The quantization of singular Lagrangian systems has been studied with increasing interest and treated first by Dirac [1] [2] , for quantizing physical systems. Following Dirac’s, researchers have developed a theory for investigating these systems using the canonical formalism [3] . The set of HJPDEs is constructed, and they wrote equations of motion as total differential equations. Then, the WKB approximation and path integral technique are quantized by using this formalism [4] - [8] .

Fractional calculus with singular systems has been treated with more interest and importance [9] - [18] . Recently, the Euler-Lagrange formalism is analyzed for second-order Lagrangian systems within fractional calculus and the fractional Hamilton-Jacobi formalism for these systems is discussed [15] [16] . More recently, authors have constructed a formalism using the canonical method for quantizing singular systems for first-order derivatives [17] [18] . In this paper, we would like to extend our work for Lagrangians having second-order derivatives.

The most important definitions of fractional calculus [9] are:

1) The left Riemann-Liouville fractional derivative

${}_{a}D{}_t^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\left(\frac{\text{d}}{\text{d}t}\right)}^n{\displaystyle \underset{a}{\overset{t}{\int }}{\left(t-\tau \right)}^{n-\alpha -1}}f\left(\tau \right)\text{d}\tau$. (1)

2) The right Riemann-Liouville fractional derivative

${}_{t}D{}_b^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\left(-\frac{\text{d}}{\text{d}t}\right)}^n{\displaystyle \underset{t}{\overset{b}{\int }}{\left(\tau -t\right)}^{n-\alpha -1}}f\left(\tau \right)\text{d}\tau$. (2)

where $\Gamma$ is the Euler gamma function, $n\in N$, $n-1\le \alpha <n$, $\alpha$ is an integer and these derivatives can be defined as follows:

${}_{a}D{}_t^{\alpha }f\left(t\right)={\left(\frac{\text{d}}{\text{d}t}\right)}^{\alpha }f\left(t\right)$, ${}_{t}D{}_b^{\alpha }f\left(t\right)={\left(-\frac{\text{d}}{\text{d}t}\right)}^{\alpha }f\left(t\right)$, (3)

Definition: Given a function $f:\left[0,\infty \right)\to \Re$. Then, for all $t>0$, $\alpha \in \left(0,1\right)$, let

$D^{\alpha }\left(f\right)\left(t\right)=\underset{\epsilon \to 0}{\lim }\frac{f\left(t+\epsilon t^{1-\alpha }\right)-f\left(t\right)}{\epsilon }$. (4)

where $D^{\alpha }$ is called the conformal fractional derivative of $f$ of order of $\alpha$ [17] .

In this work, we aim to construct the formalism for quantizing singular Lagrangian systems with second-order derivatives in fractional form.

Following Hasan [19] , we will use a formalism for second-order fractional singular Lagrangian systems to be applicable for quantizing these systems using the path integral approach. The Lagrangian formalism of second-order derivatives in fractional form is given by [19] :

$L=L\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,D^{2\alpha }{q}_i,t\right)$. (5)

where $D^{\alpha }{q}_i$ are the conformal fractional derivatives of the coordinates $q_i$ [17] .

The Lagrangian and Hamiltonian formalism for second-order derivatives have been studied by Ostrogradski [20] and these derivatives have been treated as coordinates. Therefore, we can treat these derivatives $D^{\alpha -1}{q}_i$ and $D^{\alpha }{q}_i$ as coordinates. Thus, the Poisson brackets can be defined as:

$\left\{A,B\right\}\equiv \frac{\partial A}{\partial D^{\alpha -1}{q}_i}\frac{\partial B}{\partial p_i}-\frac{\partial A}{\partial p_i}\frac{\partial B}{\partial D^{\alpha -1}{q}_i}+\frac{\partial A}{\partial D^{\alpha }{q}_i}\frac{\partial B}{\partial {\pi }_i}-\frac{\partial A}{\partial {\pi }_i}\frac{\partial B}{\partial D^{\alpha }{q}_i}$. (6)

where, the functions $A$ and $B$ are described in term of the canonical variables $D^{\alpha -1}{q}_i$, $D^{\alpha }{q}_i$, $p_i$ and ${\pi }_i$. Thus, the generalized momenta $p_i$ and ${\pi }_i$ are conjugated to the generalized coordinates $D^{\alpha -1}{q}_i$ and $D^{\alpha }{q}_i$ respectively.

