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 , i = 1,2, …, N, with the Hessian matrix of rank N-R, R < N.
3. Ostrogradski′s Formalism of Higher-Order Lagrangians
With this procedure, the phase space, described in terms of the canonical variables qn,s and pn,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
5. The Canonical Path Integral Quantization
6. Examples
In this section, the procedure described throughout this paper will be illustrated by the following two examples.
6.1. Example 1
6.2. Example 2
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 is replaced by a new function qn,i+1, 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.