Selected Papers on

ELECTRODYNAMICSEdited by Julian Schwinger

The development of quantum mechanics during the first quarter of this century pro-duced a revolution in physical thought even more profound than that associated withtne theory of relativity. Nowhere is this more evident than in the area of the theo-retical and experimental investigations centering about the properties and the inter.actions of the electromagnetic f ield, or, as it is otherwise known, electrodynamics.In this volume the history of quantum electrodynamics is dramatically unfolded throughthe original words of its creators. lt ranges from the initial successes, to the firstsigns of crisis, and then, with the stimulus of experimental discovery, to new triumphsleading to an unparalleled quantitative accord between theory and experiment. lt term-inates with the present position of quantum electrodynamics as part of the largersubject of theory of elementary particles, faced with fundamental problems and thefuture prospect of even more revolutionary discoveries.Physicists, mathematicians, electromagnetic engineers, students of the history andphilosophy of science wil l f ind much of permanent value here. The techniques ofquantum electrodynamics are not l ikely to be substantially altered by future develop.ments, and the subject presents the simplest physical i l lustration of the challengeposed by the "basic inadequacy and incompleteness of the present foundations oftheoretical physics."Papers are included by Bethe, Bloch, Dirac, Dyson, Fermi, Feynman, Heisenberg, Kusch,Lamb, 0ppenheimer, Pauli, Schwinger, Tomonaga, Weisskopf, Wigner, and others. Thereare a total of 34 papers, 29 of which are in English, I in French, 3 in German, ano1 in ltalian.Preface and historical commentary by the editor. xvii * 423pp. 6t/s x 9%.Paperbound.

A DOVER EIIITI()N DESIGNEO F()R YEARS ()T USE!We have made every effort to make this the best book possible. Our paper is opaque,with minimal show-through; it wil l not discolor or become britt le with age. Pages aresewn in signatures, in the method traditionally used for the best books, and wil l notdrop out, as often happens with paperbacks held together with glue. Books open flatfor easy reference. The binding wil l not crack or split. This is a permanent book.

d.ISBN 0-486-604+4-6 $7.50 in U.S.A.

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Edited by JuJion SchwingerProfessor of Physics,Horvard University

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Preface

PAPERS

CONTENTS

vll

PAGE

P.A.M. Dirac rHE qUANTUM THEoRy oF THE EMrssroN AND ABsoRprroNOF RADIATION

Proceedings of the Royal Society of London, Series A, Vol. l14,p.243 (re27)

Enrico Fermi sopRA r-'nlrnrrRoDINAMrcA euANTrsrrcAAtti della Reale Accademia Nazionale dei Lincei. Vol. 12.p. a3l (1930).

P.A.M. Dirac, V. A. Fock, and Boris Podolsky oN euANTUMELECTRODYNAMICS

P hy s i halis c h e Z e its c hr if t de r S ow j e tuni o n, Band 2, Heft 6 (1932)P. Jordan and E. Wigner iiern oes pAULrscHE Aqurver-rNzvERBor

Zeitschrift f i ir Physik,Yol. 47, p. 631 (1928)W. Heisenberg uern orr Mrr DER ENTSTEHUNG voN MATERTE AUS

sTRAHLUNG vnnrNijprt:sN LADUNGSscHwANKUNGENSachsiche Akademie der Wissenschaf ten, VoI..86, p. 317 (193a)

V. S. Weisskopf oN THE sELF-ENERGv AND THE ELEcTRoMAGNETTcFIELD OF THE ELECTRONPhysical Reaiew, Vol. 56, p.72 (1939)

P.A.M. Dirac THEoRIE DU PosrrRoNRapport du 7" Conseil Solaay de Physique, Structure etProprietes des Noyaux Atomiques, p.203 (I934)

24

29

4 l

62

68

| l l

82

t v C o n l e n l s

V. S. Weisskopf usER mr ELEKTRoDYNAMIK DEs vAKUUMS AUF GRUNDDER QUANTE,NTHEORIE DES ELEKTRONS

K on geli ge D anshe V i d,e iskab ern es S els kab, M ath e matis h-fy sis heMeddelelser X.IZ, No. 6 (1936)

F. Bloch and A. Nordsieck NorES oN THE RADIATIoN FIELD oF THEELECTRoN Physical Reaiew,Yol.52, p. 5a (1937)

H. M. Foley and P. Kusch oN THE INTRINSIC MoMENT oF THE

ELECTRoN Physical Reaiew,YoL 73, p' 412 (1948)Willis E. Lamb, Jr. and Robert C. Retherford FINE STRUCTURE

OF THE HYDROGEN ATOM BY A MICROWAVE METHOD

Physical Reaiew,V o1. 7 2, p. 2+L Q9a7)H. A. Bethe rHE ELECTRoMAGNETTc sHrFT oF ENERGY LEvELS

Physical Reuiew,YoI.72, p. 339 (1947)Julian Schwinger oN QUANTUM-ELECTRoDYNAMICS AND THE

MAGNETIC MOMENT OF THE ELECTRONPhys ica t Reu iew, Yo l .73 , p . 416 (1948)

Julian Schwinger oN RADIATIVE coRRECTToNS To ELECTRoNscATTERING Physical Reaiew, Vol. 75, p. 898 (1949)

J. R. Oppenheimer ELECTRoN THEoRYRapports du 8" Conseil de Physir1ue, Soluay, p' 269 (1950)

S. Tomonaga oN A RELATIvISTICALLY INvARIANT FoRMULATIoN

OF THE QUANTUM THEORY OF WAVE FIELDSProgress of Theoretical Physics, Vol. I, p.27 (1946)

Julian Schwinger QUANTUM ELECTRoDYNAMICS, III: THEELECTROMAGNETIC PROPERTIES OF THE ELECTRON-

RADIATIVE CORRECTIONS TO SCATTERINGPhysical Reaieu, Vol. 76, P. 790 (1949)

S. Tomonaga oN INFINITE FIELD REAcTIoNS IN QUANTUM FIELD

rHEoRy Physical Reaieu,Vol .74, P.224 (1948)W. Paul'i and F. Villars oN THE INvARIANT REGULARIZATIoN IN

RELATIVISTIC QUANTUM THEORYReuiews of Modern Physics, Vol. 21, p.434 (1949)

Julian Schwinger oN GAUGE INVARIANCE AND vAcuuM PoLARIZATIoN

Physical Reuiew, Vol. 82, p. 664 (1951)R. P. Feynman rHE THEoRY oF PoSITRoNS

Physical Review, Vol . 76, p.749 (1949)

92

r29

r35

136

139

r42

143

145

156

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il

t 2

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198

209

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22 R. P. Feynman spACE-TrME AppRoACH To quANTUM ELECTRo-DyNAMrcs Physical Review, Vol. 26, p. 269 (lg.4g) 296

23 R. P. Feynman MATHEMATTcAL FoRMULATToN oF THE euANTUMTHEoRy oF ELEcTRoMAGNETTc rNrnnaciroNPhysical Reaiew, Vol. 80, p.440 (1950) 2b7

24 F. J. Dyson rHE RADIATToN THEoRTES or-.r'oMONAGA, scHwrNGER,AND FEvNMAN Physical Reaiew,Yol.Tb, p. 486 ( lg4g) 27b

25 F. J. Dyson rHE s-MATRrx rN quANTUM ErFcrRoDyNAMrcsPhysical Reuiew,Yol.75, p. 1736 (1949) 2gz

26 P,A.M. Dirac rHE LAGRANGTAN rN quANTUM MECHANTcsPhysikal ische Zeitschri f t der Soutjetunion,Band B, Heft I ( lg3g) Zl2

27 R. P, Feynman spAcE-TrME AppRoAcH To NoN-RELATrvrsrrc queNtu-MEcHANrcs Reaiews of Modern Physics, Vol. 20, p.267 (lg49) g2l

28 Julian Schwinger rHE THEoRy oF euANTrzED FTELDS, r.Physical Review, Vol. 82, p. 9t4 ( lgbl) Z4Z

29 Julian Schwinger rHE THEoRy oF quANTrzED FTELDS, rr.Phys ica l Rea iew, Vo l . 91 , p .7 lg ( lgbg) Bb6

30 W. Pauli THE coNNEcrroN BETwEEN sr'rN AND srATrsrrcsPhysical Reuiew, Vol. 58, p. 716 (1940) g7Z

3l Julian Schwinger oN THE GREEN's FUNcrroNS oF quANTrzEDFTELDS, r. Proceedings of the National Academy of Sciences,Yol. 37, p. 452 (1951) 279

32 Robert Karplus and Abraham Klein ELEcrRoDyNAMrc DTsrLACE-MENT OF ATOMIC ENERGY LEVELS, III: THE HYPERFINE STRUCTUREoF posrrRoNruM Physical Reaiew, Vol. 87, p. 848 (lgb2) Zg7

33 G, Kiillen oN THE MAGNTTuDE oF THE RENoRMALTzATToNCONSTANTS IN qUANTUM ELECTRODYNAMICSKongelige Danske Videnshabernes Selshab, yol 27,No. 12 (1953) s98

34 Norman M. Kroll and Willis E. Lamb, Jr. oN THE sELF-ENERGvoF A BoUND ELEC RoN Physical Reuiew,yol. Tb, p. Bgg (1949). 414

*This paper should properly appear following paper 12, but for reasons beyondmy editorial control, it appears as the last paper. JS

PREFACE

ANv sn'rncrroN of important contributions from the extensiveliterature of quantum electrodynamics necessarily reflecm a par-ticular viewpoint concerning the significance of those works,both historically and intheir implications for the future progressof the subject. The folrowing brief commentary is intended toindicate that viewpoinr, and to supply a setting for the indi-vidual paperu. The ratter are referied to by coisecutive num-bers; and appear in the same order, which does not alwayscorrespond to the historical one. A fer,v papers, which wereomitted only because of limitations on the-siie of the volume,are mentioned explicitly in the text.

