Fortschritte der Physik / Progress of Physics: Band 26, Heft 9 1978 [Reprint 2021 ed.] 9783112519165, 9783112519158

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HEFT 9 • 1978 • BAND 26

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B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 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 I , Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortsehritte der Physik* 1 Herausgeber: Prof. Dr. Frank Kaschluhn, P r o f . Dr. Artur Lasche, Prof. Dr. Rudolf Ritsehl, Prof. D r . Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36 221 und 22 36229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion PhyBik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Cesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R • 74 Altenburg, Carl-von-Ossietzky-Stra8e 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 B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t IS,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/26/9. © 1978 by Akademie-Verlag Berlin. Printed in the German Democratie Republio. AN (EDV) S7618

ISSN 0 0 1 5 • 8 2 0 8 Fortschritte der Physik 26, 4 5 5 - 5 0 0 (1978)

Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time V . P . FROLOV*)

International

Centre for Theoretical Physics, Trieste,

Italy

Contents I.

Introduction

II.

Preliminaries 2.1. Massless fields in flat space-time: Asymptotic properties 2.2. Quantum theory in Minkowskian space 2.3. Asymptotically flat space-time: Definitions and properties

456

I I I . Null Surface Quantization in Curved Space-Time 3.1. Null surface geometry . 3.2. Null surface quantization of scalar field: Schwinger approach 3.3. Quantization of the electromagnetic field 3.4. Quantization of gravitational perturbations

457 457 461 . 465 467 467 470 474 477

TV. Null Infinity Quantization and Scattering Theory in Asymptotically Flat Space-Time . . 481 4.1. Null infinity quantization 481 4.2. Asymptotic symmetry group and asymptotic invariants 486 4.3. iS-matrix in asymptotically simple space-time 488 4.4. Some remarks on black hole evaporation 492 Appendix

496

References

500

Abstract A quantum theory of the free scalar, electromagnetic and gravitational fields in a curved asymptotically flat space-time is developed. It is shown that the Penrose conformal technique makes it possible to reformulate the null infinity quantization as a problem of the quantization on the proper null surface in the corresponding Penrose space. The Schwinger dynamical principle is exploited to derive the corresponding null surface commutation relations. The general covariant and gauge-independent form of the commutation relations is also given. The existence of the asymptotic symmetry (BMS) group in the asymptotically flat space-time is used to define uniquely the "in" and "out" vacuum states. The explicit expressions for the iS'-matrix operator and for the ¿(-matrix elements in the asymptotically simple space-time are given. The functional integration method is used to find the *) On leave of absence from P. N. Lebedev Physical Institute, Leninsky prospect 53, 117 924 Moscow, USSR. 35

Zeitschrift „Fortschritte der Physik", Heit 9

456

V . P . FROLOV

expression for the density matrix describing the observations at •/+ in the weakly asymptotically simple space-time when the information loss due to the event horizons or the existence of bare singularities is possible. The application of the developed approach to the problem of quantum evaporation of black holes (Hawking effect) is briefly discussed.

1. Introduction The problem of field quantization in a given (external) metric is of particular interest at present. This interest was greatly increased b y H A W K I N G ' S discovery [ 1 ] of the possibility of vacuum instability in the strong gravitational field of black holes. T h e quantization procedure is known to imply the solution of the following t w o problems: the construction of the commutation relations and the realization of their representation in some state space. W h i l e the solution of the first problem does not meet any principal difficulties [2] the second problem (which can be reduced to the vacuum state definition problem) is in the general case much more complicated. F r o m the physical point of view, the difficulty in the vacuum state definition reflects an ambiguity in separation of the particles into real and virtual ones in an external (gravitational) field. Formally, this uncertainty is caused b y the ambiguity (in space-time without a global timelike K i l l i n g vector field) in decomposing the field solutions into positive and negative frequency components. The problem on the vacuum-state definition can be satisfactorily solved in the physically interesting case of asymptotically f l a t space-times. I n such a space-time the geometry deflection f r o m a flat one at large distances f r o m the source becomes negligibly small, and the existence of the asymptotic symmetry group [3] allows one to define unambiguously the asymptotic energy-momentum observables. The state without incoming or outgoing particles (in- or out-vacuums) can be introduced as the lowest asymptotic energy states. T h e conformal transformation techniques and the definition of the asymptotically f l a t space-time proposed b y P E N R O S E [4, 5 ] gave a v e r y convenient mathematical tool to deal with the radiation problem of the classical massless fields [6]. I n this paper the corresponding consideration of the quantum theory of massless fields in asymptotically flat space-time is given (see also [7, 5]). The paper is organized as follows. The next three sections are preliminary and contain the reformulation of the known results on the massless scalar field in a f l a t space-time quantization in an equivalent f o r m but more convenient for generalizations (in terms of J ± quantities). The definition of the asymptotically flat space-time and some properties that we use in the sequel are also given. The second part of the paper is devoted to the problem of the null surface quantization in a curved space-time. W e find it convenient to use systematically the Schwinger dynamical, principle approach \9], which allows us to develop the quantization procedure of the scalar, electromagnetic and gravitational fields in a similar manner. The developed null surface quantization and the obtained (null surface) commutation relations are not only the first step in solving the problem of the null infinity quantization but have their own importance. I n the third part of the paper, the scattering theory of massless fields in asymptotically flat space-time is constructed. Using the asymptotic symmetry group, we define the asymptotic in- and out-vacuum states and find the explicit expression for the iS-matrix operator and /S-matrix elements in asymptotically simple space-time. The problems arising when some internal "singularities" (e.g. black holes) are present are discussed and the expression for the corresponding density matrix describing the scattering states in this case is given in the last section. Some useful formulas and some technical details are given in the appendix.

Null Surface Quantization

457

II. Preliminaries 2.1. Massless fields in flat space-time: Asymptotic properties I n this section some known results of interest in the following discussions will be collected. We begin by considering the properties of the solutions of the Klein-Gordon equation, • p=0, (2.1) in Minkowskian space. We use the approach which can easily be generalized to the case of asymptotically flat space-time and the consideration of this simple example allows us to demonstrate the main ideas of the method developed later in the paper. a) Scattering problem Different statements of the problem for Eq. (2.1) are possible. To define the solution