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8/12/2019 Story Dirac

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GENERAL I ARTICLEThe Discovery of Dirac Equation and its Impact

on Present-day Physics

Rajasekaran is a theoretical physicist at theInstitute of Mathematical

Sciences Chennai. Hisfield of research is

quantum field theory andhigh energy physics.

Currently he is working inthe area of neutrino

physics.

Reproduced with permissionfrom Dirac nd Feynman: Pio-neers in Quantum MechanicsEdited by Ranabir Duff and AsimK Ray Wiley Eastern limited1993. New Age India Ltd.

KeywordsDirac equation Klein-Gordonequation spinors hole theorystandard model.

G ajasekaranThe major events in the discovery of the Diracequation and its interpretation are traced. Thesubsequent role it has played in the developmentof quantum field theory culminating in the Standard Model of present-day high energy physicsis discussed.The relativistic wave equation of the electron ranks anlOng the highest achievements of 20th century science.Dirac's two papers on the subject published in 1928 arethe following:-P.A.M. Dirac, Proc Roy Soc Al17 610, 1928P.A.M. Dirac, Proc Roy Soc Al18, 351, 1928Dirac himself is supposed to have reuw.rked that. tIl 1/2 . Enormous amount of work has beendone on higher spins, following the footsteps of Dirac,but the subject is plagued with many inconsistencies.The following is a brief Progress Report.SpinThere is no problem at least at the level of the free fieldequation, after the reinterpretation by Pauli and Weisskopf. Interactions treated perturbatively in renormalizable quantum field theory do not cause any problem.But there are hints of difficulties in a complete (nonperturbative) theory.

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GENERAL ARTICLE

ory of strong and electroweak forces. This is now calledthe Standard Model of High Energy Physics and is thebasis of all that is known in the physical world exceptgravity.Standard Model of 1988L leo: co: 1Wa W a 1B B-4 JLV JLV - 4 JLV JLV - 4 JLV JLV

+ i L ifL,JLn

.. \ : : T a a .91 ) nOJL + t932 GJL + t92 2WJL + t(fBJL qL+ i ~ ui {yl (81 + g3 > 2 C + g1 B I ' u;+ i ~ . J : R Y I (81 + g3 >.; C - ~ 9 I B I ) rr+ . -n L (:l . T awa i B ) Int L UJL + t92 2 L - 2 91 L L+ i L en,JL(OJL - i91 B JL)eR

n1(81 + g2 a w; + g l B I ) 4>12

J-t2+ - .-\ +)2 - L r ~ n q C U R + ~ n q d Rr:nnlI:eR + h.c.)+ gravity.

m n

The explicitly written part of the above Lagrangian describes the SU(3) x SU(2) x U l) theory of strong andelectroweak forces. The equations of motion followingfrom this Lagrangian are the present-day successors ofthe Dirac and Maxwell equations of the 1928 StandardModel. This is a measure of the progress that hasbeen made in 60 years. We shall not attempt to explainthe various terms in the Lagrangian, for that will requireanother article. See for instance Rajasekaran (1989) orCheng and Li (1984). It is more important to draw at-tention to the term gravity (added at the end of the

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GENER L I RTICLEabove equation) which sticks out like a sore thumb ;we do not yet know how to add gravity to the rest ofphysics in a consistent manner.16. Two Repetitions of the Dirac TrickIn attempting to go beyond the present-day standardmodel, many brilliant discoveries have been made, someof which are directly inspired by Dirac's work. We shallmention two such discoveries.(i) s we already saw, Dirac discovered the anticommuting'Y matrices by taking the square root of p 2 :

yp = Y L PJL By taking the square root of YJLPJL, the anticommutingsupersymmetry generators Q were discovered (See forinstance Ferrara, 1987):

Or more exactly,1YtPJL = 2{Qi Qj}

where i and j are the indices of the Y matrix.(ii) Ramond's (Ramond, 1971) repetition of the Diractrick on the two-dimensional world sheet of the relativistic string produced the fermionic string and this iswhat led to the supersymmetric string or SuperstringSupersymmetry and Superstrings form crucial ingredients in present-day research.17. The Present-day ProblemAs we already indicated, the combining of relativity andquantum mechanics pioneered by Dirac showed the existence of infinite degrees of freedom which was successfully handled by quantum field theory. However the success has been partial only, because of the divergences or

y taking thesquare root of r. PJI IJIthe anticommutingsupersymmetrygenerators Q werediscpvered.

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Address or CorrespondenceG Rajasekaran

Institute of MathematicalSciences

Chennai 6 113 India.

GENERAL I ARTICLEinfinities arising from summing over arbitrarily high energies and momenta in the intermediate states which isan inevitable consequence of local relativistic quantumfield theory.In the treatment of the strong and electroweak forces,the divergences turn out to be of a mild variety, namely,renormalizable divergences) and hence a temporary solution of the divergence problem has been possible.But gravity has proved a hard nut to crack. The marriage of quantum mechanics with gravity has not yetbecome possible. The divergences arising in quantumgravity are not even renormalizable. The constructionof a finite or renormalizable theory of quantum gravityis the most important problem in Physics at present.t is in this context, that the emergence of the super

string theory assumes significance Green et al., 1987).Superstring theory is claimed to be a finite theory ofquantum gravity. But much work is needed before thatclaim can be proved.To sum up,Special Relativity Quantum Mechanics -- QuantumField Theory; General Relativity Quantum Mechanics-- Superstring Theory?

Suggested Reading[1] T P Cheng and L FLi Gauge ThsoryofElementaryPanicle Physics, Clarendon

Press, Oxford, 1984.[2] M 8 Green, J H Schwarz and E Witten, Superstring theory, Cambridge

University Press, 1987.[3] S Ferrara, Supersymmewy, North Holland and World Scientific, 1987.[4] A Pais, Inward Bound, Oxford University Press, New York, 1986.[S] G Rajasekaran,BuildinguptheSttnulardGaugeModelo/HighEnergyPhysics,

in Grtmiunion Gauge Theories and Early Universe 8 R Iyer et a1 eds.),Kluwer Academic Publishers 1989, p 18S

[6] P Ramond,Phys.Ref ., D3, 241S 1971).[7] S D Rindani and M Sivakumar J. Phys., G 12 1986) 133S; Phys. Ref . D37

1988) 3S43 ; Mod. Phys. Lett. A4 1989) 1237; Prepr int of Physical ResearchLaboratory, Ahmedabad PRL-TH-90/1.

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