The development of quantum mechanics in the years 1925and 1926 had produced rures for the description of systems ofmicroscopic particles, which involved promoting the funda-mental dynamical variabres of a corresponding clissical s.ysreminto operators with specified commutators. By this means. asystem, described initially in classical particle language, acquirescharacteristics associated with the comple*"rrru.i clissical wavepicture. It was also known that electromagnetic radiation con-tained in an enclosure, when considered as i crassicar dvnamicalsystem, was equivalent energeticaly ro a d.enumerabry infinitenumber of harmonic oscillators. with the application of thequantization process to these fictitious oscillatlrs the classicalradiation field assumed characteristics describable in the com-

P r e f q c e

plementary classical particle language. The ensuing theory ofiight qrruntum emission and absorption by atomic systems [l]m"urkei the beginning of quantum electrodynamics, as thetheory of the quanrum dynamical system formed by the electro-magneric field in inreraction with charged particles (in a nar-,or.,i"r sense, the lightest charged particles) ' The quantizationprocedure could be transferred from the variables of the fic-

titious oscillators to the components of the field in three-dimen-sional space, basecl uPon thl classical analogy between. a field

specifiei within small sPatial cells, and equivalent particle sys-

tlms. When it was attempted to quantize the cornplete electro-

magnetic field tW. Heisenberg and W' Pauli, Zerts' f' Physik

5 6 , " 1 ( 1 9 2 9 ) l , r a t h e r t h a n t h e r a d i a t i o n f i e l d t h a t r e m a i n s a f t e rthe coulomb interaction is separated, difficulties were en-

countered that stem from the gauge ambiguity of the potentialsthat appear in the Lagrangian formulation of the Maxwell

equations. The only t.uidyrturnical degrees of freedom are those

of the radiation part of the fielrl' Yet one can'employ additional

degrees of freedom which are suPpressed finally by imposing a

.oisistent resrricrion on the admissible states of the system [2] 'To make more evident the relativistic invariance of the scheme,

other equivalent forms rvere given to the theory by introducing

different time coordinates for each of a fixed number of charged

particles coupled to the electromagnetic field t3l' This formal

period of quantization of the electromagnetic field lvas ter-

-irrut.d by a critical analysis of the limitations in the accuracy

of simultaneous measurements of trvo field strengths' produced

by the knorvn quantum restrictions on the simultaneous meas-

urability of properties of material test bodies [N' Bohr andL. Rosenfeld, Kgl' Danske Vid' Sels', Math'-fys' Medd' 12'

No. 8 (1933) 1. The complete agreement of these considerationswith the torrrnt implications of the operator commutation rela-

tions indicated the necessity and consistency of applying the

quantum mechanical description to all dynamical sYstems' The

synthes iso f thecomplemerr ta ryc lass ica lpar t i c leandf ie ld lan-guages in the .o.r."p, of the quantized field' as exemplified in

the treatment of the electromagnetic field, was found to be of

P r e f q c e , i x

general applicability to systems formed by arbitrary numbersof identical particles, although the rules of field quantizationderived by analogy from those of particle mechanics were toorestrictive, yielding only systems obeying the Bose-Einsteinstatistics. The replacement of commutators by anti-commutatonwas necessary to describe particles, like the electron, that obeythe Fermi-Dirac statistics t4l. In the latter situation there is norealizable physical limit for which the sysrem behaves as aclassical field.

But, from the origin of quantum electrodynamics in thedu*il1l theory of point charges came a legacy of difficurties. Thecoupling of an elecrron with the electromagnetic field imptiedan infinite energy displacement, and, indeed, an infinite shift ofall spectral lines emitted by an atomic system IJ. R. Oppen_heimer, Phys. Rev. gb,46l (lgg0) l; in the reaction of the elec_tromagneric field stimulated by the presence of the elecrron,arbitrarily shorr wave lengths play a disproportionare and di-vergent role. The phenomenon of .le.t.orr-poritron pair crea_tion, which finds a natural place in the relativistic eleciron fieldtheory, contributes to this situation in virtue of the fluctuatingdensities of charge and current that occur evep in the vacuumstate [5] as the matter-field counterpart of the fluctuations inelectric and magnetic field strengrhs. In compuring the energyof a single electro' relative to that of the vac'um state, it is ofsignificance that the presence of the electron tends to suppressthe charge-current fluctuations induced by the fluctuating elec-tromagnetic field. The resulting electron energ'y, while stilldivergent in its dependence upon the contriu.,iiorrs of arbi-trarily shorr wave lengths, exhibits only a logarithmic infinity[6]; the combination of quanrum and relativistic effects hasdestroyed all correspondence with the crassical theory and itsstrongly structure-dependent electromagnetic mass. The exist-ence of current fluctuations in the vacuum has other implica-tions, since the introduction of an electromagnetic field inducescurrents that tend to modify the initial field; the "vacuum,' actsas a polarizable medium [71. New non-linear electromagneticphenomena appear, such as the scamering of one light beam by

P r e f q c e

another, or by an electrostatic field. But, in the calculation ofthe current induced by weak fields, there occurred terms thatdepended divergently uPon the contributions of high-energyelectrorr-positron pairs. These were generally considered to becompletely without physical significance, although it was no-ticed 181 that the contribution to the induced charge densitythat is proportional to the inducing density, with a logarith-mically divergent coefficient, would result in an effective reduc-tion oi all densities by a constant factor which is not observableseparately under ordinary circumstances. In contrast with the

divergences at infinirely high energies, another kind of divergentsituarion was encountered in calculating the total probabilitythat a photon be emitted in a collision of a charged particle'Here, lio*.lr"r, the deficiency was evidently in the approximatemerhod of calculation; in any deflection of a charged particleit is certain that "zero" frequency quanta shall be emitted, whichfact must be taken into account if meaningful questions are tobe asked. The concentration on photons of very low energy per-mitted a sufficiently accurate treatnent to be developed [9], inwhich it was recognized that the correct quantum descriptionof a freely moving charged particle includes an electromagneticfield that accompanies the particle, as in the classical picture. Italso began to be appreciated that the quantum treatment ofradiation processes was inconsistent in its identification of themass of the electron, when decoupled from the electromagneticfield, with the experimentally observed mass. Part of the effectof the electromagneric coupling is to generate the field that ac-companies the charge, and which reacts on it to produce anelectromagnetic mass. This is familiar classically, where the sumof the two mass contributions aPpears as the effective electronmass in arr equation of motion which, under ordinary condi-tions, no longer refers to the detailed structure of the electron.Hence, it was concluded that a classical theory of the latter typeshould be the correspondence basis for a quantum electro-dynamics tH. A. Kramers, Quantentheorie des Elektrons undder Strahlung, Leipzig, 19381.

Further progress came only with the spur of experimentaldiscovery. Exploiting the wartime development of electronic

P r e f q c e

and microwave techniques, delicate measurements disclosed thatthe electron possessed an intrinsic magnetic moment slightlygreater than that predicted by the relativistic quanrum theoryof a single particle [10], while anorher prediction of the lattertheory concerning the degeneracy of states in the excited levelsof hydrogen was contradicted by observing a separation of thestates [111. (Historically, the experimental stimulus came en-tirely from the latter measurement; the evidence on magneticanomalies received its proper interpretation only in conse-quence of the theoretical prediction of an additional spin mag-netic moment.) If these new electron properties weie to beunderstood as electrodynamic effects, the theory had to berecast in a usable form. The parameters of mass and chargeassociated with the elecrron in the formalism of electrodynamicsare not the quantities measured under ordinary conditions. Afree electron is accompanied by an electromagnetic field whicheffectively alters the inertia of the system, and an electromag-netic field is accompanied by a current of electron-positron pairswhich effectively akers rhe strength of the field and of allcharges. Hence a process of renormalization must be carried out,in which the initial parameters are eliminated in favor of thoselvith immediate physical significance. The simplest approximatemethod of accomplishing this is ro compure the electrodynamiccorrections to some property and then subtract the effect of themass and charge redefinitions. While rhis is a possible non-relativistic procedure 1121, it is not a satisfactory basis for rela-tivistic calculations where the difference of two individuallydivergent terms is generally ambiguous. It was necessary to sub-ject the conventional Hamiltonian electrodynamics to a trans-formation designed to introduce the proper description ofsingle electron anci photon states, so that the interactions amongthese particles would be characterized from the beginning byrxperimental parameters. As the result of this calculation [13],performed to the first significant order of approximarion in theelectromagnetic coupling, the electron acquired new electro-dyriamic properties, which were completely finire. These in-cluded an energy displacement in an external magnetic fieldcorresponding to an additional spin magnetic moment, and a

xl

x i i P r e f q c e

displacement of energy levels in a Coulomb field. Both predic-tions were in good accord with experiment, and later refine-ments in experiment and theory have only emphasized thatagTeement. However, the Coulomb calculation disclosed a seri-ous flaw; the additional spin interaction that appeared in anelectrostatic field was not that expected from the relativistictransformation properties of the supplementary spin magneticmoment, and had to be artificially corrected [14, footnote 5],[15]. Thus, a complete revision in the computational techniquesof the r-elativistic theory could not be avoided. The electro-dynamic formalism is invariant under l-orentz transformationsand gauge transformations, and the concePt of renormalizationis ir accord with these requirements. Yet, in virtue of thedivergences inherent in the theory, the use of a particular co-ordinate system or gauge in the course of computation couldresult in a loss of covariance. A version of the theory rvas neededthat manifested covariance at every stage of the calculation. Thebasis of such a formulation was found in the distinction betrveenthe elemenrary properries of the individual uncoupled fields,and the efiects produced by the interaction between them [16]'[J. Schwinger, Phys. Rev. 74, 1439 (1948) ]'

'fhe application

o1 these methods to rhe problems of vacuum polarization, elec-tron mass, and the electromagnetic ProPerties of single electronsnow gave finite, covariant results which justified and extendedthe earlier calcularions 1171. Thus, to the first approximationat least, the use of a'covariant renormalization technique hadproduced a rheory thar rvas devoid of divergences and in agree-ment .r.vith experience, all high energy difficulties being isolatedin the renormalization constants. Yet, in one asPect of thesecalculations, the preservation of gauge invariance, the utmostcaurion was required 1181, and the need was felt for less delicatemethods of evaluation. Extreme cale rvould not be necessary if.by some device, the various divergent integrals could be ren-dered convergent while maintaining their general covariantfearures. This can be accomplished by substituting, for the massof the particle, a suitably weighted sPectrum of masses; rvhereall auxiliary masses eventually tend to infinity [19]. Such a Pro-cedure has no rneaning in terms of physically realizable Particles.

P r e f q c e

It is best understood, and replaced, by a description of the elec-tron r,vith the aid of an invariant proper-time parameter. Di\:ergences appear only when one integrates over this parameter,and gauge invariant, Lorentz invariant results are automatica[yguaranteed merely by reserving this integration to the end ofthe calculation [20].

