# Canonical Quantization of  Higher-Order Lagrangians

Authors: Khaled I. Nawafleh
Journal: Journal of Applied Mathematics
 (2011)

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## 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.


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## Body

## 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 Section2, 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 LagrangianL=L(qi,q̇i), i=1,2,…,N, with the Hessian matrix ∂2L/∂q̇i∂q̇j of rank N-R,R<N.The canonical formulation [3] gives the set of the Hamilton-Jacobi partial differential equations as(2.1)H0′=p0+H0≡∂S∂t+H0(qβ,qa,pa=∂S∂qa)=0,Hμ′=pμ+Hμ≡∂S∂qμ+Hμ(qβ,qa,pa=∂S∂qa)=0,a=1,…,N-R,μ=N-R+1,…,N,
where p0 and qμare the momenta conjugate to t andqμ, respectively,(2.2)p0=∂S(qi,t)∂t,pμ=∂S(qi,t)∂qμ.
The canonical Hamiltonian H0 is given by(2.3)H0=paq̇a+pμq̇μ-L.
The equations of motion are obtained as total differential equations in many variables as follows: (2.4)dqa=∂H0′∂padt+∂Hμ′∂padqμ,dpi=-∂H0′∂qidt-∂Hμ′∂qidqμ,(2.5)dz=-H0dt-Hμdqμ+pa∂H0′∂padt+pa∂Hμ′∂padqμ,
where z=S(t,qa,qμ). The set of equations (2.4) and (2.5) is integrable if and only if(2.6)dH0′=0,∂Hμ′=0
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 ofN generalized coordinates qn(t):(3.1)L0(qn,q̇n,…,qn(m)),m≥1,
where    qn(s)=dsqn/dts, s=0,1,…,m  and  n=1,…,N.The Euler-Lagrange equations of motion are obtained as [12](3.2)∑s=0m(-1)sdsdts(∂L0∂qn(s))=0.Theories with higher derivatives, which have been first developed by Ostrogradski [12], treat the derivatives  qn(s)(s=0,…,m-1)  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,m-1 and qn,s-1, (s=1,…,m-1) read as(3.3)pn,m-1≡∂L0∂qn(m),pn,s-1≡∂L0∂qn(s)-ṗn,s,s=1,…,m-1.
Therefore, the canonical Hamiltonian is given by (3.4)H0(qn,0,…,qn,m-1;pn,0,…,pn,m-1)=∑s=0m-2pn,sqn,s+1+pn,m-1q̇n,m-1-L0(qn,0,…,qn,m-1,q̇n,m-1).
Hamilton's equations of motion are written using Poisson bracket as [5, 6](3.5)q̇n,s=∂H0∂pn,s={qn,s,H0},(3.6)ṗn,s=∂H0∂qn,s={pn,s,H0},
where {,} is the Poisson  bracket defined as(3.7){A,B}=∑s=0m-1∂A∂qn,s∂B∂pn,s-∂B∂qn,s∂A∂pn,s.
The fundamental Poisson brackets are(3.8){qn,spn′,s′}=δnn′δss′,{qn,s,qn′,s′}={pn,s,pn′,s′}=0,
where n,n′=1,…,N,ands,s′=0,…,m-1.With this procedure, the phase space, described in terms of the canonical variablesqn,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,m-1,qn,i,i=0,1,…,m-2) such that the following recursion relations would hold [13, 14]:(4.1)q̇n,i=qn,i+1.
Clearly, the variables (qn,m-1,qn,i), would then correspond to the time derivatives (qn(m-1),qn(i)) respectively, that is,(4.2)qn(0)=qn,0,q̇n=qn,1,…,qn(m-1)=qn,m-1,qn(m)=q̇n,m-1.
Equation (4.1) represents relations between the new variables. In order to enforce these relations for independent variables (qn,m-1,qn,i), additional Lagrange multipliers λn,i(t) are introduced [14]. The variables (qn,m-1,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 (4.3)LT(qn,i,qn,m-1,q̇n,i,q̇n,m-1,λn,i)=L0(qn,i,qn,m-1,q̇n,m-1)+∑i=0m-2λn,i(q̇n,i-qn,i+1).
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:(4.4)pn,m-1=∂LT∂q̇n,m-1,(4.5)pn,i=∂LT∂q̇n,i=λn,i=-Hn,i,(4.6)πn,i=∂LT∂λ̇n,i=0=-Φn,i,
the canonical Hamiltonian can be obtained as(4.7)H0(qn,i,qn,m-1,pn,m-1,λn,i)=pn,m-1q̇n,m-1+∑i=0m-2pn,iq̇n,i+∑i=0m-2πn,iλ̇n,i-LT(qn,i,qn,m-1,q̇n,i,q̇n,m-1,λn,i),
Equations (4.5) and (4.6) represent primary constraints [1, 2]. Their Hamilton-Jacobi partial differential equations can be obtained as(4.8)H0′=p0+H0(qn,i,qn,m-1,pn,m-1,λn,i)=0,(4.9)Φn,i′=πn,i=0,(4.10)Hn,i′=pn,i-λn,i=0.
The equations of motion can be written as total differential equations in many variables as follows: (4.11)dqn,j=dqn,j,(4.12)dqn,m-1=∂H0′∂pn,m-1dt,(4.13)dpn,j=-∂H0′∂qn,jdt,dpn,m-1=-∂H0′∂qn,m-1dt,dλn,j=dλn,j,dπn,j=-∂H0′∂λn,jdt+dqn,j,j=0,1,…,m-2.
The total differential equations are integrable if and only if(4.14)dH0′=dp0-dH0=0,dHn,j′=dpn,j-dλn,j=0,dΦn,j′=dπn,j=0.


