Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2011, Article ID 375838, 11 pages
Research Article

Canonical Quantization of Higher-Order Lagrangians

Department of Physics, Mu'tah University, Karak 61710, Jordan

Received 4 July 2011; Accepted 22 August 2011

Academic Editor: Nicola Guglielmi

Copyright © 2011 Khaled I. Nawafleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [710].

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 [111].

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(𝑞𝑖,̇𝑞𝑖), 𝑖=1,2,,𝑁, with the Hessian matrix 𝜕2𝐿/𝜕̇𝑞𝑖𝜕̇𝑞𝑗 of rank 𝑁-𝑅,𝑅<𝑁.

The canonical formulation [3] gives the set of the Hamilton-Jacobi partial differential equations as𝐻0=𝑝0+𝐻0𝜕𝑆𝜕𝑡+𝐻0𝑞𝛽,𝑞𝑎,𝑝𝑎=𝜕𝑆𝜕𝑞𝑎𝐻=0,𝜇=𝑝𝜇+𝐻𝜇𝜕𝑆𝜕𝑞𝜇+𝐻𝜇𝑞𝛽,𝑞𝑎,𝑝𝑎=𝜕𝑆𝜕𝑞𝑎=0,𝑎=1,,𝑁𝑅,𝜇=𝑁𝑅+1,,𝑁,(2.1) where 𝑝0 and 𝑞𝜇are the momenta conjugate to 𝑡 and𝑞𝜇, respectively,𝑝0=𝑞𝜕𝑆𝑖,𝑡𝜕𝑡,𝑝𝜇=𝑞𝜕𝑆𝑖,𝑡𝜕𝑞𝜇.(2.2) The canonical Hamiltonian H0 is given by𝐻0=𝑝𝑎̇𝑞𝑎+𝑝𝜇̇𝑞𝜇𝐿.(2.3) The equations of motion are obtained as total differential equations in many variables as follows: 𝑑𝑞𝑎=𝜕𝐻0𝜕𝑝𝑎𝑑𝑡+𝜕𝐻𝜇𝜕𝑝𝑎𝑑𝑞𝜇,𝑑𝑝𝑖=𝜕𝐻0𝜕𝑞𝑖𝑑𝑡𝜕𝐻𝜇𝜕𝑞𝑖𝑑𝑞𝜇,(2.4)𝑑𝑧=𝐻0𝑑𝑡𝐻𝜇𝑑𝑞𝜇+𝑝𝑎𝜕𝐻0𝜕𝑝𝑎𝑑𝑡+𝑝𝑎𝜕𝐻𝜇𝜕𝑝𝑎𝑑𝑞𝜇,(2.5) where 𝑧=𝑆(𝑡,𝑞𝑎,𝑞𝜇). The set of equations (2.4) and (2.5) is integrable if and only if𝑑𝐻0=0,𝜕𝐻𝜇=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 qa and momenta pi as functions of qμ and t.

3. Ostrogradski's Formalism of Higher-Order Lagrangians

Consider a higher-order Lagrangian system of 𝑁 generalized coordinates 𝑞𝑛(𝑡):𝐿0𝑞𝑛,̇𝑞𝑛,,𝑞𝑛(𝑚),𝑚1,(3.1) where 𝑞𝑛(𝑠)=𝑑𝑠𝑞𝑛/𝑑𝑡𝑠, 𝑠=0,1,,𝑚 and 𝑛=1,,𝑁.

The Euler-Lagrange equations of motion are obtained as [12]𝑚𝑠=0(1)𝑠𝑑𝑠𝑑𝑡𝑠𝜕𝐿0𝜕𝑞𝑛(𝑠)=0.(3.2)

