The Electromagnetic Form of the Dirac Electron Theory
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1 0 The Electrmagnetic Frm f the Dirac Electrn Thery Aleander G. Kyriaks Saint-Petersburg State Institute f Technlgy, St. Petersburg, Russia* In the present paper it is shwn that the Dirac electrn thery can be represented with the special frm f the Mawell thery, if the Dirac wave functin is identified with the plane electrmagnetic wave in sme specific way. This representatin allws us t see new pssibilities in the cnnectin f classical and quantum electrdynamics. Keywrds: quantum electrdynamics, classical electrdynamics, quantum mechanics interpretatin Intrductin W.J. Archibald [1] was the first wh paid attentin t the pssibility f representing the Schrödinger equatin in electrmagnetic frm. Fr the Dirac electrn equatins this pssibility was mentined in the bk []. The ther frms (quaternin, biquaternin, etc.) f the Dirac equatin and the separate mathematical aspects f this theme were cnsidered in many articles []. But an understanding f the physical meaning f these representatins has been lacking until nw. * Present address: Athens, Greece, agkyriak@yah.cm
2 1 The Dirac equatin has many particularities. In the mdern interpretatins these particularities are cnsidered mathematical features that d nt have a physical meaning. Fr eample [4] (sectin 4-4), it can prve that all the physical cnsequences f Dirac s equatin d nt depend n the special chice f Dirac s matrices. They wuld be the same if a different set f fur 44 matrices... had been chsen. In particular it is pssible t interchange the rles f the fur matrices by unitary transfrmatin. S, their differences are nly apparent. The mathematical prperties f the Dirac matrices are well knwn: they are anti-cmmutative and Hermitian; they cmpse a grup f 16 matrices; the bilinear frms f these matrices have defined transfrmatin prperties; it has been repeatedly pinted ut that in classical physics these matrices describe the vectr rtatins, etc. Belw we will shw that all the Dirac electrn equatin particularities can be entirely cnnected with the particularities f the equatins f classical electrdynamics. We will als shw why the physical cnsequences f Dirac s equatin d nt depend n the special chice f Dirac s matrices..0. Electrdynamics frm f the Dirac equatin The Dirac bispinr cntains fur functins, while an electrmagnetic field in the general case cntains si functins. As it is knwn, the number f the functins in these theries is nt cnnected in any way t the space dimensin (see e.g. [5]). Obviusly, in this case it is impssible t reduce ne anther. We will shw that such a pssibility eists nly in a specific case, when Dirac s spinr is identified with the fields f the plane electrmagnetic wave.
3 .1. The spinr frm f the Dirac equatin As it is knwn, there are tw mathematical descriptin frms f the representatin f the Dirac electrn equatin: spinr and bispinr. The Dirac equatins in the spinr frm [-6] are the fllwing: εϕ cσ pχ mc ϕ (.1) εχ cσ pϕ mc χ where σ are Pauli matrices, and ϕ and χ are the s-called spinrs, represented by the fllwing matrices: ϕ 1 χ 1 ϕ =, χ =, (.) ϕ χ.. The bispinr frm f the Dirac equatin Mre ften the Dirac equatin is described in the bispinr frm. Entering the functin: ψ ϕ χ ψ ψ ψ ψ = = called bispinr, the equatins (.1) can be written in ne equatin. There are tw bispinr Dirac equatin frms [6]: ( ) c p mc 1 4 (.) αε α β ψ = 0, (.4) ψ ( αε ) cα p β mc = 0, (.5)
4 which crrespnd t the tw signs f the relativistic epressin f the energy f the electrn: 4 ε =± c p mc, (.6) Here ε = iħ, p = iħ are the peratrs f the energy and t mmentum, ε, p are the electrn energy and mmentum, c is the light velcity, m is the electrn mass, ψ is the wave functin (ψ is the Hermitian-cnjugate wave functin) named bispinr and α = 1, α, α β are the Dirac matrices: i i 0 α 1 =, α =, i i (.7) α =, α 4 = It is als knwn that fr each sign f the equatin (.6) there are tw Hermitian-cnjugate Dirac equatins. We will cnsider the electrdynamics meaning f all these equatins.
