THE SPINOR FIELD THEORY OF THE PHOTON
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1 Romnin Rports in Physics, Vol. 66, No., P. 9 5, 4 THE SPINOR FIELD THEORY OF THE PHOTON RUO PENG WANG Pking Univrsity, Physics Dprtmnt, Bijing 87, P.R. Chin E-mil: rpwng@pku.du.cn Rcivd Octobr 8, Abstrct. I introduc spinor ild thory or th photon. Th thr-dimnsionl vctor lctromgntic ild nd th our-dimnsionl vctor potntil r componnts o this spinor photon ild. A spinor qution or th photon ild is drivd rom Mxwll s qutions, th rltions btwn th lctromgntic ild nd th our-dimnsionl vctor potntil, nd th Lorntz gug condition. Th covrint quntiztion o r photon ild is don, nd only trnsvrs photons r obtind. Th vcuum nrgy divrgnc dos not occur in this thory. A covrint positiv rquncy condition is introducd or sprting th photon ild rom its complx conjugt in th prsnc o th lctric currnt nd chrg. Ky words: photon, spinor.. INTRODUCTION Th lctromgntic intrction is th bst studid on mong th our known undmntl intrctions. In th rm o quntum thory, th photon is th quntum o th lctromgntic ild. Thr r svrl wys to rprsnt th lctromgntic ild: by th thr-dimnsionl lctric ild nd mgntic ild vctors, by th 4 4 lctromgntic tnsor, or by th our-dimnsionl potntil vctor [, ]. Th lctric nd mgntic ilds r dirctly rltd to th nrgy dnsity o th lctromgntic ild. Th our-vctor potntil is dirctly rltd to th Lgrngin dnsity o intrction. But howvr, non o ths ilds cn b rgrdd s th photon ild, bcus th photon dnsity cn not b xprssd s innr products o ths ilds thir djoint ilds. In this ppr, I introduc th spinor photon ild tht stisis spinor qution similr to th Dirc qution or th lctron. Th thr-dimnsionl vctor lctric ild, th thr-dimnsionl mgntic vctor ild nd th our-dimnsionl vctor potntil r componnts o this spinor photon ild. Th spinor qution or th photon ild is bsd on th Mxwll qutions, th rltions btwn th lctromgntic ild nd th ourdimnsionl vctor potntil, nd th Lorntz gug condition. Th Lgrngin dnsitis or th r photon ild nd or th photon ild in intrction th
2 Ruo Png Wng mttr r stblishd. Covrint quntiztion o photon ild is crrid out, nd only trnsvrs photons rg rom th quntiztion procdur. Th vcuum stt o th photon ild is ound to hv null nrgy. Th solution or th photon ild in th prsnc o th lctric currnt nd chrg is ound, nd covrint positiv rquncy condition is introducd or sprting th photon ild rom its complx conjugt. Th Mxwll qutions, th rltions btwn th lctromgntic ild nd th our-dimnsionl vctor potntil, nd th Lorntz gug condition r rwrittn s two ight-componnt spinor qutions in Sc.. Th spinor photon ild is introducd in Sc.. Th quntiztion o th photon ild is trtd in Sc. 4, nd th photon ild in th prsnc o lctric currnt nd chrg is nlysd in Sc. 5.. THE SPINOR EQUATION FOR THE ELECTROMAGNETIC FIELD Mxwll s qutions or th lctric nd mgntic ilds E nd H in th prsnc o chrg dnsity ρ nd currnt dnsity j cn b writtn in th ollowing orm: x x ε E = µ H µ j, () ( µ ) = ( ε ) H E, () = ( µ H ), () ( E ), (4) = ε ε ρ whr ε is th vcuum prmittivity, µ is th mgntic prmbility o th vcuum, nd x =ct. Bcus th lctric ild, th mgntic ild, th currnt dnsity nd th chrg dnsity r rl quntitis, thy r compltly dscribd by thir positiv rquncy componnts. An ltrntiv wy or writing th bov qutions is to introduc n ight componnts spinor lctromgntic ild nd n ight componnts spinor lctric currnt dnsity dind by = T (5) ε E ε E ε E µ H µ H µ H ( ) nd
