Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics

The following sections are included:

• Dedication
• Preface
• Preface to the First Edition
• Contents
• List of Figures
• List of Tables

Path integrals deal with fluctuating line-like structures. These appear in nature in a variety of ways, for instance, as particle orbits in spacetime continua, as polymers in solutions, as vortex lines in superfluids, as defect lines in crystals and liquid crystals. Their fluctuations can be of quantum-mechanical, thermodynamic, or statistical origin. Path integrals are an ideal tool to describe these fluctuating line-like structures, thereby leading to a unified understanding of many quite different physical phenomena. In developing the formalism we shall repeatedly invoke well-known concepts of classical mechanics, quantum mechanics, and statistical mechanics, to be summarized in this chapter. In Section 1.5, we emphasize open problems of operator quantum mechanics in spaces with curvature and torsion. These problems will be solved in Chapters 10 and 11 by means of path integrals.

The operator formalism of quantum mechanics and quantum statistics may not always lead to the most transparent understanding of quantum phenomena. There exists another, equivalent formalism in which operators are avoided by the use of infinite products of integrals, called path integrals. In contrast to the Schrödinger equation, which is a differential equation determining the properties of a state at a time from their knowledge at an infinitesimally earlier time, path integrals yield the quantum-mechanical amplitudes in a global approach involving the properties of a system at all times.

Important information on every quantum-mechanical system is carried by the correlation functions of the path x(t). They are defined as the functional averages of products of path positions at different times, x(t₁). Quantities of this type are observable in simple scattering experiments. The most efficient extraction of correlation functions from a path integral proceeds by adding to the Lagrangian an external source term disturbing the system, and by studying the response to the disturbance. In this chapter, the procedure is developed for the harmonic action treated in the last chapter. The source term couples the path x(t) linearly to a so-called current j(t). Such a term does not destroy the solvability of the path integral. The resulting amplitude is a simple functional of j(t). Its functional derivatives supply all correlation functions of the system. It is called the generating functional of the theory. It serves to derive the celebrated Wick rule for calculating the correlation functions of an arbitrary number of x(t). This forms the basis for perturbation expansions of anharmonic theories.

The path integral approach renders a clear intuitive understanding of quantum-mechanical effects in terms of quantum fluctuations, exhibiting precisely how the laws of classical mechanics are modified by these fluctuations. In some limiting situations, the modifications may be small, for instance, if an electron in an atom is highly excited. Its wave packet encircles the nucleus in almost the same way as a point particle in classical mechanics. Then it is relatively easy to calculate quite accurate quantum-mechanical amplitudes by expanding them around classical expressions in powers of the fluctuation width.

Most path integrals cannot be performed exactly. It is therefore necessary to develop approximation procedures which allow us to approach the exact result with any desired accuracy, at least in principle. The perturbation expansion of Chapter 3 does not serve this purpose since it diverges for any coupling strength. Similar divergencies appear in the semiclassical expansion of Chapter 4…

The path integral representations of the time evolution amplitudes considered so far were derived for orbits x(t) fluctuating in a euclidean space with Cartesian coordinates. Each coordinate runs from minus infinity to plus infinity. In many physical systems, however, orbits are confined to a topologically restricted part of a Cartesian coordinate system. This changes the quantum-mechanical completeness relation and with it the derivation of the path integral from the time-sliced time evolution operator in Section 2.1. We shall consider here only a point particle moving on a circle, in a half-space, or in a box. The path integral treatment of these systems is the prototype for any extension to more general topologies.

A realistic physical system usually contains many orbits of indistinguishable particles such as identical atoms or electrons, to be labeled by x^(v)(t) with ν = 1,2,3, … , n. Their Hamiltonian is invariant under the group of all n! permutations of the orbital indices ν. The Schrodinger wave functions can then be classified according to the irreducible representations of the permutation group…

Many physical systems possess rotational symmetry. In operator quantum mechanics, this property is of great help in finding wave functions and energies of a system. If a rotationally symmetric Schrödinger equation is transformed to spherical coordinates, it separates into a radial and several angular differential equations. The latter are universal and have well-known solutions. Only the radial equation contains specific information on the dynamics of the system. Being an ordinary one-dimensional Schrödinger equation, it can be solved with the usual techniques…

The fundamental quantity calculated from a path integral is the time evolution amplitude or propagator of a system (x_bt_b|x_at_a). In this chapter we give a general method of extracting the Schrödinger wave functions from (x_bt_b|x_at_a).

The path integral of a free particle in spherical coordinates has taught us an important lesson: In a euclidean space, we were able to obtain the correct time-sliced amplitude in curvilinear coordinates by setting up the sliced action in Cartesian coordinates xⁱ and transforming them to the spherical coordinates q^μ = (r,θ,φ). It was crucial to do the transformation at the level of the finite coordinate differences, Δxⁱ → Δq^μ. This produced higher-order terms in the differences Δq^μ which had to be included up to the order (Δq)⁴/ε. They all contributed to the relevant order ε. It is obvious that as long as the space is euclidean, the same procedure can be used to find the path integral in an arbitrary curvilinear coordinate system q^μ, if we ignore subtleties arising near coordinate singularities which are present in centrifugal Barriers, angular barriers, or Coulomb potentials. For these, a special treatment will be developed in Chapters 12-14…

We now use the path integral representation of the last chapter to find out which Schrödinger equation is obeyed by the time evolution amplitude in a space with curvature and torsion. If there is only curvature, the result establishes the connection with the operator quantum mechanics described in Chapter 1. In particular, it will properly reproduce the energy spectra of the systems in Sections 1.6 and 1.7—a particle on the surface of a sphere and a spinning top—which were quantized there via group commutation rules. If the space carries torsion also, the Schrödinger operator emerging from our formulation will be a prediction. Its correctness will be verified in Chapter 13 by an application to the path integral of the Coulomb system which can be transformed into a harmonic oscillator by a nonholonomic mapping involving curvature and torsion.

