Fortschritte der Physik / Progress of Physics: Band 29, Heft 5 [Reprint 2022 ed.] 9783112655924

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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 P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 5 • 1981 . B A N D 29

A K A D E M I E -

31728

V E R L A G

EVP 1 0 , - M

.

B E R L I N

ISSN 0 0 1 5 - 8 2 0 8

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Zeitschrift „Fortschritte der Physik 14 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 1080 Berlin, Leipziger Straße 3 - 4 ; Femruf: 2236221 und 2236229; Telex-Nr.: 114420; Bank: Staatsbank der DDR, Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lulz Kothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzuglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15,— M (Preis für die DDR: 1 0 , - M). Bestellnummer dieses Heftes: 1027/29/5. © 1981 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 29, 187—218 (1981)

Quantum Electrodynamics in Curved Space-Time I. L . BUCHBINDER

Pedagogical Institute, Tomsk,

USSR

E . S. FRADKIN

Lebedev Physical Institute, Moscow,

USSR

D . M . GITMAN

Pedagogical Institute, Tomsk,

USSR

Contents Introduction 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

187

Lagrangian and Representaion Picture Free Spinor Field Free Electromagnetic Field Amplitudes of Quantum Processes. Generalized Normal Product Feynman's Rules in Curved Space-Time Reduction Formulae Equations for Exact Green's Functions Calculation of Mean Values Representation of the Generating Functionals by Means of Functional Integrals Some Consequences of Unitary Condition of the S-Matrix References

188 190 194 198 202 203 205 209 . . . . 213 . 216 217

Introduction The quantum field theory in curved space-time where a gravitational field is described b y classical metrics, and all the other physical fields being quantized, is an important stage in the construction of the quantum gravity theory. The interest to the quantum theory with an external gravitational field may be explained by its applications to black hole physics and cosmology. Modern achievements of the quantum field theory in curved space-time are given in review papers by D E W I T T [1] and P A B K E R [2, 3]. It is generally accepted that the quantum field theory in curved space-time is a rather good model for t h e description of many quantum gravitational effects. However, most of the investigations are carried out in the theory of free fields 1 ). The problems of the fields theory interacting in curved space-time have been studied in papers [4—16], where mainly the particular aspects have been considered. The important features of the quantum theory with an external gravitational field are connected with the ambiguity of a vacuum state and first of all lead to the effect of*) We call free fields the ones interacting with an external gravitational field only. 1

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 5

188

I. L. Buchbinder, E. S. Fradkin, D. M. Gitman

particles creation from vacuum. For the correct description of a particle creation effect the interaction with external gravitational fields should be taken into account exactly. On the base of perturbation theory one should consider the interactions of fields with each other and selfinteraction. The possibility of particles creation from vacuum points out that the standard Feynman's rules for the calculation of quantum processes amplitudes should be modificated. Note that the quantum theory with an external gravitational field has two types of matrix elements in comparison with the quantum field theory in a flat space. First, while processes amplitudes calculation matrix elements appear, vacua of initial and final states do not coincide. Second, the theory contains the matrix element being the mean values relative to the initial vacuum. Such matrix elements are, for example, a number of particles created from vacuum or energy-momentum tensor of created matter. Generally speaking, Feynman's rules for the calculation of two types indicated above matrix elements should be different. The suggested paper is devoted to the construction of formalism of quantum electrodynamics in curved space-time concerning the possibility of particle creation by a strong gravitational field. The same problems appear in quantum electrodynamics with an external electromagnetic field too, where they are chiefly solved [17—19]. This paper is the paper [17—19] method generalization concerning curved space-time. The lay out of the paper is as follows. In section 1 the Lagrangian of quantum electrodynamics in curved space-time is constructed and the interaction picture taking into account the external gravitational field exactly is introduced. (This is the analog of the Furry picture in quantum electrodynamics). The transform from the Heisenberg picture to the interaction picture is carried out in a manifestly covariant way [20]. In sections 2 , 3 the properties of free spinor and electromagnetic quantum fields are discussed and conditions under which initial and final creation and annihilation operators are connected by unitarity transformation are indicated. Sections 4, 5 deal with the derivation of Feynman's rules for quantum processes amplitudes calculation on the base of generalized normal product of operators. In section 6 the way of reduction formulae derivation is indicated and the suitable Green's functions are introduced. In section 7 a generating functional for this Green's function is defined and the system of functional equations for them is obtained. Section 8 is devoted to Feynman's rules derivation to calculate mean values relative to the initial state. Green's functions which appear are shown to be different from those introduced for the processes amplitudes calculation. The generating functional for mean values calculation is introduced and a functional equation for it is obtained. In section 9 the representation of different generating functional s by means of functional integrals is introduced. In section 10 some consequences of ^-matrix unitary condition is considered which leads to the generalization of the optic theorem. 1. Lagrangian and Representation Picture Let us consider spinor electrodynamics in curved space-time.The Lagrangian fields system has the form: I = Here

+

+ Hint.

(1)

is Lagrangian of a free spinor field: = 1/-?(*)

y(*) fa"* (*)

- m] y>(x)

(2)

where y^(x) = ePa{x) y", y" is Dirac's matrices, e^a(x) — tetrad, D^ = d^ip + 1 l2coIMJboabip covariant spinor derivative, o j ^ spin connection, o ab = l/'i(yttyb — yby") (see e.g. [21]).

Quantum Electrodynamics in Curved Space-Time

189

Jf 2 is Lagrangian of a free electromagnetic field in the Lorentz gauge: 1

F

"

(

Z

)

+ 2 (P„4*(z))«J (3)

Fuy(x) =

-

Jf i n t is interaction Lagrangian: •Tint = -i-g(x)

M »

j"(z) A„(X)

W

/"(«) = y tvO*)» y"^) v M l • The following equations of motion are taken from Lagrangian (1.4) iy"(x)

— mip(x) = ey^{x) f(x)

A^x)

iD/Jji(x) y^(x) + myi(x) = —eyj(x) y^(x) A^x)

• where

M*)

(5)

=

• 4,=

The commutation relations for operators ip(x), fix), A^(x) can be written in a manifestly covariant form. Let a(x) — const be a spacelike surface, f^ an arbitrary vector, {w(x') /(*'))(,}

a

(6)

*(x) + mxp(x) = 0 .

Let us consider the description of quantum processes for noninteracting fermions. For the calculations of amplitudes processes one should give the initial and final states. Suppose that space-time is globally hyperbolic [22] and consider two spacelike hypersurfaces a{x) = oi- Initial states will be given on the surface crj and final states on the surface