Throughout these developments the basic vierv of eiectro-magnetism \vas that originated by Maxr,vell and Lorentz-theinteraction betr.veen charges is propagated through the field bylocal action. In its quantum mechanical transcription it leads toformalisms in rvhich charged parricles and field appear on rhesame footing dynamically. But anorher approach is also familiarclassically; the field produced by arbitrarily moving charges canbe evaluated, and the dynamical problem reformulated as thepurely mechanical one of particles interacting rvith each other,and themselves, through a propagated ac[ion at a distance. Thetransference of this line of thought into quantum language t2ll,122], {23} rvas accompanied by another shift in emphasis rela-tive to the previously described rvork. In the latter, the effecton the particles of the coupling rvith the electromagnetic fieldwas expressed by additional energy terms rvhich could then beused to evaluate energy displacements in bound states, or tocompute corrections to scattering cross-sections. Now the funda-mental viervpoint rvas that of scattering, and in its approximateversions led to a detailed space-time description of the variousinteraction mechanisms. The two approaches are equivalent;the formal integration of the differential equations of onemethod supplying the starting point of the other t241. But ifone excludes the consideration of bound states, it is possible toexpand the elements of a scattering rnatrix in powers of thecoupling constant, and examine the effect of charge and massrenormalization, term by term, to indefinitely high porvers. Itappeared that, for any process, the coefficient of each porver inthe renormalized coupling constant rvas completely finite ;251.This highly satisfacrory result did nor mean, hor,vever, that theact of renormalizafion had, in itself, produced a more correcttheory. The convergence of the porver series is not established,

x tv P r e f q c e

and the series doubtless has the significance of an asymPtoticexpansion. Yet, for prac.tical purposes, in which the smallness ofthe coupling parameter is relevant, this analysis gave assurancethat calculations of arbitrary precision could be performed-

The evolutionary process by which relativistic field theoryrvas escaping from the confines of irs non-relativistic heritageculminated in a complete reconstruction of the foundations ofquantum dynamics. The quantum mechanics of particles hadbeen expressed as a set of operator prescriptions superimposedupon the structure of classical mechanics in Hamiltonian form.When extended to relativistic fields, this approach had the dis-advantage of producing an unnecessarily great asymmetrybetween time and space, and of placing the existence of Fermi-Dirac fields on a purely empirical basis. But the Hamiltonianform is not the natural starting point of classical dynamics.Rather, this is supplied by Hamilton's action principle, andaction is a relativistic invariant. Could quantum dynamics bedeveloped independently from an action principle, which, beingfreed from the limitations of the correspondence principle,might automatically produce two distinct types of dynamicalvariables? The correspondence relation between classical action,and the quantum mechanical description of time develoPmentby a transformation function, had long been knorvn t261. It hadalso been observed that, for infinitesimal time intervals and suf-ficiently simple systems, this asymptotic connection becomessharpened into an identity of the phase of the transformationfunction with the classically evaluated action i271. The generalquantum dynamical principle was found in a difierential char-acterization of transformation functions, involving the variationof an action operator t281. When the action operator is chosento produce first order differential equations of motion, or fieldequations, it indeed predicts the existence of nvo types of dy-namical variables, with operator properties described by com-mutators and anti-commutatorc, respectively t291. Furthermore,rhe connection benveen the statistics and the spin of the particlesis inferred from invariance requirements, which strengthens the

P r e f o c e

previous arguments based upon properties of non-interactingparticles t301. The practical util ity of this quantum dynamicalprinciple stems from its very nature; it supplies differentialequations for the construction of the transformation functionsthat contain all the dynamical properries of the sysrem. It leadsin particular to a concise expression of quantum electro-dynamics in the form of coupled differential equations for elec-tron and photon propagation functions t3ll. such functionsenjoy the advantages of space-time pictorializability, combinedwith general applicability to bound systems or scatrering situa-tions. Among these applications has been a treatment of thatmost electrodynamic of systems-positronium, the metastableatom formed by a positron and an electron. The agreementbetween theory and experiment on the finer details of thissystem is another quantitative triumph of quantum electro-dynamics [32].

The post-war developments of quantum electrodynamics havebeen largely dominated by questions of formalism and tech-nique, and do not contain any fundamental improvement inthe physical foundations of the theory. such a situarion is notnew in the history of physics; ir took the labors of more than acentury to develop the methods that express fully the mechani-cal principles laid down by Newton. Bur, we may ask, is therea fatal fault in the srrucrure of field rheory? Could it not be thatthe divergences-apparent symproms of malignancy-are onlyspurious byproducts of an invalid expansion in powers of thecoupling constant and that renormalization, which can changeno physical implication of the theory, simply recrifies thismathematical enor? This hope disappears on recognizing thatthe observational basis of quantum electrodynamics is self-con-tradictory. The fundamenral dynamical variables of the elec-tron-positron field, for example, have meaning only as symbolsof the localized creation and annihilation of charged particles,to which are ascribed a definite mass without reference to theelectromagnetic field. Accordingly it should be possible, in prin-ciple, to confirm these properties by measurements, which, if

xvl P r e f o c e

they are to be uninfluenced by the coupling of the particles tothe electromagnetic field, must be performed instantaneously.But there appears to be nothing in the formalism to set a stand-ard for arbitrarily short times and, indeed, the assumption thatover sufficiently small intervals the two fields behave as thoughfree from interaction is contradicted by evaluating the sup-posedly small effect of the coupling. Thus, although the startingpoint of the theory is the independent assignment of propertiesto the two fields, they can never be disengaged to give thoseproperties immediate observational signi{icance. It seems thatwe have reached the limits of the quantum theory of measure-ment, which asierts the possibility of instantaneous observations,without reference to specific agencies. The localizat\on of chargewith indefinite precision requires for its realization a couplingwith the electromagnetic field that can attain arbitrarily largemagnitudes. The resulting appearance of divergences, and con-tradictions, serves to deny the basic measurement hypothesis.\{e conclude that a convergent theory cannot be formulatedconsistently within the framework of present space-time con-cepts. To limit the magnitude of interactions while retainingthe customary corirdinate description is contradictory, since nomechanism is prbvided for precisely localized measurements.

In attempting to account for the properties of electron andpositron, it has been natural to use the simplified form of quan-tum electrodynamics in which only these charged particles areconsidered. Despite the apparent validity of the basic assump-tion that the electron-positron field experiences no appreciableinteraction with fields other than electromagnetic, this physi-cally incomplete theory suffers from a fundamental limitation.It can never explain the observed value of the dimensionlesscoupling constant measuring the electron charge. Indeed, sincecharge renormalization is a property of the electromagneticfield, and the latter is influenced by the behavior of every kindof fundamental particle with direct or indirect electromagneticcoupling, a full understanding of the electron charge can existonly when the theory of elementary particles has come to a stage

P r e f q c e xv t l

of perfection that is presently unimaginable. It is not likelythat future developments will change drastically the practicalresults of the electron theory, which gives contemporary quarr-tum electrodynamics a cerrain enduring value. yet the real sig-nificance of the work of the past decade lies in the recognitionof the ultimate problems facing electrodynamics, the problemsof conceptual consistency and of physical completeness. No finalsolution can be anticipated until physical science has met theheroic challenge to comprehend the structure of the sub-micro-scopic world that nuclear exploration has revealed.

Jurrex ScnwrNcrRCambridge, Mass.1956

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Poper I

The Quantum, Theory of the Emission and, Abso,t"lpti;on ofRad,iatiotz..

By P. A. M. Dmlc, St. John's College, Cambridge, and Institute forTheoretical Physics, Copenhagen.

(Communicatecl by N. Bohr, X'or. Mem. R.S.-Received. February 2, L927.)

$1. Introd,untion and, Summ,ary.The new quantum theory, based on the assumption that the dynamical

variables do not obey the commutative law of multiplication, has by now beendeveloped sufficiently to form, a fairly complete theory of dynamics. One cantreat mathematically the problem of any dynamical system composed of anumber of particles with instantaneous forces acting between them, providecl itis describable by a Hamiltonian function, and one can interpret the mathematicsphysically by a quite definite general method. On the other hand, hardlyanybhing has been done up to the present on quantum electrodynamics. Thequestions of the correct treatment of a system in which the forces are propa-gatecl with the velocity of light instead of instantaneously, of the proiluction ofan electromagnetic field by a moving electron, and of the reaction of this fielclon the electron have not yet been touched. In adclition, there is a seriousilifficulty in making the theory satisfy all the requirements of the restricted

244 P. A. M. Dirac.

principle of relativity, since a Hamiltonian* function can no longer be used.This relativity question is, of course, connected with the previous ones, and itrryill be impossible to answer any one question completely without at the sametime answering them all. However, it appears to be possible to build up afairly satisfactory theory of the emission of radiation and of the reaction ofthe radiation field on the emitting systern on the basis of a kinematics anddynamics which are not strictly relativistic. This is the main object of thepresent paper. The theory is non-relativistic only on account of the timebeing counted throughout as a c-number, instead of being treated symmetricallywith the space co-ordinates. The relativity variation of mass with velocityis taken into account without difficulty.

The underlying ideas of the theory are very simple. Consid,er an atom inter-aoting with a field of rad.iation, which we may suppose for definiteness to beconfined in an enclosure so as to have only a discrete set of d.egrees of freedom.Resolving the radiation into its X'ourier components, we can consider the energyaud. phase of each of the components to be dynamical variables describing theradiation field. Thus if E, is the energy of a component labelled r anil 0"is the corresponcling phase (defined as the time since the wave was in a standard,phase), wecansuppose each E, and 0, to form a pair of canonically conjugatevariables. In the'abs"o"" of any interaction between the fielct and the atom,the whole system of fielcl plus atom will be descibable by the Hamiltonian

H: X"E, * Ho (1)equal to the total energy, Ho being the Hamiltonian for the atom alone, sincethe variables 8,, 0, obviously satisfy their canonical equations of motion

When there is interaction between the fielcl and the atom, it could be taken intoaccount on the classical theory by the acldition of an interaction term to theHamiltonian (1), which would be a function of the variables of the atom and ofthe variables 8,, 0, that describe the field. This interaction term would givethe efiect of the rad.iation on the atom, and also the reaction of the atom on theradiation field.

In order that an analogous method. may be used. on the quantum theory,it is necessary to assume that the variables Er, 0n are q-numbers satisfyingthe stand.ard. quantum oonditions 0rE, - Erg, : ih, etc., where h is (2rc)-rtimes the usual Planek's constant, like the other dynamical variables of theproblem. This assumption immed.iately gives light-quantum properties to

E,:-ffi:0, 6,:ffi:t.

Em,iss'ion and, Absorlttion of Rad,ia,tion, 245

the rafiation.* X'or if v" is the frequency of the component r, 2rvr0, is anaugle variable, so that its canonical conjugate E,l2nv, can only &ssume adiscrete set of values tlifiering by multiples of. h, which means that En canchange only by integral multiples of the quantum (Znh) v,. rf we now add aninteraction term (taken over from the clasical theory) to the Hamiltonian (1),the problem can be solved zsselding to the rules of quantum mechanics, andwe would expect to obtain the correct results for the action of the radiationand the atom on one another. It will be shown that we actually get the eorrectlaws for the emission and absorption of radiation, and the correct values forEinstein's A's and. B's. In the author's previous theory,f where the energiesand phases of the eomponents of radiation were c-numbers, only the B's could.be obtained, and the reaction of the atom on the rafiation could not be takeninto account.