Theories with higher derivatives, which have been first developed by Ostrogradski [12], treat the derivatives 𝑞𝑛(𝑠)(𝑠=0,,𝑚1) as independent coordinates. Therefore, we will indicate this by writing them as 𝑞𝑛(𝑠)=𝑞𝑛,𝑠. In Ostrogradski's formalism, the momenta conjugated, respectively, to 𝑞𝑛,𝑚1 and 𝑞𝑛,𝑠1, (𝑠=1,,𝑚1) read as𝑝𝑛,𝑚1𝜕𝐿0𝜕𝑞𝑛(𝑚),𝑝𝑛,𝑠1𝜕𝐿0𝜕𝑞𝑛(𝑠)̇𝑝𝑛,𝑠,𝑠=1,,𝑚1.(3.3) Therefore, the canonical Hamiltonian is given by 𝐻0𝑞𝑛,0,,𝑞𝑛,𝑚1;𝑝𝑛,0,,𝑝𝑛,𝑚1=𝑚2𝑠=0𝑝𝑛,𝑠𝑞𝑛,𝑠+1+𝑝𝑛,𝑚1̇𝑞𝑛,𝑚1𝐿0𝑞𝑛,0,,𝑞𝑛,𝑚1,̇𝑞𝑛,𝑚1.(3.4) Hamilton's equations of motion are written using Poisson bracket as [5, 6]̇𝑞𝑛,𝑠=𝜕𝐻0𝜕𝑝𝑛,𝑠=𝑞𝑛,𝑠,𝐻0,(3.5)̇𝑝𝑛,𝑠=𝜕𝐻0𝜕𝑞𝑛,𝑠=𝑝𝑛,𝑠,𝐻0,(3.6) where {,} is the Poisson bracket defined as{𝐴,𝐵}=𝑚1𝑠=0𝜕𝐴𝜕𝑞𝑛,𝑠𝜕𝐵𝜕𝑝𝑛,𝑠𝜕𝐵𝜕𝑞𝑛,𝑠𝜕𝐴𝜕𝑝𝑛,𝑠.(3.7) The fundamental Poisson brackets are𝑞𝑛,𝑠𝑝𝑛,𝑠=𝛿𝑛𝑛𝛿𝑠𝑠,𝑞𝑛,𝑠,𝑞𝑛,𝑠=𝑝𝑛,𝑠,𝑝𝑛,𝑠=0,(3.8) where 𝑛,𝑛=1,,𝑁,and𝑠,𝑠=0,,𝑚1.