Now, the fractional of the Hessian matrix is defined as [19] :

$W_{ij}=\frac{{\partial }^{2}L}{\partial D^{2\alpha }{q}_i\partial D^{2\alpha }{q}_j}$, $i,j=1,2,\cdots ,N$ (7)

The fractional Lagrangian is called regular if it’s rank is $N$ otherwise the Lagrangian is singular $N-R$, $R<N$. Dirac showed in his formalism for investigating Lagrangians having singular nature that the number of degrees of freedom can be reduced from $N$ to $N-R$ due to the constraints [1] [2] . Thus, we can define the momenta ${\pi }_i$ conjugated to the coordinates $D^{\alpha }{q}_i$ as [19] :

${\pi }_a=\frac{\partial L}{\partial D_{}^{2\alpha }{q}_a}$, $a=1,2,\cdots ,N-R$ (8)

${\pi }_{\mu }=\frac{\partial L}{\partial D_{}^{2\alpha }{q}_{\mu }}$, $\mu =N-R+1,\cdots ,N$. (9)

Also, the momenta $p_i$ conjugated to the coordinates $D^{\alpha -1}{q}_i$ can be defined as [19] :

$p_a=\frac{\partial L}{\partial D^{\alpha }{q}_a}-\frac{\text{d}}{\text{d}t}\left(\frac{\partial L}{\partial D^{2\alpha }{q}_a}\right)$; (10)

$p_{\mu }=\frac{\partial L}{\partial D^{\alpha }{q}_{\mu }}-\frac{\text{d}}{\text{d}t}\left(\frac{\partial L}{\partial D^{2\alpha }{q}_{\mu }}\right)$, (11)

where

${\pi }_{\mu }=-H_{\mu }^{\pi }\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,p_a,{\pi }_a\right)$ (12)

and

$p_{\mu }=-H_{\mu }^p\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,p_a,{\pi }_a\right)$ (13)

Thus, Equations (12) and (13) represent primary constraints [1] [2] and can be written as:

${H^{\prime }}_{\mu }^p\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,p_i,{\pi }_i\right)=p_{\mu }+H_{\mu }^p=0$; (14)

${H^{\prime }}_{\mu }^{\pi }\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,p_i,{\pi }_i\right)={\pi }_{\mu }+H_{\mu }^{\pi }=0$. (15)

We can calculate the Hamiltonian $H_{\circ }$ in fractional form as:

$\begin{array}{l}{H}_{\circ }=-L\left(D^{\alpha -1}{q}_i,D{}^{2\alpha }q{}_{\mu },D^{\alpha }{q}_i,D^{2\alpha }{q}_a\right)+p_a{D}^{\alpha }{q}_a+{\pi }_a{D}^{2\alpha }{q}_a\\ -D^{\alpha }{q}_{\mu }{H}_{\mu }^p-D^{2\alpha }{q}_{\mu }{H}_{\mu }^{\pi },\mu =1,\cdots ,R;a=R+1,\cdots ,N.\end{array}$ (16)

A natural of singular Lagrangian indicates that the momenta $p_{\mu }$ and ${\pi }_{\mu }$ are not independent of $p_a$ and ${\pi }_a$. Thus, we can write the set of fractional (HJPDEs) as:

${H^{\prime }}_{\circ }=p_{\circ }+H_{\circ }\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,\frac{\partial S}{\partial D^{\alpha -1}{q}_a},\frac{\partial S}{\partial D^{\alpha }{q}_a}\right)=0$; (17a)

${H^{\prime }}_{\mu }^p=p_{\mu }+H_{\mu }^p\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,\frac{\partial S}{\partial D^{\alpha -1}{q}_a},\frac{\partial S}{\partial D^{\alpha }{q}_a}\right)=0$; (17b)

${H^{\prime }}_{\mu }^{\pi }={\pi }_{\mu }+H_{\mu }^{\pi }\left(D^{\alpha -1}{q}_i,D^{\alpha }{q}_i,\frac{\partial S}{\partial D^{\alpha -1}{q}_a},\frac{\partial S}{\partial D^{\alpha }{q}_a}\right)=0$. (17c)

where $S=S\left(D^{\alpha -1}{q}_a,D^{\alpha -1}{q}_{\mu },D^{\alpha }{q}_a,D^{\alpha }{q}_{\mu },t\right)$ is the fractional Hamilton’s principal function. Considering the definitions of the generalized momenta as $p_a=\frac{\partial S}{\partial D^{\alpha -1}{q}_a},p_{\mu }=\frac{\partial S}{\partial D^{\alpha -1}{q}_{\mu }},{\pi }_a=\frac{\partial S}{\partial D^{\alpha }{q}_a},{\pi }_{\mu }=\frac{\partial S}{\partial D^{\alpha }{q}_{\mu }}$ and $p_{\circ }=\frac{\partial S}{\partial t}$.

