Feynman Path Integrals: Proceedings of the International Colloquium Held in Marseille, May 1978

445 Pages·1979·5.788 MB·English-French

by  S. Albeverio, R. Hoegh-Krohn (auth.), S. Albeverio, Ph. Combe, R. Høegh-Krohn, G. Rideau, M. Sirugue-Collin, M. Sirugue, R. Stora (eds.)

Lecture Notes
ni Physics
Edited by .J Ehlers, MQnchen, .K Hepp, Z~irich
.R Kippenhahn, M~inchen, .H .A Weidenm~iller, Heidelberg
and .J Zittartz, nISK
Managing Editor: .W Beiglb6ck, Heidelberg
601
namnyeF Path slargetnI
Proceedings of eht International Colloquium
Held ni Marseille, yaM 8791
Edited yb
.S Albeverio, .hP Combe, .R ,nhorK-hgeeH
.G Rideau, .M Sirugue-Collin, .M Sirugue dna .R Stora
galreV-regnirpS
nilreB Heidelberg New kroY 91 7 9
Editors
S. Albeverio Ph. Combe
Fakult~t f(Jr Mathematik M. Sirugue-Collin
der UniversitAt Bielefeld M. Sirugue
D-4800 Bielefeld 1 C.N.R.S. - Luminy - Case 907
Centre de Physique Theorique
F-13288 Marseille Cedex 2
.R Heegh-Krohn
Matematisk Institut
Universite G. Rideau
N-Blindern-Oslo 3 Universite de Paris VII
Laboratoire de Physique
Theorique et Math6matique
.R Stora
Tour 33-43
C.N.R.S. - Luminy - Case 907
2, place Jussieu
Centre de Physique Th6orique
F-75221 Paris Cedex 05
F-13288 Marseille Cedex 2
and
C.ER.N.
Division Theorique
1121-HC Gen6ve 23
ISBN 3-540-09532-2 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-09532-2 Springer-Verlag New York Heidelberg Berlin
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- DROWEROF -
ehT circle of ideas which generated the so called namnyeF path
integrals is contained in work by Dirac in the thirties and Feynman in the
forties. Especially in the latter work the representation of transition amplitudes
in terms of heuristic integrals no "path space" - or their approximations - saw
developed into na alternative general formulation of quantum dynamics, equivalent
to the previous formulations (Heisenberg, Schr~dinger) and the more recent versions
(Schwinger) in certain cases (e.g. non relativistic quantum mechanics with Lagran-
gians at most quadratic in the velocities). Mathematically, the heuristic integrals
introduced by Feynman being integrals over spaces of functions, have their natural
place in the realm of functional integration, a discipline which sah its classical
roots in work connected with the calculus of variations (Volterra) and dah a first
great impact in analysis through Wiener's introduction (1923) of Wiener integrals
(on a space of continuous functions) to handle the problem of Brownian motion and
heat diffusion. Whereas Wiener integrals, and more generally the integrals intro-
duced later no in connection with the study of stochastic processes, are integrals
with respect to positive finite measures, the heuristic expression for namnyeF
path integrals is in terms of a complex formal density which does not define
a measure. Thus the mathematical definition of the objects understood under the
eman of Feynman path integrals posed genuine wen problems. Such problems have
been attacked by different methods since the fifties.
It is important to realize (and ew hope this will eb conveyed also
by these Proceedings) that the ideas connected with Feynman path integrals have
a vast range of implications, and that different mathematical realizations of
these are possible, according to the type of applications eno is aiming to.
eW should hope that all present points of views are well represented in these
Proceedings. Part of the contributions deal with various mathematical definitions
of Feynman path integrals and the development of the mathematical tools needed
to make Feynman integrals work. The contributions concerned with these mathematical
problems in view of applications to non relativistic quantum mechanics sa well
sa scalar quantum field theory ("commutative Feynman path integrals") are gathered
in Section I.
In order that quantization by Feynman path integrals might eb a true
alternative formulation of quantum dynamics, the problems given by the existence
of half integer-valued spin particles should also eb handled. Mathematical approa-
ches to "non commutative Feynman integrals" are gathered in Section II.
VI
Since their inception the namnyeF path integrals have prompted
connections (Kac, Dynkin, Gelfand, Minlos, Yaglom, Nelson, Symanzik) with integrals
associated with stochastic processes, no the basis of the natural observation that
eno nac get solutions of Schr~dinger equations by analytic continuation from those
of the corresponding diffusion equations. Such connections have received several
wen impulses in the last fifteen years, through a successful Euclidean approach
to quantum fields, dna through the discovery of the connection between stochastic
mechanics dna quantum mechanics. These aspects are represented in Section III of
these Proceedings.
enO beautiful feature of namnyeF path integrals is that they contain
in a simple direct yaw the relation of quantum dynamics with classical dynamics.