5 4.. Electrdynamics frm f the Dirac equatin withut mass Let us cnsider, fr eample, the plane electrmagnetic wave mving n y - ais: i( t± ky ) E = Ee, (.8) i( t± ky) H = He, In the general case it has tw plariatins and cntains the fllwing field vectrs: E, E, H, H ( Ey = Hy = 0) (.9). The directin f the plane electrmagnetic wave is defined by the Pynting vectr [7]: i j k c c S = E H E Ey E 4π =, (.10) 4π H Hy H where i, jk, are the unit vectrs f y-,, aes. Fr the wave, which mves alng y - ais we have: S = jeh ( EH), (.11) y Let us enter the Dirac spinrs as electrmagnetic waves in the fllwing way: where ϕ = ϕ( y) and χ = χ( y). ϕ E H =, χ = i E H In this case the bispinr ψ ψ ( y) (.1) = will have the fllwing frm:
6 5 ψ 1 E ψ E ψ = =, ψ = ( E E ih ih ), (.1) ψ ih ψ 4 ih Using (.1), we can write the equatin f the electrmagnetic wave, mving alng the y - ais, in the frm: ( ε c p ) ψ (.14) The equatin (.14) can als be written in the fllwing frm: ( ) αε ( ) c α p ψ = 0, (.15) In fact, taking int accunt that ( αε ) = ε, ( α p) = p, (.16) we see that the equatins (.14) and (.15) are equivalent. Factriing (.15) and multiplying it frm the left n the Hermitian-cnjugate functin ψ we get: ψ αε cα p αε cα p ψ = 0, (.17) ( ) ( ) The equatin (.17) may be disintegrated n tw Dirac equatins withut mass: ψ αε cα p = 0, (.18) ( ) ( c p) αε α ψ = 0, (.19)
7 6 It is nt difficult t shw (using (.1) that the equatins (.18) and (.19) are the Mawell equatins f the electrmagnetic waves: retarded and advanced..4. Electrdynamics frm f Dirac equatin with mass Let us cnsider first tw Hermitian-cnjugate equatins, crrespnding t the minus sign f the epressin (.6): ( ) c p mc αε α β ψ = 0, (.0 ) ψ ( αε ) cα p β mc = 0, (.0 ) Using (.1) frm (.0 ) and (.0 ) we btain: 1 E i E c t y c 1 E i E c t y c 1 E i H c t y c 1 E i H c t y c (.1),
8 where Apeirn, Vl. 11, N., April E i E c t y c 1 E i E c t y c 1 E i H c t y c 1 E i H c t y c 7 (.) mc =. The equatins (.1) and (.) are Mawell ħ equatins with cmple currents. (It is interesting that alng with the electrical current, the magnetic current als eists here. This current is equal t er by Mawell s thery, but its eistence accrding t Dirac des nt cntradict the quantum thery). As we see, the equatins (.1) and (.) differ by the current directins. We culd fresee this result befre the calculatins, since the functins ψ and ψ differ by the argument signs: i t ψ = ψ0e i t and ψ = ψ0e. Let us cmpare nw the equatins that crrespnd t bth plus and minus signs f (.6). Fr the plus sign f (.5) we have the fllwing tw equatins: ( ) c p mc αε α β ψ = 0, (.) ψ ( αε ) cα p β mc = 0, (.4) The electrmagnetic frm f the equatin (.) is:
9 1 E i E c t y c 1 E i E c t y c, (.5) 1 E i H c t y c 1 E i H c t y c Obviusly, the electrmagnetic frm f the equatin (.4) will have the ppsite signs f the currents cmparatively t (.5). Cmparing (.5) and (.1) we can see that the equatin (.5) can be cnsidered as the Mawell equatin f the retarded wave. If we dn t want t use the retarded wave, we can transfrm the wave functin f the retarded wave t the frm: ψ E E ret = ih ih 8, (.6) Then, cntrary t the system (.5) we get the system (.). The transfrmatin f the functin ψ ret t the functin ψ adv is called, in quantum mechanics, the charge cnjugatin peratin..5. Electrdynamics sense f bilinear frms Enumerate the main Dirac matrices [6]: 1) α4 β is the scalar,
10 Apeirn, Vl. 11, N., April 004 ) α = { α, α} { α, α, α, α } µ α = α α α α is the pseudscalar. 9 is the 4-vectr, ) Using (.1) and taking int accunt that ψ ψ ( y) = it is easy t btain the electrdynamics epressins f the bispinrs, crrespnding t these matrices: ψ α ψ = E E H H = E H = 8π I, where I 1 1) ( ) ( ) 4 1 is the first scalar f the Mawell thery; ) ψ αψ = E H = 8π U and ψ αψ y = 8π cg y. Thus, the electrdynamics frm f the 4-vectr bispinr value is the 1 energy-mmentum 4-vectr f the Mawell thery: U, g. c ) ψ α 5 ψ = ( EH EH ) = ( E H) is the pseudscalar and E H = I is the secnd scalar f the electrmagnetic field thery. ( ) As it is knwn, frm the Dirac equatin the prbability cntinuity equatin can be btained: Ppr ( rt, ) div Spr ( rt, ) (.7) t Here P (, ) pr rt = ψ α0 ψ is the prbability density, and S (, ) pr rt = cψ αψ is the prbability flu density. Using the abve P rt, = 8π U and S = cg = 8π S. Then results we can btain: ( ) pr in the electrmagnetic frm the equatin (.7) has the frm: U div S = 0, (.8) t pr