## 5. The Canonical Path Integral Quantization

If the coordinatest,qn,i,λn,i are denoted by tα, that is,(5.1)tα=t,qn,i,λn,i,
then the set of primary constraints (4.8), (4.9), and (4.10) can be written in a compact form as(5.2)Hα′=0,
where      (5.3)Hα′=H0′,Hn,i′,Φn,i′.
Making use of [7], the canonical path integral for the extended Lagrangians can be obtained as(5.4)K(qn,m-1′,qn,i′,λn,i′,t′;qn,m-1,qn,i,λn,i,t)=∫qn,m-1qn,m-1′∏n=1N(Dqn,m-1Dpn,m-1)exp⁡[iℏ∫tαtα′(-H̅α+pn,m-1∂Hα′¯∂pn,m-1)dtα],n=1,…,N,i=0,…,m-2.
Note that (4.12) gives(5.5)∂Hα′∂pn,m-1dtα=∂H0′∂pn,m-1dt+∂Φn,i′∂pn,m-1dλn,i+∂Hn,i′∂pn,m-1dqn,i=dqn,m-1.
Therefore, (5.4) can be written as(5.6)K(qn,m-1′,qn,i′,λn,i′,t′;qn,m-1,qn,i,λn,i,t)=∫qn,m-1qn,m-1′∏n=1N(Dqn,m-1Dpn,m-1)exp⁡[iℏ∫tαtα′(-H̅αdtα+pn,m-1dqn,m-1)].
However, according to (4.6) and (4.7), we get(5.7)Hn,i=-λn,iΦn,i=0,
so, it can bee found that (5.8)Hαdtα=H0dt+Hn,idqn,i+Φn,idλn,i=H0dt-λn,idqn,i.
Then the transition amplitude can be written in the final form as(5.9)K(qn,m-1′,qn,i′,λn,i′,t′;qn,m-1,qn,i,λn,i,t)=∫qn,m-1qn,m-1′∏n=1N(Dqn,m-1Dpn,m-1)exp⁡[iℏ∫tαt′α(-H̅0dt+λn,idqn,i+pn,m-1dqn,m-1)].
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:(6.1)L0=12(q̈12-q̇12-q12).
If (4.2) is used, we can write(6.2)q1(0)=q10,q̇1=q11,q̈1=q̇11,
Hence, the Lagrangian (6.1) becomes(6.3)L0=12(q̇112-q112-q102).
Upon using (4.1), the recursion relation is q̇10=q11. And with the aid of (4.3), the extended Lagrangian is simply  (6.4)LT=12(q̇112-q112-q102)+λ10(q̇10-q11).
The conjugate momenta can be obtained as (6.5)P11=∂LT∂q̇11=q̇11,P10=∂LT∂q̇10=λ10,π10=∂LT∂λ̇100.
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:(6.6)H0=12(P112+q102+q112)+λ10q11.
Accordingly, the set of HJPDE's can be written as(6.7)H0′=P0+H0=0,H10′=P10-λ10=0,Φ10′=π10=0.
From (5.9), the canonical path integral quantization for this system is          (6.8)K(q11′,q10′,λ10′,t′;q11,q10,λ10,t)=∫Dq11Dp11exp⁡[iℏ∫(-H0dt+p11dq11+λ10dq10)],
where Dq11=limk→∞∏j=1k-1dq11j;Dp11=limk→∞∏j=0k-1(dp11/2πℏ).(6.9)K=∫Dq11Dp11exp⁡[iℏ∫((-12(P112+q102+q112)-λ10q11)dt+p11dq11+λ10dq10)].
Equation (6.9) can be written in a compact form as(6.10)K=∫Dq11Dp11exp⁡[iℏ∫(-12(P112+q102+q112)-λ10q11+p11q̇11+λ10q̇10)dt].
Upon changing the integration over dt  to summation, we have(6.11)K=∫Dq11∏j=0k-1(dp11j2πℏ)exp⁡[iεℏ∑j=0k-1(-p11j22-q10j22-q11j22+p11jq̇11j+λ10j(q̇10j-q11j))].
The p11j-integration can be performed using the Gaussian integral:(6.12)K=∫Dq111(2πℏ)k(2πℏiε)k/2exp⁡[iεℏ∑j=0k-1(q̇11j22-q10j22-q11j22+λ10j(q̇10j-q11j))]=(12πℏiε)k/2∫Dq11exp⁡[iℏ∫(q̇1122-q1022-q1122+λ10(q̇10-q11))dt]=(12πℏiε)k/2∫Dq11exp⁡[iℏ∫LTdt].