Namely the detailed quantum behaviour nac eb obtained from the detailed classical
behaviour by a method of stationary phase (the classical behaviour being dictated
by the fundamental variational principles) following the original proposals of
Dirac, Feynman, Pauli dna Schwinger. In this yaw eno gets connections of the
namnyeF path integrals with the theory of oscillatory integrals dna Fourier
integral operators (aspects of these relations are illustrated also in Section
I . Conversely, namnyeF path integrals, in sa hcum as they ydobme a "quantization
procedure" nac eb expected to have close relations, worth while to eb studied in
details, with other quantization procedures. In particular it is natural to try
to understand better the connections with such quantization procedures sa the
"geometric quantization" dna the method of "Poisson brackets", both of which
are being geometric in nature, closely related with the symplectic dna differential
geometric features of the above mentioned methods of stationary phase. (Inciden-
tally, there are here connections also with problems of group representations
(Kirillov, Kostant, Auslander, e.g.)). emoS of these "geometrical aspects" are
presented in Section IV.
Since namnyeF path integrals give a wen definition of quantum dynamics,
it is interesting to apply them, even no a heuristic level, to domains where the
usual approaches to quantization meet difficulties, for instance general relati-
vity. In particular a definition of namnyeF path integrals no curved spaces is
needed. .C eD Witt discusses the possibility of such a definition in analogy with
the work done for the Wiener integrals no manifolds. Another domain where both
the usual dna the namnyeF quantization procedures meet difficulties is afforded by
Lagrangians involving higher than quadratic terms in the velocities. Such problems
are discussed in Section .V
Another domain where ideas connected with namnyeF path integrals have
played na important role, in laying down lines of research, is concerned
with gauge fields. nO the basis of the general heuristic principle (justified
mathematically in several cases, as mentioned above) according to which namnyeF
path integrals contain the classical dynamics and give rise to asymptotic expan-
sions around the classical solutions, na intensive study of classical gauge
fields sah been made in the last few years. This area is represented is Section VI
where both some classical and some quantum situations are studied with the hope
to come closer eventually to a construction of quantized gauge fields.
The study of the asymptotic series themselves that are obtained by
a formal expansion around the classical limit is na interesting object of study
in general, not only for gauge fields. This is discussed in Section VII.
eW hope that the above remarks might help in making understandable
to the reader that the diversity of aspects and approaches which is reflected
in these Proceedings is just a sign of the richness of the approach to quantiza-
tion given by Feynman path integrals. The emas diversity should also make clear
that the subject of Feynman path integrals should not eb considered sa a closed
one, no the contrary, much work is needed, no the conceptual mathematical dna
physical level, in order to bring to fruition all the beautiful potentialities
contained in those ideas. Also the presence of heuristic suggestions in emos of
the contributions should act sa a stimulus to wen mathematical efforts to give
mathematical form to the different parts of the building.
eW have collected in these Proceedings the invited lectures, ordered
along the lines of above "themes", as well sa some contributed communications.
eW would like to express our gratitude to Professor demmahoM Mebkhout,
Directeur ed I'UER Pluridisciplinaire ed Luminy, dna Professor Claude Mesliand,
President ed l'Universit@ ed Provence, for their kind interest in this Colloquium.
eW are very grateful to the secretaries of the Centre ed Physique
Th@orique dna to Mrs. .G Niard for their assistancy during the meeting. Our special
thanks are eud to Maryse Cohen-Solal whose experience sah been of constant inva-
luable help throughout all stages. eW particularly thank her for her patience and
skill in the preparation of these Proceedings.
eW also gratefully acknowledge the financial support of the Universit@
ed Paris VII, of the Universit@ ed Provence dna the Universit@ d'Aix-Marseille II,
as well as the Centre ed Physique Th~orique ed Marseille.
