Canonical Quantization of Higher-Order Lagrangians

Abstract

After reducing a system of higher-order regular Lagrangian into first-order singular Lagrangian using constrained auxiliary description, the Hamilton-Jacobi function is constructed. Besides, the quantization of the system is investigated using the canonical path integral approximation.

1. Introduction

The efforts to quantize systems with constraints started with the work of Dirac [1, 2], who first set up a formalism for treating singular systems and the constraints involved for the purpose of quantizing his field, with special emphasis on the gravitational field. In Dirac’s canonical quantization method, the Poisson brackets of classical mechanics are replaced with quantum commutators.

A new formalism for investigating first-order singular systems-, the canonical-, was developed by Rabei and Guler [3]. These authors obtained a set of Hamilton-Jacobi partial differential equations (HJPDEs) for singular systems using Caratheodory’s equivalent-Lagrangian method [4]. In this formalism, the equations of motion are obtained as total differential equations and the set of HJPDEs was determined. Recently, the formalism has been extended to second- and higher-order Lagrangians [5, 6]. Depending on this method, the path-integral quantization of first-and higher-order constrained Lagrangian systems has been applied [7–10].

Moreover, the quantization of constrained systems has been studied for first-order singular Lagrangians using the WKB approximation [11]. The HJPDEs for these systems have been constructed using the canonical method; the Hamilton-Jacobi functions have then been obtained by solving these equations.

The Hamiltonian formulation for systems with higher-order regular Lagrangians was first developed by Ostrogradski [12]. This led to Euler's and Hamilton's equations of motion. However, in Ostrogradski's construction the structure of phase space and in particular of its local simplistic geometry is not immediately transparent which leads to confusion when considering canonical path integral quantization.

In Ostrogradski's construction, this problem can be resolved within the well-established context of constrained systems [13] described by Lagrangians depending on coordinates and velocities only. Therefore, higher-order systems can be set in the form of ordinary constrained systems [14]. These new systems will be functions only of first-order time derivative of the degrees of freedom and coordinates which can be treated using the theory of constrained systems [1–11].

The purpose of the present paper is to study the canonical path integral quantization for singular systems with arbitrary higher-order Lagrangian. In fact, this work is a continuation of the previous work [15], where the path integral for certain kinds of higher-order Lagrangian systems has been obtained.

The present work is organized as follows: in Section 2, a review of the canonical method is introduced. In Section 3, Ostrogradski's formalism of higher-order Lagrangians is discussed. In Section 4, the formulation of the canonical Hamiltonian is reviewed briefly. In Section 5, the canonical path integral quantization of the extended Lagrangian is applied. In Section 6, two illustrative examples are investigated in detail. The work closes with some concluding remarks in Section 7.

2. Review of the Canonical Method

The starting point is a singular Lagrangian 𝐿=𝐿(𝑞_𝑖,𝑞_𝑖̇), 𝑖=1,2,…,𝑁, with the Hessian matrix 𝜕²𝐿/𝜕𝑞_𝑖̇𝜕𝑞_𝑗̇ of rank 𝑁-𝑅,𝑅<𝑁.