It will also be shown that the Hamiltonian which describes the interactionof the atom and the electromagnetic waves carl be mad.e identical with theHamiltonian for the problem of the interaction of the atom with an assemblyof partioles moving with the velocity of light and satisfying the Einstein-Bosestatistics, by a suitable choice of the interaction energy for the particles. Themrmber of particles having any specified direction of motion and energy, whichcan be used. as a dynamical variable in the Hamiltonian for the particles, isequal to the number of quanta of energy in the correspond.ing wave in theHamiltonian for the waves. There is thus a complete harmony between thewave and light-cluantum d.escriptions of the interaction. We shall actuallybuild up the theory from the light-quantum point of view, and. show that theHamiltonian transforms naturally into a form which resembles that for thewaves.

The mathematical development of the theory has been mad.e possible by theauthor's general transformation theory of the quantum matrices.f owingto the fact that we count the time as a c-number, we are allowed to use the notionof the value of any dynamical variable at any instant of time. This value is

* Similar assumptions have been qsed by Born and Jordan [.2. f. physik,, vol. B4,p. 886 ( 1925)l for the purpose of taking over the classical formula for the emission oJ rad.iationby a dipole into the quantum theory, and by Born, Heisenberg and Jordan ['Z, f. physik,'vol. 35, p. 606 (r925)l for calculating the energy fluctuations in a field of black-bod.yradiation.

t ' Roy. Soc. Proc.,'A, vol. ll2, p. 661, $ b (f926). This is quoteil later by, loc. cit.,I.1 ' Roy. Soc. Proc.,' A, vol. ll3, p. 62I (1927). This is quotort later by lnc. ci,t,,Il.. An

essentially equivalent theory has been obtained inclependently by Jordan [,2. f. physik,'vol. 40, p. 809 (1927)1. See also, X'. London, ,2. f. physik,, vol. 40, p. t9B (1926).

216 P. A. M. Dirac.

a q-number, capable of being represented by a generalised " rnatrix " accordingto many fifierent matrix schemes, some of whioh may have continuous langesof rows and columns, andl may require the matrix elements to involve certainkincls of infnities (of the type given bythe 8 functions*). A matrix scheme canbe found in which any clesired set of constants of integration of the dynamicalsystemthatcommuteare represented by diagonal matrices, or in which a set ofvariables that commute are represented by matrices that a,re diagonal at aspecified time.t The values of the diagonal elements of a diagonal matrixrepresenting any q-number are the characteristic values of that q-number. ACartesian co-ordinate or momentum will in general have all characteristic valuesfrom - o to * co , while an action variable has only a discrete set of character-istio values. (We shall make it a rule to use unprimed letters to denote thedynamioal variables or q-numbers, and- the same letters primed or multiplyprimed to denote their oharacteristic values. Transformation fu:rctions or eigen-functions are functions of the characteristic values and not of the q-numbersthemselves, so they shoulcl always be written in terms of primed variables.)

If /((, tl) is any funotion of the canonical variables lp, r1a, t'he matrix repre-senting/at any time I in the matrix scheme in which the [a at time I are diagonalmatrices may be written down without any trouble, since the matrices repre-senting the

r and r1e themselves at time I are linown, namely,

t*('1") :1t'8 (1'1"),nr,(e'{): -ih8 (Lt'-(t") ... I ({o-t' -2,,-t") 8' (E*'-1*') 8 (Er+r'- r+r")"'Thus if the Hamiltonian H is given as a function of the ei, and 47,, we can atonce write down the matrix H(1' 4\. We can then obtain the transformationfunotion, (l'la') say, which transforms to a matrix scheme (a) in which theHamiltonian is a d.iagonal matrix, as (l' l0-') must satisfy the integral equation

fl ]J(qe') d1" (1" 1n'): W (oc') .(1'ln'), (3)

J

of which the characteristic values trV(oc') are the energy levels. This equationis just schr

Ent,ission and, Absotption of Radiation. 247

ieand.vleon account of the special equations (2) for the matrices representinge and r;e. Equation (3) may be written in the more general form

in which it can be applied to systems for which the Hamiltonian involves thetime explicitly.

one may have a dynamical system specified by a Hamiltonian H whichcannot be expressed as an algebraic function of any set of canonical variables,bnt udich can allthe same be represented by a matrix H(8,,). such a problemcan still be solved by the present method., since one can still use equation (B)to obtain the energy levels and eigenfunctions. 'w,e shall find that the Hamilto_nianwhich describes theinteraction of a light-quantum andanatomicsystem isof this more general type, so that the interaction can be treated mathematically,although one cannot talk about an interaction potential energy in the usualsense.

rt should be obs:erved. that there is a difierence between a light-wave and thede Broglie or schrrifinger w'aye associated with the light-quanta. x'irstly, thelight-wave is always real, while the cle Broglie wave associated with a light-quantum moving in a definite direction must be taken to involve an imaginaryexponential. A more important difierence is that their intensities are to beinterpreted in different ways. The number of tight-quanta per u:rit volumeassociated with a monochromatic light-wave equals the energy per unit volumeof the wave divitled by the energy (2nh)v o{ a single light-quantum. on theother hand a monochromatic d.e Broglie wave of amplitude a (multiplied intothe imaginary exponential factor) must be interpreted as representing az light-quanta per unit volume for all frequencies. This is a special case of the generalrule for interpreting the matrix analysis,* accorfing to which, if. 1l;1a,1 or*o (*') is the eigenfunction in the variables [p of the state oc, of an atomicsystem (or simple particle), r*o (tr')lz is the probability of each(ahaving thevalue 17,' ,lor I {; (Ei ) 12 dll dr' . . . is the probability of each

* lyrng betweenthe values (7r' and.Er' * dr', whenthe (ahavecontinuous ranges of character-istic values] on the assumption that all phases of the system are equally probable.The r-ave whose intensity is to be interpreted in the frst of ihure two waysappea* in the theory only when one is dealing with an assembly of the associatedparticles satisfying the Einstein-Bose statistics. There is thus no such waveassociated with electrons.

I,,r't', d,l' 6'la!) : th o gq, la,)lot, (3',)

* Loc. cit., II, g$ 6, ?.

248 P. A. M. Dirac.

g2. Tlw Perturbqtion of an Assembl'y of Ind'epenilent Bystems'

we shall now oonsider the transitions produced in an atomic system by an

arbitrary perturbation. The method we shall aclopt will be that previouslygiven by the author,f which leails in a simple way to equations which determineihe probability of the system being in any stationary state of the unpertr:rbedsyrt"* at any time.f This, of cour$e, gives immediately the probable numbero1 systems in that state at that time for an assembly of the systems

that are indepenilent of one another and are all perburbecl in the same way.

The object of the present section is to show that the equations for the ratesof change of these probable numbers can be put in the Hamiltonian form in a

simple manner, which will enable further developments in the theory to be

made.Let I{o be the Hamiltonian for the unperturbed system and Y the perburbing

energy, which can be an arbitrary function of the dynamical variables and may

or may not involve the time explicitly, so that the Hamiltonian for the perturbed

system is H : Ho * Y. The eigenfunctions for the perturbed system must'

satisfv the wave equation i 'ha,s13t: (Ho + Y) ,1,,where (Iro * Y) is an operator. If (l : >,erQ, is the solution of this equationthat satisfies the proper initial conditions, where the tf,'s are the eigenfunctionsfor the unperturbed system, each assooiatecl with one stationary state labellecl

by the suffix r, and the Q,,'s atefunctions of the time only, then lo, l2 is the prob-ability of the system being in the state r at any time' The o"s must be nor-

malised initially, and will then always remain normalised.. The theory will

applydirectlytoanassemblyofNsimilarinclepenclentsystemsifwemultiply

"u"n of these ar's by NL so as to make E, I a, lz : N' S'e shall now have that'

lo, 12 is the probable number of systems in the state r'The equation that d'etermines the rate of change of the a,'s is$

'ihd,: XJ',*,, (4)where the V,*'s ate t'he elements of the matrix representing V' The coniugate

imaginary equation t _nuu,*: xrvrr*or* : xpr*v.. $,)

I lroc. adt.I.t Th" tn"ory has recently been extended by Born [,2. f. Physik,' vol. 40,

p. 167 (1926)]go as to take into account tho adiabatic changes in the stationary stat'es that may be

produced by tho perturbation as well as the transitions. This extension is not used in

the Present PaPer.$ Loc. cit., I, equation (25).

E'mission cr,nd Absorption of Radiation. 249Ifwe regard a, and ih ao* as canonical conjugates, equations (4) and (4') takerhe Hamiltonian form with the Hamiltonian function X', : Xrron*vrro'namely,

on't6",*

., ila.* aF,t Xll --;.- : - :---:

dt da,

lve can transform to the canonical variables N, d, by the contact trans-Jormation

a, : fi t"-i6'1n, an* - fr *gt$":h,

This transformation makes the new variables I{, and S,real, N" being equalio a,e,,*: lu,lz, the probable number of systems in the state r, and, $,lhbeing the phase of the eigenfunction that represents them. The HamiltonianF, now becomes

X'3. : )"rVrrNr+$r*9i (d'-6';72,

and the equations that d.etermine the rate at which transitions occur have thecanonical form

\ r - 8 F t r 0 F ,'" : - 4l' 9": N'A more convenient way of putting the transition equations in the Hamiltonian

{orm may be obtained with the help of the quantitiesbr: &re-iw'tlh, br* : gr*

"iw,tlh,\Y,beingthe energy of the state r. 'We have l4ls equal to la,lz,the probablenumber of systems in the state r. X,or b" we find

ih'b, : W,b, +, ihh,s-;w sft- Wrb, | 2"Yrrb"ei (w,--w,)4,1

rviththe help of (a). If rve putY,r:p*si(w,-w"Itth,sothato* is a constantwhen V does not involve the time explicitly, this red.uces to

ih 'b , :W,b ,+X,0" ,6":8"Hr"6", (5)

rvhere Hru : trl-" 8r* f or' which is a matrix element of the total HamiltonianH : Ho f v with the time factor ei(w'-w)'/l removed, so that Hr" is a constantrvhen H does not involve the time explicitly. Equation (b) is of the same formas equation (4), and may be put in the Hamiltonian form in the same way.

rt should be noticed that equation (b) is obtained directly if one vrrites downthe schriidinger equation in a set of variables that specify the statiouary statesof the unperturbed. system. rf these variables arc (0, and if H(E'e') d.enotes

vol,. cxlv.-a.

dan 1dt 'ih

250 P. A. M. Dirac.

a matrix element of the total Hamiitonian H in the ([) scheme, thisSchriidinger equation woulcl be

i,h A,! G') lAt: Er- H (4'e') ,l/ (F.'), (6)like equation (3,). This differs from the previous equation (5) only in thenotation, a single suffix r being there used to denote a stationary state insteadof a set of numerical values l*' fot the variables 11,, and. fu being used insteadof + (g'). Equation (6), ancl therefore also equation (5), can still be used whenthe Hamiltonian is of the more general type which cannot be expressed as analgebraic function of a set of canonial variables, but can still be representedby a matrix H(Z'\') or H,.".

