Volume 68A, number5,6 PHYSICSLETTERS 30 October1978

DIRAC’S CLASSICAL THEORY OF ELECTRONS IN THE UNITARY

FIELD THEORY OF EINSTEINWITH TELE-PARALLELISM

H.-J.TREDERZentralinstitutjllr Astrophysik,AkademiederWissenschaftenderDDR,Potsdam-Babelsberg,DDR

and

W. YOURGRAUFoundationsofPhysics,UniversityofDenver, Denver, CO 80210, USA

Received17 August 1978

Onecanunify theEinstein—Maxwellfield equationswithin theframeworkof Einstein’sriemanniangeometrywith tele-parallelismto obtain4 X 4 vectorfield equationsby meansofthe4 X 4 tetradshAt.Thesenewunitary field equationscor-respond— in themacroscopiclimit — to thegeneral-relativisticequationswith Dirac’s classicalelectronicfield. This crossingto thelimit signifies that Eddington’snumberfor electrons,(Gm

2/e2)~ i041, consideredto bepracticallyzero.

Accordingto Einstein’sactualintention,aunifica- £ — hB ‘RF FAmn\ ( A = 0, 1, 2, 3,tion of gravitationandelectromagnetismrequiresmore — Amn m,n 0, 1, 2, 3than geometrizationof theEinstein—Maxwellfield (1)theory.The generalizations(andmodifications)of theEinstein—Maxwellequationsconnectedwith sucha where I hB

1I (—g)1/2 andais a numericalconstant.

unificationshouldfurnish— following Einstein The referencetetradshA1,determiningthetele-paral-

non-classicalmatterfields, andcorrespondinglysources lelism of the riemannianspace—timeV4, are linkedof gravitationandelectromagnetismascorrectionsto with theriemannianmetricgf~by meansof thepseudo.thegeneral-relativisticfield equations. orthonormalizationcondition

Only recently,oneof theauthors[6] proposedan — hA ~ — I kinterpretationof Einstein’sunitary“field theorywith ~fk — i k’

7AB’ 7?AB— hA hB ~ (2)tele-parallelizedRiemannspaces”[2] which indeed wherenAB is the Minkowski tensorof theLorentzcorrespondsto Einstein’sconception.The represen- tangent-spaceat eachpointV

4, andg~denotesthetationof gravitationalandelectromagneticfields by gravitationalpotentialof GRT.TheFAik are Einstein’sthetetradsh”k, i.e. referencesystemsdefiningtele- anholonomicobjectsof thetele-paralleltransportparallelism,automaticallyleadsto theclassicalelec- [2—41,viz.,trodynamicswith additionalscalarelectronfields —

suggestedby Dirac [11. FAik alhAk—8kk4I - (3)

Within the frameworkof this theory,we consider ForvanishingFAik, thespace—timeV4 is pseudo-

a Lagrangedensity£ which isthe sumof the density euclidean,that is,~ is given by(—g)

112Rof GRT andthe generalizedLarmordensity(_g)lI2F~f~FAikknown from the Maxwell theory~1: * .2isaspecialcaseof thegeneralLagrangedensityformed

J�~hBll(aR+bFAjkFAik+dhAl;lhA&k),~ Seeright column, wherea, b, andd arenumericalconstants,

4i 5

Volume68A, number5, 6 PHYSICSLETTERS 30 October1978

glk = a~P’~ak~PBnAB . (4) specify to this endthereferencesystemsuchthat theelectromagneticvectorpotentialA. coincideswith the

Thefield equationsstemmingfrom the lagrangian time-like tetradh0

1via h0

1= (e/mc~)A1.In this way,(1), written in the matrix (4 X 4) form of Maxwell’sequations,readasfollows [6]: thevectorpotentialby necessitysatisfiesthe gauge

conditionof Dirac’s classicaltheoryof electrons[1],FA1k;k = — ~ah~k [R~ — ~

6kR viz., A1A1= (m

2c4)/e2.TheMaxwell field tensorbe-comesthenan anholonomicobject [6]:

+(2/a)(~~51kF FBmn FBImFBkm)]. (5)BmnFolk = (e/mc2)Flk = (e/mc2)(alAk — akAI) . (10)