Marseille, November 1978
S. Albeverio, Ph. Combe, .R H~egh-Krohn,
.G Rideau, .M Siru~ue-Collin, .M Sirugue, .R Stora
- STNETNOC -
Foreword ................................................... III
Contents ................................................... VII
Participants ............................................... IX
Section I .................................................. 1
S. ALBEVERIO, R. NHORK-HGEOH - Feynman Path Integrals and the
Corresponding Method of Stationary Phase ...... 3
A.M. CHEBOTAREV, V.P. VOLSAM - Processus de sauts et leurs
applications dans la m~canique quantique ...... 58
A. NAMURT - The Polygonal Path Formulation of the Feynman
Path Integral ................................. 73
Section II .................................................. 103-104
Ph. COMBE, R. RODRIGUEZ, .M SIRUGUE-COLLIN, .M SIRUGUE -
Weyl Quantization of Classical Spin Systems.
Quantum Spins and Fermi Systems ............... 105
P. EERK - Feynman Path Integral and Theory of Forms ......... 120
Section III .................................................. 137-138
Ph. DRAHCNALB - Caract~risation de processus par la m~thode
des specifications locales .................... 139
P. COLLET - Renormalization Group Approach to the
Hierarchical Model ............................. 149
D. DOHRN, F. GUERRA, P. RUGGIERO, Spinning Particles and
Relativistic Particles in the Framework of
Nelson's Stochastic Mechanics .................. 165
R. GIELERAK, .W ,IKSWOWRAK L. STREIT - Construction of
a Class of Characteristic Functionals .......... 182
J. RAUCH, D.N. WILLIAMS - Topics on Euclidean Classical
Field. Equations with Unique Vacuua ............ 189
L. STREIT - Null plane fields and automodel random processes.203
Section IV .................................................. 207-208
A. ZCIWORENHCIL - D~formations et Quantification ............ 209
D.J. SIMMS - Geometric Quantisation and the Feynman Integral. 220
J.M. SOURIAU - Alg6bres Tierces ............................. 224
Section .V .................................................. 225-226
C. DEWITT-MORETTE -A Reasonable Method for Computing Path
Integrals on Curved Spaces ..................... 227
IIIV
.M .M MIZRAHI - Correspondence Rules and Path Integrals .... 234
J. TARSKI - Feynman-Type Integrals defined in Terms of
General Cylindrical Approximations ........... 254
Section VI ................................................. 280-281
H. HOGREVE, R. SCHRADER, R. SEILER - Bounds on the Euclidean
Functional Determinant ....................... 282
A. VONVALS - Application of Path Integrals to Non-Pertur-
bative Study of Massive Yang-Mills Theory .... 289
J.MADORE, J.L.RICHARD, R.STORA , F = • F, a review ........ 304
Section Vll ................................................ 335-336
R. BALIAN, .G PARISI, A. SOROV - Quartic Oscillator ........ 337
C. ITZYKSON - Perturbation Theory at Large Orders .......... 361
L.C. O'RAIFEARTAIGH, G. PARRAVICINI - Anomalous Behaviour
of the Effective Potential ................... 374
Short Communications ....................................... 389-390
J. BERTRAND, .M GINOCCHIO - Non-Affine Path Algorithm in the
Functional Integral Calculus of Schr~dinger
Kernels ...................................... 391
J. BERTRAND, .M IRAC - Non-Uniqueness in writing Schr~dinger
Kernel as a Functional Integral .............. 398
G. BURDET, C. MARTIN, M. PERRIN - About the Conformal
Properties of Yang-Mills Fields .............. 403
.G CURCl, R. FERRARI - Infrared Problem and Zero-Mass Limit
in a Model of Non-Abelian Gauge Theory ....... 410
H.M. FRIED - Unitary Restrictions on Semi-Classical Appro-
xiamtions to Certain Functional Integrals .... 418
L. GARRIDO, M. NAS MIGUEL - nO the Fokker Planck Lagrangian. 423