'We now lake b, and''ihb,* to be canonically conjugate variables instead of

a, and. 'iha,*. The equation (5) anct its conjugate imaginary equation willnow take the Hamiltonian form rvith the Hamiltonian function

X' : Xrrbr* Hrr6".

Proceeding as before, we make the contact transformation

br :N j s -de ' lh , bf - Nrt d!o'tk,

to the new canonical variables N", 0r, where N, is, as before, the probabletrumber of systems in the state r, and 0, is a new phase. The llamiltonian n'

will now become'F : Xru H", Nr* N s+ e!(e'-e')lk,

and the equations for the rates of change of N" and 0, will take the canonicalform

' r aF ;0Fl\": - D-0, ur : aN;.The Hamiltonian may be written

n': xrw"N, * xr.?r"N"*Nslet(0"-o''r&. (9)The first term X,Wf{" is the total proper energy of the assembly, and theseoond may be regardecl as the additional enelgy d"ue to the perturbation. Iftheperturbation is zero, the phases 0n would increase linea,rly with the time,while the previous phases {, would in this case be constants'

$3. Ihe Perturbution of an Assembty sati,sfui,ng the Ei,nstei,n-Bose Stati,sti,cs.According to the preceding section we oan describe the efiect of a perturba-

tion on an assembly of independent' systems by means of canonical variablesand Hamiltonian equations of motion. The development of the theory which

(7)

(8)

Emission and Absorption of Rad,iation. 25Lnaturally suggests itself is to make these canonical variables q-numbers satisfy-ing the usual quantum conditions instead of c-numbers, so that their lramilto-nian equations of motion become true quantum equations. The Hamiltonianfunction will now provide a Schr

l o

252 P. A' M' I)irac'

The Hamiltonian (7) now becomesn' : Er.b"*Hrrb, : xrrNrfot0'/lEf" (N' { L1t

"-;e'1tt: l,.,H,,N,l (N, + 1 - $,,)*s.t(4"-e")1h

inwhichtheH,,arest i l lc-numbers.Wemaywri tethisn. intheformcorre.sponding to (9)

x' : x,wN" * x",o",Ni (N, +1 - 8',)* dt@'-o'tth

in which it is again composed of a proper energy term X'W'N' and an inter-

action energY term.The wave equation written in terms of the variables N' ist

,ih&+ (Nr', Nr', N"' '..) : F.l, (Nr', N2', N3' "'), (12)

where T is au operator , each 0, occurring in n' being interpreted to mean ih a /aNr' "

If we applythe operator e+il'lhto any functiou -f(Nr" N," "'N"' "') of thevariables N1', N2', ... the result is

e+io, lk l (Nl ,Nr ' , . . . Nr ' , . . . ) : 6+a/ut" '7(N1' , Nz ' , . " Nr ' " ' )-

-f (Nr', Nr', " 'N"' + 1' " ' ) 'If we use this rule in equation (12) and use the expression (11) for x' we obtainf

m! l ,(N,' , Nn', Nr' . . ' )d t " '

: ! r ,H" ,Nr ' , l (Nr ' , + 1 - 8^)+t(Nr" Nr ' , ' . .Nr ' , -1 , " 'Nr ' , + 1, " ' ) ' (13)

we see from the right-hand side of this equation that in the matrix leple-

senting X', the term in n' involving ed?'-t')lk will contribute -only to

t h o s e m a t r i x e l e m e n t s t h a t r e f e r t o t r a n s i t i o n s i n w h i c h N , d e c r e a s e sby unity and N, increases by unitv,

'i'e', t'o matrix elements of the type

u(o i , f r r ' . . .N , ' . . .N , ' ; N1 ' , Nz ' " ' N , ' - 1 " 'N" ' + 1 " ' ) ' r f we f ind a*oloiioo +(or,, Ni ...) of equation (13) that is normalisecl li.e., one for whichXN,,,*,,...i,trtN.', Nr' . '.)lt:1] and that satisfies the proper initial con-aitioo., then I Q (N1" Nz' . ..) l 2 will be the probability of that distribution inwhich Nr' systems are in state 1, Nr'in state 2' "' at any time'

Considerf i rst thecasewhenthereisonlyonesystemintheassembly.Theprobability of its being in the state g is determined by the eigenfunction

fWearesupposingfordefinitenesst.hat,thelabelrofthestat ionarystatest,akesthevalues I, 2,3, . . ' .

t Whens : r ,{r (N1', Nr' . . 'N' ' - 1 " ' l f* ' f l ) is to be taken to mean rf (N"N" " 'N"'" ' ) '

(11)

(11',)

l l

Emission and, Absoryttion oJ Racl,iation. 253

{(\', Nr', ...) in which all the N,'s are put equal to zero except \,, which isput equal to unity. This eigenfunction we shall denote by Q {q}. 'when it issubstituted in the lefb-hancl side of (18), all the terms in the summation onthe right-hand side vanish except those for which r : {, and we are left with

4

,ih *+{q}: E"Ho,{{s},

which is the same equation as (5) with ..l., {q} playrng the part of bo. This estab-lishes the {act that the present theory is equivalent to that of the precedingsection vrhen there is only one system in the assembly.

Now take the general case of an arbitrary number of systems in the assembly,and assume that they obey the Einstein-Bose statistical mechanics. Thisrequires that, in the ordinary treatment of the probrem, only those eigen-functions that are symmetrical between all the systems must be taken intoaccount, these eigenfunctions being by themselves suffcient to give a completequantum solution of the problem.f we shall now obtain the equation for therate of change of one of these symmetrical eigenfunctions, and show that it isidentical with equation (13).

rf we label each system with a number n, t]o,en the Hamiltonian for theassembly will be He : X,H (zr), where H (n) is the H of $2 (equal to Ho f V)expressed in terms of the variables of the ruth system. A stationary state ofthe assembly is defined by the numbers t11 r 2 .. . r, . . . which are the labels of thestationary states in which the separate systems lie. The Schriiclinger equationfor the assembly in a set of variables that specify the stationary states will beof theform(6) fwith 116 instead of H], and we can write it in the notation ofequation (5) thus:-

ihb@rr r . . . ) : Xo , , , , . . .F .e .@{r . . . ! s rsz . . . )b (s rs r . . . ) , (14)whereH^(rrre...i sr$2...)is the general matrix element of Hafwiththetimefactor removed]. This matrix element vanishes when more than one s, d.ifiersfoom the corresponfing rr; equals Hr."^ when s. differs trom r* and everyother s' equals rn; and. equals }n]tr*, when every s, equals r". substitutingthese values in (14), we obtaini ,h 'b(rrrr . . . ) : x*xu,, *r f i r , , " ,h(r{r . . . rm-lsmrm+r.. . ) * )rHr. r^b(r{2.. . ) . ( tb)we must now restriot b (ry, ...\ to be a syn,metrical function of the variables

11, r2... in order to obtain the Einstein-Bose statistics. This is permissiblesince if b (rtrr...) is symmetrical at anytime, then equation (1b) shows that

I Loc. ci,t.,I, g 3.

t 2

254 P. A. M. Dirac.

b1rrr, ...) is also symmetrical atthat time, so that b ('t, "') rrill remainsymmetrical.

Let N, denote the number of systems in the state r. Then a stationary stateof the assembly describable by a symmetrical eigenlunction may be specifiedby the numbers \, N, .'. N,... just as well as by the numbers tv t2 "' ra "'tand we shall be able to transform equation (15) to the variables N1, Nr ....'We

cannot aotually take the new eigenfunction b (N1, Nz "') equal to the pre-vious one b (rrrr...), but must take one to be a numerical multiple of theother in order that each may be correctly normalised. with respect to itsrespeotive variables. We must have, in fact,

\ ru r , . . lb ( r r r r . . . ) l ' : 1 : Xu , ,N, . . . ib (Nr , N , " ' ) l ' ,and hence we must take lb(Nr, N, ...) l2 equal to the sum of lb(rtrr "')12 forallvalues of thenumberst1,rs... such that there are N. of themequal to 1,Ntequal to 2, etc. There are N l/Nl ! N2 ! ... terms in this sum, where N : XNnis the total number of systems, and they are all equal, since b(rtrr"') is asymmetrical function of its variables t1, 12 ."' Hence we must have

b (N1, N2 ' . . ) : (N ! /Nr ! Nz ! . . . )* a ( ' { , " ' ) '

If we make this substitution in equation (15), the left-haud side will becomeift (\ ! N, ! ... /N !)* i (Nr, N, ...). The term H,,n,*b (r{, "'r*-1s',,nr7aar "')in the first summation on the right-hand sid'e will become

lN1 ! N2 ! . . . (N'-1) ! . . . (N' + 1) ! . . . /N l l l H,ub (N1, N2 . . . N,- 1 " 'N'+ 1" ' ) , (16)

where we have written r tor r* and s for s*. This term must be summed- for

all values of s except r, and must then be summed for r taking each of the valuest1t rz .... Thus each term (16) gets repeated by the summatiou process untilit occurs a total of Nn times, so that it contributes

N, [N, ! N2 ! ... (N' - 1) ! ... (N,+1) ! ... lN !]] H,,b (N1, Nz ..'N,-1 "' N' + 1"'):N , * (N, * 1 )+(N, ! N21. . . /N ! ) 'H"b(N1 'Nz. ' . N" -1 . . . N"+ 1 . . . )

to the right-hancl sitle of (15). n'inally, the term I.o}J,,,*b (rr, 12 "'\ becomes

>J'TJI, , .b(r{r . . . ) :xN,Hnn'(N, ! N, l " 'A'T !)+ b ( \ ' N, " ' ) '

Henoe equation (15) becomes, with the removal of the faotor (N, ! N2 !... /N !)+,,ihb (N L,N, ...) : x,x,*, N"* (N"+1)+ H^b (Nr, N, ... N'-1 "' N, + 1 "')

+>N"H""b(N1, N2.. .) ' (17)

t 3

Emission and, Absorgttion of Rcrd,itttion. 255

which is identical with (18) fexcept for the fact that in (17) the primes havebeen omitted from the N's, which is permissible when we do not require to referto the N's as q-numbers]. we have thus established that the Hamiltonian(11) describes the efiect of a perturbation on an assembly satisfying the Einstein-Bose statistics.

