In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.
In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
Several aspects of the application of path integrals to quantum mechanics are considered.in chapter one., the path integral is used to form gauge invariant variables [Mandelstam, 1962] and these are used in the specific calculation of the non-relativistic scattering of charged scalar particles from a cla ssica l Aharonov-Bohm solenoid.This allow s the calculation to be done without an explicit gauge choice.Chapter two continues with the application of the path integral on a multiply connected space, and forms a constructive derivation for exact scalar diffraction from an impenetrable two dimensional wedge, firs t investigated by Sornmerfeld [1896], Chapter three is more speculative; and represents a preliminary attempt to obtain a path integral, invariant under general (time dependent) canonical transformations.This would lead the way to a consistent quantization scheme, and a quantum mechanical application of the Hamilton-Jacobi philosophy [Goldstein, 1980; Schulman, 1980] Chapter four is a review of the ambiguities in the Dirac quantization scheme from a minimum set of postulates.[introduction) §1 Quantization C lassical mechanics is a description of nature using commuting variables that Is well formulated by the Hamilton least action principle.Butcla ssica l mechanics is not a complete model of nature; leaving several features unexplained, in fact c la ssica l electrodynamics predicts non-causa!behavior for an accelerating charge [Jackson, 19751 and might be considered incomplete on these grounds alone, it might even be speculated that there is only one se lf consistent theory of everything; that might then be deduced on consistency grounds without resort to observing the real thing.Some of the discrepancies in the classica l theory have been overcome in the guantum generalization.Dirac [1925, 1958] proposed how a guantum theory might be Induced from the classica l theory.He postulated that the c la ssica l commuting variables become non-commuting operators and suggested how the Quantum dvnamics be obtained from the cla ssica l Hamilton eouations of motlc)n .Since the quantum theory is the rlore f undamental theorv (the cla ss leal predl ctions follow inq in the j~\ -£ ( } imit) it might be argued that K • ; ; n should b? the start!ng point.U ! i dC s set erne, howev er has the advar tage of s tart 1 nq from a well up r j p p c f f od th sory.In this scheme to edCi : cla ssica l system corr esponds many qijantum generaliza Pons, (eact yield 1 ng the sa me classica l prediction s for 1 Dirac's metfiOd howe V c ; ■ s ambiguous In not generating one unlgue membei generalizations, some f i irf Xnr 1 Ul iCi specification (sijeh as norma'i oi uei ] ng} i e g u m e u *.o c o n i p« e « _ e * y' speeir y the quantum theory (these [edtures di reviewed in more de ^ n 1 1 i r -\ L d i ! 1 1 !chapter four). 1 i! i S i 111Q H t not seern orSadvantags; out uue lu tn is dniuiquiLV, l ids s il a i cecnniquec-soon as ine P ~i(!p j 9 {Introduction} §11 Translating between formalisms Although no formalism of quantum mechanics is more fundamental than any other, each is supposed to he equivalent and so one should he derivable from another.Assuming a knowledge of traditional quantum mechanics one may deduce the path integral formalism.Starting from the position to position amplitude for Heisenberg eigenstates: H h A -V (This may be generalized to the transition between any two states.)Inserting a set of position resolutions of unity; f dq Iq X q !leads to:t(N)iqfN-i)ft(N-i)><q(N-i.),t(N-ni ... Iq(i),tn)><q(i),t(i)iq(0},t(0i> where: qh = w U ~ uu.i and are not Intearated over.f -F L M .dq n iil,.^K ,<q(ki;t(k)iak~i);t(k-i)> Recall that for a Heisenberg eigenstate: iLRk),t(ki/ -expLiH(q,p)t/ n] iqck)/ where iqck)> Is the state at time zero and H(q,p) is the Hamiltonian operator.Pace 14 /op It ' in P a th in teg ra tio n fin iro d u ciio n ) So: < n ksl;N< quo! exp[-iH<q,p )At/ft] io(k-u> continue by inserting momentum resolutions of unity: n i=ljN^pv'.;n k=1 N < q(k)!p(k)><p(k)l exp[-iH(q,p)At/'h] !q(k-n> or [ •••/ -n , 11HdqU) n 1 , 1NdP(i) C -J jif.it.i i f . 2 nk=1 N<qfk)| exp[HH(q,p)At/ti]!pfk)><p(k)!qrk--!)