The canonical formulation [3] gives the set of the Hamilton-Jacobi partial differential equations as𝐻^0=𝑝₀+𝐻₀≡𝜕𝑆𝜕𝑡+𝐻⁰𝑞^𝛽,𝑞_𝑎,𝑝_𝑎=𝜕𝑆𝐻𝜕𝑞_𝑎=0,_𝜇=𝑝_𝜇+𝐻_𝜇≡𝜕𝑆𝜕𝑞_𝜇+𝐻_𝜇𝑞^𝛽,𝑞_𝑎,𝑝_𝑎=𝜕𝑆𝑎=1,…,𝑁−𝑅,𝜇=𝑁−𝑅+1,…,𝜕𝑁𝑞,_𝑎=0,(2.1) where 𝑝₀ and 𝑞_𝜇are the momenta conjugate to 𝑡 and𝑞_𝜇, respectively,𝑝₀=𝜕𝑆𝑞^𝑖,𝑡𝜕𝑡,𝑝^𝜇=𝜕𝑆𝑞^𝑖,𝑡𝜕𝑞_𝜇.(2.2) The canonical Hamiltonian H₀ is given by𝐻₀=𝑝_𝑎𝑞_𝑎̇+𝑝_𝜇𝑞_𝜇̇−𝐿.(2.3) The equations of motion are obtained as total differential equations in many variables as follows: 𝑑𝑞_𝑎=𝜕𝐻^0𝜕𝑝_𝑎𝑑𝑡+𝜕𝐻^𝜇𝜕𝑝_𝑎𝑑𝑝_𝑖𝑑𝑞_𝜇,=−𝜕𝐻^0𝜕𝑞_𝑖𝑑𝑡−𝜕𝐻^𝜇𝜕𝑞_𝑖𝑑𝑞_𝜇,(2.4)𝑑𝑧=−𝐻₀𝑑𝑡−𝐻_𝜇𝑑𝑞_𝜇+𝑝_𝑎𝜕𝐻^0𝜕𝑝_𝑎𝑑𝑡+𝑝_𝑎𝜕𝐻^𝜇𝜕𝑝_𝑎𝑑𝑞_𝜇,(2.5) where 𝑧=𝑆(𝑡,𝑞_𝑎,𝑞_𝜇). The set of equations (2.4) and (2.5) is integrable if and only if𝑑𝐻^0=0,𝜕𝐻^𝜇=0(2.6) are identically satisfied. If they are not, one could consider them as new constraints and again should test the consistency conditions. Thus, in repeating this procedure one may obtain a new set of conditions. Equations (2.4) then can be solved to obtain the coordinates q_a and momenta p_i as functions of q_μ and t.

3. Ostrogradski's Formalism of Higher-Order Lagrangians

Consider a higher-order Lagrangian system of 𝑁 generalized coordinates 𝑞_𝑛(𝑡):𝐿₀𝑞^𝑛,𝑞_𝑛̇,…,𝑞^𝑛(𝑚),𝑚≥1,(3.1) where 𝑞^𝑛(𝑠)=𝑑^𝑠𝑞_𝑛/𝑑𝑡^𝑠, 𝑠=0,1,…,𝑚 and 𝑛=1,…,𝑁.

The Euler-Lagrange equations of motion are obtained as [12]^𝑚_𝑠=0(−1)^𝑠𝑑^𝑠𝑑𝑡^𝑠𝜕𝜕𝑞𝐿^0𝑛(𝑠)=0.(3.2)

Theories with higher derivatives, which have been first developed by Ostrogradski [12], treat the derivatives 𝑞^𝑛(𝑠)(𝑠=0,…,𝑚−1) as independent coordinates. Therefore, we will indicate this by writing them as 𝑞^𝑛(𝑠)=𝑞_𝑛,𝑠. In Ostrogradski's formalism, the momenta conjugated, respectively, to 𝑞_{𝑛,𝑚−1} and 𝑞_{𝑛,𝑠−1}, (𝑠=1,…,𝑚−1) read as𝑝_{𝑛,𝑚−1}≡𝜕𝐿⁰𝜕𝑞^𝑛(𝑚)𝑝_{𝑛,𝑠−1},≡𝜕𝐿⁰𝜕𝑞^𝑛(𝑠)−𝑝̇_𝑛,𝑠,𝑠=1,…,𝑚−1.(3.3) Therefore, the canonical Hamiltonian is given by 𝐻₀𝑞_𝑛,0,…,𝑞_{𝑛,𝑚−1};𝑝_𝑛,0,…,𝑝_{𝑛,𝑚−1}=^𝑚−2_𝑠=0𝑝_𝑛,𝑠𝑞_𝑛,𝑠+1+𝑝_{𝑛,𝑚−1}𝑞̇_{𝑛,𝑚−1}−𝐿₀𝑞_𝑛,0,…,𝑞_{𝑛,𝑚−1},𝑞̇_{𝑛,𝑚−1}.(3.4) Hamilton's equations of motion are written using Poisson bracket as [5, 6]𝑞̇_𝑛,𝑠=𝜕𝐻⁰𝜕𝑝_𝑛,𝑠𝑝̇_𝑛,𝑠=𝑞_𝑛,𝑠,𝐻₀,(3.5)=𝜕𝐻⁰𝜕𝑞_𝑛,𝑠=𝑝_𝑛,𝑠,𝐻₀,(3.6) where {,} is the Poisson bracket defined as{𝐴,𝐵}=^𝑚−1_𝑠=0𝜕𝐴𝜕𝑞_𝑛,𝑠𝜕𝐵𝜕𝑝_𝑛,𝑠−𝜕𝐵𝜕𝑞_𝑛,𝑠𝜕𝐴𝜕𝑝_𝑛,𝑠.(3.7) The fundamental Poisson brackets are𝑞_𝑛,𝑠𝑝_{𝑛′,𝑠′}=𝛿_𝑛𝑛^′𝛿_𝑠𝑠′,𝑞_𝑛,𝑠,𝑞_{𝑛′,𝑠′}=𝑝_𝑛,𝑠,𝑝_{𝑛′,𝑠′}=0,(3.8) where 𝑛,𝑛^=1,…,𝑁,and𝑠,𝑠′=0,…,𝑚−1.