$4. The Reaction of the Assembl,y on the perturbi,ng System.up to the present we have considered onry perturbations that can be repre-

sented by a perturbing energy v added to the Hamiltonian of the perburbedsystem, Y being a function only of the dynamical variables of that system andperhaps of the time. The theory may readily be extend.ed to the case whenthe perturbation consists of interaction with a perturbing dynamical system,the reaction of the perturbed system on the perturbing system being takeni:nto account. (The distinction between the perturbing system and the per-turbed system is, of co'rse, uot real, but it will be kept up for convenience.)

-w-e now consider a perturbing system; d.escribed, say, by the canonical

variables Jr, or,,, the J's being its first integrals when it is alone, interactingvith an assembly of perturbed systems with uo mutual interaction, that satisfvthe Einstein-Bose statistics. The total Hamiltonian will be of the form

H, : H" (J) f X,H (n),where H" is the Hamiltonian of the perturbing system (a function of the J,sonly) ancl H (zr) is equal to the proper energy Ho (n) prus the perturbation energyv(za) of the nth system of the assembly. H (n) is a function only of the variablesof the zlth system of the assembly and of the J's and zo's, and d.oes not involvethe time explicitly.

The schriidinger equ.ation corresponding to equation (I4) is nowihb g', rtr z ...): Es- Er,, r, ... H, (J,, rtrz... ; J,,, srsr.,,) b (J,,srsa ...),

in which the eigenfunction b involves the additional variables J6,. The matrixelement Hr(J', r{2.,.; J',srsr...) isnowalways a constant. As before,itvanishes when more than one s, difiers from the corresponding r*. whens. difiers lromr* and every other s, equals r, it reduoes to H (J'r*; J,,s*),which is the (J'r*i tr"s*) matrix element (with the time factor removed) ofH : Ho f Y, the proper energy plus the perburbation energy of a singlesystem of the assembly; while when every s, equars ro, it has the valueH, (J') 8.r,y,* 2*H (J'r*i J"r*). ff, as before, we regtrict the eigenfunctions

EARROLT COLLEGE LIBRI\R?ITFT FNIA AANNTANA 6OANI

l 4

256 P. A. M. Dirac'

to be symmetrical in the variable$ t11 12 "', w can again transform to thevariables N1, Nz . ' ., which will lead, as before, to the result

dhb Q', N1', Ne' ...) : Hr (J1) b (J', N'r, Nn' ...)g)s,E".$r'*(Ns',+1-8n,)+Ir1J',r; J',s) b(J",Nr"Nz',.. 'N,',-' l.. 'N,',* 1"') (18)

This is the schr

t 5

representing the incident electron, approaching the atomic system, which arescatbered or difiracted in all directions. The square of the amplitude of thewaves scattered in any direction with any frequency is then ur.,i*"d by Bornro be the probability of the electron being scattered in that direction withrhe corresponding energy.

This method does not appear to be capable of extension in any simple marinerro the general problem of systems that make transitions from one state to othersof the same energy. Arso there is at present no very direct and certain wayof interpreting a periodic sorution of a wave equation to appry to a,non-periodicphysical phenomenon such as a colrision. (The more definite method thats-ill now be given shows that Born's assumption is not quite right, it beingnecessary to multiply the square of the amplitude by u

""rtuio factor.)A,' alternative method of solving a collision probrem is to find a non-peri,od,,icsolution of the wave equation which consists initialry simply of prane wavesmoving over the whole of space in the necessary direction with the necessarytrequency to represent the incident electron. fn course of time waves movingia other directions must appear in order that the wave equation may remainsatisfied. The probability of the electron being scattered ii any direction withany energy will then be determinecr by the rate of growth of the correspondingharmonic component of these wa",,e'. The way the mathematics is to beinterpreted is by this method quite definite, being the same as that of thebeginning of 92.

we shall apply this method to the generar problem of a system which makestransitions from one state to others of the same energy under the action of aperturbation. Let Ho be the rramiltonian of the unperturbed system and.Y the perturbing energy, which must not involve the Jime explici'y. If wetake the case of a continuous range of stationary states, specified by the firstintegrals, q'k say' of the unperturbed motion, then, following the method of$ 2, we obtain

Emission and, Absorptiun of Eqd,iation. 257

(2rli,h a (u') : fV @,n,) i[,x,, . a (a!,),

corresponding to equation (4). The probability of lhe system being in a statefor which each ae lies between oca, and. aa'{ d,ue, at any time isla (u,)12 d,ar, . fur, ...wtren a (a') is p'roperly normalised and satisfi.es the proper initial conditions.rf initially the system is in the state a0, we must take the iuitiar value of a (a,)to be of the form o0 ' 8 (oc'- a0). 'we shalr keep oo arbitrary, as it would beinconvenientto normalise a(u') inthe present case. x'or a first approximation

l 6

258 P. A. lL Dilac.

wemaysubstitutefora(oc,)intheright-hand-sideof(21)itsinit ialvalue'Thisgives

i'ha(a-') - aoY(ct'ao) : xoa(u'uo\'{[\Y(o')-w(ao)Jtift'where o (oc'oco) is * ooo*tuot anil W (ot') is the energy of the state a"

Hence

i,ha(a"'): a08 (o! - no) | aoa(x'a.o)#ffi ' PzjX'or values of the a1' such that W (or') difiers appreciably from W(c(0)' o(a')is a periodic function of the tirne whose amplitucle is small when-the

perburbing

enelgyVissmall,sothattheeigenfunct,ionscorrespondingtothesestationarystates are not excited' to any appreciable extent' On the other hancl'

for values'

of the xp' such that W (a')-: W (oco) ' antl d'l * v"Lo for some k' a (*') increasesuni{ormlywithrespecttothetime,sothattheprobahilityofthesystembeinginthestateu'atatytimeincreasesproportionallywiththesquareofthetime'Physically,theprobabilityofthe'y*tt*beinginastatewithexactlythesameproper energy as the initial'prcper energy W (oco) is of no importance' beingin f in i tes ima l , .Weare in te res ted"on ly in the in tegra lo f theprobab i l i t ythrough a small range of proper energy values about the initiai proper

energy'

which, as we shall find, increases linearly with the time, in agreement with the

ordinary ProbabilitY laws'We transform from the variables 0(1, 0(2 " ' *u r'o a set of variables that are

arbitrary independent fu:rctions of the oc's such that one of them is the proper

energy W, say, the variables W, Yr, \2, "' n(u-.' The probability at any timeof the system lying inastationarystate for whioh each 1r lies bet'ween^r&'andl

yn' * d,yr,'is now (apart from the normalising factor) equal to

d^(r' . d\r' ... d^(u-,'i I otot') f ffiffi dw'' (23)Torat imethat is largeoomparedwiththeperiodsofthesystemweshal l f inclthat practically the i,not" of the integral in (23) is contributed by values ofW' verY close to W-o : W (oco)' Put

a (u') :o (W', y') and ? (ar' , uz' . . .. n,') 13 (W', T" "' Tu-r') : J (W" "y')'Then for the integral in (23) we findl, with the help of (22) (provided'Tk' *\kafor some k)I ir(w', T')lt J (w', v') dw'

: I a0 l, J i, {w', v' ; w0, yo) l' J (w', r', [eo t*'-*"r'ro -j]!:o !:':t'4:11 dw'

- 2 I *01' [ 1, 1W',v' ; Wo, yo) l' J (W',T') [1 - cos (W' -Wo) t lh] | (W' -W0)2' dW'J

: 2 I ao V, ft . IL, (Wo I hn lt,y' ;Wo, ^ro) p J (W0 f hr lt,^O (L - cos r) lnz' d'n'

1 7

Emission and, Absorption of Rad,icttiotr,. 259if one makes the substitution (w'-w0)t1h:*. n'or large values of f thisreduces to

2lao lz tlh .lu(W0, y' ; 1V0, T0) j, J (Wo, y,) f' (t-" os n) lrz . d,n: zrc I aolz tlh. f r,#, r, i wo, yo) l, J(W, y,).

The probability per unit time of a transition to a state for which each y7, liesbetween ya' and yr,' * dT*, is thus (apart from the normalising factor)

Znl ao lLlh. l, (Wo, y' i W0, yo) l, J (Wo, T,) ilTr, .i l^(r, ... d,y*_t,, (24)which is proportional to the square of the matrix element associated with thattransition of the perturbing energy.

To apply this result to a simpre collision problem, we take the cr,s to be theeomponents of momentuu p*, pa, 7t" of the colliding electron and the y,s tobe 0 and /, the angles which determine its direction of motion. If, taking therelativity change of mass with velocity into account, we let p denote theresultant momentum, equal to (p|*pf*p*)t,and E the energy, equal to(mzc4a-Pcz1t, of the electron, mheingits rest-mass, we find for the Jacobian

J : 0 : ( P , , P u , ' P " \ : E P - ' - - ^"

- TE;o;;) : ;, "o u'Thus the J (Wo, y') of the expression (24) has the value

J (Wo, T') : E'P' sin 0'/c2, (25)where E' and P' refer to that value for the energy of the scattered electron whichnrakes the total energy equartheinitial energywo (i,.e.,to that varue resuiredby the conservation of energy).

We must now interpret the initial value of a(x,), namely, a0 g(a,_ao),which we did not normalise. According to $ 2 the wave function in terms of thevariables ae is 6 (a'): a (a') s-iw'tln, so that its initial value is

o0 8(oc'- d.o)e-iw'tth:ao E(p,, -p,o)g(pr, - puo) g(p,, _,p:)e-i\\ ' tth.If we use the transformation function*

(r' I P' ) : (2Th)-3l2si>*"P"' r' lh,and the transrormati;;;

l{*,rn) rr(p,) d,p' d,pu,d,sti,we obtain for bhe initial wave fu4ction in the co-ord.inates n, ao z the value

ao (%t6; z t z "i>

q,p,or' I h e- i'w't I h .'t The symbol c is used for brevity to detrote n, y, z,

l 8

26A P. A. M. Dirac.

