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Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Proper Dirac Quantization of Free Particle on D-Dimensional Sphere

Hagen KLEINERT and Sergei V. SHABANOVgif gif


We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically incorrect.

1. Quantization of a free point particle in curved space is a long-standing and controversial problem in quantum mechanics. Dirac has emphasized that canonical quantization rules are consistent only in a Cartesian reference frame [1]. Attempts to generalize these rules to curved space run into the notorious operator-ordering problem of momentum and coordinates, making the Hamiltonian operator non-unique. Podolsky[3] avoided this problem by postulating that the Laplacian in the free Schrödinger operator tex2html_wrap_inline471 should be replaced by the Laplace-Beltrami operator tex2html_wrap_inline473 , where tex2html_wrap_inline475 are the partial derivatives with respect to the D-dimensional curved-space coordinates, and g is the determinant of the metric tex2html_wrap_inline481 . This postulate has generally been accepted as being correct since it yields, for a D-dimensional sphere of radius R embedded in a D+1 -dimensional Cartesian space with coordinates tex2html_wrap_inline489 , an energy tex2html_wrap_inline491 . Here tex2html_wrap_inline493 with tex2html_wrap_inline495 are the unique quantum-mechanical differential operator representation of the D(D+1)/2 generators tex2html_wrap_inline499 of the rotation group SO(D+1) in flat space. If we take a to label the index pairs ij with tex2html_wrap_inline507 , then tex2html_wrap_inline499 are the matrices tex2html_wrap_inline511 , whose nonzero commutation rules are


The canonical commutation rules tex2html_wrap_inline513 transfer the Lie algebra (1) to the operators tex2html_wrap_inline515 . The square of the total angular momentum tex2html_wrap_inline517 is the Casimir operator of the orthogonal group SO(D+1), with the eigenvalues l(l+D-1), tex2html_wrap_inline521 [4]. Thus, by quantizing classical angular momentum rather than canonical variables (i.e., by putting hats on L's rather than on p's) operator-ordering problems are avoided, making this energy spectrum the most plausible one [5].

Doubts on the correctness of this spectrum have been raised by DeWitt[6] in his first attempt to quantize the system by a straightforward generalization of Feynman's time-sliced path integral to curvilinear coordinates. He found an extra energy proportional to the Riemannian scalar curvature tex2html_wrap_inline527 of in the Schrödinger operator:


with a proportionality constant tex2html_wrap_inline529 . Various successors have presented modifications of DeWitt's procedure leading to other proportionality factors tex2html_wrap_inline531 [7] and tex2html_wrap_inline533 [8].

For the above sphere, the extra term produces an extra constant energy equal to tex2html_wrap_inline535 which would contradict the previous results. At fist sight, this contradiction seems to be rather irrelevant. Common experiments are only capable of detecting energy differences, in which this constant drops out. Cosmology, however, is sensitive to an additive constant, which would change the gravitational energy of interstellar rotating two-atomic gases. Apart from such somewhat esoteric applications, a correct quantization of this simplest non-Euclidean system is certainly of fundamental theoretical interest.

It is therefore important to find physical systems for which the tex2html_wrap_inline527 -term is not a constant, so that it can distort the energy spectrum of the Schrödinger operator (2). Such a system has recently been found, although in a somewhat indirect way. When solving the path integral of the hydrogen atom, a two-step nonholonomic mapping is needed which passes through an intermediate non-Euclidean space [9, 10]. The existence of an extra tex2html_wrap_inline527 -term in (2) would modify significantly the known level spacings in the hydrogen spectrum. Thus experiment eliminates an extra curvature scalar, in agreement with Podolsky and with the quantization of angular momentum. The successful procedure was generalized into a simple nonholonomic mapping principle, which allows to map classical and quantum-mechanical laws in flat space into correct laws in curved space [9, 10, 11].

The absence of an tex2html_wrap_inline527 -term in (2) is therefore a crucial test for any quantum theory in curved spaces. It is the purpose of this note to show that this absence can be established for a sphere also within the operator approach to quantum mechanics, after suitably preparing the description for an application of Dirac's theory of constrained systems.

