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IL NUOVO CIMENTOVOL. 112 B, N. 1 Gennaio 1997

The Feynman path integral of constrained systems

S. MUSLIH and Y. GU¨LER Department of , Middle East Technical University - 06531 Ankara, Turkey

(ricevuto l’11 Giugno 1996; approvato il 23 Luglio 1996)

Summary. — The Feynman path integral for constrained systems is constructed using the canonical formalism introduced by Güler. This approach is applied to a free relativistic particle and Christ-Lee model. PACS 03.65 – mechanics. PACS 11.10.Ef – Lagrangian and Hamiltonian approach.

1. – Introduction

The quantization of a classical system can be achieved by the method [1]. If we ignore the ordering problems, it consists in replacing the classical , by quantum when classically all the states on the phase space are accessible. This is no longer correct in the presence of constraints. An approach due to Dirac [2] is widely used for quantizing the constrained Hamiltonian systems [3-5]. The path integral is another approach used for the quantization of constrained systems. This approach was formulated by Faddeev [6]. Faddeev and Popov [7] handle constraints in the path integral formalism by quantizing singular theories with first-class constraints in the canonical gauge. The generalization of the method to theories with second-class constraints is given by Senjanovic [8]. Fradkin and Vilkovisky [9,10] rederived both results in a broader context, where they improved Faddeev’s procedure mainly to include covariant constraints; also they extended this procedure to the Grassman variables. When the dynamical system possesses some second-class constraints there exists another method given by Batalin and Fradkin [11]: the BFV-BRST quantiza- tion method. One enlarges the phase space in such a way that the original second-class constraints become converted into the first-class ones, so that the number of physical degrees of freedom remains unaltered. These quantization schemes have the properties that by using them one can easily control important properties of quantum theory such as unitarity and positive-

97 98 S. MUSLIH and Y. GU¨LER definiteness of the metric. Besides, relativistically covariant formulation of quantum theory is obtained by the quantization schemes. Now we would like to make a brief review of the path integral formulation.

2. – The Feynman path integral formulation

The path integral quantization is defined by the Feynman kernel [12, 13]. i In the operator version of canonical quantization one turns the functions q , pi into × i × operators q and pi , which satisfy the relations

× k × 4 k (1) [q , pr ] id l .

Eigenstates defined by the eigenvalue equations

(2) q× i Nqb 4q i Nqb , p× i Npb 4p i Npb , form an orthonormal system

.aq 8Nqb4d(q82q), ap8Npb4d(p82p), (3) / N N4  N N4 ´ dq qbaq 1, dp pbap 1.

This transition may be performed at any time. States at arbitrary times are obtained by means of unitary transformation generated by the Hamiltonian × (4) Nq, tb 4exp [itH]Nqb , where Nqb is assumed to be an eigenstate at t40. The (Feynman kernel) for the c(q, t) 4 aa, tNcb is thus

.D(q 8, t 8, q, t) 4 aq 8, t 8Nq, tb, (5) / × ´D(q8, t8, q, t)4aq8Nexp [2i(t 82t) H] Nqb .

In this case the Feynman kernel connects the Schrödinger wave function in two different times as

(6) c(q 8, t 8) 4 dqD(q8, t8, q, t)c(q, t).

There are many prescriptions to define the Feynman path integral. This freedom reflects the fact that a classical Hamiltonian does not uniquely determine a quantum Hamiltonian—there is an operator ambiguity. Different path integral definitions correspond to different quantum operator orderings. In our calculations we will use a specific one, the Weyl ordering [14-18], which will be discussed very briefly. Let us define the momentum and the position operators as

dp (7) aq 8NpNqb4 pexp [ip(q 82q) ] , aq 8NqNqb4qd(q82q). 2p THE FEYNMAN PATH INTEGRAL QUANTIZATION OF CONSTRAINED SYSTEMS 99

The Weyl ordering is defined in the following way:

×× 4 1 ××1 ×× (8) (pq)w (pq qp), 2

1 ×× 3 4 ×× 3 1 ××× 2 1 × 2 ××1 × 3 × (9) (pq )w (pq qpq q pq q p), etc. , 4

! all possible orders (10) General expression4 . total number of possible orders

The above treatment leads us to obtain 81 8N× N 4 dp 82 q q (11) aq Hw qb exp [ip(q q) ] H p, , 2p g 2 h × × × where Hw is the Weyl transform of the Hamiltonian operator H(q, p). Thus, the Weyl order is specified to be the mid-point prescription. To clarify the situation we consider the path integral in curvilinear coordinates. Consider the following point :

(12) x a Kq a 4f a (x), a41, R, D ,

2 4 a 2 4 a b (13) ( ds) ! ( dx ) ! dq dq Mab , a ab where the matrix Mab is given by ˇ ˇ 4 xc xc (14) Mab . ˇqa ˇqb The volume element in the two representations is given by

(15) dax4g daq , where

4 1O2 a 4 a 4 (16) g det (Mab ) , d x dx1 R dxD , d q dq1 R dqD . In the q system, since g is evaluated at the mid-point, it cannot be used to make the a 4 volume element d q dq1 RdqD an invariant, so it is convenient to eliminate the Jacobian factor in the volume element [14-18]. Thus we introduce

1 1 (17) axNtb 4 aqNtb and axNab 4 aqNab . kg kg

Hence

(18) f(q, t) 4 aqNtb 4kg c(x, t),

4 N 4 (19) f a (q) aq ab kg c a (x). 100 S. MUSLIH and Y. GU¨LER

Thus the Weyl transform of the Hamiltonian H is defined as

× 1 (20) H4kg (q×) H(q×, p×) , kg(q×) 4 × × × 1 × (21) H H(q, p)w DVw (q). In theories with the Lagrangian given in the form

1 4 . a . b 2 (22) L Mab q q V(q), 2 × × × × the Hamiltonian H(q, p)w and DVw (q) are defined as 1 × × × 4 × × 21 1 × × 21 × 1 × 21 × × 1 × (23) H(q, p)w (pa pb Mab 2pa pb Mab pb pb Mab pa pb ) V(q), 8

1 ˇ ˇq ˇ ˇq (24) DV (q) 4 b a . w k ˇ g ˇ hlk ˇ g ˇ hl 8 qa xc qb xc Now the path integral representation of the propagator in the phase space is defined as × . (25) aq 8Nexp [2i(t 82t) H] Nqb 4 ( Dq)( Dp) exp ki k(pq2H)ldtl, × where H is the Weyl transform of H(q×, p×). In order to obtain the path integral expression in configuration space, we perform p integration in (25)

8N 2 82 × N 4   . 2 (26) aq exp [ i(t t) H] qb ( Dq)(g) exp ki k (L(qi , qi , t) DVw (q))l dtl . For the quantization of singular systems, Faddeev [18] incorporated the Dirac formalism into the Hamiltonian form of the Feynman integral. Now we would like to discuss his formulation briefly. Consider a system with n degrees of freedom. It may have r first-class constraints f a , but no second-class constraints. Let us choose r gauge constraints x a , then 2r constraints fulfil

(27) ]f a, f b (40, a, b41, 2, R, r ,

(28) det N]f a , x b (Nc0 on the hypersurface defined by f a 40, xa40, where ] , ( denotes the Poisson bracket. The path integral representation is given as

1Q × . (29) q 8Nexp [2i(t 82t) H ] Nq 4 » d (q , p ) exp i  dt (p p 2H ) , a 0 b m j j y { j j 0 }z t 2Q

j41, R, n THE FEYNMAN PATH INTEGRAL QUANTIZATION OF CONSTRAINED SYSTEMS 101 where the measure of integration is given as

r n 4 N] a b (N a a j (30) dm (q, p) det f , f » d(x ) d(f ) » dq dpj , a41 j41

K6Q and the trajectories q(t) coincide at t with the solutions qin (t) and qout (t) of the equations describing the asymptotic . The expression (29) can be written in an equivalent form,

1Q × . (31) q 8Nexp [2i(t 82t) H ] Nq 4 » dq *dp* exp i  dt (p *q*2H *) . a 0 b y { }z 2Q