With this procedure, the phase space, described in terms of the canonical variables 𝑞_𝑛,𝑠and𝑝_𝑛,𝑠, is obeying the equations of motion that are given by (3.5) and (3.6), which are first-order differential equations.

4. Formulation of the Canonical Hamiltonian

Recall the higher-order Lagrangian given in (3.1), and let us introduce new independent variables (𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖,𝑖=0,1,…,𝑚−2) such that the following recursion relations would hold [13, 14]:𝑞̇_𝑛,𝑖=𝑞_𝑛,𝑖+1.(4.1) Clearly, the variables (𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖), would then correspond to the time derivatives (𝑞^{𝑛(𝑚−1)},𝑞^𝑛(𝑖)) respectively, that is,𝑞^𝑛(0)=𝑞_𝑛,0,𝑞_𝑛̇=𝑞_𝑛,1,…,𝑞^{𝑛(𝑚−1)}=𝑞_{𝑛,𝑚−1},𝑞^𝑛(𝑚)=𝑞̇_{𝑛,𝑚−1}.(4.2) Equation (4.1) represents relations between the new variables. In order to enforce these relations for independent variables (𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖), additional Lagrange multipliers 𝜆_𝑛,𝑖(𝑡) are introduced [14]. The variables (𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖,𝜆_𝑛,𝑖), thus, determine the set of independent degrees of freedom of the extended Lagrangian system. The extended Lagrangian of this auxiliary description of the system is given by 𝐿_𝑇𝑞_𝑛,𝑖,𝑞_{𝑛,𝑚−1},𝑞̇_𝑛,𝑖,𝑞̇_{𝑛,𝑚−1},𝜆_𝑛,𝑖=𝐿₀𝑞_𝑛,𝑖,𝑞_{𝑛,𝑚−1},𝑞̇_{𝑛,𝑚−1}+^𝑚−2_𝑖=0𝜆_𝑛,𝑖𝑞̇_𝑛,𝑖−𝑞_𝑛,𝑖+1.(4.3) The new Lagrangian in (4.3) is singular, and one can use the standard methods of singular systems like Dirac's method or the canonical approach to investigate this Lagrangian.