This application is, however, not straight-forward. A D-dimensional sphere is most simply described by embedding it into a D+1 -dimensional Cartesian x-space via a constraint tex2html_wrap_inline549 . Within Dirac's classification scheme [2], such a constraint is of second class. We shall demonstrate that Dirac's quantization rules for such systems produce wrong energy levels. A correct quantization becomes possible only by making use of a recently developed conversion [13] of second-class constraints to first-class constraints, which instead of configuration space restrict the quantum-mechanical Hilbert space [2]. This conversion requires an extension of the phase space of the initial Cartesian system to a larger auxiliary Cartesian phase space, where the correct quantization is known. The operators representing the first-class constraints are generators of gauge transformations, and the physical states are all found by going into the gauge invariant subspace of the Hilbert space.

Constraints associated with gauge symmetry have first been mastered in quantum electrodynamics (Coulomb's law), and are now a standard tool in the quantization of gauge theories [12].

2. We begin by quantizing the system with the help of Dirac's theory for second-class constraints. A free point particle with a Hamiltonian tex2html_wrap_inline551 moving in a flat Euclidean D+1-dimensional is restricted to the surface of a D-dimensional sphere by the primary constraint in configuration space [2]


The dynamical consistency condition tex2html_wrap_inline557 leads to an additional secondary constraint in phase space


It expresses the fact that a motion on a sphere has no radial component. Although the canonical variables are Cartesian, the canonical quantization is not applicable, since the constraints (3), (4) cannot be enforced for the associated operators -- the conditions tex2html_wrap_inline559 would be in conflict with the commutation relation tex2html_wrap_inline561 . To resolve this difficulty, Dirac replaced the Poisson symplectic structure tex2html_wrap_inline563 by the so-called Dirac brackets


where tex2html_wrap_inline565 is a matrix tex2html_wrap_inline567 formed from all primary and secondary constraints tex2html_wrap_inline569 . This matrix is assumed to be non-degenerate, a defining property of second-class constraints [2]). The Dirac brackets provide us with an antisymmetric operation which is distributive and associative, i.e., which satisfies Leibnitz rule and Jacobi identity, thus forming a symplectic structure which is as good as Poisson's.

The removal of the inconsistency is ensured by the automatic property


Hamiltonian equations of motion generated by the Dirac bracket tex2html_wrap_inline571 coincide with those generated by the original Poisson bracket tex2html_wrap_inline573 on the surface of constraints tex2html_wrap_inline569 . Thus, Dirac brackets produce classically the same equations of motion as Poisson brackets. Quantization proceeds by the replacement tex2html_wrap_inline577 . The property (6) allows us to replace the constraints by operator equations tex2html_wrap_inline579 without the earlier contradictions. Moreover, since the constraint operators tex2html_wrap_inline581 commute with any other operator, they can be given any c-number value, for instance zero. For the surface of the sphere, the quantized Dirac brackets (5) read


The operator ordering problem occurring in the right-hand side of (9) is uniquely resolved under the condition that the algebra (7)-(9) satisfies the Jacobi identity. The operator tex2html_wrap_inline585 commutes with all canonical operators and is therefore a c-number, which can be set equal tex2html_wrap_inline589 .

To find the spectrum of the Hamiltonian tex2html_wrap_inline591 , we make use of the identity


Inserting this into tex2html_wrap_inline593 , we obtain the Hamiltonian


where tex2html_wrap_inline595 as before and tex2html_wrap_inline597 is a radial momentum operator. Note that the operator (11) is determined without operator-ordering ambiguities. The identity (10) can equally well be inserted to the left and the right of the momentum operators tex2html_wrap_inline599 , with a unique result.

The radial operator tex2html_wrap_inline601 commutes with all canonical operators and is therefore some complex number c. It has necessarily an imaginary part, due to the obvious relation tex2html_wrap_inline605 implied by (8). Thus we may decompose tex2html_wrap_inline607 , where tex2html_wrap_inline609 is an arbitrary real number. The constant tex2html_wrap_inline609 is determined by expressing tex2html_wrap_inline601 in terms of the operator of the second constraint (4). The ordering of tex2html_wrap_inline615 and tex2html_wrap_inline617 in it is undetermined, so that tex2html_wrap_inline619 may be written as tex2html_wrap_inline621 with an arbitrary imaginary part tex2html_wrap_inline623 . Inserting the above constant for tex2html_wrap_inline601 and setting tex2html_wrap_inline619 equal to zero is solved uniquely by tex2html_wrap_inline629 and tex2html_wrap_inline631 .