3 Th spinor ild thory o th photon whr jx ( c, ) whr W hv j = µ j j j j T, (6) = ρ j is th our-vctor currnt dnsity. Th Mxwll qutions (-4) now cn b writtn s = α j, (7) x σ i I σ i I α =, α =, iσ I iσ I iσ σ i α = σ i iσ i σ = nd I =. i α α n +α n α = δ mn, mn, =,,. () Th lctromgntic ild cn b dscribd by th our-vctor potntil Ax = ( φ/ c, A ). Th rltion btwn th lctric nd mgntic ilds nd th our-vctor potntil, nd th Lorntz gug condition cn b writtn s: x ( A ) (8) (9) ε A = ε ε E, () c = εa µ H, () c x Th rltions ( ) cn b rwrittn s ( ε A ) = ( ε ) A. ()
4 Ruo Png Wng 4 = α, (4) x c whr th spinor potntil ild ( x ) is dind by ε T =. (5) x A x A x A x A x On my obsrv tht th qution (4) lso holds i w rplc th Lorntz gug condition th ollowing on: x ( A ) ε = ε A 8, (6) c whr 8 is n rbitrry sclr constnt. In this cs, th spinor lctromgntic ild tks th ollowing orm = T (7). ( ε E x ε E x ε E x µ H x µ H x µ H x 8 x ) According to proprtis o th lctric ild E, mgntic ild H, nd ourvctor potntil Ax undr continuous spc-tim trnsormtions, w hv th ollowing rltion or ( x ) nd ( x ) undr Lorntz trnsormtion nd ( x ) = xp φ l, (8) ( ) ( x ) = xp φ α l, (9) whr v v v φ = ln + ln. () v c c Undr rottion chrctrizd by th rottion ngl φ, xprssd s n xil vctor, w hv nd ( s) ( x ) = xp iφ, () ( s) ( x ) = xp iφ, ()
5 5 Th spinor ild thory o th photon nd Σ iσ s=, l=, Σ iσ i i i i Σ =, Σ =, Σ =. i i Th ollowing commuttion rltion holds or s : () (4) n m nmp p p= [ s, s ] = i ε s. (5) Equtions (7) nd (4) r invrint undr continuous spc-tim trnsormtions (S th Appndics).. THE PHOTON FIELD On cn us ( x ) or x to rprsnt th lctromgntic ild, but it is not possibl to xprss th photon dnsity s n innr product o ( x ) or its djoint ild. Thror nithr ( x ), nor x cn b rgrdd s th photon ild. Th concpt o photon is closly rltd to monochromtic lctromgntic pln wvs, so w considr monochromtic pln wv k k, xp( ikx) (6) + k in th bsnc o lctric currnt nd chrg. Lt ( x + k ) nd b th positiv rquncy prts o k ( x ) nd k. W ind tht th innr product + k + k is qul to th tim vrg o nrgy dnsity, which should qul to, in th cs o monochromtic wv, th product o th photon dnsity nd th photon nrgy ck. On othr hnd, w hv nd k* k k* k k + k ε E E =µ H H = (7)
6 4 Ruo Png Wng 6 E A A E E E, (8) + k* + k + k* + k + k* + k i x x = i x x = x x ck thus th photon dnsity is qul to W hv i + i. (9) + k + k + k + k + k x + k + k + k + k + k + k i + i = ( ) τ, + k () whr iβ I 4 τ = β =, () β i I 4 nd I 4 is th 4 4 unit mtrix. Bsd on th rltion (), w din th photon ild s ( x ) x k ( ik( x x) ). () π ( x ) 4 d xp 4 ( ) k > On my obsrv tht th condition k > is covrint or th r photon ild, bcus it dos not contin Fourir componnts k < k. According to Eqs. (7) nd (4), th r photon ild stisis th ollowing qution: i i = i α w β, () x c α α w =, β =, (4) α I 8 whr I 8 is th 8 8 unit mtrix. Th invrinc o Eq. () undr continuous spc-tim trnsormtions is ssurd by th invrinc o Eqs. (7) nd (4). W hv ( x ) = xp φ Λ, (5) l Λ = α l (6)
7 7 Th spinor ild thory o th photon 5 or Lorntz trnsormtions, nd ( s ) ( x ) = xp iφ, (7) undr spc rottions, whr s s =. (8) s Eq. () is invrint lso undr spc invrsion nd tim rvrsl. It is sy to vriy tht τ ( x, x ) nd τ ( x, x ) stisy th sm spinor qution Eq. () s ( x, x ), whr β β τ =, τ =. (9) β β Th qution or th r photon ild cn b drivd rom th ollowing Lgrngin dnsity L = i + cα w + i β t, (4) whr = τ is th djoint ild. On my obsrv tht thr r totl o 5 componnt qutions or photon ild. Among ths 5 qutions only qutions r indpndnt, nd th othr 4 qutions (corrsponding to Eqs. () nd ()) r dirct conclusion o ths qutions. By mns o vritionl clcultion, ll ths indpndnt qutions cn b obtind. Th conjugt ild o is L =. (4) π = i Th Hmiltonin now cn b clcultd: H ( L ) = d x π = = d x ( i cαw iβ ). Th Lgrngin dnsity (4) is invrint undr globl phs chng o th photon ild. This implis th consrvtion o th photon numbr N or r photon ild: (4)