In (29) ( pj , qj) are any set of coordinates while in (31) ( p *, q*) are canonical coordinates and H* denotes a Hamiltonian written in terms of q*’s and p*’s. In order to prove (31), one changes, coordinates from the set ( p, q) to ( p *, pa, Q*, qa) with 4 ( pa , qa) “redundant variables”, such that pa x a . One gets rid of the redundant variables with delta-functions, and the residue is detN]f a , f b(N. In fact the functional integral representation (31) is an integration over the independent variables q*, p*. 8 If there exist additional 2r second-class constraints u m , the path integral representation is given by Senjanovic [8] as

r 8N 2 82 × N 4 N a b (N a a (32) aq exp [ i(t t) H0 ] qb » det f , f d(x ) d(f )Q a

1Q 2r8 . » ( ) det] , (N1O2 »dqj dp exp i  dt (p q 2H ) . Q d um u a u b j y { j j 0 }z m j 2Q

Another approach on the Feynman path integral quantization of constrained systems is discussed by Blau [18]. In this approach, Blau writes down the Feynman path integrals as follows: given a classical Hamiltonian, one constructs a quantum Hamiltonian by the usual procedure of promoting the position and the momentum functions to quantum operators. In the presence of constraints, the quantum Hamiltonian acts on a restricted . In this case one can reexpress the Hamiltonian in terms of canonical position and momentum operators in the restricted Hilbert space. Then one can make a correspondence between these canonical operators and the classical functions which appear in the path integral. In this case the path integral representation is given as

q 8 t 8 8N 2 82 × N 4   . 2 8 (33) aq exp [ i(t t) H] qb ( Dq *)( Dp *) exp y{i (p *q* H )dt}z, q t where q*, p* are the canonical phase space and H8 denotes a Hamiltonian written by q*’s and p*’s. 102 S. MUSLIH and Y. GU¨LER

3. – The canonical formulation

The canonical formulation [19-21] gives the set of Hamilton-Jacobi partial- differential equation (HJPDE) as ˇ ˇ 8 S S 4 4 2 1 4 2 (34) H t , qa , , 0, a, b 0, n r 1, R, n , a 1, R, n r , a g b ˇ ˇ h qa ta where 84 1 (35) Ha Ha(tb, qa, pa) pa , and H0 is defined as . . . (36) H 42L(t, q , q , q 4w )1p w 1q p N 42 , 4 0, n2r11, R, n . 0 i n a a a a m m pn Hn n The equations of motion are obtained as total differential equations in many variables as follows: ˇH 8 ˇH 8 ˇH8 4 a 4 a 42 a 4 (37) dqa dta , dpa dta , dpm dta , m 1, R, r , ˇpa ˇpa ˇtm ˇH 8 (38) dz4 2H 1p a dt , g a a ˇ h a pa 4 where z S(ta , qa). The set of eqs. (37), (38) is integrable if 84 (39) dH0 0, 84 4 (40) dHm 0, m 1, R, r . If conditions (39) and (40) are not satisfied identically, one considers them as new constraints and again tests the consistency conditions. Thus, repeating this procedure one may obtain a set of conditions. Now we would like to give the Feynman path integral formulation in the canonical method. The canonical formalism leads us to obtain the set of canonical phase space coordinates qa and pa as functions of ta , besides the canonical integral is obtained 8 in terms of the canonical coordinates. Ha can be interpreted as the infinitesimal generators of the canonical transformation given by parameters ta , respectively. In this case, the propagator for the constrained system is given as

8 8 qa ta ˇH 8 8 4  a a  2 1 a (41) D(qa , t ; qa , t ) ( Dq )( Dp ) exp i H pa dt , a a y { g a ˇp h a}z q a a ta or, in equivalent form, 8 8 qa ta (42) D(q 8, t 8; q , t ) 4 ( Dq a )( Dp a ) exp i (2H dt 1p dq ) , a a a a y { a a a a }z q a ta a41, R, n2r, a40, n2r11, R, n , × × × where Ha are Weyl-ordered transform of Hamiltonian operators Ha (tb , qa , pa). THE FEYNMAN PATH INTEGRAL QUANTIZATION OF CONSTRAINED SYSTEMS 103

In the following two sections we will work out the Feynman path integral for two singular systems: the free relativistic particle and the Christ-Lee model. . 3 1. The Feynman path integral for a free relativistic particle. – As a first example let us consider a free relativistic particle of non-zero , moving in D-dimensional Minkowski space described by the usual parametrization-invariant action [3]. The action is given as . . 42  2 m 1O2 4 2 (43) S m ( xmx ) dt , m 0, 1, R,,D 1.