With this procedure, the phase space, described in terms of the canonical variables 𝑞𝑛,𝑠and𝑝𝑛,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 (𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝑖=0,1,,𝑚2) such that the following recursion relations would hold [13, 14]:̇𝑞𝑛,𝑖=𝑞𝑛,𝑖+1.(4.1) Clearly, the variables (𝑞𝑛,𝑚1,𝑞𝑛,𝑖), would then correspond to the time derivatives (𝑞𝑛(𝑚1),𝑞𝑛(𝑖)) respectively, that is,𝑞𝑛(0)=𝑞𝑛,0,̇𝑞𝑛=𝑞𝑛,1,,𝑞𝑛(𝑚1)=𝑞𝑛,𝑚1,𝑞𝑛(𝑚)=̇𝑞𝑛,𝑚1.(4.2) Equation (4.1) represents relations between the new variables. In order to enforce these relations for independent variables (𝑞𝑛,𝑚1,𝑞𝑛,𝑖), additional Lagrange multipliers 𝜆𝑛,𝑖(𝑡) are introduced [14]. The variables (𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,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 𝐿𝑇𝑞𝑛,𝑖,𝑞𝑛,𝑚1,̇𝑞𝑛,𝑖,̇𝑞𝑛,𝑚1,𝜆𝑛,𝑖=𝐿0𝑞𝑛,𝑖,𝑞𝑛,𝑚1,̇𝑞𝑛,𝑚1+𝑚2𝑖=0𝜆𝑛,𝑖̇𝑞𝑛,𝑖𝑞𝑛,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:𝑝𝑛,𝑚1=𝜕𝐿𝑇𝜕̇𝑞𝑛,𝑚1,𝑝(4.4)𝑛,𝑖=𝜕𝐿𝑇𝜕̇𝑞𝑛,𝑖=𝜆𝑛,𝑖=𝐻𝑛,𝑖𝜋,(4.5)𝑛,𝑖=𝜕𝐿𝑇𝜕̇𝜆𝑛,𝑖=0=Φ𝑛,𝑖,(4.6) the canonical Hamiltonian can be obtained as𝐻0𝑞𝑛,𝑖,𝑞𝑛,𝑚1,𝑝𝑛,𝑚1,𝜆𝑛,𝑖=𝑝𝑛,𝑚1̇𝑞𝑛,𝑚1+𝑚2𝑖=0𝑝𝑛,𝑖̇𝑞𝑛,𝑖+𝑚2𝑖=0𝜋𝑛,𝑖̇𝜆𝑛,𝑖𝐿𝑇𝑞𝑛,𝑖,𝑞𝑛,𝑚1,̇𝑞𝑛,𝑖,̇𝑞𝑛,𝑚1,𝜆𝑛,𝑖,(4.7) Equations (4.5) and (4.6) represent primary constraints [1, 2]. Their Hamilton-Jacobi partial differential equations can be obtained as𝐻0=𝑝0+𝐻0𝑞𝑛,𝑖,𝑞𝑛,𝑚1,𝑝𝑛,𝑚1,𝜆𝑛,𝑖Φ=0,(4.8)𝑛,𝑖=𝜋𝑛,𝑖𝐻=0,(4.9)𝑛,𝑖=𝑝𝑛,𝑖𝜆𝑛,𝑖=0.(4.10) The equations of motion can be written as total differential equations in many variables as follows: 𝑑𝑞𝑛,𝑗=𝑑𝑞𝑛,𝑗,(4.11)𝑑𝑞𝑛,𝑚1=𝜕𝐻0𝜕𝑝𝑛,𝑚1𝑑𝑡,(4.12)𝑑𝑝𝑛,𝑗=𝜕𝐻0𝜕𝑞𝑛,𝑗𝑑𝑡,𝑑𝑝𝑛,𝑚1=𝜕𝐻0𝜕𝑞𝑛,𝑚1𝑑𝑡,𝑑𝜆𝑛,𝑗=𝑑𝜆𝑛,𝑗,𝑑𝜋𝑛,𝑗=𝜕𝐻0𝜕𝜆𝑛,𝑗𝑑𝑡+𝑑𝑞𝑛,𝑗,𝑗=0,1,,𝑚2.(4.13) The total differential equations are integrable if and only if𝑑𝐻0=𝑑𝑝0𝑑𝐻0=0,𝑑𝐻𝑛,𝑗=𝑑𝑝𝑛,𝑗𝑑𝜆𝑛,𝑗=0,𝑑Φ𝑛,𝑗=𝑑𝜋𝑛,𝑗=0.(4.14)

