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American Journal of Physics

Path-integral quantization of constrained Hamiltonian dynamical system: A pedagogical example

Manan Sengupta

Published Nov 1, 1986• 1 citation
Publisher

Abstract

The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.

DOI
10.1119/1.14816

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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 ... 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•Jan 1, 1987•0 citations
The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
Having gained some insight into the concept of 'actual and virtual paths' in a phase-space formalism (Sobouti and Nasiri 1993 Int. J. Mod. Phys. B 7 3255, Nasiri et al 2006 J. Math. Phys. 47 092106), in the present paper we address the question of 'extended' phase-space stochastic quantization of Hamiltonian systems with first class holonomic constraints. We present the appropriate Langevin equations, which quantize such constrained systems, and prove the equivalence of the stochastic quantization method with the conventional path-integral gauge measure of Faddeev–Popov quantization.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.
The Hamiltonian treatment of constrained systems in Gu¨ler′sG\ddot{u}ler'sGu¨ler′s formalism leads us to the total differential equations in many variables. These equations are integrable if the corresponding system of partial differential equations is a Jacobi system. The main aim of this paper is to investigate the quantization of the finite dimensional systems with constraints using the canonical formalism introduced by Gu¨lerG\ddot{u}lerGu¨ler. This approach is applied for two systems with constraints and the results are in agreement with those obtained by Dirac's canonical quatization method and path integral quantization method.
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/