11 which is the energy cnservatin law f the electrmagnetic field The electrdynamics sense f the matrices chice.1. The electrdynamics sense f the transpsitin f the matrices As we saw abve, the matri sequence ( α 1, α, α ) agrees with the electrmagnetic wave, which has y -directin. But herewith nly the α -matri is wrking, and the ther tw matrices d nt give the terms in the equatin. The verificatin f this fact is the Pynting vectr calculatin: the bilinear frms f α 1, α-matrices are equal t er, and nly the matri α gives the right nn-er cmpnent f the Pynting vectr. A questin arises: hw t describe the waves, which have and -directins. It is nt difficult t see that the matrices sequence is nt determined by sme special requirements. In fact, this matrices sequence can be changed withut breaking any quantum electrdynamics results [4, 6]. Intrducing the aes indees, which indicate the electrmagnetic wave directin, we can write three grups f the matrices, each f which crrespnds t ne and nly ne wave directin: ( α 1, α y, α ), ( α, α y, α 1,), ( α, α1 y, α ). Let us chse nw the wave functin frms, which give the crrect Mawell equatins. We will take as the initial frm that f the y - directin, which we have already used. Frm it, by means f the indees transpsitin arund the circle, we will get the frms f the and y - directins.
12 41 Since in the initial case the Pynting vectr has the minus sign, we can suppse that the transpsitin must be cunterclckwise. Let us eamine the suppsitin, checking the Pynting vectr values: 1) Fr ( α 1, α y, α ) we have E E ψ = ψ( y), ψ = ih ih, ψ = ( E E ih ih ) ih ih ψαψ 1 = ( E ) 0 E ih ih =, E E ψ α ψ ( EH ) y EH = = E H, ψ α ψ = 0, (.1) ) Fr ( α, α y, α 1,) we have E Ey ψ = ψ( ), ψ = ih ih y ψ α ψ = E H ) Fr ( α, α1 y, α ) we have y, ( E Ey ih ih y) ψ = ; ψ α ψ = 0, ψαψ 1 = 0, (.), y
13 4 Ey E ψ = ψ( ), ψ = ih, ψ = ( Ey E ihy ih ) ; y ih ψ α ψ = 0, ψαψ 1 = 0, ψ α y ψ = E H, (.) As we see, we tk the crrect result: by the cunterclckwise indees transpsitin the wave functins describe the electrmagnetic waves, which mve in a negative directin with regard t the crrespnding c-rdinate aes. We can suppse that, by the clckwise indees transpsitin, the wave functins will describe the electrmagnetic waves, which mve in a psitive directin alng the c-rdinate aes. Let us prve this: 1) Fr ( α 1, α y, α ) we have E E ψ = ψ( y),, ψ =, ψ = ih ( E E ih ih ) ; ih ψαψ 1 ψ α ψ = E H y, ψ α ψ = 0, (.4) ) Fr ( α, α y, α 1,) we have y
14 Ey E ψ = ψ( ), ψ = ih y ih ψ α ψ = E H ) Fr ( α, α1 y, α ) we have Apeirn, Vl. 11, N., April 004, ( Ey E ihy ih ) ψ = ;, y 4 ψ α ψ = 0, ψαψ 1 = 0, (.5) E Ey ψ = ψ( ), ψ = ih, ψ = ( E Ey ih ih y) ; ih y ψ α ψ = 0, ψαψ 1 = 0, ψ α y ψ = E H, (.6) As we see, nce again we get the crrect results. Nw we will prve that the abve chice f the matrices gives the crrect electrmagnetic equatin frms. Using the bispinr Dirac equatin (.) as an eample and transpsing the indees clckwise we btain fr the psitive directin f the electrmagnetic wave the fllwing results fr the, y, -directins crrespndingly:
15 1 Ey H 1 E H = i E,, y = i E c t c c t y c 1 E H y 1 E H = i E, i E, c t c = c t y c 1 y E 1 H E = i Hy, = i H, c t c c t y c 1 E y = i H, 1 H E, c t c = i H c t y c 1 E H y = i E, c t c 1 Ey H i Ey, c t = c 1 E y = i H, c t c 1 y E i Hy. c t = c 44 (.7) As we can see, we have btained three equatin grups, each f which cntains fur equatins, as is necessary fr the descriptin f all electrmagnetic wave directins. In the same way fr all ther frms f the Dirac equatin analgue results can be btained. Obviusly, it is pssible via cannical transfrmatins t chse the Dirac matrices in such a way that the electrmagnetic wave will have any directin.