5. The Canonical Path Integral Quantization

If the coordinates 𝑡,𝑞𝑛,𝑖,𝜆𝑛,𝑖 are denoted by 𝑡𝛼, that is,𝑡𝛼=𝑡,𝑞𝑛,𝑖,𝜆𝑛,𝑖,(5.1) then the set of primary constraints (4.8), (4.9), and (4.10) can be written in a compact form as𝐻𝛼=0,(5.2) where 𝐻𝛼=𝐻0,𝐻𝑛,𝑖,Φ𝑛,𝑖.(5.3) Making use of [7], the canonical path integral for the extended Lagrangians can be obtained as𝐾𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖,𝑡;𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖=,𝑡𝑞𝑛,𝑚1𝑞𝑛,𝑚1𝑁𝑛=1𝐷𝑞𝑛,𝑚1𝐷𝑝𝑛,𝑚1𝑖exp𝑡𝛼𝑡𝛼𝐻𝛼+𝑝𝑛,𝑚1𝜕𝐻𝛼𝜕𝑝𝑛,𝑚1𝑑𝑡𝛼,𝑛=1,,𝑁,𝑖=0,,𝑚2.(5.4) Note that (4.12) gives𝜕𝐻𝛼𝜕𝑝𝑛,𝑚1𝑑𝑡𝛼=𝜕𝐻0𝜕𝑝𝑛,𝑚1𝑑𝑡+𝜕Φ𝑛,𝑖𝜕𝑝𝑛,𝑚1𝑑𝜆𝑛,𝑖+𝜕𝐻𝑛,𝑖𝜕𝑝𝑛,𝑚1𝑑𝑞𝑛,𝑖=𝑑𝑞𝑛,𝑚1.(5.5) Therefore, (5.4) can be written as𝐾𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖,𝑡;𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖=,𝑡𝑞𝑛,𝑚1𝑞𝑛,𝑚1𝑁𝑛=1𝐷𝑞𝑛,𝑚1𝐷𝑝𝑛,𝑚1𝑖exp𝑡𝛼𝑡𝛼𝐻𝛼𝑑𝑡𝛼+𝑝𝑛,𝑚1𝑑𝑞𝑛,𝑚1.(5.6) However, according to (4.6) and (4.7), we get𝐻𝑛,𝑖=𝜆𝑛,𝑖Φ𝑛,𝑖=0,(5.7) so, it can bee found that 𝐻𝛼𝑑𝑡𝛼=𝐻0𝑑𝑡+𝐻𝑛,𝑖𝑑𝑞𝑛,𝑖+Φ𝑛,𝑖𝑑𝜆𝑛,𝑖=𝐻0𝑑𝑡𝜆𝑛,𝑖𝑑𝑞𝑛,𝑖.(5.8) Then the transition amplitude can be written in the final form as𝐾𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖,𝑡;𝑞𝑛,𝑚1,𝑞𝑛,𝑖,𝜆𝑛,𝑖=,𝑡𝑞𝑛,𝑚1𝑞𝑛,𝑚1𝑁𝑛=1𝐷𝑞𝑛,𝑚1𝐷𝑝𝑛,𝑚1𝑖exp𝑡𝛼𝑡𝛼𝐻0𝑑𝑡+𝜆𝑛,𝑖𝑑𝑞𝑛,𝑖+𝑝𝑛,𝑚1𝑑𝑞𝑛,𝑚1.(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=12̈𝑞21̇𝑞21𝑞21.(6.1) If (4.2) is used, we can write𝑞1(0)=𝑞10,̇𝑞1=𝑞11,̈𝑞1=̇𝑞11,(6.2) Hence, the Lagrangian (6.1) becomes𝐿0=12̇𝑞211𝑞211𝑞210.(6.3) Upon using (4.1), the recursion relation is ̇𝑞10=𝑞11. And with the aid of (4.3), the extended Lagrangian is simply 𝐿𝑇=12̇𝑞211𝑞211𝑞210+𝜆10̇𝑞10𝑞11.(6.4) The conjugate momenta can be obtained as 𝑃11=𝜕𝐿𝑇𝜕̇𝑞11=̇𝑞11,𝑃10=𝜕𝐿𝑇𝜕̇𝑞10=𝜆10,𝜋10=𝜕𝐿𝑇𝜕̇𝜆100.(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=12𝑃211+𝑞210+𝑞211+𝜆10𝑞11.(6.6) Accordingly, the set of HJPDE's can be written as𝐻0=𝑃0+𝐻0𝐻=0,10=𝑃10𝜆10Φ=0,10=𝜋10=0.(6.7) From (5.9), the canonical path integral quantization for this system is 𝐾𝑞11,𝑞10,𝜆10,𝑡;𝑞11,𝑞10,𝜆10=,𝑡𝐷𝑞11𝐷𝑝11𝑖exp𝐻0𝑑𝑡+𝑝11𝑑𝑞11+𝜆10𝑑𝑞10,(6.8) where 𝐷𝑞11=lim𝑘𝑘1𝑗=1𝑑𝑞11𝑗;𝐷𝑝11=lim𝑘𝑘1𝑗=0(𝑑𝑝11/2𝜋).𝐾=𝐷𝑞11𝐷𝑝11𝑖exp12𝑃211+𝑞210+𝑞211𝜆10𝑞11𝑑𝑡+𝑝11𝑑𝑞11+𝜆10𝑑𝑞10.(6.9) Equation (6.9) can be written in a compact form as𝐾=𝐷𝑞11𝐷𝑝11𝑖exp12𝑃211+𝑞210+𝑞211𝜆10𝑞11+𝑝11̇𝑞11+𝜆10̇𝑞10𝑑𝑡.(6.10) Upon changing the integration over 𝑑𝑡 to summation, we have𝐾=𝐷𝑞11𝑘1𝑗=0𝑑𝑝11𝑗2𝜋exp𝑖𝜀𝑘1𝑗=0𝑝211𝑗2𝑞210𝑗2𝑞211𝑗2+𝑝11𝑗̇𝑞11𝑗+𝜆10𝑗̇𝑞10𝑗𝑞11𝑗.(6.11) The 𝑝11𝑗-integration can be performed using the Gaussian integral:𝐾=𝐷𝑞111(2𝜋)𝑘2𝜋𝑖𝜀𝑘/2exp𝑖𝜀𝑘1𝑗=0̇𝑞211𝑗2𝑞210𝑗2𝑞211𝑗2+𝜆10𝑗̇𝑞10𝑗𝑞11𝑗=12𝜋𝑖𝜀𝑘/2𝐷𝑞11𝑖exṗ𝑞2112𝑞2102𝑞2112+𝜆10̇𝑞10𝑞11=1𝑑𝑡2𝜋𝑖𝜀𝑘/2𝐷𝑞11𝑖exp𝐿𝑇.𝑑𝑡(6.12)