Upon introducing the canonical momenta:𝑝_{𝑛,𝑚−1}=𝜕𝐿^𝑇𝜕𝑞̇_{𝑛,𝑚−1}𝑝,(4.4)_𝑛,𝑖=𝜕𝐿^𝑇𝜕𝑞̇_𝑛,𝑖𝜋=𝜆_𝑛,𝑖=−𝐻_𝑛,𝑖,(4.5)_𝑛,𝑖=𝜕𝐿^𝑇𝜕𝜆̇_𝑛,𝑖=0=−Φ_𝑛,𝑖,(4.6) the canonical Hamiltonian can be obtained as𝐻₀=𝑞𝑝_𝑛,𝑖,𝑞_{𝑛,𝑚−1𝑛,𝑚−1},𝑝_{𝑛,𝑚−1},𝜆_𝑛,𝑖𝑞̇_{𝑛,𝑚−1}+^𝑚−2_𝑖=0𝑝_𝑛,𝑖𝑞̇_𝑛,𝑖+^𝑚−2_𝑖=0𝜋_𝑛,𝑖𝜆̇_𝑛,𝑖−𝐿_𝑇𝑞_𝑛,𝑖,𝑞_{𝑛,𝑚−1},𝑞̇_𝑛,𝑖,𝑞̇_{𝑛,𝑚−1},𝜆_𝑛,𝑖,(4.7) Equations (4.5) and (4.6) represent primary constraints [1, 2]. Their Hamilton-Jacobi partial differential equations can be obtained as𝐻^0=𝑝₀+𝐻₀Φ𝑞_𝑛,𝑖,𝑞_{𝑛,𝑚−1},𝑝_{𝑛,𝑚−1},𝜆_𝑛,𝑖=0,(4.8)_𝑛,𝑖𝐻=𝜋_𝑛,𝑖=0,(4.9)_𝑛,𝑖=𝑝_𝑛,𝑖−𝜆_𝑛,𝑖=0.(4.10) The equations of motion can be written as total differential equations in many variables as follows: 𝑑𝑞𝑑𝑞_{𝑛,𝑗𝑛,𝑚−1}=𝑑𝑞_𝑛,𝑗,(4.11)=𝜕𝐻^0𝜕𝑝_{𝑛,𝑚−1}𝑑𝑝_𝑛,𝑗𝑑𝑡,(4.12)=−𝜕𝐻^0𝜕𝑞_𝑛,𝑗𝑑𝑝_{𝑛,𝑚−1}𝑑𝑡,=−𝜕𝐻^0𝜕𝑞_{𝑛,𝑚−1}𝑑𝜆_𝑛,𝑗𝑑𝑡,𝑑𝜋_𝑛,𝑗=𝑑𝜆_𝑛,𝑗,=−𝜕𝐻^0𝜕𝜆_𝑛,𝑗𝑑𝑡+𝑑𝑞_𝑛,𝑗,𝑗=0,1,…,𝑚−2.(4.13) The total differential equations are integrable if and only if𝑑𝑑𝐻𝐻^0_𝑛,𝑗=𝑑𝑝₀−𝑑𝐻₀=0,𝑑Φ_𝑛,𝑗=𝑑𝑝_𝑛,𝑗−𝑑𝜆_𝑛,𝑗=0,=𝑑𝜋_𝑛,𝑗=0.(4.14)

