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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 375838, 11 pages
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.
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 . These authors obtained a set of Hamilton-Jacobi partial differential equations (HJPDEs) for singular systems using Caratheodory’s equivalent-Lagrangian method . 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 . 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 . 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  described by Lagrangians depending on coordinates and velocities only. Therefore, higher-order systems can be set in the form of ordinary constrained systems . 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 , 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 , , with the Hessian matrix of rank .
The canonical formulation  gives the set of the Hamilton-Jacobi partial differential equations as where and are the momenta conjugate to and, respectively, The canonical Hamiltonian H0 is given by The equations of motion are obtained as total differential equations in many variables as follows: where . The set of equations (2.4) and (2.5) is integrable if and only if 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 generalized coordinates : where , and .
The Euler-Lagrange equations of motion are obtained as 
Theories with higher derivatives, which have been first developed by Ostrogradski , treat the derivatives as independent coordinates. Therefore, we will indicate this by writing them as . In Ostrogradski's formalism, the momenta conjugated, respectively, to and , read as Therefore, the canonical Hamiltonian is given by Hamilton's equations of motion are written using Poisson bracket as [5, 6] where is the Poisson bracket defined as The fundamental Poisson brackets are where .
With this procedure, the phase space, described in terms of the canonical variables , 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 () such that the following recursion relations would hold [13, 14]: Clearly, the variables (), would then correspond to the time derivatives () respectively, that is, Equation (4.1) represents relations between the new variables. In order to enforce these relations for independent variables (), additional Lagrange multipliers are introduced . The variables (), 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 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: the canonical Hamiltonian can be obtained as Equations (4.5) and (4.6) represent primary constraints [1, 2]. Their Hamilton-Jacobi partial differential equations can be obtained as The equations of motion can be written as total differential equations in many variables as follows: The total differential equations are integrable if and only if
5. The Canonical Path Integral Quantization
If the coordinates are denoted by , that is, then the set of primary constraints (4.8), (4.9), and (4.10) can be written in a compact form as where Making use of , the canonical path integral for the extended Lagrangians can be obtained as Note that (4.12) gives Therefore, (5.4) can be written as However, according to (4.6) and (4.7), we get so, it can bee found that Then the transition amplitude can be written in the final form as Equation (5.9) represents the canonical path integral quantization of higher-order regular Lagrangians as first-order singular Lagrangians.
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: If (4.2) is used, we can write Hence, the Lagrangian (6.1) becomes Upon using (4.1), the recursion relation is . And with the aid of (4.3), the extended Lagrangian is simply The conjugate momenta can be obtained as It is obvious that the second and third equations are constraints. Therefore, the coordinates and represent the restricted coordinates.
Using (4.4), the canonical Hamiltonian takes the form: Accordingly, the set of HJPDE's can be written as From (5.9), the canonical path integral quantization for this system is where . Equation (6.9) can be written in a compact form as Upon changing the integration over to summation, we have The -integration can be performed using the Gaussian integral:
6.2. Example 2
As a second example, consider the three-dimensional second-order regular lagrangian: If we put then the above Lagrangian can be written as Here the recursion relations are Accordingly, the extended Lagrangian can be given as: The corresponding momenta are calculated as Therefore, the canonical Hamiltonian reads Thus, the set of HJPDE's can be written as Then, the canonical path integral quantization for this system is constructed as where . Changing the integration over to summation and integrating over , and k times we get
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 is replaced by a new function , which is led to a constraint equation, , 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.
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