6.2. Example 2

As a second example, consider the three-dimensional second-order regular lagrangian:𝐿0=12̈𝑞21+̈𝑞22+̈𝑞2312̇𝑞21+̇𝑞23.(6.13) If we put𝑞1(0)=𝑞10,𝑞2(0)=𝑞20,𝑞3(0)=𝑞30,̇𝑞1=𝑞11,̇𝑞2=𝑞21,̇𝑞3=𝑞31,̈𝑞1=̇𝑞11,̈𝑞2=̇𝑞21,̈𝑞3=̇𝑞31.(6.14) then the above Lagrangian can be written as𝐿0=12̇𝑞211+̇𝑞221+̇𝑞23112𝑞211+𝑞231.(6.15) Here the recursion relations arė𝑞10=𝑞11;̇𝑞20=𝑞21,̇𝑞30=𝑞31.(6.16) Accordingly, the extended Lagrangian can be given as:𝐿𝑇=12̇𝑞211+̇𝑞221+̇𝑞23112𝑞211+𝑞231+𝜆10̇𝑞10𝑞11+𝜆20̇𝑞20𝑞21+𝜆30̇𝑞30𝑞31.(6.17) The corresponding momenta are calculated as𝑝11=̇𝑞11,𝑝10=𝜆10,𝜋10𝑝=0,21=̇𝑞21,𝑝20=𝜆20,𝜋20𝑝=0,31=̇𝑞31,𝑝30=𝜆30,𝜋30=0(6.18) Therefore, the canonical Hamiltonian reads𝐻0=𝑝2112+𝑝2212+𝑝2312+12𝑞211+𝑞231+𝜆10𝑞11+𝜆20𝑞21+𝜆30𝑞31.(6.19) Thus, the set of HJPDE's can be written as𝐻0=𝑃0+𝐻0Φ=0,10=𝜋10Φ=0,20=𝜋20Φ=0,30=𝜋30𝐻=0,10=𝑝10𝜆10𝐻=0,20=𝑝20𝜆20𝐻=0,30=𝑝30𝜆30=0.(6.20) Then, the canonical path integral quantization for this system is constructed as𝐾𝑞𝑛1,𝑞𝑛0,𝜆𝑛0,𝑡;𝑞𝑛1,𝑞𝑛0,𝜆𝑛0=,𝑡3𝑛=1𝐷𝑞𝑛1𝐷𝑝𝑛1𝑖exp𝐻0𝑑𝑡+𝜆𝑛0𝑑𝑞𝑛0+𝑝𝑛1𝑑𝑞𝑛1,(6.21) where 𝑛=1,2,3.𝐾=3𝑛=1𝐷𝑞𝑛1𝐷𝑝𝑛1𝑖exp𝑝2𝑛12𝑞2112𝑞2312+𝜆𝑛0̇𝑞𝑛0𝑞𝑛1+𝑝𝑛1̇𝑞𝑛1.𝑑𝑡(6.22) Changing the integration over 𝑑𝑡 to summation and integrating over 𝑝11,𝑝21 and 𝑝31k times we get 1𝐾=2𝜋𝑖𝜀3𝑘/23𝑛=1𝐷𝑞𝑛1𝑖exṗ𝑞2𝑛12𝑞2112𝑞2312+𝜆𝑛0̇𝑞𝑛0𝑞𝑛1=1𝑑𝑡2𝜋𝑖𝜀3𝑘/2𝐷𝑞11𝐷𝑞21𝐷𝑞31𝑖exp𝐿𝑇.𝑑𝑡(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 ̇𝑞𝑛,𝑖 is replaced by a new function 𝑞𝑛,𝑖+1, which is led to a constraint equation, 𝑞𝑛,𝑖+1̇𝑞𝑛,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.


