References

1

B.S. DeWitt, Rev. Mod. Phys. 29, 377 (1957).

2

B. Podolsky, Phys. Rev. 32, 812 (1928).

3

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 A 101, 241 (1981).
A good survey over similar attempts is given by
M.S. Marinov, Phys. Rep. 60, 1 (1980).

4

Among the most widely discussed procedures was a postpoint discretization due to Ito and a midpoint discretization due to Stratonovich, with different mathematical advantages. For a detailed discussion see the textbooks
H. Risken, The Fokker-Planck Equation, second edition, Springer, 1983, Vol. 18;
R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, Springer, Berlin 1985.

A recent description of the relation between time slicing and Ito versus Stratonovich calculus can be found in
H. Nakazato, K. Okano, L. Schülke, and Y. Yamanaka, Nucl. Phys. B 346, 611 (1990).
Stochastic differential equations in curved spaces are developed in
K.D. Elworthy, Stochastic differential equations on manifolds, Cambridge Univ. Press, 1982;
M. Emery, Stochastic calculus in manifolds, Springer, Berlin 1989.

5

R. Graham, Z. Phys. B 26, 397 (1977).

6

K.D. Elworthy, Path Integration on Manifolds, in Mathematical Aspects of Superspace, eds. H.-J. Seifert, C. Clarke, and A. Rosenblum, Reidel, 1984.

7

L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965.

8

D.J. Simms and N. M. J. Woodhouse, Lectures on geometric quantization, Springer, Berlin 1976;
J. Sniatycki, Geometric quantization and quantum mechanics, Springer, Berlin 1980;
P.L. Robinson and J.H. Rawnsley, The metaplectic representation, Mpc structures, and geometric quantization, publ. by the American Mathematical Society in the series Memoirs of the American Mathematical Society no. 410 0065-9266, Providence, R.I., 1989.

9

For details and more references see H. Kleinert, Gauge Fields in Condensed Matter, Vol. I Superflow and Vortex Lines, pp. 1-744, and Vol. II Stresses and Defects, World Scientific, Singapore 1989, pp. 744-1443.

10

K. Kondo, in: Proc. 2nd Japan Nat. Congr. Applied Mechanics, Tokio, 1952
B.A. Bilby, R. Bullough and E. Smith, Proc. R. Soc. London A 231, 263 (1955);
E. Kröner, in: Physics of defects, Les Houches summer school XXXV, North-Holland, Amsterdam 1981.

11

H. Duru and H. Kleinert, Phys. Lett. B 84, 185 (1979); Fortschr. d. Phys. 30, 401 (1982).

12

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics,, second edition, World Scientific, Singapore 1995.

13

H. Kleinert, Mod. Phys. Lett. A 4, 2329 (1989).

14

H. Kleinert, Phys. Lett. B 236, 315 (1990).

15

P. Fiziev and H. Kleinert, New Action Principle for Classical Particle Trajectories In Spaces with Torsion, Europh. Lett. 35, 241 1996 (hep-th/9503074 and http://www.physik.fu-berlin.de/~kleinert/kleiner_re219/newvar.html).

16

A. Pelster and H. Kleinert, FU-Berlin preprint, May 1996 (gr-qc/9605028 and http://www.physik.fu-berlin.de/~kleinert/kleiner_re243/preprint.html).

17

Our notation for the geometric quantities in spaces with curvature and torsion is the same as in
J.A. Schouten, Ricci Calculus, Springer, Berlin 1954.

18

P. Fiziev and H. Kleinert, Euler Equations for Rigid-Body -- A Case for Autoparallel Trajectories in Spaces with Torsion, Berlin preprint 1995 (hep-th/9503075 and http://www.physik.fu-berlin.de/~kleinert/kleiner_re224/euler.html).

19

H. Kleinert, Collective Quantum Fields,
Lectures presented at the First Erice Summer School on Low-Temperature Physics, 1977, Fortschr. Physik 26, 565-671 (1978).
See also the predecessors:
H. Kleinert, Field Theory of Collective Excitations-- A Soluble Model,
Phys. Lett. B 69, 9 (1977),
as well as the derivation of an SU(3) SU(3) chirally invariant field theory of mesons from a quark theory in
H. Kleinert, Hadronization of Quark Theories and a Bilocal form of QED, Phys. Lett. B 62, 429 (1976);
H. Kleinert, On the Hadronization of Quark Theories, Lectures presented at the Erice Summer Institute 1976, in
Understanding the Fundamental Constituents of Matter,
Plenum Press, New York, 1978, A. Zichichi ed., pp. 289-390.