Here, x m are functions of arbitrary parameter t describing the displacement of the particle along its world line and the Lagrangian is . . 42 2 m 1O2 (44) L m( xmx ) . The Lagrangian L is singular since its Hessian

ˇ2L m . . . (45) A 4 42 (g x2 2x x ), mn . . . mn m n ˇxm ˇxn (2x2)3O2 has rank D21. m, n41, 2, R, D21 and the metric convention is “mostly plus”. m The generalized momenta pm conjugate to the coordinate x are defined as . ˇL mx (46) p 4 4 m . m . . ˇxm (2x2)1O2 Therefore the zeroth component is . mx (47) p 4 0 , 0 . (2x2)1O2 and the i-th component is . mx (48) p 4 i , i41, R, D21. i . (2x2)1O2 . Since the rank of the Hessian matrix is D21, one can solve (48) for xi in terms of p i . and x0 . In fact . . p i x0 (49) xi 4 4v i . 2 1 2 1O2 (m pi ) Substituting (49) in (47), one has . mx0 p 0 4 N (50) . O .i4 i . (2x2)1 2 x v

Thus, we obtain 0 4 2 1 2 1O2 (51) p (m pi ) . 104 S. MUSLIH and Y. GU¨LER

Hence, the primary constraint is 84 1 4 (52) H0 p0 H0 0, where H0 is defined as 4 2 1 2 1O2 (53) H0 (m pi ) . Besides, the canonical Hamiltonian is defined as

42 0 i .0 i 2 21 2 1O2 .01 i (54) H L(x , x , x , v ) (m pi ) x pi v . Calculations show that H vanishes identically. The canonical method [19-21] leads us to obtain the set of Hamilton Jacobi partial differential equations as [22] (55) H 840, 84 1 (56) H0 p0 H0 , where H0 is defined as 4 2 1 2 1O2 4 2 (57) H0 (m pi ) , i 1, R, D 1. Making use of (35), (37) and (55), (56), the phase space coordinates x i and p i are obtained in terms of x 0 . Besides, the canonical action is calculated as

x 09 ˇH 8 (58) z4  2H 1p 0 dx 0 , g 0 i ˇ h pi x 08 or

x 09 4 2 1 . 0 (59) z  ( H0 pi xi )dx . x 08 Making use of (42) and (59), the path integral for a single relativistic particle is expressed as

x 09 . 9 09N 8 08 4 D21 D21  2 1 0 (60) ax , x x , x b d xd pexpyi { ( H0 pi xi )dx }z, x 08 × where H0 is the Weyl transform of the Hamiltonian H0 and it is given as 4 2 1 2 1O2 (61) H0 (m pi ) . Note that, in four-dimensional Minkowski space, one has

t 9 . 9 9N 8 8 4 3 3  2 21 2 1O2 (62) ax , t x , t b d xd pexpyi { (px (m p ) )dt}z, t 8 in agreement with [23]. THE FEYNMAN PATH INTEGRAL QUANTIZATION OF CONSTRAINED SYSTEMS 105 . 3 2. The Feynman path integral for the Christ-Lee model. – As a second example we consider the Christ-Lee problem [14, 15, 24], which is described by the singular Lagrangian

1 . . (63) L4 (r2 1r 2 (u2l)2 )2V(r). 2

Here, r and u are polar coordinates, l is another generalized coordinate. V(r) is the central potential of the system. The generalized momenta read as

4 . (64) pr r , . 4 2 2 2 (65) pu r (u l) , 4 (66) pl 0. Since the rank of the Hessian matrix is two, we have only one primary constraint as 84 (67) Hl 0.