This corresponds to an initial distribution of I o0 12 (2tch)*3 electrons per unitvolume. since their velocity is Pocz/Eo, the number per unit time striking a

unit surface at right-angles to their direction of motion is lo0l2P0c2/(2nh)sBo'Dividing this into the expression (24) we obtain, with the help of (25)'

4n2(%ch)2E$t, 0t' ; po)itSrio 0' d,0' d+'. (26)- ' ' c n

t

This is the efiective area that must be hit by an electron in order that it shall

be scattered. in the solid. angle sin 0' d0'd{'withthe energy E'' This resultdifiers by the factot (Zrch)zlumv', . P',/Po from Born's.* The necessity for thelactor P'/Po in (26) could have been predicted" from the principle of detaileclbalancing, as the factor I o: (p' ; Po)l 2 is symmetrical between the clirect andTeverse processes.f

$ 6. Appli,cation to Light-Qrtanta'We shall now apply the theory of $ 4 to the case when the syst'ems of the

assernbly are light-quanta, the theory being applicable to this case since light-

quanta obey the Einstein-Bose statistics and have no mutual interaction. A

light,quantum is in a stationary state when it is moving with constant momen-

tum in a straight line. Thus a stationary state r is flxed by the three com-

ponents of momentum of the light-quantum anil a variable that specifies its

Jate of polarisation. We shali work on the ass*mption that there are a finite

number of these stationary states, lying very close to one anothel' as it would.

be inconvenient to use continuous ranges. The interaction of the light-quanta

with an atomic system will be described by a Hamiltonian of the form (20),in rrhich H, (J) is the Hamiltonian for the atomic system alone, and thecoefficients ,t)rs ate for the present unlic0.own. we shall show that this formlor the Hamiltonian, with the o", arbitrary, leads to Einstein's laws for the

emission and absorption of radiation.The light-quantum has the peculiarity that it apparently ceases to exist

when it is in one qf its stationary states, namely, the zero state, in which its

moment'um, and therefore also its energy, ate zetl, When a light-quantum

is absorbed it can be consiclered to jump into this zero state, and when one isemitted it can be considered to jump from the zero state to one in which it is

* In a more recent paper ('Nachr. Gesell. d. Wiss',' Gottingen, p' 146 (1926)) Born hasobtained a, result in agreement with that of the present paper tor non-relativit'y mechanics,by using an intorpretation of the analysis baseil. on the conservation theorems. I amindebtecl to Prof. N. Bohr for soeing an atlvance copy of this work'

f See Klein and Rosseland, '2. f. Physik,' vol' 4, p' 46, equation (4) (1921)'

t 9

Emission antl Absorpttion of Rad,iation. 261::rsically in evidence, so that it appears to have been createcl. since trrere is- -' limil to the number of right-quanta that may be createcl in this way, rve must!'.pose that there are an infinite number of light-quanta in the zero srate, so::at the Nu of the Hamiltonian (20) is infinite. we must now have 0u, the,,-:riable canonically conjugate to No, a constant, since

Oo : afTaNo : Wo f terms involving No-l o, (No 1 1;-;'"d wo is zero. rn order that the Hamiltonian (20) may remain finite it is-.eessary for the coeffi.cients or0, oor to be infinitely smali. We shall suppose:iat they are infinitely small in such a way as to make oreNo+ and zr6rNoi:nite, in ord.er that the transition probability coeffcients may be finite. Th's-e put

o"o (No * l)!' e-ieotn - oD ustNr!6i0oih : ur*,

rhere u, and 'i.r'+ are finite and. conjugate imaginaries. we may consicler the'' and u,* to be functions only of the J's and zo's of the atomic system, since:heir factors (Nn

* l)t s-tea'n and Nodrteo/a are practically constants, the rate-'f change of No being very small compared with Nr. ihe Hamiltonian (20)now becomes

n' : Hr(J) + >,W,N, lZ,asfu,Njglo * o,o(N, r tf "_te,1n1

*X"*oX"*o?r,,N,+(N" + I - 8,,)* fto,.-e;tr,. e7)The probability of a transition in which a light-quantum in the state r is

absorbed is proportional to the square of the modulus of that matrix element ofthe Hamiltonian which refers to this transition. This matrix eiemenr, mustcome from the term arN,ls.an in the Hamiltonian, and must therefore beproportional to N"'* where N,' is the number of light-quanta in state r beforethe process. The probability of the absorptioo p"or.*, is thus proportionalto Nn'' rn the same way the probability oi a light-quantum in state r beingemitted is proportional to (N,, f l), and the probability of a light_quantum instate r being scatteredinto state s is proportional to N,, (ti,, it). Radiativeprocesses of the more generar type considered by Einstein and Ehrenfest,f inwhich more than one light-quantum take part simultaneously, are not ailowedon the present theory.

To establish a convrsslien between the number of light-quanta per stationarystate and the intensity of the radiation, we consider an enclosure of finitevolume, A say, containing the radiation. The number of stationary statesfor light-quanta of a

_given type of polarisation whose frequency lies in the

| 'Z. f. Physik,' vol. lg, p. BOf (f923).

20

262 P. A. M. Dirac.

rarge v?. to v, { dv, and whose d.irection of motion lies in the solid angle d'a,about the direction of motion for state r will now be Lv,2d,v,cla,l&. The energyof the iight-quanta in these stationary states is thus Nr' .Znhv, . Av,2d'vrd'arlC'This must equal Ac-lr durt^,, where I, is the intensity per unit frequency

ranse of the radia.tion about the state r. Hence

I, : N,' (Zrh)v,t f c2, (28)so that N,'is propoqtional to I, and (N,' { 1) is proportional to I, f (2nh)v,3lcz'-We

thus obtain that the probability of an absorption process is proportional to

I,, the incident intensity per unit frequency range, and that of an emissionprocess is proportional to I,l (Znh)v,s/cz, which are just Einstein's laws.*In the same way the probability of a process in which a light-quantum is scattered

from a state r to a state s is proportional to r, [I" | (2nh)v,3 lcl, which is Pauli'slaw for the scattering of radiation by an electron.t

57. The Probabi,l,i,ty Coffici,ents for Em'ission and' Absorption'

We shall now consider the interaction of an atom and. radiation from the wave

point of view. we resolve the radiation into its xtourier components,. and

suppose that their number is very large but finite. Let each component be

labelletl by a suffix r, andl supposo there ale o,r eomponents associated with the

radiation of a definite type of polarisation per unit solid' angle per unit fre-

quency range about the component r. Each component r can be desoribed by

a vector potential k, chosen so as to make the scalar potential zero. The

perturbation term to be adcled to the Hamiltonian will now be, according to

the classical theory with neglect of relativity mechanics, c-L\, rc, XD where X,

,is the component of the total polarisation of the atom in the direction of rn,

which is the direction of the electric vector of the component r''we

can, as explained in $ 1, suppose the field to be described bythe canonicalvariables Nn, 0n, of which N, is the number of quanta of energy of the oom-

ponent r, and 0, is its canonically conjugate phase, equal to 2rchv" times the0, of $1.

'We shall nowhave Kr:&raos \rlh, where o, is the amplitudeof

.rn, rvhich can be oonnected with N, as follows:-The flow of energy per unit

area per unit time for the component 7 is uftc-r ar2v,2. Hence the intensity

* The ra,tio of stimulated to spontaneous emission in the present theory is just twice itsvalue in Einstein's. This is because in the present theory either polarised. component ofthe incident radiation can stimulate only radiation polarised in the same way, while inEinstein's the two polarised components are treated together' This remark applies also.to the scattering Process.

f Pauli, '.2. f. ?hysik,' vol' 18, p.272 (7923).

2 l

Eqn';ssion cud, Absorpti,on of Rad,ia,tion. 263

per unit frequency range of the radiation in the neighbourhood of the com-ponent r is I":72rc-\d,2v,26,. Comparing this with equation (28), we obtaina, : 2 (hv, I co")l Nr}, and hence

u,,:2(hv,lco")l N,+ cos O"/fr.The Hamiltonian for the whole system of atom plus radiation would now be,

according to the classical theory,F : Hr, (J) + >" lZihvSN, f 2c-rX" (hv,lco,)l X"N,! cos 0"/1, (2g)

where He (J) is the Hamiltonian for the atom alone. On the quantum theorywe must make the variables N" and 0" canonical q-numbers like the variablesJp, w,that describe the atom. We must now replace th.e l{"} cos O,.fh in (2g)by the real q-nurnber

! {N,.+ B:e,tt, + e-i?rthN i}:1r{y7;i otortt + (N,. + 1),, s-i,erfty.so that the Hamiltonian (29) becomes

F : Hr, (J) + >, (2nhv,) N" a /a+ 6- ; ), ( v,/o,)l X, {IrI,} rta'lr' f (N, f 1)r e

This is of the form (27), wiihar: ur* - ht c-; (vr/o,.)l X"

-ierlh\) '(30)

(31)and nrr:() (r, s I 0).The wave point of view is thus consistent with the light-quantum point of viewand gives values for the unknown interaction coeffi.cient o," in the light-quantum theory. These values are not such as would. enable one to expressthe interaction energy as an algebraic function of canonical variables. Sincethe wave theory gives ,u"r: 0 for r, s f 0, it would seem to show that there areno direct scattering processes, but this may be due to an incompleteness inthe present wave theory.

lVe shall now show that the Hamiltonian (30) leads to the correct expressionsfor Einstein's A's and B's. we mustfirst modifyslightly the analysis of $bso as to apply to the case when the system has a large number of discrete station-ary states instead of a continuous range. rnstead of equation (2I) we shallnow have

i,h a (a' ) : 2^', Y (a' a") a (a").If the system is initially in the state oc', we must take the initial value of a (a')to be 8.,'0, which is now correctly normalised. This gives for a first approxi-mation

i 'h a (a.'): Y (oc'0c0,) : u(a'al) "i' l 'er

(a')-N (ao))tlh,which leads to

,d[W(c)-W(ao)]l lh _ I' ih a(u'): $.,on { u(u'*o)r,fW(oc')-W (no)llh'

22

264 P. A. M. Dirac.

corresponding ro Q2)- If, as before, we transform to the variables w, Tr,Tz ...\u-r, we obtain lwhen 1' # Yo)

o (IM'Y') : o (W', y' ; W0, To) [1-ei(w'-sr94h]/(W'-W0)'The probability of the'system being in a state for which eaoh 1r equals 11'is Er,v, lo (W' y')lt. If the stationary states lie close together and if the time Iis not too great, we can replace this sum by the integral ( AWr-t | | o (W'T') l' dW' ,where AW is the separation between the energy levels. Evaluating this integralas before, we obtain for the probability per unit time of a transition to a statefor which each y* : yo'

2nlhN[. lu (Wo, y'; Wo, yo) 12. (321In applying this result'we oan take the 1's to be any set of variables that areindependent of the total proper energy'v[ and that together with w definea stationary state.