5. The Canonical Path Integral Quantization

If the coordinates 𝑡,𝑞_𝑛,𝑖,𝜆_𝑛,𝑖 are denoted by 𝑡_𝛼, that is,𝑡_𝛼=𝑡,𝑞_𝑛,𝑖,𝜆_𝑛,𝑖,(5.1) then the set of primary constraints (4.8), (4.9), and (4.10) can be written in a compact form as𝐻^𝛼=0,(5.2) where 𝐻^𝛼=𝐻^0,𝐻^𝑛,𝑖,Φ^𝑛,𝑖.(5.3) Making use of [7], the canonical path integral for the extended Lagrangians can be obtained as𝐾𝑞=^{𝑛,𝑚−1},𝑞^𝑛,𝑖,𝜆^𝑛,𝑖,𝑡^;𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖,𝜆_𝑛,𝑖,𝑡_{𝑞′𝑛,𝑚−1𝑞𝑛,𝑚−1𝑁}_𝑛=1𝐷𝑞_{𝑛,𝑚−1}𝐷𝑝_{𝑛,𝑚−1}exp𝑖ℏ^𝑡′𝛼_𝑡𝛼−𝐻_𝛼+𝑝_{𝑛,𝑚−1}𝜕𝐻^𝛼𝑛=1,…,𝑁,𝑖=0,…,𝜕𝑚𝑝−2_{𝑛,𝑚−1}.𝑑𝑡^𝛼,(5.4) Note that (4.12) gives𝜕𝐻^𝛼𝜕𝑝_{𝑛,𝑚−1}𝑑𝑡_𝛼=𝜕𝐻^0𝜕𝑝_{𝑛,𝑚−1}𝑑𝑡+𝜕Φ^𝑛,𝑖𝜕𝑝_{𝑛,𝑚−1}𝑑𝜆_𝑛,𝑖+𝜕𝐻^𝑛,𝑖𝜕𝑝_{𝑛,𝑚−1}𝑑𝑞_𝑛,𝑖=𝑑𝑞_{𝑛,𝑚−1}.(5.5) Therefore, (5.4) can be written as𝐾𝑞=^{𝑛,𝑚−1},𝑞^𝑛,𝑖,𝜆^𝑛,𝑖,𝑡^;𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖,𝜆_𝑛,𝑖,𝑡_{𝑞′𝑛,𝑚−1𝑞𝑛,𝑚−1𝑁}_𝑛=1𝐷𝑞_{𝑛,𝑚−1}𝐷𝑝_{𝑛,𝑚−1}exp𝑖ℏ^𝑡′𝛼_𝑡𝛼−𝐻_𝛼𝑑𝑡_𝛼+𝑝_{𝑛,𝑚−1}𝑑𝑞_{𝑛,𝑚−1}.(5.6) However, according to (4.6) and (4.7), we get𝐻_𝑛,𝑖=−𝜆_𝑛,𝑖Φ_𝑛,𝑖=0,(5.7) so, it can bee found that 𝐻_𝛼𝑑𝑡_𝛼=𝐻₀𝑑𝑡+𝐻_𝑛,𝑖𝑑𝑞_𝑛,𝑖+Φ_𝑛,𝑖𝑑𝜆_𝑛,𝑖=𝐻₀𝑑𝑡−𝜆_𝑛,𝑖𝑑𝑞_𝑛,𝑖.(5.8) Then the transition amplitude can be written in the final form as𝐾𝑞=^{𝑛,𝑚−1},𝑞^𝑛,𝑖,𝜆^𝑛,𝑖,𝑡′;𝑞_{𝑛,𝑚−1},𝑞_𝑛,𝑖,𝜆_𝑛,𝑖,𝑡_{𝑞′𝑛,𝑚−1𝑞𝑛,𝑚−1𝑁}_𝑛=1𝐷𝑞_{𝑛,𝑚−1}𝐷𝑝_{𝑛,𝑚−1}exp𝑖ℏ^𝑡𝛼_𝑡𝛼−𝐻₀𝑑𝑡+𝜆_𝑛,𝑖𝑑𝑞_𝑛,𝑖+𝑝_{𝑛,𝑚−1}𝑑𝑞_{𝑛,𝑚−1}.(5.9) Equation (5.9) represents the canonical path integral quantization of higher-order regular Lagrangians as first-order singular Lagrangians.

6. Examples

In this section, the procedure described throughout this paper will be illustrated by the following two examples.

6.1. Example 1

As a first example, let us consider a one-dimensional second-order regular lagrangian of the form:𝐿₀=12𝑞²¹̈−𝑞²¹̇−𝑞²¹.(6.1) If (4.2) is used, we can write𝑞𝑞₁̇¹⁽⁰⁾=𝑞₁₀,𝑞₁̈=𝑞₁₁,=𝑞̇₁₁,(6.2) Hence, the Lagrangian (6.1) becomes𝐿₀=12𝑞̇²¹¹−𝑞²¹¹−𝑞²¹⁰.(6.3) Upon using (4.1), the recursion relation is 𝑞̇₁₀=𝑞₁₁. And with the aid of (4.3), the extended Lagrangian is simply 𝐿_𝑇=12𝑞̇²¹¹−𝑞²¹¹−𝑞²¹⁰+𝜆₁₀𝑞̇₁₀−𝑞₁₁.(6.4) The conjugate momenta can be obtained as 𝑃₁₁=𝜕𝐿^𝑇𝜕𝑞̇₁₁=𝑞̇₁₁,𝑃₁₀=𝜕𝐿^𝑇𝜕𝑞̇₁₀=𝜆₁₀,𝜋₁₀=𝜕𝐿^𝑇𝜕𝜆̇₁₀0.(6.5) It is obvious that the second and third equations are constraints. Therefore, the coordinates 𝑞₁₀ and 𝜆₁₀ represent the restricted coordinates.