The canonical Hamiltonian H0 reads as

p 2 p 2 4 r 1 u 1 1 (68) H0 lp V(r). 2 2r 2 u

Equations (67) and (68) lead to the set of Hamilton-Jacobi partial-differential equations 84 1 4 (69) H0 p0 H0 0, 84 4 (70) Hl pl 0. Making use of (42) and (69), (70), the path integral for this system is obtained as

t 8 . . (71) r 8, 8, 8, t 8; r, , , t 4 Dr D Dp Dp exp i (2H 1p r1p )dt , a u l u l b u r u y { 0 r u u }z t where H0 is the Weyl-ordered transform of the Hamiltonian H0 which can be obtained as

4 × × × × × 1 (72) H0 kr H0 (pr , pu , r, u, l) , kr

p 2 p 2 1 4 r 1 u 1 1 2 (73) H0 lp V(r) . 2 2r 2 u 8r 2

An important point to be specified here is that from the set of (HJPDE) and the 8 8 equations of motion, the Hamiltonians H0 and Hl are interpreted as infinitesimal generators of canonical transformations for two parameters t, l, respectively. Although 84 l is introduced as a coordinate in the Lagrangian, the integrability conditions dH0 0 84 and dHl 0 force us to treat it as a parameter like t. 106 S. MUSLIH and Y. GU¨LER

Equation (71) may be expressed as

q 8 N dr(n)du(n)dpr(n)dpu(n) (74) ar 8, u8, l8, t 8; r, u, l, tb 4 lim  » Q eK0 n41 (2 )2 q p

2 2 . prn 1 . pu n Qexp i p r 2 2V(r )1 1p ( 2 )2 , y e { rn n n 2 u n u l 2 }z 2 8rn 2rn where 2 1 4] ( . 4 qn11 qn 4 qn11 qn .qn rn, un , qn , qn , (75) / e 2 4 8 4 82 ´qN11 q , Ne t t .

Integrating over pu we obtain

q 8 N r dr(n)du(n)dpr(n)dpu(n) (76) ar 8, u8, l8, t 8; r, u, l, tb 4 lim  » Q eK0 n41 (2 ) q p

1O2 2 1 ie . . pr 1 exp [r( 2 ) ]2 exp i p r2 2V(r )1 . Q u l y e r n 2 z g2ipe h k 2 l { 2 8r } As was specified previously that t, l are two independent parameters, we will evaluate the path integral for a given value of l. For simplicity let us take l40. In this case (76), after integration over u, will give

q 8 N dr(n)dp(n) 2 8 8 4  r . 2 pr 2 1 1 (77) ar , t ; r, tb lim » exp ie pr r V(r) . eK0 n41 (2 ) k m 2 8r 2 nl q p

Now integrating over pr we obtain . r2 1 (78) ar 8, t 8; r, tb 4 Dr exp yi 2V(r)1 dtz , { 2 8r 2 } This result is in complete agrement with the results given in ref. [15, 18, 23].

4. – Conclusion

In this work we have followed the canonical method to construct the Feynman path integral for constrained systems. This treatment leads us to the equations of motion as total differential equations in many variables. If the system is integrable, then one can construct the canonical variables qa and pa in terms of ta . Besides the action integral is obtained in terms of the canonical variables with the Weyl transform of the × Hamiltonian operators Ha . It is obvious from eq. (42) that our approach has two advantages: first, we avoid to solve explicitly the higher-order generation constraints. Second, in the case of THE FEYNMAN PATH INTEGRAL QUANTIZATION OF CONSTRAINED SYSTEMS 107 relativistic theory, the exponent in the path integral is manifestly invariant, while secondary- or higher-generation constraints usually spoil manifestly Lorentz invariance. In the relativistic-particle example, since the integrability conditions dH840, 84 dH0 0 are satisfied identically, the system is integrable. Hence, the canonical phase i 0 space coordinates x and pi are obtained in terms of the parameter (x ). The path integral is then followed directly as given in (60). In usual formulation [23], one has to fix a gauge and to integrate over the extended phase space and after integration over the redundant variables, one can arrive at the result (62). 8 8 The Christ-Lee system is integrable. Hence, H0 and Hl can be interpreted as infinitesimal generators of canonical transformations given by the parameters t and l. Although l is introduced as a coordinate in the Lagrangian, the presence of constraints and the integrability conditions force us to treat it as a parameter like t. In this case the path integral is obtained as an integration over the canonical phase space coordinates r, u, pr , pu . Other treatments [15, 23] need a gauge-fixing condition to obtain the path integral over the canonical variables.

R E F E R E N C E S

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