we now return to the problem d.efined by the llamiltonian (30) ancl consideran absorption process in which the atom jumps from the state J0 to the stateJ' with the absorption of a iight-quantum fromstate r. we take the variablesy, to be the variables J', of the atom together with variables that define thedirection of motion and state of polarisation of the absorbed quantum, butnot its energy. The matrix element o (W0, Y' ; W0, Yn) is now

hu2 c- B t 2 $, f o,)r I 2 Xi (JoJ' )N"0,where X,1J0J') is the ordinary (JoJ',) matrix element of X". Hence from (32) theprobability per unit time of the absorption process is

2 n h v r , i , r rftfrW ff,\*,(JoJ')lzN,o'

To obtain the probability for the process when the light-quantum comes foomany direction in a solid angle do, we must multiply this expression by the numberof possible directions for the light-quantum in the solid angle d

23

Em,ission and, Absorption of Rad'iation, 265

The present theory, since it gives a proper account of spontaneous emission;must presumably give the efiect of radiation reaction on the emitting system,and enable one to calculate the natural breadths of spectral lines, if one canovercome the mathematical difficulties involved in the general solution of thewave problem corresponding to the Hamiltonian (30). ilso the theory enablesone to understand how it comes about that there is no violation of the law of theconservation of energy when, say, a photo-electron is emitted from anatomunder the action of extremely weak incident radiation. The energy of inter-action of the atom and. the rad.iation is a q-number that does not commute withthe frst integrals of the motion of the atorn alone or with the intensity of theradiation. Thus one cannot specify this energy by a c-number at the sametime that one specifi.es the stationary state of the atom and the intensity of theradiation by c-numbers. In particular, one cannot say that the interactionenergy tends to zero as the intensity o{ the incident radiation tends to zero.There is thus always an unspecifiable amount of interaction energy whichcan supply the energy for the photo-electron.

r would like to express my thanks to Prof. Niels Bohr for his interest in thiswork and for much friendly discussion about it.

Summ,tr,ry.The problem is treated of an assembly of similar systems satisfying the

Einstein-Bose statistical mechanics, which interact with another difierentsystem, a Hamiltonian {unction being obtained. to describe the motion. ,Thetheory is applied to the interaction of an assembly of light-quanta with anordinary atorn, and. it is shown that it gives Einstein's laws for the emissionand absorption of radiation.

The interaction of an atom with electromagnetic waves is then considered,and it is shown that if one takes the energies ancl phases of the waves to beq-numbers satisfying the proper quantum conditions instead of c-numbers,the Hamiltonian function takes the same form as in the light-quantum treat-ment. The theory leads to the correct expressions for Einstein's A's and B,s.

24 Poper 2

RENDICONTI DELLA R . ACCADEMIA NAZIONALE DEI L INCEIClasse di Scienze fisiche. matematiche e naturali.

Estratto dal vol. XlI, serie 6', zo sem., fasc. 9. - Roma, novembre rgJo-vrrr.

Fisica. - Sopra I'elettrodinami.ca quantistica. Nota II (') diE. Frnur, presentata dal Socio O. M. ConerNo.

In una Nota pubblicata recentemente in questi < Rendiconti 1 (z) [escritto in forma quantistica le equazioni dell'elettrodinamica; ciod le equa-zioni del sistema costituito dal campo elettronragnetico e da un numeroqualunque di cariche elettriche puntiformi. Le equazioni scritte allora siiiferivano al caso non relativistico; presupponevano ciob che Ia velocitldelle cariche uon fossero molto elevate. Esse possono tuttavia senza alcunadifficolti essere scritte in forma relativistica, basandosi sopra la teoria diDirac dell'elettrone rotante. E noto che recentemente anche W. Heisenberge W. Pauli (r) lx6le triittato il problema dell'elettrodinamica quantistica.Siccome pero i metodi seguiti da questi autori sono essenzialmente diversidai miei, credo non inutile pubblicare anche i miei risultati.

La forma definitiva in cui ver-ranno espressi i risultati di questo lavorob particohrmente semplice. Troveremo infatti che la Hamiltoniana che, nelsenso del principio di corrispondenza, rappresenta la naturale traduzionequantistica dell'elettrodinamica classica, si ottiene senrplicemente aggiun-gendo alla Hamiltoniana clella teoria dell ' irradiazione di Dirac un termineche rappresenta l'energia elettrostatica del sistema di cariche elettriche; permodo che, nella presente forma, l'elettrodinamica quantistica viene a nonessere in alcun modo piri complicata della teoria di Dirac deli'irradiazione.Questa semplificazione si pub raggiungere come vedremo mediante unaopportuna espressione della condizione

( ' ) +t#+di 'u-oche lega tra di loro i potenziali scalare e vettore e che, anche nella teoriadi Heisenberg e Pauli costituisce uno degli elementi pir\ caratteristici del-I'elettrodinamica quantistica.

Nella Nota I abbiarno trovata I 'espressione (zI) che rappresenta I 'Ha-miltoniana del nostro sistema, Se, invece della meccanica classica, vogliamorappresentare il moto dei punti per mezzo della Hamiltoniana di Dirac,

(r) Pervenuta all'Accademia il z9 settembre rgjo,(21 E, Fnnut, ,9,88r, 1929. Citata nel seguito con I.13) W. HcrsrNnnnc und W. Peurt, , 56, r, ry29; tg, r5o, r93o,

25

possi4mo verilicare facilmentel' Hamiltoniana seguente :

- 4 3 2 -

che, al posto della (zr) I dobbiamo usare

(") H - - ,2t,X p,, -)8,m, r. +1t,t lp) e, cos t,; +

+ 6: - P?) + zrr"u? (*?, + w?, + x?- a)]Le notazioni sono quelle della Nota I; 8, e y, rappresentano un q-sca-

lare e un { - vettore tali che 8;, f;r ,T;f ,Ti< sono i quattro operatori, rap-presentabili con matrici del quarto ordine, che intervengono nella Hamil-toniana relativa all ' i .esimo punto materiale I natr-rrahneute le y e la I rel;rt ivea uno dei punti sono permutabili con le y e Ia 8 relative a un altro deiPunu.

Osservianro in particolare che dall'Hamiltoniana. ALI

\Q,: f f i : - t ' i P,:l aH, a r : a d * : 4 , i G ) r :

da cuid _,-

cos Ii; - 2rc\ts\i \ a, sin lr; .

Da questa equazione e dalle (3) risulta subito che anche dalla nuovaHamiltoniana (z) deriva l 'equazione (lB)I; e quindi clre se l 'espressione(t9) I si annulla insieme alla sua derivata prima all ' istante zero, essa restasernpre nulla in virtu delle equazioni differenziali. E rest:r quindi verificatala condiz ione ( r ) , equivalente a l la ( r9) I . Per tnezzo del le (3) , Ia ( r9) Isi pu6 scrivere(+) 2ft\ts ls - P, : Oe la sua deriv:rta, a meno di un fattore costante

(t) 6s - 27rvs a, * # l*1',cos rs; : o.

. ?l+(

26- 433 -

In una interpretazione classica potfemmo dunque dire che l'elettrodi-namica ordinaria si ottiene integrando le eqr-razioni canoniche dedotte dalla(z) e imponendo (4) e (5) come condizioni iniziali; ci6 basta, poichd sid detto che se (4) e (5) sono verificate all'istaute zero, esse lo sotro ancheautomaticamente a un istante qualsiasi.

Per tradr.rrre tutto questo nel linguaggio della meccanica quantistica,osserviamo che, affinchd le due grand,ezze (+)

"

(i) possalro avere simul-taneanrente il valore zero, E necessario che esse siano commutabili, poichdaltrirrenti il fatto che una clelle due grandezze ha un valore determinatorenderebbe di necessiti indeterrninato il valore dell'altra. Ora si verifica facil-mente, in b;rse alle ordinarie regoie di conrmutazione, che i primi membridi (a) e (5) sono effettivamente comtnutabili; si puo quindi anche quanti-sticamente attribuire ad essi allo stesso istante il valore determinato zero.

Al procedimento classico di integrazione delle equazioni canoniche convalori arbitrari delle costanti di integrazione, corrisponde, nella nreccanicaondulatoria f integrazione dell'equazione di Schroedinger corrispondente allaHamiltoniana (z), scegliendo arbitrariamente la funzione che raPpresentalo scalare di campo(6) * : Q(t , h tQ; tw, , t ' t t ) , ' , X, , , Q' )

(ot rappresenta simbolicamente la coordinata interna < spincoordinate > del-l ' i .esimo corpuscolo) per i l valore t: o del terl. lPo. Se vogliamo invecesoddisfare Ie condizioni (+)

"

(l) nort possiamo pir\ lasciare arbitraria questafunzione; resta invece determinato il modo secondo cui essa dipende dallevari:rbili ls e Qs . Siccome infatti 6, , conittgata di 2g' , deve avere, secondol . ( l ) i l va lore

6 s . ^ c r / B: 27r\s Q - rr*a |/ "# ? e; cos fs;

risulta che tf deve dipendere da p nel fattore

0)Dalla (4) segue

fattore

(8)

il quale b del restozione corrispondente

,"# *, ('*, a' - . *\ EV ",'o' r,;).

in modo simile che Q'

z x i

2 h 2 f r l tu t i / t

gii contenuto nel fattorealle condizioni (a) e (5)

deve intervenire soltanto nel

(Z). In conclusione la solu-deve avere la forma:

(s) * : lU ," I j ' ' ( "n,e, -

.* i f f i4", ' """ , , ) ] e(t ,x; , ,oi ,w,, ,w,,) .

-43+-Dobbiamo 'ra dimostrare che effettivamente si pro soddisfare I'equa-

zione di Schroedinger per tnezzo della posizione (9). L'eqr.razione cli Schroe-dinger dedotta dall 'Hamiltoniarra (z) d

-*#:"qdove H d naturalmente interpretato corne un operatore. sostituendo nella(to) al posto di { l 'espressione (9) si trova, con calcoli non diff ici l i laseguente equazione a cui deve soddisfare g:

( ro)

dove R rappresenta il seguente

+ .;)

oPeratore

{)e'clf?r,X (A,,2u,, I A,,w,,)sinr,, *- ) 8 i m i c ": ) R : - t1T ;Xp tf r

+ ) l : ( c o '' a l z ' "

-*#: on( r t )

+ 2n2v2 @:, + *:)]+#?n(?,' .o, r,,) .

:1*r>+ (> e; cos r,;) : lDU .T E s r . i v ; \ - / z f , r ; i

A prescindere dall 'ult imo termine, R coincide co' I 'Hamiltoniana dellateoria dell ' irradiazione cli Dirac, in cui si trascura