It is now easy to find the spectrum of the first term in the Hamiltonian operator (11). The modified canonical rules (7)-(9) transfer the Lie algebra (1) to the operators tex2html_wrap_inline633 in just the same way as ordinary canonical commutation rules, so that the energy levels of angular momentum l are


The additional constant energy has a different D-dependence than the tex2html_wrap_inline527 -term proposed by DeWitt and others [7, 8]. In particular, it is nonzero even for a particle on a circle (D=2), where nobody ever expected such a term.

Thus the modified symplectic structure proposed by Dirac, although esthetically appealing and yielding a unique result for a particle on the surface of a sphere, must be rejected on physical grounds as contradicting the established quantization via angular momentum operators and nonholonomic mapping principle.

3. In second-class constrained systems, not every phase space variable can be made an operator. Dirac's new symplectic structure (5) represented by (7)-(9) accounts for this fact via its degeneracy in the embedding phase space. This permits the canonical variables of fluctuations transverse to the manifold on which the particle moves to remain c-numbers. There is, however, a defect in Dirac's procedure: Although physical excitations of the transverse degree of freedom are eliminated, the system maintains a memory of the forbidden motion by a nonzero c-number valued transverse energy tex2html_wrap_inline647 . From the physical point of view, the existence of such a memory must be rejected. After all, the embedding space is only an artifact for the introduction Euclidean canonical commutation rules. It does not belong to the manifold where the particle moves. An alternative approach must therefore be found where a memory of the embedding space is absent.

Such an alternative is offered by gauge theories. In these, equations of motion exhibit symmetries which are local in time with the consequence that the dynamics of some degrees of freedom is not specified by the equations of motion. Such local symmetries are gauge symmetries, and the undetermined degrees of freedom correspond to pure gauge configurations. Upon quantization, non-physical degrees of freedom are removed by restricting the Hilbert space to a physical subspace formed by gauge invariant states.

Thus, rather than restricting the particle motion to the surface of a sphere via constraints as above, we consider a free motion in a Cartesian coordinate system, and impose the condition that the physical states in Hilbert space are invariant under arbitrary time-dependent rescalings of the radial size of the system. The transverse momentum tex2html_wrap_inline601 can be made a generator of gauge transformations, and we may require tex2html_wrap_inline651 to be zero for physical states.

To achieve this goal, we invoke the method of abelian conversion that allows one to transform a second-class constrained system into an abelian gauge theory [13]. Dynamics of physical (gauge invariant) degrees of freedom in the effective abelian theory is the same as dynamics of physical degrees of freedom in the original second-class constrained system. In general, the abelian conversion proceeds as follows. Given a set of second-class constraints tex2html_wrap_inline653 ), one extends the original phase space by extra independent canonical variables tex2html_wrap_inline655 ). The extended phase space is equipped with the canonical symplectic structure tex2html_wrap_inline657 and tex2html_wrap_inline563 (all other brackets are zero). With the help of abelian conversion, quantum dynamics on a manifold can be formulated independently of the parametrization of the manifold [14].

An equivalent set of abelian first-class constraints tex2html_wrap_inline661 is constructed in such a way that it satisfies


This amounts to solving first-order differential equations with the boundary condition tex2html_wrap_inline663 . Given the new constraints tex2html_wrap_inline661 , the original system Hamiltonian H(x,p) is converted into a Hamiltonian on the extended phase space tex2html_wrap_inline669 by solving the equation


with the boundary condition tex2html_wrap_inline671 . The extrema of the associated extended action tex2html_wrap_inline673 determine equations of motion in the extended phase space. These depend on 2M arbitrary functions of time tex2html_wrap_inline677 as a manifestation of the gauge freedom. There exists a choice for tex2html_wrap_inline677 such that the auxiliary phase space variables tex2html_wrap_inline681 and tex2html_wrap_inline683 vanish at all times, whereas tex2html_wrap_inline685 and tex2html_wrap_inline687 solve the original equations of motion of the second-class constrained system [13].

