Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 2
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Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965: “Over the succeeding years, ... Dr.Feynman’s approach to teaching the subject of quantum mechanics evolved somewhat away fromthe initial path integral approach.”2H. Kleinert, Fortschr. Phys. 6 , 1, (1968), and Group Dynamics of the Hydrogen Atom, Lectures presented at the 1967 Boulder Summer School, published in Lectures in Theoretical Physics,Vol. X B, pp. 427–482, ed.
by A.O. Barut and W.E. Brittin, Gordon and Breach, New York, 1968.3See my 1976 Erice lectures, Hadronization of Quark Theories, published in Understanding theFundamental Constituents of Matter , Plenum press, New York, 1978, p. 289, ed. by A. Zichichi.4H. Kleinert, Phys. Lett. B 69 , 9 (1977); Fortschr. Phys. 26 , 565 (1978); 30 , 187, 351 (1982).xiiixiv“square root coordinates” (to be explained in Chapters 13 and 14).5 These transformations led to the correct result, however, only due to good fortune. In fact, ourprocedure was immediately criticized for its sloppy treatment of the time slicing.6A proper treatment could, in principle, have rendered unwanted extra terms whichour treatment would have missed.
Other authors went through the detailed timeslicing procedure,7 but the correct result emerged only by transforming the measureof path integration inconsistently. When I calculated the extra terms according tothe standard rules I found them to be zero only in two space dimensions.8 Thesame treatment in three dimensions gave nonzero “corrections” which spoiled thebeautiful result, leaving me puzzled.Only recently I happened to locate the place where the three-dimensional treatment went wrong. I had just finished a book on the use of gauge fields in condensedmatter physics.9 The second volume deals with ensembles of defects which are defined and classified by means of operational cutting and pasting procedures on anideal crystal.
Mathematically, these procedures correspond to nonholonomic mappings. Geometrically, they lead from a flat space to a space with curvature andtorsion. While proofreading that book, I realized that the transformation by whichthe path integral of the hydrogen atom is solved also produces a certain type oftorsion (gradient torsion). Moreover, this happens only in three dimensions.
In twodimensions, where the time-sliced path integral had been solved without problems,torsion is absent. Thus I realized that the transformation of the time-sliced measurehad a hitherto unknown sensitivity to torsion.It was therefore essential to find a correct path integral for a particle in a spacewith curvature and gradient torsion. This was a nontrivial task since the literaturewas ambiguous already for a purely curved space, offering several prescriptions tochoose from.
The corresponding equivalent Schrödinger equations differ by multiplesof the curvature scalar.10 The ambiguities are path integral analogs of the so-calledoperator-ordering problem in quantum mechanics. When trying to apply the existingprescriptions to spaces with torsion, I always ran into a disaster, some even yieldingnoncovariant answers. So, something had to be wrong with all of them. Guided bythe idea that in spaces with constant curvature the path integral should produce thesame result as an operator quantum mechanics based on a quantization of angularmomenta, I was eventually able to find a consistent quantum equivalence principle5I.H.
Duru and H. Kleinert, Phys. Lett. B 84 , 30 (1979), Fortschr. Phys. 30 , 401 (1982).G.A. Ringwood and J.T. Devreese, J. Math. Phys. 21 , 1390 (1980).7R. Ho and A. Inomata, Phys. Rev. Lett. 48 , 231 (1982); A. Inomata, Phys. Lett. A 87 , 387(1981).8H.
Kleinert, Phys. Lett. B 189 , 187 (1987); contains also a criticism of Ref. 7.9H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989, Vol. I, pp.1–744, Superflow and Vortex Lines, and Vol. II, pp. 745–1456, Stresses and Defects.10B.S. DeWitt, Rev. Mod. Phys. 29 , 377 (1957); K.S. Cheng, J. Math.
Phys. 13 , 1723 (1972),H. Kamo and T. Kawai, Prog. Theor. Phys. 50 , 680, (1973); T. Kawai, Found. Phys. 5 , 143(1975), H. Dekker, Physica A 103 , 586 (1980), G.M. Gavazzi, Nuovo Cimento 101 A, 241 (1981);M.S. Marinov, Physics Reports 60 , 1 (1980).6H. Kleinert, PATH INTEGRALSxvfor path integrals in spaces with curvature and gradient torsion,11 thus offering alsoa unique solution to the operator-ordering problem.
This was the key to the leftoverproblem in the Coulomb path integral in three dimensions — the proof of the absenceof the extra time slicing contributions presented in Chapter 13.Chapter 14 solves a variety of one-dimensional systems by the new techniques.Special emphasis is given in Chapter 8 to instability (path collapse) problems inthe Euclidean version of Feynman’s time-sliced path integral. These arise for actionscontaining bottomless potentials.
A general stabilization procedure is developed inChapter 12. It must be applied whenever centrifugal barriers, angular barriers, orCoulomb potentials are present.12Another project suggested to me by Feynman, the improvement of a variationalapproach to path integrals explained in his book on statistical mechanics13 , founda faster solution. We started work during my sabbatical stay at the University ofCalifornia at Santa Barbara in 1982. After a few meetings and discussions, theproblem was solved and the preprint drafted.
Unfortunately, Feynman’s illnessprevented him from reading the final proof of the paper. He was able to do thisonly three years later when I came to the University of California at San Diego foranother sabbatical leave. Only then could the paper be submitted.14Due to recent interest in lattice theories, I have found it useful to exhibit thesolution of several path integrals for a finite number of time slices, without goingimmediately to the continuum limit. This should help identify typical lattice effectsseen in the Monte Carlo simulation data of various systems.The path integral description of polymers is introduced in Chapter 15 wherestiffness as well as the famous excluded-volume problem are discussed.
Parallels aredrawn to path integrals of relativistic particle orbits. This chapter is a preparationfor ongoing research in the theory of fluctuating surfaces with extrinsic curvaturestiffness, and their application to world sheets of strings in particle physics.15 I havealso introduced the field-theoretic description of a polymer to account for its increasing relevance to the understanding of various phase transitions driven by fluctuatingline-like excitations (vortex lines in superfluids and superconductors, defect lines incrystals and liquid crystals).16 Special attention has been devoted in Chapter 16 tosimple topological questions of polymers and particle orbits, the latter arising bythe presence of magnetic flux tubes (Aharonov-Bohm effect). Their relationship toBose and Fermi statistics of particles is pointed out and the recently popular topicof fractional statistics is introduced.
A survey of entanglement phenomena of singleorbits and pairs of them (ribbons) is given and their application to biophysics isindicated.11H. Kleinert, Mod. Phys. Lett. A 4 , 2329 (1989); Phys. Lett. B 236 , 315 (1990).H. Kleinert, Phys. Lett. B 224 , 313 (1989).13R.P. Feynman, Statistical Mechanics, Benjamin, Reading, 1972, Section 3.5.14R.P. Feynman and H. Kleinert, Phys. Rev. A 34 , 5080, (1986).15A.M. Polyakov, Nucl.
Phys. B 268 , 406 (1986), H. Kleinert, Phys. Lett. B 174 , 335 (1986).16See Ref. 9.12xviFinally, Chapter 18 contains a brief introduction to the path integral approachof nonequilibrium quantum-statistical mechanics, deriving from it the standardLangevin and Fokker-Planck equations.I want to thank several students in my class, my graduate students, and my postdocs for many useful discussions. In particular, T. Eris, F. Langhammer, B. Meller,I. Mustapic, T. Sauer, L. Semig, J. Zaun, and Drs. G. Germán, C.
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