In Chapter 8 we have seen that for systems with a centrifugal barrier, the euclidean form of Feynman's original time-sliced path integral formula diverges for certain attractive barriers. This happens even if the quantum statistics of the systems is well defined. The same problem arises for a particle in an attractive Coulomb potential, and thus in any atomic system…

One of the most important successes of Schrödinger quantum mechanics is the explanation of the energy levels and transition amplitudes of the hydrogen atom. Within the path integral formulation of quantum mechanics, this fundamental system has resisted for many years all attempts at a solution. An essential advance was made in 1979 when Duru and Kleinert recognized the need to work with a generalized pseudotime-sliced path integral of the type described in Chapter 12. After an appropriate coordinates transformation the path integral became harmonic and solvable. A generalization of this two-step transformation has meanwhile led to the solution of many other path integrals to be presented in Chapter 14. The final solution of the problem turned out to be quite subtle due to the nonholonomic nature of the subsequent coordinate transformation which required the development of a correct path integral in spaces with curvature and torsion. Only this made it possible to avoid unwanted fluctuation corrections in the Duru-Kleinert transformation of the Coulomb system, a problem in all earlier attempts.

The first consistent solution was presented in the first edition of this book in 1990.

The combination of a path-dependent time reparametrization and a compensating coordinate transformation, used by Duru and Kleinert to transform the Coulomb path integral into a harmonic-oscillator path integral, can be generalized to relate a variety of path integrals to each other. In this way, many unknown path integrals can be solved by their relation to known path integrals. In this chapter, the method is explained for a typical sample of one-dimensional path integrals as well as for a more involved three-dimensional system. The latter describes a generalization of the Coulomb system consisting of two particles which carry both electric and magnetic charges. It is commonly referred to as the dionium atom (in analogy to the positronium atom, the bound state between electron and positron). We also discuss further possible generalizations of the solution method.

The use of path integrals is not confined to the quantum-mechanical description point particles in spacetime. An important field of applications lies in polymer physics where they are an ideal tool for studying the statistical fluctuations of line-like physical objects.

In the previous chapter, the binary interaction potential between the polymer elements was approximated by a 6-function. Quantum-mechanically, this potential is not completely impenetrable. Correspondingly, a polymer with such an interaction has a finite probability of self-intersections. This is only a rough approximation to the situation in nature where the atomic potential is of the hard-core type and self-intersections are extremely rare. In a grand-canonical ensemble, a polymer is often entangled with itself and with others. It can be disentangled only if it has open ends. Macroscopic fluctuations are required to achieve this in the form of worm-like creeping processes. Compared with local fluctuations, these take a very long time. For closed polymers, disentangling is impossible without breaking bonds at the cost of large activation energies. In order to study such entanglement phenomena in their purest form, it is useful to idealize the strongly repulsive interaction potential, as in Section 15.8, to a topological constraint of the type discussed in Chapter 6…

Tunneling processes govern the decay of metastable atomic and nuclear states, as well as the transition of overheated or undercooled thermodynamic phases to a stable equilibrium phase. Path integrals are an important tool for describing these processes theoretically. For high tunneling barriers, the decay proceeds slowly and its properties can usually be explained by a semiclassical expansion of a simple model path integral. By combining this expansion with the variational methods of Chapter 5, it is possible to extend the range of applications far into the regime of low barriers…

The theoretical tools developed up to this point are able to describe all thermodynamic equilibrium properties of quantum systems. In contact with a thermal reservoir they maintain a fixed temperature and other intrinsic thermodynamic parameters. Partition function and the density matrix can be calculated by continuing quantum-mechanical time evolution amplitudes to an imaginary time t_b – t_a = –ih/k_BT. In this chapter we want to go beyond equilibrium physics and develop a path integral formalism capable of describing nonequilibrium time-dependent phenomena of quantum systems in contact with a thermal reservoir. Note that the tunneling phenomena discussed in Chapter 17 were nonequilibrium processes whose full understanding requires the theoretical framework to be presented now. This was circumvented by considering only certain quasi-equilibrium questions. The equilibrium formalism was applicable at positive coupling constants, and the quasi-equilibrium results were reached by an analytic continuation to negative coupling constants…

Particles moving at large velocities near the speed of light are called relativistic particles. If such particles interact with each other or with an external potential, they exhibit quantum effects which cannot be described by their orbital fluctuations alone. At very short interaction times, additional particles or pairs of particles and antiparticles are created or annihilated, and the total number of particle orbits is no longer invariant. Ordinary quantum mechanics which always assumes a fixed number of particles cannot describe such processes. The associated path integral has the same problem since it is a sum over a given set of particle orbits. Thus, even if relativistic kinematics is properly incorporated, a path integral cannot yield an accurate description of relativistic particles. An extension becomes necessary which includes an arbitrary number of mutually linked and branching fluctuating orbits…

The following sections are included:

• Index