.W KERLER - Distribution Definition of Path Integrals ...... 429
H. LESCHKE - Functional Integral Representations and Ine-
qualities for Bose Partition Functions ....... 435
A. TEUOR - Renormalization of Yang-Mills Theory develope d
around an Instanton .......................... 444
Closing Address ............................................ 448
- STNAPICITRAP -
T. EGREBAA Universit~ ed NeuchAtel
.S OIREVEBLA Universit~t Bielefeld ; SRNC,TPC Marseille
.E RAMA Universit~ ed Paris VII
.H YRCAB Universit~ d'Aix-Marseille II, Luminy
CPT, SRNC Marseille
J. DNARTREB Universit~ ed Paris VII
.hP DRAHCNALB Universit~t Bielefeld
.O ILETTARB CPT, SRNC Marseille
.G TEDRUB Universit~ ed Dijon
.R ANOMRAC Universit~ d'Aix-Marseille II, Luminy
.P TELLOC Universit6 ed ev~neG
.hP EBMOC Universit~ d'Aix-Marseille II, Luminy
,TPC SRNC Marseille
I. ANAD Technion, Haifa
.C ETTEROM-TTIWED University of Texas at Austin
.P SOLCUD Centre Universitaire ed Toulon
,TPC SRNC Marseille
.hC LAVUD Universit~ d'Aix-Marseille II, Luminy
CPT, SRNC Marseille
J. DADAHLE Universit~ ed Provence ; SRNC,TPC Marseille
K.D. YHTROWLE University of Warwick
.M AIRAF Universit~ ed Provence ;CPT, SRNC Marseille
.R IRARREF Universita di Pisa
.R FIGARI Universit~ di Napoli
H.M. DEIRF Brown University, Providence
.W IKSNYZCRAG University of Wroclaw
L. ODIRRAG University of Barcelona
.F DAHRAF-ISSUOBAHG Universit~t grubmaH
.P ZEHG Centre Universitaire ed Toulon
,TPC SRNC Marseille
.M OIHCCONIG Universit~ ed Paris VII
.N ININNAVOIG Universit~ ed ev~neG
A. NNAMSSORG CPT, SRNC Marseille
.F ARREUG Universita di Salerno
.H NHAH Techn. Universit~t Braunschweig
.R NHORK-HGE~H ,TPC SRNC Marseille, dna Oslo University
J,-CI. DRAUOH Universit~ ed Paris VII
.G IMMIRZI Universit~ di Napoli
,B MUHCOI Universit~ ed Provence ;CPT, SRNC Marseille
.M CARI Universit~ ed Paris VII
.C NOSKYZTI NEC Saclay
A. CEIBUKAJ University of wasraW
.W IKSWOWRAK University of Wroclaw
.K RELLEK Universit~t Dortmund
.W RELREK Philipps Universit~t, Marburg
A. RUOPKAHK University of Teheran
.P EERK Universit~ ed Paris Vl
A. TREBMAL Universit~ d'Aix-Marseille II, Luminy
CPT, SRNC Marseille
.F EHCUOGNAL University of Leuwen, Heverlee
J. YAREL Coll~ge ed France, Paris
.H EKHCSEL Universit~t Dortmund
A. ZCIWORENHCIL Coll~ge ed France, Paris
A. IRAWHSEHAM University of Mysore
.M -UOSSELONAM
UOCITAMMARG Ecole Polytechnique, Palaiseau
.hC NITRAM Universit~ ed Dijon
M.M. MIZRAHI Center for Naval Analyses, Arlington
R.P. INIADNOM ICTP, Trieste
J, STYUN Universit~ ed snoM ; ,TPC SRNC Marseille
.E IRFONO Universit~ di amraP
L.C. HGIATRAEFIAR'O Dublin Institute for Advanced Studies
.C IREIMLAP Universit~ di Napoli
.M NIRREP Universit~ ed Dijon
.O TEUGIP Universit~t KarlsrUhe
J.L. DRAHCIR CPT, SRNC Marseille
.G UAEDIR Universit~ ed Paris VII
J.M. AREVIR Advanced School of Physics~ Trieste
.R ZEUGIRDOR Universit~ d'Aix-Marseille II, Luminy
CPT, SRNC Marseille
.D STREAKEOR University of Leuwen, Heverlee
A. TEUOR CPT, SRNC Marseille
S. ARABIKAKAS HTWR Aachen
.M NAS LEUGIM ZABIUR University of Barcelona
.W REDIENHCS Brown Boveri Research Center, Baden-D~ttwil
.R RELIES Freie Universit~t, Berlin
D.J. SMMIS Trinity College, Dublin
.M EUGURIS CPT, SRNC Marseille
.M NILLOC-EUGURIS Universit~ ed Provence ; CPT, SRNC Marseille
A. VONVALS NEC Saclay
J.M. UAIRUOS Universit~ ed Provence ; CPT, SRNC Marseille
.R AROTS CPT, SRNC Marseille
.N OBAZS Universit~ ed ev~neG
.M NOLAT Universit~ ed Montpellier
J. IKSRAT Universit~t Clausthal
A. NAMURT Heriot Watt University, Edinburgh
J.W.F. ELLA~ Syracuse University
A. SOROV NEC Saclay
D.N. SMAILLIW University of Michigan, Ann Arbor
A. NHOSFLUW nepO University, Milton Keynes
.K YAJIMA HTE Z~rich
.K ADIHSOY Universita di Salerno