IL NUOVO CIMENTOVOL. 112 B, N. 1 Gennaio 1997 The Feynman path integral quantization of constrained systems S. MUSLIH and Y. GU¨LER Department of Physics, 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 – Quantum mechanics. PACS 11.10.Ef – Lagrangian and Hamiltonian approach. 1. – Introduction The quantization of a classical system can be achieved by the canonical quantization method [1]. If we ignore the ordering problems, it consists in replacing the classical Poisson bracket, by quantum commutators 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 operator 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 commutator 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 propagator (Feynman kernel) for the wave function 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 canonical transformation: (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 motion. 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.