16 45.. The electrdynamics sense f cannical transfrmatins f Dirac s matrices and bispinrs As is knwn [,8], the transitin frm sme independent variables t thers is made by means f the unitary peratr, which is called the cannical transfrmatin peratr. Actually the chice (.7) f the matrices α, made by us is nt unique. In this case there is a free transfrmatin f a kind: α = SaS ' ', (.8) where S is a unitary matri. The last ne crrespnds t functins transfrmatin: If we chse matrices ψ = S, (.9) α ' as: i i α' 1 =, α' =, i i 0, (.10) α ' =, α' 4 = then the functins ψ will be cnnected t functins accrding t the relatinship: ψ ' ψ =, ψ =, ψ =, ψ =,(.11)
17 46 The unitary matri S, which crrespnds t this transfrmatin, is equal t: S =, (.1) It is nt difficult t check that by means f this transfrmatin we will als receive the equatins f the Mawell thery. Actually, using (.1) and (.11) it is easy t receive: ψ ' = E, = E, = ih, = ih,(.1) whence: ( E ih ) ( E ih ) =, ( E ih ) ( E ih ) (.14) Substituting these functins in the Dirac equatin we will receive the crrect Mawell equatins fr the electrmagnetic waves (in duble quantity). It is pssible t assume, that the functins crrespnd t the electrmagnetic wave, mving under the angle f 45 degrees t bth crdinate aes. Thus, we see that actually any chice f the Dirac matrices changes nly the directin f the electrmagnetic wave.
18 4.0. The electrmagnetic frm f the electrn thery Lagrangian As it is knwn [7], the Lagrangian f the Mawell thery in the case f the electrmagnetic waves is: 1 LM = ( E H ), (4.1) 8π and as a Lagrangian f the Dirac thery can take the epressin [6]: L = ψ ε cα p β mc ψ, (4.) D ( ) Fr the electrmagnetic wave mving alng the (4.) can be written: L D y 47 -ais the equatin 1 ψ ψ mc = ψ ψ α y i ψ βψ, (4.) c t y ħ Transferring each term f (4.) in the electrdynamics frm we btain the electrmagnetic frm f the Dirac thery Lagrangian: U L = s div S i ( E H ), t 4π (4.4) (Let us nte that in the case f the variatin prcedure we must * distinguish the cmple cnjugate field vectrs E *, H and E, H ). The equatin (4.4) can als be written in anther frm. Using the cmple electrical and magnetic currents: e jτ = i E and m jτ = i H we take: π π U e m Ls = div S ( jτ E jτ H), (4.5) t
19 It is interesting that since L s = 0 thanks t (.4), we can take the equatin: U div S e m ( j τ E j τ H ) = 0, (4.6) t which has the frm f the energy-mmentum cnservatin law fr the Mawell equatin with current. Cnclusin The abve results shw that the Dirac thery can be written in the electrmagnetic frm as cnsistently as in the usual spinr frm. Such representatin makes the new interpretatin f the quantum electrdynamics pssible [9]. References 1. W.J. Archibald. Canad. Jurn. Phys.,, 565 (1955). A.I. Akhieer, W.B. Berestetskii. Quantum electrdynamics. Mscw, 1959 (Interscience publ., New Yrk,1965).. T. Kga. Int. J. Ther. Phys., 1, N 6, p (1975); A.A. Camplattr. Int. J. Ther. Phys. 19, N, p (1980); W.A. Rdrigues, Jr. arxiv: math-ph/0104 v1. 4. E. Fermi. Ntes n quantum mechanics. (The University f Chicag press, 1960). 5. H.A. Bethe. Intermediate Quantum Mechanics L.T. Schiff. Quantum Mechanics. (nd editin, McGraw-Hill Bk Cmpany, Inc, New Yrk, 1955). 7. M.-A. Tnnelat. Les Principes de la Therie Electrmagnetique et de la Relativite. (Massn et C. Editeurs, Paris, 1959). 8. V. Fck. The principles f the quantum mechanics. Leningrad, A.G. Kyriaks. 48
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