### 6.2. Example 2

As a second example, consider the three-dimensional second-order regular lagrangian:(6.13)L0=12(q̈12+q̈22+q̈32)-12(q̇12+q̇32).
If we put(6.14)q1(0)=q10,q2(0)=q20,q3(0)=q30,q̇1=q11,q̇2=q21,q̇3=q31,q̈1=q̇11,q̈2=q̇21,q̈3=q̇31.
then the above Lagrangian can be written as(6.15)L0=12(q̇112+q̇212+q̇312)-12(q112+q312).
Here the recursion relations are(6.16)q̇10=q11;q̇20=q21,q̇30=q31.
Accordingly, the extended Lagrangian can be given as:(6.17)LT=12(q̇112+q̇212+q̇312)-12(q112+q312)+λ10(q̇10-q11)+λ20(q̇20-q21)+λ30(q̇30-q31).
The corresponding momenta are calculated as(6.18)p11=q̇11,p10=λ10,π10=0,p21=q̇21,p20=λ20,π20=0,p31=q̇31,p30=λ30,π30=0
Therefore, the canonical Hamiltonian reads(6.19)H0=p1122+p2122+p3122+12(q112+q312)+λ10q11+λ20q21+λ30q31.
Thus, the set of HJPDE's can be written as(6.20)H0′=P0+H0=0,Φ10′=π10=0,Φ20′=π20=0,Φ30′=π30=0,H10′=p10-λ10=0,H20′=p20-λ20=0,H30′=p30-λ30=0.
Then, the canonical path integral quantization for this system is constructed as(6.21)K(qn1′,qn0′,λn0′,t′;qn1,qn0,λn0,t)=∫∏n=13(Dqn1Dpn1)exp⁡[iℏ∫(-H0dt+λn0dqn0+pn1dqn1)],
where n=1,2,3.(6.22)K=∫∏n=13(Dqn1Dpn1)exp⁡[iℏ∫(-pn122-q1122-q3122+λn0(q̇n0-qn1)+pn1q̇n1)dt].
Changing the integration over dt  to summation and integrating over p11,p21 and p31k times we get (6.23)K=(12πℏiε)3k/2∫∏n=13Dqn1exp⁡[iℏ∫(q̇n122-q1122-q3122+λn0(q̇n0-qn1))dt]=(12πℏiε)3k/2∫Dq11Dq21Dq31exp⁡[iℏ∫LTdt].