FormulatedasEinstein’sequations,eqs.(5) maybestatedas Invoking this ansatz,the Lagrangedensity£ now

furnishes,fora -~ oo, thefollowing systemof fieldRik—~g

1~R= — (2/a)~g F FAmn— FA~FAkm) equations:1k Amn

— (Fjj/; 1~k+ FAk’; 1!i’

1

1). (6) FIk;k = _2Ak(Ri~~— ~gikR + (8irG/c4)Tth), (11)

Theseequationshaveyet to be completedby the 6 (R1 — ~ökR + (8~G/c4)T1k)hk~= 0,symmetryconditions:

FAI1lhAk= FAk11hA1. (7) where17 = 1, 2, 3 and TJ~is Maxwell’s energy—momen-

tumtensor,that is,In eachcase,eqs.(5) or (6) and(7), onehas 16 fieldequationsfor the4 X 4 componentshA1.Thismeans T/c = ~ k~’ ~mn FimF’~m. (12)I mnthat the referencetetradsh’

41(x

1)are specifiedup tothe rigid (global) Lorentzrotationsin thetangent The 13 equations(11) togetherwith Dirac’s gaugespacesdescribedby thetransformations conditionfor A

1 representa systemof 14 equations.Accordingto eqs.(11), the combinedEinstein—

= ~ wABo,~B= &A~ (8) Maxwell equationsof GRT,with alcd’B= 0. The local Lorentzcovarianceis thusviolated,in accordancewith the genuineconception R,k= (8lrG/c

4)TIk, Flk;k = 0, (13)of tetradtheory [5]. Eqs.(5) statethat thevariationof the lagrangian(1) withrespectto thehA~vanishes: aresatisfiedin theentireV

4, if andonly if theyholdon anarbitraryspace-likehypersurfacex

0 = const.(2 IhB

1I)_162/~hA= 0. Eqs. (11) in generaldescribea combinedEinstein—

Fora —~~ theseequationsbecomeequivalentto Maxwell field with Dirac’s classicalelectroniccurrentEinstein’svacuumequations,viz., XA

1 asthe field source:eqs.(11) togetherwith Dirac’s

hA~’(R1k— ~ 6kg) = 0. (9) gaugecondition*2 are equivalentto thegeneral-relativistic field equations

(Fora -÷ ~o, lagrangian(1) is indeedtantamountto(—g)1/2R.) Rjk — ~ = —(8lrG/c4)(TIk+ XAIAk) (14)

Wewish, however— in thesenseof Einstein’s Fik;k =

intention— to achievethat thetele-parallelizedRiemannspace—timerepresentsaunified geometric wherethe densityof the Dirac currentXA1 is giventheoryof gravitationandelectromagnetism.Forthis by [6]purpose,we introducethe factoraasthevery great, = (e2/8irGm2)R. (15)

butfinite, Eddingtonnumbera= e2/4i~Gin2 1041,wheree is thecharge,andm the mass,of electrons.Accordingly,we interpretthehA

1 asthe commonpotentialof gravitationalandelectromagneticfields. *2 DiracaddsthegaugeconditionAlA1 = m

2c4/e2by means

Abandoningalso the globalLorentzinvariance,we of themultiplier ?~to theLarmordensity.

416

Volume68A, number5, 6 PHYSICSLETTERS 30 October1978

References [5] H.-J.Treder,Gravitationstheorieund Aquivalenzprinzip(Akademie-Verlag,Berlin, 1971;Atomizdat,Moscow,1973).

[1] P.A.M. Dirac, Proc. Roy. Soc. (London)A209 (1951)291.[2] A. Einstein,Sitzungsber.Akad, Wiss. (Berlin) (1928) [6] H.-J. Treder,Ann. PhysikLeipzig 35 (1978)to bepublished.

217,226;(1929) 2, 156;(1930) 18. [71R. WeitzenbOck,Sitzungsber.Akad. Wiss. (Berlin) (1928)460.

[3] A. Einstein,Math. Ann. 102 (1930)685.

141 T. Levi-Civita, Sitzungsber.Akad. Wiss. (Berlin) (1929)No. IX.

417