The coherent‐state representation of quantum‐mechanical propagators as well‐defined phase‐space path integrals involving Wiener measure on continuous phase‐space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self‐adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self‐adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin‐operator Hamiltonian as well‐defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.

1.
A representative sample of references is the following:
I. M.
Gel’fand
and
A. M.
Yaglom
,
J. Math. Phys.
1
,
48
(
1960
);
D. G.
Babbitt
,
J. Math. Phys.
4
,
36
(
1963
);
E.
Nelson
,
J. Math. Phys.
5
,
332
(
1964
);
K. Itô, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (California U.P., Berkeley, 1967), Vol. 2, Part 1, pp. 145–161;
J.
Tarski
,
Ann. Inst. H. Poincaré
17
,
313
(
1972
);
K.
Gawedzki
,
Rep. Math. Phys.
6
,
327
(
1974
);
A.
Truman
,
J. Math. Phys.
17
,
1852
(
1976
);
S. A. Albeverio and R. J. Hoegh‐Krohn, Mathematical Theory of Feynman Path Integrals (Springer, Berlin, 1976);
V. P.
Maslov
and
A. M.
Chebotarev
,
Theor. Math. Phys.
28
,
793
(
1976
);
C.
DeWitt‐Morette
,
A.
Maheshwari
, and
B.
Nelson
,
Phys. Rep.
50
,
255
(
1979
);
D.
Fujiwara
,
Duke Math. J.
47
,
559
(
1980
);
P.
Combe
,
R.
Hoegh‐Krohn
,
R.
Rodriguez
, and
M.
Sirugue
,
Commun. Math. Phys.
77
,
269
(
1980
);
F. A.
Berezin
,
Sov. Phys. Usp.
23
,
763
(
1980
);
T.
Ichinose
,
Proc. Jpn. Acad. Ser. A
58
,
290
(
1982
);
I.
Daubechies
and
J. R.
Klauder
,
J. Math. Phys.
23
,
1806
(
1982
).
2.
E.
Nelson
,
J. Math. Phys.
5
,
332
(
1964
).
3.
See. e.g., L. Shulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
4.
An announcement of our results appears in
J. R.
Klauder
and
I.
Daubechies
,
Phys. Rev.
52
,
1161
(
1984
).
5.
J. R. Klauder, in Path Integrals, edited by George J. Papadopoulos and J. T. Devreese (Plenum, New York, 1978).
6.
R. H.
Cameron
,
J. Anal. Math.
10
,
287
(
1962/63
).
7.
E.
Lieb
,
Commun. Math. Phys.
31
,
327
(
1973
).
8.
See. e.g, E. J. McShane, Stochastic Calculus and Stochastic Models (Academic, New York, 1974).
9.
V.
Bargmann
,
Commun. Pure Appl. Math.
14
,
187
(
1961
).
10.
J. E.
Moyal
,
Proc. Camb. Philos. Soc.
45
,
99
(
1949
);
M. S.
Bartlett
and
J. E.
Moyal
,
Proc. Camb. Philos. Soc.
45
,
545
(
1949
).
11.
I.
Daubechies
and
J. R.
Klauder
,
J. Math. Phys.
23
,
1806
(
1982
).
12.
S.
Girvin
and
T.
Jach
,
Phys. Rev. B
29
,
5617
(
1984
).
13.
T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966).
14.
I.
Daubechies
and
J. R.
Klauder
,
Lett. Math. Phys.
7
,
229
(
1983
).
15.
J. R.
Klauder
,
J. Math. Phys.
23
,
1797
(
1982
).
16.
See, e.g., I. M. Gel’fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, New York, 1983);
or N. Ja. Vilenkin, Special Functions and the Theory of Group Representations (A. M. S., Providence, RI, 1968), Vol. 22.
17.
P.
Chernoff
,
J. Funct. Anal.
2
,
238
(
1968
).
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