Applying this procedure to our particular second-class constraints (3) and (4) yields


while the extended Hamiltonian assumes the form


where tex2html_wrap_inline689 are the classical components of the angular momentum (for a=ij, tex2html_wrap_inline693 ). The extended phase space variables p,x,P,Q are Cartesian and satisfy the standard Poisson bracket relations. They can be turned into hermitian operators in the usual way by the replacement tex2html_wrap_inline697

Physical states are invariant with respect to transformations generated by tex2html_wrap_inline699 : tex2html_wrap_inline701 and tex2html_wrap_inline703 , where tex2html_wrap_inline705 are parameters of the gauge transformations [14]. The first is an arbitrary time-dependent rescaling of the size of the system, which is geometrically equivalent to the initial restriction of the motion to the surface of the sphere, while the second implies that the auxiliary degree of freedom is a pure gauge. The first-class constraints restrict the physical Hilbert space to the gauge-invariant sector by the Dirac conditions [2]


The general solution has the form tex2html_wrap_inline707 , where f(x,Q) is some fixed function, whereas tex2html_wrap_inline711 are wave functions on the D-sphere. In the physical Hilbert space, we can set tex2html_wrap_inline699 to zero in the Hamilton operator (16). Thus we find the energy values


rather than (12). There is no additional constant energy, in agreement with [3, 5, 9].

This result is obtained without ordering problems. Although ordering ambiguities are not absent altogether, they do not affect the final result since they occur only in the quantization of the gauge generator tex2html_wrap_inline717 , where they modify only the explicit form of the physically irrelevant function tex2html_wrap_inline719 , but not the physical Hilbert space described by the wave function tex2html_wrap_inline711 , nor the spectrum (18). In fact, in the simple system at hand the ordering ambiguities in tex2html_wrap_inline723 produce only a multiplicative renormalization of the physical states tex2html_wrap_inline725 .

4. The above quantization of a particle on a sphere via abelian conversion is of course applicable to arbitrary homogeneous spaces. But what about arbitrary manifolds? When attempting a straight-forward generalization we run once more into operator-ordering problems for the extended Hamiltonian (16), and these require a new strategy. The quantum Hamiltonian tex2html_wrap_inline591 has no ordering ambiguity and is again adopted as a starting point. Let n(x) be a unit vector normal to the manifold in the embedding space (if this space has a dimension higher than D+1, more normal vectors are needed to specify transverse directions). The condition tex2html_wrap_inline733 removes the transverse motion and offers itself as a generator of gauge transformations. This, however, is not consistent for two reasons. First, since tex2html_wrap_inline735 depends on position, tex2html_wrap_inline737 is nonzero, so that the free-particle Hamiltonian is not gauge invariant. Second, the operator tex2html_wrap_inline739 is not hermitian making finite gauge transformations non-unitary.

The first problem can be resolved via an abelian conversion method performed immediately at the quantum level [13]. Here one uses operator versions of second-class constraints tex2html_wrap_inline741 and tex2html_wrap_inline743 to restrict the motion to a manifold specified by F(x)=0, the forbidden direction being specified by the normal vector tex2html_wrap_inline747 . Then one extends the system by two extra canonical operators tex2html_wrap_inline749 and tex2html_wrap_inline751 obeying standard Heisenberg commutation relation, and commuting with tex2html_wrap_inline617 and tex2html_wrap_inline615 . The conversion of the second-class constraint operators tex2html_wrap_inline581 is enforced by solving equations (13) and (14) with Poisson brackets replaced by commutators. The abelian gauge generators have the same basic structure as those in (15), only that the tex2html_wrap_inline617 -dependent factor of tex2html_wrap_inline749 in tex2html_wrap_inline723 depends now on tex2html_wrap_inline765 . This construction solves the first problem of finding a gauge invariant Hamiltonian operator.

The hermiticity problem for the generator tex2html_wrap_inline723 must be solved in a way that dynamics of physical degrees of freedom does not depend on the embedding procedure. For this we observe that the operator tex2html_wrap_inline769 is just as good as tex2html_wrap_inline723 itself to eliminate excitations of the transverse modes. It has the advantage of being hermitian. In addition, it commutes with tex2html_wrap_inline773 , so the abelian gauge algebra (13) is retained by such a choice of the second generator. Thus we modify the conversion method by taking tex2html_wrap_inline775 as the constraint operator tex2html_wrap_inline619 rather than tex2html_wrap_inline739 . At the classical level, the new constraint tex2html_wrap_inline781 is certainly equivalent to tex2html_wrap_inline783 . But at the operator level, it is superior by generating unitary gauge transformations. With the new generator, the conversion equation (14) gives rise to a new Hamiltonian operator in the extended Cartesian coordinate system.

It is not hard to verify that the physical states (17) have the form tex2html_wrap_inline785 where tex2html_wrap_inline787 , that is, the kinetic energy of the transverse motion is strictly zero for physical states with our choice of the gauge generators.

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Hagen Kleinert
Sat Feb 1 08:24:59 MET 1997