## 6.1. Example 1

As a first example, let us consider a one-dimensional second-order regular lagrangian of the form:(6.1)L0=12(q̈12-q̇12-q12).
If (4.2) is used, we can write(6.2)q1(0)=q10,q̇1=q11,q̈1=q̇11,
Hence, the Lagrangian (6.1) becomes(6.3)L0=12(q̇112-q112-q102).
Upon using (4.1), the recursion relation is q̇10=q11. And with the aid of (4.3), the extended Lagrangian is simply  (6.4)LT=12(q̇112-q112-q102)+λ10(q̇10-q11).
The conjugate momenta can be obtained as (6.5)P11=∂LT∂q̇11=q̇11,P10=∂LT∂q̇10=λ10,π10=∂LT∂λ̇100.
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:(6.6)H0=12(P112+q102+q112)+λ10q11.
Accordingly, the set of HJPDE's can be written as(6.7)H0′=P0+H0=0,H10′=P10-λ10=0,Φ10′=π10=0.
From (5.9), the canonical path integral quantization for this system is          (6.8)K(q11′,q10′,λ10′,t′;q11,q10,λ10,t)=∫Dq11Dp11exp⁡[iℏ∫(-H0dt+p11dq11+λ10dq10)],
where Dq11=limk→∞∏j=1k-1dq11j;Dp11=limk→∞∏j=0k-1(dp11/2πℏ).(6.9)K=∫Dq11Dp11exp⁡[iℏ∫((-12(P112+q102+q112)-λ10q11)dt+p11dq11+λ10dq10)].
Equation (6.9) can be written in a compact form as(6.10)K=∫Dq11Dp11exp⁡[iℏ∫(-12(P112+q102+q112)-λ10q11+p11q̇11+λ10q̇10)dt].
Upon changing the integration over dt  to summation, we have(6.11)K=∫Dq11∏j=0k-1(dp11j2πℏ)exp⁡[iεℏ∑j=0k-1(-p11j22-q10j22-q11j22+p11jq̇11j+λ10j(q̇10j-q11j))].
The p11j-integration can be performed using the Gaussian integral:(6.12)K=∫Dq111(2πℏ)k(2πℏiε)k/2exp⁡[iεℏ∑j=0k-1(q̇11j22-q10j22-q11j22+λ10j(q̇10j-q11j))]=(12πℏiε)k/2∫Dq11exp⁡[iℏ∫(q̇1122-q1022-q1122+λ10(q̇10-q11))dt]=(12πℏiε)k/2∫Dq11exp⁡[iℏ∫LTdt].


## 6.2. Example 2

As a second example, consider the three-dimensional second-order regular lagrangian:(6.13)L0=12(q̈12+q̈22+q̈32)-12(q̇12+q̇32).
If we put(6.14)q1(0)=q10,q2(0)=q20,q3(0)=q30,q̇1=q11,q̇2=q21,q̇3=q31,q̈1=q̇11,q̈2=q̇21,q̈3=q̇31.
then the above Lagrangian can be written as(6.15)L0=12(q̇112+q̇212+q̇312)-12(q112+q312).
Here the recursion relations are(6.16)q̇10=q11;q̇20=q21,q̇30=q31.
Accordingly, the extended Lagrangian can be given as:(6.17)LT=12(q̇112+q̇212+q̇312)-12(q112+q312)+λ10(q̇10-q11)+λ20(q̇20-q21)+λ30(q̇30-q31).
The corresponding momenta are calculated as(6.18)p11=q̇11,p10=λ10,π10=0,p21=q̇21,p20=λ20,π20=0,p31=q̇31,p30=λ30,π30=0
Therefore, the canonical Hamiltonian reads(6.19)H0=p1122+p2122+p3122+12(q112+q312)+λ10q11+λ20q21+λ30q31.
Thus, the set of HJPDE's can be written as(6.20)H0′=P0+H0=0,Φ10′=π10=0,Φ20′=π20=0,Φ30′=π30=0,H10′=p10-λ10=0,H20′=p20-λ20=0,H30′=p30-λ30=0.
Then, the canonical path integral quantization for this system is constructed as(6.21)K(qn1′,qn0′,λn0′,t′;qn1,qn0,λn0,t)=∫∏n=13(Dqn1Dpn1)exp⁡[iℏ∫(-H0dt+λn0dqn0+pn1dqn1)],
where n=1,2,3.(6.22)K=∫∏n=13(Dqn1Dpn1)exp⁡[iℏ∫(-pn122-q1122-q3122+λn0(q̇n0-qn1)+pn1q̇n1)dt].
Changing the integration over dt  to summation and integrating over p11,p21 and p31k times we get (6.23)K=(12πℏiε)3k/2∫∏n=13Dqn1exp⁡[iℏ∫(q̇n122-q1122-q3122+λn0(q̇n0-qn1))dt]=(12πℏiε)3k/2∫Dq11Dq21Dq31exp⁡[iℏ∫LTdt].


## 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 termq̇n,i is replaced by a new function qn,i+1, which is led to a constraint equation, qn,i+1-q̇n,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.



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*Source: 375838-2011-10-20.xml*