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.
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.
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.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in 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 somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
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/
From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.
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.