Fortschritte der Physik / Progress of Physics: Band 27, Heft 11/12 1979 [Reprint 2021 ed.] 9783112522806, 9783112522790

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R PHYSIKALISCHEN

GESELLSCHAFT

DER I) EUTSC11EN D EMOKRATISC H EIS R I . PUBLIK VON F. KASCHI.UIIN. A. LÖSCHE. R. RITSC1IL U M ) R. ROMPE

H E F T 11/12 • 1979 • B A N D 27

A K A D E M I E -

V E R L A G EVP 2 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDK an das Zeitungsvertriebsamt, an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3 - 4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4 — 6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 701 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der Physik 4 * Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2 2 3 6 2 2 9 ; Telex-Nr. 114420; B a n k : Staatsbank der DDR, Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizcnznumraer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamther Stellung: VEB Druckhaus „Maxim Gorki' 4 , DDR - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15t— M (Preis für die DDR: 1 0 , - M) Bestellnummer dieses Heftes: 1027/27/11/12. (C) 1979 by Akademie -Verlag Berlin. Printed in the German Democratic Repubiic. AN (EDV) 57618

ISSN 0 0 1 5 • 8 2 0 8 Fortschritte der Physik 27, 5 1 1 - 5 4 5 (1979)

Dynamics on the Group Manifold and Path Integral M. S. M a r i n o v 1 ) , M. V. T e r e n t y e v

Institute of Theoretical and Experimental Physics, Moscow 117259, USSR Abstract Classical and quantal dynamics on the compact simple Lie group and on a sphere of arbitrary dimensionality are considered. The accuracy of the semiclassical approximation for Green's functions is discussed. Various path integral representations of Green's functions are presented. The special features of these representations due to the compactness and curvature are analysed. Basic results of the theory of Lie algebras and Lie groups used in the main text are presented in the Appendix. Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Material point on a circle: the ?7(1) group Classical dynamics on the group manifold The quantal dynamics Semiclassical approximation The path integral Path integral for sphere of arbitrary dimensionality Conclusion

Appendices A. Some results of the theory of Lie groups and Lie algebras A.l. The Lie algebras A.2. Matrix representations A.3. The regular representation A.4. Compact groups A.5. Integration on a group A.6. Unitary groups B. Group as a Riemannian space C. Maximal torus References

511 515 518 520 522 524 527 531 532 532 532 533 534 536 537 538 540 543 545

1. Introduction

Dynamics of any quantal system is described by the evolution operator, and its coordinate-space matrix element is the Green's function M(qt, q0;t) = (qk| exp ( - J ^ ) l«o> On leave from ITEP since June 1979 40

Zeitschrift „Fortschritte der Physik", Heft 1 1 - 1 2 / 7 9

(1-1)

512

M . S . MARINOV, M . V . TERENTYEV

where q is the coordinate vector and H is the Hamiltonian operator. The Green's function has a fundamental property

X(qt, qo-,h + h) = J X(qt, q, t2) X{q, q0; t j dq

(1.2)

where d q is an integration measure. In particular, it follows from (1.2) that

lim-X(q„ q0;t) = d(qt - q0).

(1.3)

f->0

A consequence of the definition (1.1) is that Green's function may be represented by the spectral expansion

W{qt, q0; t ) = £ Wn{qt) Wn*(q0) exp y> (q)

Ie\

H, Hip

E yi .

(1.4) tp„

where n is an eigen-function of the operator functions are n = n n The assumed to be orthonormal with the measure d q , while the Hamiltonian H is hermitian with this measure. The function (1.1) satisfies the Schroedinger equation, in both of its arguments, with the initial condition (1.3). A representation of Green's function in the closed form of the path integral is useful for physical applications. This representation is especially appropriate for the perturbative expansion and for the [semiclassical approximation. However, the Feynman original form [1, 2] r ri t (1.5)

X{q„ q»;t)= J 3q exp - f l(q, q) dr

where Jf is the classical Lagrangian, and the functional measure ¿¿q is defined as a limit of correspondingly normalized multiple integral, is known to be directly applicable only to the simplest (though the most interesting) case, k l when the coordinate space is infinite, and the Lagrangian is of the for, 1 k with kl independent of For a more complicated Lagrangian with, for instance, the metric tensor gkl depending on q (the motion in a Riemannian space) Eq. (1.5) needs a modification involving an additional term in the Lagrangian, proportional to h 2 . This approach was developed by D E W I T T [3], the corrections to Jf were also discussed by M C L A U G H L I N and S C H U L M A N [4]. The reasons for modification of the path integral in case of curvilinear coordinates were also considered by E D W A B D S and G U L Y A E V [5], Arthurs [6] (polar coordinates) and in more recent works by G E R V A I S and J E V I C K I [7] and S A L O M O N S O N [ + 1) E

C('(cos 0)

(7.17)

and to use the asymptotical expansion of the Bessel function J ( 1 / r ) at large values of its argument and index (I -f- v) ( / / , - , É„) may be chosen in such a way t h a t the set of the commutators for the basis elements acquires the most simple form [ # „ £ » ] = 0;

[tìpÉa]

=iXjÉ»

(A.4) [Éa,É-x]

=

itifìi;

[Éx, Ép]

=

N(a,

¡ì)

.

Here 1 k ^ r, where r is the rank of the algebra, i.e. the number of the commuting basis elements. The commuting elements of the Lie algebra form a subalgebra, named the Cartan subalgebra. At linear transformations of the subalgebra the numbers tXj behave as the components of a vector. This r-dimensional vector a is named the root of the algebra. I n view of the Jacobi identity, the numbers óò are related to ocj (see further Eq. (A.6)). The number function N(a, ,'i) does not vanish only if a, ft, and a + ft are roots of the algebra ; the explicit expression may be found in the book by GILMORE [26], p. 280. The introduction of the canonical basis is, in fact, a reduction of the system of C1,- n X nmatrices (j = 1 , . . . , r) with elements equal to the structure constants C°b, to a diagonal form. If there is a basis for which all the structure constants are real (in this case the Lie

Dynamics on the Group Manifold and Path Integral

533

algebra is named real), the matrices Cj are skew-Hermitean, the eigen-values are pure imaginary, and the root vectors are real. If a is a root, —a is also a root, so that the total number of roots is even, n — r = 2p. The canonical basis of a real algebra is reduced to a real form by means of the linear transformation £±IX = j„ :f iK%. For the canonical basis the Killing tensor is separated into two sectors Gjk = £

a*, = GJ(a + 0);

Xjock;

G„ = Gh = 0 (A.5)

G, = 2(«,•«») + X N(«> «') a'

N(a

+ «'.

—«)

while ¡x' — GaGikxk. The canonical commutators are covariant under linear transformations of the Cartan subalgebra, as well as under gauge transformations of the elements j X j t « where is a number. Using this freedom, one may reduce the Killing tensor to isotropic form Gjk = Adjk,

Ga=

A,

&> = ... — am ... — «(1> while w{i) = 0 for the Cartan subspace, and wu) = a for any a. The highest weight is the highest root. The structure constants are imaginary in the canonical basis; it is easily seen that Cf+=-Cf;

Ca+=-