Researchers in Reference [5] wrote the equations of motion and Hamilton’s principal function as total differential equations, also we can write these equations in fractional form as follows [5] :

$\text{d}{D}^{\alpha -1}{q}_a=\frac{\partial {H^{\prime }}_{\circ }}{\partial p_a}\text{d}t+\frac{\partial {H^{\prime }}_{\mu }^p}{\partial p_a}\text{d}{D}^{\alpha -1}{q}_{\mu }+\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial p_a}\text{d}{D}^{\alpha }{q}_{\mu },$ (18)

$\text{d}{D}^{\alpha }{q}_a=\frac{\partial {H^{\prime }}_{\circ }}{\partial {\pi }_a}\text{d}t+\frac{\partial {H^{\prime }}_{\mu }^p}{\partial {\pi }_a}\text{d}{D}^{\alpha -1}{q}_{\mu }+\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial {\pi }_a}\text{d}{D}^{\alpha }{q}_{\mu },$ (19)

$-\text{d}{p}_i=\frac{\partial {H^{\prime }}_{\circ }}{\partial D^{\alpha -1}{q}_i}\text{d}t+\frac{\partial {H^{\prime }}_{\mu }^{{}^p}}{\partial D^{\alpha -1}{q}_i}\text{d}{D}^{\alpha -1}{q}_{\mu }+\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial D^{\alpha -1}{q}_i}\text{d}{D}^{\alpha }{q}_{\mu },$ (20)

$-\text{d}{\pi }_i=\frac{\partial {H^{\prime }}_{\circ }}{\partial D^{\alpha }{q}_i}\text{d}t+\frac{\partial {H^{\prime }}_{\mu }^{{}^p}}{\partial D^{\alpha }{q}_i}\text{d}{D}^{\alpha -1}{q}_{\mu }+\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial D^{\alpha }{q}_i}\text{d}{D}^{\alpha }{q}_{\mu },$ (21)

$\begin{array}{c}\text{d}S=\left(-H_{\circ }+p_a\frac{\partial {H^{\prime }}_{\circ }}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\circ }}{\partial {\pi }_a}\right)\text{d}t+\left(-H_{\mu }^p+p_a\frac{\partial {H^{\prime }}_{\mu }^p}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\mu }^p}{\partial {\pi }_a}\right)\text{d}{D}^{\alpha -1}{q}_{\mu }\\ +\left(-H_{\mu }^{\pi }+p_a\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial {\pi }_a}\right)\text{d}{D}^{\alpha }{q}_{\mu }.\end{array}$ (22)

The set of Equations (18)-(22) are integrable if the total derivative of Equation (17) is zero [3] [5] ,

$\text{d}{H^{\prime }}_{\circ }=0$; $\text{d}{H^{\prime }}_{\mu }^p=0$; $\text{d}{H^{\prime }}_{\mu }^{\pi }=0$. (23)

Thus, the degrees of freedom are reduced from $N$ to $N-R$, and the canonical phase space coordinates have been reduced from $\left\{D^{\alpha -1}{q}_i,p_i,D^{\alpha }{q}_i,{\pi }_i\right\}$ to $\left\{D^{\alpha -1}{q}_a,p_a,D^{\alpha }{q}_a,{\pi }_a\right\}$. Therefore, the path integral approach can be represented in the fractional form as:

$\begin{array}{l}K\left(D^{\alpha -1}{q}_a,D^{\alpha -1}{q}_{\mu },D^{\alpha }{q}_a,D^{\alpha }{q}_{\mu },t\right)\\ ={\displaystyle \int {\displaystyle \underset{a=1}{\overset{N-R}{\prod }}\text{d}{D}^{\alpha -1}{q}_a\text{d}{D}^{\alpha }{q}_a\text{d}{p}_a\text{d}{\pi }_a}}\exp i[{\displaystyle \int \left(-H_{\circ }+p_a\frac{\partial {H^{\prime }}_{\circ }}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\circ }}{\partial {\pi }_a}\right)\text{d}t}\\ +{\displaystyle \int \left(-H_{\mu }^p+p_a\frac{\partial {H^{\prime }}_{\mu }^p}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\mu }^p}{\partial {\pi }_a}\right)\text{d}{D}^{\alpha -1}{q}_{\mu }}\\ +{\displaystyle \int \left(-H_{\mu }^{\pi }+p_a\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial p_a}+{\pi }_a\frac{\partial {H^{\prime }}_{\mu }^{\pi }}{\partial {\pi }_a}\right)\text{d}{D}^{\alpha }{q}_{\mu }}].\end{array}$ (24)

We will discuss an example of second-order fractional singular Lagrangian has primary and secondary constraints:

$\begin{array}{c}L=\frac{1}{2}\left({\left(D^{2\alpha }{q}_1\right)}^2+{\left(D^{2\alpha }{q}_2\right)}^2\right)-\frac{1}{2}\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\\ +\frac{1}{2}{D}^{\alpha -1}{q}_3+D^{\alpha }{q}_3{D}^{2\alpha }{q}_3.\end{array}$ (25)

The generalized momenta read:

$p_1=-D^{\alpha }{q}_1-D^{3\alpha }{q}_1$; (26a)

$p_2=-D^{\alpha }{q}_2-D^{3\alpha }{q}_2$; (26b)

$p_3=0=-H_3^p$; (26c)

${\pi }_1=D^{2\alpha }{q}_1$; (26d)

${\pi }_2=D^{2\alpha }{q}_2$; (26e)

${\pi }_3=D^{\alpha }{q}_3=-H_3^{\pi }$. (26f)

Here, Equations (26c) and (26f) can be written as:

${H^{\prime }}_3^p=p_3=0$; (27)

${H^{\prime }}_3^{\pi }={\pi }_3-D^{\alpha }{q}_3=0$. (28)

and represent as primary constraints [1] [2] .

We calculate the fractional Hamiltonian $H_0$ as:

$\begin{array}{c}{H}_{\circ }=p_1{D}^{\alpha }{q}_1+p_2{D}^{\alpha }{q}_2+\frac{1}{2}\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\\ -\frac{1}{2}{\left(D^{\alpha -1}{q}_3\right)}^2+\frac{1}{2}\left({\pi }_1^2+{\pi }_2^2\right).\end{array}$ (29)

The set of HJPDEs, are written as:

$\begin{array}{l}{H^{\prime }}_{\circ }=p_{\circ }+H_{\circ }=p_{\circ }+p_1{D}^{\alpha }{q}_1+p_2{D}^{\alpha }{q}_2+\frac{1}{2}\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\\ -\frac{1}{2}{\left(D^{\alpha -1}{q}_3\right)}^2+\frac{1}{2}\left({\pi }_1^2+{\pi }_2^2\right).\end{array}$ (30)

${H^{\prime }}_3^p=p_3=0$; (31)

${H^{\prime }}_3^{\pi }={\pi }_3-D^{\alpha }{q}_3=0$. (32)

Using the fundamental Poisson brackets $\left\{D^{\alpha -1}{q}_i,p_j\right\}\equiv {\delta }_{ij}$ and $\left\{D^{\alpha }{q}_i,{\pi }_j\right\}\equiv {\delta }_{ij}$, $\left\{D^{\alpha -1}{q}_i,D^{\alpha -1}{q}_j\right\}=\left\{D^{\alpha }{q}_i,D^{\alpha }{q}_j\right\}=0=\left\{D^{\alpha }{q}_i,D^{\alpha -1}{q}_j\right\}=\left\{p_i,{\pi }_i\right\}$, where $i,j=1,\cdots ,N$.

Thus, $\{{H^{\prime }}_3^p,{H^{\prime }}_{\circ }\}=D^{\alpha -1}{q}_3={H^{″}}_3^p$. It gives secondary constraint [1] [2] :

${H^{″}}_3^p=D^{\alpha -1}{q}_3=0$. (33)

Following Dirac’s classification [1] [2] , the constraint (32) is first-class and (31) and (33) are second-class. There are no further constraints.