>Considering the firs t alternative (the consequences of the second are discussed later) and looking at the factor: <p(k)| e x p [-iH (q p )A t/n ] !q{k-n> define a function H through: exp[-1 H(q,p)At/ h] =<p! exp[-tH (q,p)A t/h] !q>/<p|q>Even trough it is intended to take the At->0 lim it; one must work to order At, since there are N such terms, where NAt = T (the 'time of flight").In general a physical Hamiltonian hides (A t)-1 contributions (discussed later) and so agreement between all terms of the expanded exponentials is •'eauired.Agreement to lowest order yields: HsQ,p ; -U u;H \Q,p )iqx / \ u !Q/ 5o for a Hermitean Hamiltonian: < p lq X q!p> H2(q,p) = < plH (q,p )jq>< q|H (q,p )ip> integrating over q and using the resolution of unity for q: [" Ciq \ PiqX \ qip> H \*■_ !,p ) -j Qq k ujH \q,p )H *•.q,p j iq/ \ Qip/
This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum system coupled to a magnetic field, and diffusion as described by the Fokker-Planck equation. In both examples, the Hamiltonian contains products of the position and momentum operators. A quantization problem then arises since these operators do not commute, and the correspondence principle is no longer sufficient to determine the quantum Hamiltonian. The order between quantum operators is determined by additional conditions, such as hermiticity or conservation of probabilities. The calculation of the corresponding path integral then suffers from ambiguities, directly related to this quantization problem. The continuum limit is no longer unique, but depends on the limiting process. This chapter also considers a situation where space has a nontrivial topology, in this case a circle, and shows how this influences the calculation of the path integral.
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the Rimannian configurational spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for non-invariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between coordinates and momenta of a prticle. A possible cosmological consequence of the deformation is presented in short. Key words: quantum mechanics, Riemannian space, geodesic motion, deformation.
It is well known that the representations over an arbitrary configuration space related to a physical system of the Heisenberg algebra allow to distinguish the simply and non simply-connected manifolds [arXiv:quant-ph/9908.014, arXiv:hep-th/0608.023]. In the light of this classification, the dynamics of a quantum particle on the line is studied in the framework of the conventional quantization scheme as well as that of the enhanced quantization recently introduced by J. R. Klauder [arXiv:quant-ph/1204.2870]. The quantum action functional restricted to the phase space coherent states is obtained from the enhanced quantization procedure, showing the coexistence of classical and quantum theories, a fundamental advantage offered by this new approach. The example of the one dimensional harmonic oscillator is given. Next, the spectrum of a free particle on the two-sphere is recognized from the covariant diffeomorphic representations of the momentum operator in the configuration space. Our results based on simple models also point out the already-known link between interaction and topology at quantum level.
Abstract In this chapter, we generalize the previous path integral expression to a wider class of physical systems. To this end, we first discuss a change of variables in the path integral and the relationship to the operator formalism, and build up the polar coordinate path integral formula. Next we determine those cases where the p and q are confined within some restricted region, known as constrained systems, by applying the Faddeev-Senjanovic path integral formula to a sphere. Further we discuss quantum mechanics on a ring and the Aharonov-Bohm effect as the simplest examples of multiply connected space characterized by unshrinkable closed loops in configuration space. Also we make some comments on quantum mechanics for a more general manifold. In the next section the path integral for spin is de- rived with a newly introduced coherent state that also gives an alternative expression for the harmonic oscillator. Then, finally, cases where the WKB approximation happens to be exact are analysed.