–12 (2018)
  * [12] Mookiah, M.R.K., Hogg, S., MacGillivray, T.J., Prathiba, V., Pradeepa, R., Mohan, V., Anjana, R.M., Doney, A.S., Palmer, C.N., Trucco, E.: A review of machine learning methods for retinal blood vessel segmentation and artery/vein classification. Medical Image Analysis 68, 101905 (2021)
  * [13] Staal, J., Abràmoff, M.D., Niemeijer, M., Viergever, M.A., Van Ginneken, B.: Ridge-based vessel segmentation in color images of the retina. IEEE transactions on medical imaging 23(4), 501–509 (2004)
  * [14] Stosic, T., Stosic, B.D.: Multifractal analysis of human retinal vessels. IEEE transactions on medical imaging 25(8), 1101–1107 (2006)
  * [15] Trucco, E., Ballerini, L., Relan, D., Giachetti, A., MacGillivray, T., Zutis, K., Lupascu, C., Tegolo, D., Pellegrini, E., Robertson, G., et al.: Novel vampire algorithms for quantitative analysis of the retinal vasculature. In: 2013 ISSNIP Biosignals and Biorobotics Conference: Biosignals and Robotics for Better and Safer Living (BRC). pp. 1–4. IEEE (2013)
  * [16] Velasco, A.V., Engelmann, J., Rawlik, K., Canela-Xandri, O., Tochel, C., Lona-Durazo, F., Mookiah, M.R.K., Doney, A., Parra, E., Trucco, E., et al.: Decreased retinal vascular complexity is an early biomarker of mi supported by a shared genetic control. medRxiv (2021)
  * [17] Wang, X., Xie, L., Dong, C., Shan, Y.: Real-ESRGAN: Training real-world blind super-resolution with pure synthetic data. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 1905–1914 (2021)
  * [18] Wightman, R.: PyTorch Image Models. https://github.com/rwightman/pytorch-image-models (2019). https://doi.org/10.5281/zenodo.4414861
  * [19] Wightman, R., Touvron, H., Jégou, H.: Resnet strikes back: An improved training procedure in timm. arXiv preprint arXiv:2110.00476 (2021)
  * [20] Yalniz, I.Z., Jégou, H., Chen, K., Paluri, M., Mahajan, D.: Billion-scale semi-supervised learning for image classification. arXiv preprint arXiv:1905.00546 (2019)
  * [21] Zekavat, S.M., Raghu, V.K., Trinder, M., Ye, Y., Koyama, S., Honigberg, M.C., Yu, Z., Pampana, A., Urbut, S., Haidermota, S., et al.: Deep learning of the retina enables phenome-and genome-wide analyses of the microvasculature. Circulation 145(2), 134–150 (2022)