Using (4.4), the canonical Hamiltonian takes the form:𝐻₀=12𝑃²¹¹+𝑞²¹⁰+𝑞²¹¹+𝜆₁₀𝑞₁₁.(6.6) Accordingly, the set of HJPDE's can be written as𝐻𝐻^0=𝑃₀+𝐻₀=0,_10Φ=𝑃₁₀−𝜆₁₀=0,_10=𝜋₁₀=0.(6.7) From (5.9), the canonical path integral quantization for this system is 𝐾𝑞^11,𝑞^10,𝜆^10,𝑡′;𝑞₁₁,𝑞₁₀,𝜆₁₀,𝑡=𝐷𝑞₁₁𝐷𝑝₁₁exp𝑖ℏ−𝐻⁰𝑑𝑡+𝑝₁₁𝑑𝑞₁₁+𝜆₁₀𝑑𝑞₁₀,(6.8) where 𝐷𝑞₁₁=lim_𝑘→∞∏^{𝑘−1𝑗=1}𝑑𝑞_11𝑗;𝐷𝑝₁₁=lim_𝑘→∞∏^{𝑘−1𝑗=0}(𝑑𝑝₁₁/2𝜋ℏ).𝐾=𝐷𝑞₁₁𝐷𝑝₁₁exp𝑖ℏ−12𝑃²¹¹+𝑞²¹⁰+𝑞²¹¹−𝜆₁₀𝑞₁₁𝑑𝑡+𝑝¹¹𝑑𝑞₁₁+𝜆₁₀𝑑𝑞₁₀.(6.9) Equation (6.9) can be written in a compact form as𝐾=𝐷𝑞₁₁𝐷𝑝₁₁exp𝑖ℏ−12𝑃²¹¹+𝑞²¹⁰+𝑞²¹¹−𝜆₁₀𝑞₁₁+𝑝₁₁𝑞̇₁₁+𝜆₁₀𝑞̇₁₀𝑑𝑡.(6.10) Upon changing the integration over 𝑑𝑡 to summation, we have𝐾=𝐷𝑞_11𝑘−1_𝑗=0𝑑2𝑝𝜋ℏ^11𝑗exp𝑖ℏ𝜀^𝑘−1_𝑗=0−𝑝2^211𝑗−𝑞2^210𝑗−𝑞2^211𝑗+𝑝^11𝑗𝑞̇_11𝑗+𝜆_10𝑗𝑞̇_10𝑗−𝑞_11𝑗.(6.11) The 𝑝_11𝑗-integration can be performed using the Gaussian integral:𝐾=𝐷𝑞₁₁1(2𝜋ℏ)^𝑘2𝑖𝜋𝜀ℏ^𝑘/2exp𝑖𝜀ℏ^𝑘−1_𝑗=0𝑞̇2^211𝑗−𝑞2^210𝑗−𝑞2^211𝑗+𝜆^10𝑗𝑞̇_10𝑗=1−𝑞_11𝑗2𝜋ℏ𝑖𝜀^𝑘/2𝐷𝑞¹¹exp𝑖ℏ𝑞2̇²¹¹−𝑞2²¹⁰−𝑞2²¹¹+𝜆¹⁰𝑞̇₁₀=1−𝑞₁₁𝑑𝑡2𝜋ℏ𝑖𝜀^𝑘/2𝐷𝑞¹¹exp𝑖ℏ𝐿^𝑇𝑑𝑡.(6.12)