20

L.P. Gorkov, Sov. Phys. JETP 9, 1364 (1959).

21

V.L. Ginzburg and L.D. Landau, Eksp. Teor. Fiz. 20, 1064 (1950).

22

A.L. Leggett, Rev. Mod. Phys. 47, 331 (1975).

23

K.D. Schotte and U. Schotte, Phys. Rev. 182, 479 (1969);
see also:
S. Tomonaga, Progr. Theor. Phys. 5, 63 (1950).

24

For a review see:
D.R. Bes, R.A. Broglia, Lectures delivered at ``E. Fermi'' Varenna Summer School, Varenna, Como\ Italy, 1976. For recent studies: D.R. Bes, R.A. Broglia, R. Liotta, B.R. Mottelson, Phys. Letters B 52, 253 (1974); B 56, 109 (1975), Nuclear Phys. A 260, 127 (1976).
See also:
R.W. Richardson, J. Math. Phys. 9, 1329 (1968), Ann.\ Phys. (N.Y.) 65, 249 (1971) and N.Y.U. Preprint 1977,
as well as references therein.

25

J. Hubbard, Phys. Rev. Letters 3, 77 (1959); B. Mühlschlegel, J. Math. Phys. , 3, 522 (1962); J. Langer, Phys. Rev.\ A 134, 553 (1964); T.M. Rice, Phys. Rev. A 140 1889 (1965); J. Math. Phys. 8, 1581 (1967); A.V. Svidzinskij, Teor. Mat. Fiz. 9, 273 (1971); D. Sherrington, J. Phys.\ C 4, 401 (1971).

26

E. Witten, Commun. Math. Phys. 92, 455 (1984);
P. DiVecchia and P. Rossi, Phys. Lett. B 140, 344 (1984);
P. DiVecchia, B. Durhuus and J.L. Petersen, Phys. Lett. B 144, 245 (1984);
Y. Frishman, Phys. Lett. B 146, 204 (1984);
E. Abdalla and M.C.B. Abdalla, Nucl. Phys. B 225, 392 (1985);
D. Gonzales and A.N. Redlich, Phys. Lett. B 147, 150 (1984);
C.M. Naón, Phys. Rev. D 31, 2035 (1985);
See also the recent development by
P.H. Damgaard, H.B. Nielsen, and R. Sollacher, Nuclear Phys. B 385, 227 (1992) (hep-th/9407022);
P.H. Damgaard and R. Sollacher, Cern preprint (hep-th/9407022);
A.N. Theron; F.A. Schaposnik, F. G. Scholtz and H.B. Geyer, Nucl. Phys. B 437, 187 (1995) (hep-th/9410035);
C.P.  Burgess and F. Quevedo, Phys. Lett. B 329 (1994) 457; Nucl. Phys. B 421, 373 (1994);
C.P. Burgess, A. Lutkin, and F. Quevedo, Phys. Lett. B 336, 18 (1994);
J. Fröhlich, R.  Götschmann and P.A. Marchetti, preprint (hep-th/9406154).

27

S. Coleman, Phys. Rev. D 11, 2088 (1975);
S. Mandelstam, Phys. Rev. D 11, 3026 (1975);
B. Schroer and T.T. Truong, Phys. Rev. D 15, 1684 (1977).

28

For a semiclassical study of the model at finite times see
H. Kleinert and H. Reinhardt, Nucl. Phys. A 332, 33 (1979).

29

H. Kleinert, Nonabelian Bosonization as a Nonholonomic Transformations from Flat to Curved Field Space. FU-Berlin preprint 1996
(http://www.physik.fuberlin.de/~kleinert/kleiner_re239/preprint.html).

 
FIGURES
  
Fig. 1: {Crystal with dislocation and disclination generated by nonholonomic coordinate transformations from an ideal crystal. Geometrically, the former transformation introduces torsion and no curvature, the latter curvature and no torsion. 

=1mm

 
Fig. 2: Images under a holonomic and a nonholonomic mapping of a fundamental path variation. In the holonomic case, the paths x(t) and in (a) turn into the paths q(t) and in (b). In the nonholonomic case with , they go over into q(t) and shown in (c) with a closure failure at analogous to the Burgers vector in a solid with dislocations.}  
            =1mm