  1. P. A. M. Dirac, “Generalized Hamiltonian dynamics,” Canadian Journal of Mathematics, vol. 2, pp. 129–148, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. P. A. M. Dirac, Lectures on Quantum Mechanics, vol. 2, Belfer Graduate School of Science, Yeshiva University, New York, NY, USA, 1967.
  3. E. M. Rabei and Y. Guler, “Hamilton-Jacobi treatment of second-class constraints,” Physical Review A, vol. 46, no. 6, pp. 3513–3515, 1992. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Holden-Day, San Francisco, Calif, USA, 1967.
  5. B. M. Pimentel and R. G. Teixeira, “Hamilton-Jacobi formulation for singular systems with second-order Lagrangians,” Il Nuovo Cimento B, vol. 111, no. 7, pp. 841–854, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  6. B. M. Pimentel and R. G. Teixeira, “Generalization of the Hamilton-Jacobi approach for higher-order singular systems,” Il Nuovo Cimento B, vol. 113, no. 6, pp. 805–817, 1998. View at Google Scholar · View at Scopus
  7. S. Muslih and Y. Güler, “The Feynman path integral quantization of constrained systems,” Il Nuovo Cimento B, vol. 112, no. 1, pp. 97–107, 1997. View at Google Scholar
  8. E. M. Rabei, “On the quantization of constrained systems using path integral techniques,” Il Nuovo Cimento B, vol. 115, no. 10, pp. 1159–1165, 2000. View at Google Scholar · View at Scopus
  9. S. I. Muslih, “Quantization of singular systems with second-order Lagrangians,” Modern Physics Letters A, vol. 17, no. 36, pp. 2383–2391, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S. I. Muslih, “Path integral formulation of constrained systems with singular-higher order Lagrangians,” Hadronic Journal, vol. 24, no. 6, pp. 713–720, 2001. View at Google Scholar · View at Zentralblatt MATH
  11. E. M. Rabei, K. I. Nawafleh, and H. B. Ghassib, “Quantization of constrained systems using the WKB approximation,” Physical Review A, vol. 66, no. 2, pp. 024101/1–024101/4, 2002. View at Google Scholar · View at Scopus
  12. M. V. Ostrogradski, Mémoires de l'Académie de Saint Peters Bourg, vol. 6, p. 385, 1850.
  13. J. Govaerts, Hamiltonian Quantization and Constrained Dynamics, Leuven University Press, Leuven, Belgium, 1991.
  14. J. M. Pons, “Ostrogradski's theorem for higher-order singular Lagrangians,” Letters in Mathematical Physics, vol. 17, no. 3, pp. 181–189, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. I. Nawafleh and A. A. Alsoub, “Quantization of higher order regular lagrangians as first order singular lagrangians using path integral approach,” Jordan Journal of Physics, vol. 1, no. 2, pp. 73–78, 2008. View at Google Scholar