The equations of motion (18)-(22) can be calculated as:

$\text{d}{D}^{\alpha -1}{q}_1=D^{\alpha }{q}_1\text{d}t,$ (34)

$\text{d}{D}^{\alpha -1}{q}_2=D^{\alpha }{q}_2\text{d}t,$ (35)

$\text{d}{D}^{\alpha }{q}_1={\pi }_1\text{d}t,$ (36)

$\text{d}{D}^{\alpha }{q}_2={\pi }_2\text{d}t,$ (37)

$\text{d}{p}_1=0,$ (38)

$\text{d}{p}_2=0,$ (39)

$\text{d}{p}_3=0,$ (40)

$-\text{d}{\pi }_1=\left(p_1+D^{\alpha }{q}_1\right)\text{d}t,$ (41)

$-\text{d}{\pi }_2=\left(p_2+D^{\alpha }{q}_2\right)\text{d}t,$ (42)

$\text{d}{\pi }_3=\text{d}{D}^{\alpha }{q}_3,$ (43)

$\text{d}S=\frac{1}{2}\left(\left[{\pi }_1^2+\frac{1}{2}{\pi }_2^2\right]-\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\right)\text{d}t+{\pi }_3\text{d}{D}^{\alpha }{q}_3$ (44)

Considering ${\pi }_3=D^{\alpha }{q}_3$,

$\text{d}S=\frac{1}{2}\left(\left[{\pi }_1^2+\frac{1}{2}{\pi }_2^2\right]-\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\right)\text{d}t+D^{\alpha }{q}_3\text{d}{D}^{\alpha }{q}_3.$ (45)

Integrate Equation (45), the action function becomes:

$S={\displaystyle \int \frac{1}{2}}\left(\left[{\pi }_1^2+\frac{1}{2}{\pi }_2^2\right]-\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\right)\text{d}t+\frac{1}{2}{\left(D^{\alpha }{q}_3\right)}^2.$ (46)

Finally, by obtaining the fractional action function $S$, we can represent the path integral approach in fractional form as:

$\begin{array}{l}K\left(D^{\alpha -1}{q}_1,D^{\alpha -1}{q}_2,D^{\alpha -1}{q}_3,D^{\alpha }{q}_1,D^{\alpha }{q}_2,D^{\alpha }{q}_3,t\right)\\ ={\displaystyle \int \text{d}{D}^{\alpha -1}{q}_1\text{d}{D}^{\alpha -1}{q}_2\text{d}{D}^{\alpha }{q}_1}\text{d}{D}^{\alpha }{q}_2\text{d}{p}_1\text{d}{p}_2\text{d}{\pi }_1\text{d}{\pi }_2\\ \exp i\left[{\displaystyle \int \frac{1}{2}}\left(\left[{\pi }_1^2+{\pi }_2^2\right]-\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\right)\text{d}t+\frac{1}{2}{\pi }_3^2\right].\end{array}$ (47)

Taking into account, $D^{\alpha -1}{q}_3=0$, ${\pi }_3=D^{\alpha }{q}_3$ thus, $D^{\alpha }{q}_3=0$.

Thus, Equation (47) becomes:

$\begin{array}{l}K\left(D^{\alpha -1}{q}_1,D^{\alpha -1}{q}_2,D^{\alpha -1}{q}_3,D^{\alpha }{q}_1,D^{\alpha }{q}_2,D^{\alpha }{q}_3,t\right)\\ ={\displaystyle \int \text{d}{D}^{\alpha -1}{q}_1\text{d}{D}^{\alpha -1}{q}_2\text{d}{D}^{\alpha }{q}_1}\text{d}{D}^{\alpha }{q}_2\text{d}{p}_1\text{d}{p}_2\text{d}{\pi }_1\text{d}{\pi }_2\\ \exp i\left[\frac{1}{2}{\displaystyle \int \left(\left[{\pi }_1^2+{\pi }_2^2\right]-\left[{\left(D^{\alpha }{q}_1\right)}^2+{\left(D^{\alpha }{q}_2\right)}^2\right]\right)\text{d}t}\right].\end{array}$ (48)

In this work, we constructed a formalism for quantizing singular systems using fractional path integral technique. We wrote Hamilton’s principal function and equations of motion in fractional form as total differential equations. Calculating the fractional Hamilton’s principal function enabled us to formulate the fractional path integral technique. Then, the quantization can be carried out. Finally, we discussed a mathematical example to demonstrate our formalism.