6.2. Example 2

As a second example, consider the three-dimensional second-order regular lagrangian:𝐿₀=12𝑞²¹̈+𝑞²²̈+𝑞²³̈−12𝑞²¹̇+𝑞²³̇.(6.13) If we put𝑞𝑞₁̇¹⁽⁰⁾=𝑞₁₀,𝑞²⁽⁰⁾=𝑞₂₀,𝑞³⁽⁰⁾=𝑞₃₀,𝑞₁̈=𝑞₁₁,𝑞₂̇=𝑞₂₁,𝑞₃̇=𝑞₃₁,=𝑞̇₁₁,𝑞₂̈=𝑞̇₂₁,𝑞₃̈=𝑞̇₃₁.(6.14) then the above Lagrangian can be written as𝐿₀=12𝑞̇²¹¹+𝑞̇²²¹+𝑞̇²³¹−12𝑞²¹¹+𝑞²³¹.(6.15) Here the recursion relations are𝑞̇₁₀=𝑞₁₁;𝑞̇₂₀=𝑞₂₁,𝑞̇₃₀=𝑞₃₁.(6.16) Accordingly, the extended Lagrangian can be given as:𝐿_𝑇=12𝑞̇²¹¹+𝑞̇²²¹+𝑞̇²³¹−12𝑞²¹¹+𝑞²³¹+𝜆₁₀𝑞̇₁₀−𝑞₁₁+𝜆₂₀𝑞̇₂₀−𝑞₂₁+𝜆₃₀𝑞̇₃₀−𝑞₃₁.(6.17) The corresponding momenta are calculated as𝑝𝑝₁₁=𝑞̇₁₁,𝑝₁₀=𝜆₁₀,𝜋₁₀=0,₂₁𝑝=𝑞̇₂₁,𝑝₂₀=𝜆₂₀,𝜋₂₀=0,₃₁=𝑞̇₃₁,𝑝₃₀=𝜆₃₀,𝜋₃₀=0(6.18) Therefore, the canonical Hamiltonian reads𝐻₀=𝑝²¹¹2+𝑝2²²¹+𝑝2²³¹+12𝑞²¹¹+𝑞²³¹+𝜆₁₀𝑞₁₁+𝜆₂₀𝑞₂₁+𝜆₃₀𝑞₃₁.(6.19) Thus, the set of HJPDE's can be written as𝐻^0Φ=𝑃₀+𝐻₀=0,_10Φ=𝜋₁₀=0,_20Φ=𝜋₂₀=0,_30𝐻=𝜋₃₀=0,_10𝐻=𝑝₁₀−𝜆₁₀=0,_20𝐻=𝑝₂₀−𝜆₂₀=0,_30=𝑝₃₀−𝜆₃₀=0.(6.20) Then, the canonical path integral quantization for this system is constructed as𝐾𝑞=^𝑛1,𝑞^𝑛0,𝜆^𝑛0,𝑡′;𝑞_𝑛1,𝑞_𝑛0,𝜆_𝑛0,𝑡₃_𝑛=1𝐷𝑞_𝑛1𝐷𝑝_𝑛1exp𝑖ℏ−𝐻⁰𝑑𝑡+𝜆_𝑛0𝑑𝑞_𝑛0+𝑝_𝑛1𝑑𝑞_𝑛1,(6.21) where 𝑛=1,2,3.𝐾=³_𝑛=1𝐷𝑞_𝑛1𝐷𝑝_𝑛1exp𝑖ℏ−𝑝2^2𝑛1−𝑞2²¹¹−𝑞2²³¹+𝜆^𝑛0𝑞̇_𝑛0−𝑞_𝑛1+𝑝_𝑛1𝑞̇_𝑛1𝑑𝑡.(6.22) Changing the integration over 𝑑𝑡 to summation and integrating over 𝑝₁₁,𝑝₂₁ and 𝑝₃₁k times we get 𝐾=12𝜋ℏ𝑖𝜀^3𝑘/2³_𝑛=1𝐷𝑞_𝑛1exp𝑖ℏ𝑞2̇^2𝑛1−𝑞2²¹¹−𝑞2²³¹+𝜆^𝑛0𝑞̇_𝑛0=1−𝑞_𝑛1𝑑𝑡2𝜋ℏ𝑖𝜀^3𝑘/2𝐷𝑞¹¹𝐷𝑞₂₁𝐷𝑞₃₁exp𝑖ℏ𝐿^𝑇𝑑𝑡.(6.23)

7. Conclusion

In this work, we have investigated the canonical path integral quantization of higher-order regular Lagrangians. Where the higher-order regular Lagrangians are first treated as first-order singular Lagrangians, this means that each velocity term 𝑞̇_𝑛,𝑖 is replaced by a new function 𝑞_𝑛,𝑖+1, which is led to a constraint equation, 𝑞_𝑛,𝑖+1−𝑞̇_𝑛,𝑖=0, that is added to the original Lagrangian. The same procedure is repeated for the second and other higher order terms of velocities. Every time, a new constraint is obtained and added to the original Lagrangian. As a result to this procedure, the new constructed Lagrangian is the extended first-order Lagrangian.

Once the extended Lagrangian is obtained, it is treated using the well-known Hamilton-Jacobi method which enables us to obtain the equations of motion. Besides, the action integral can be derived and the quantization of the system may be investigated using the canonical path integral approximation.

In this treatment, we believe that the local structure of phase space and its local simplistic geometry is more transparent than in Ostrogradski's approach. In Ostrogradski's approach, the structure of phase space leads to confusion when considering canonical path integral quantization.