8 6 Ruo Png Wng 8 N = ρ d ph x (4) nd ρ ph + j ph =, (44) t whr th photon dnsity ρ ph is givn by th innr product btwn th photon ild nd its djoint : nd ρ = (45) ph j = c α (46) ph w is th photon currnt dnsity. On my obsrv tht th photon dnsity dind by Eq. (45) my tk ngtiv vlus. But howvr, whn w tlk bout photons w rr to lctromgntic ilds wll-dind rquncis. By dirct clcultion, on cn vriy tht ρ ph i hs wll-dind rquncy. According to th rltion btwn symmtris nd consrvtion lws [, 4], w my obtin th ollowing xprssions or th momntum P nd th ngulr momntum M o th r photon ild: nd P= i x (47) d [ i ] ( ) M = x x + x s (48) d d. It is clr tht s cn b intrprtd s th spin oprtor o th photon ild. According to xprssion (5), w hv n m nmp p p= [ s, s ] = i ε s, n, m=,,. (49) 4. QUANTIZATION OF THE PHOTON FIELD It is convnint to quntiz th photon ild in th momntum spc. To do this, w hv to ind irstly th pln wv solutions o th photon ild. By substituting th ollowing orm o solution xp( ikx) w( k ) (5)
9 9 Th spinor ild thory o th photon 7 into th spinor qution (), w ind iβ w k α k w =. c k (5) Eq. (5) prmits two indpndnt nontrivil solutions k = k. Thy cn b chosn s w± ( k) = ( c k ( q ± ir) c k ( q ± ir) c k ( q ± ir) c c k r iq c k r iq c k r iq (5) ( ) ( ) ( ) T ), iq ± r iq ± r iq ± r whr q nd r r two unity vctors stisying th ollowing conditions: k ˆ = k/ k. kˆ q= r, kˆ r = q, q r= kˆ, nd r k = r k, (5) w + ( k ) nd w ( k ) r orthogonl: w w w w W lso hv nd h h = h τ h =δ hh. k k k k k (54) ( ) h h k ˆ s w k = hw k, h=±, (55) = + = s w k s ( s ) w k w k. (56) h h h Thror photons r prticls o spin s =. On my lso obsrv tht th componnts in w + ( k ) nd w ( k) corrsponding to 8 r zro, so th Lorntz gug condition is r-obtind. Hving pln wv solutions o th photon ild, w my now xpnd in pln wvs ikx = wh ( k) bh ( k), k h V k ikx = wh ( k) bh ( k), k h V k (57)
10 8 Ruo Png Wng k = k. According to rltions (4) nd (57), th Lgrngin o th photon ild cn b xprssd s unction o th vribls q () hk t : (, ) L t q = q () (), h t i c k q h t h t (58) k k k = ( ω ) ω= q hk t bh k xp i t, ck. (59) Th conjugt momntum o q hk () t cn b clcultd, nd w hv () L p hk t = = i bh iωt q hk () t By pplying th quntiztion condition [, ] ollowing commuttion rltion or b ± ( k ) nd b ± ( k ) b, b ( ) =δ δ. b ± ( k ) nd b ± h h hh kk k xp. (6) q p = i δ δ, w ind th hk hk hh kk k k (6) k r just th photon nnihiltion oprtor nd th photon crtion oprtor. Th Hmiltonin o th photon ild cn lso b clcultd. W obtin H p q L b k b k (6) = hk hk = ω h h. k h k h W obsrv tht th vcuum nrgy o th photon ild is zro. Th commuttion rltions or th photon ild cn b writtn in covrint orm. According to th commuttion rltions (6) nd th xprssion (57), w hv l, m ( x ) = Dlm ( x x ), l, m=,,,8, (6) whr th 8 8 mtrix D ( x ) is givn by th ollowing xprssion Th rplcnt c D = kδ( k ) k k l+ ( k l)( k l ) (64) ( π) 4 ikx d. k > V k ( π ) d k (65)
11 Th spinor ild thory o th photon 9 ws usd in obtining th rltion (6). Undr Lorntz trnsormtions, D ( x ) trnsorms to D x = xp φ l D x xp φ l. (66) On cn vriy no diiculty tht by using th xpnsions (57), th commuttion rltions (6) cn b drivd rom commuttion rltion (6). Thror, th commuttion rltions (6) nd (6) r quivlnt. 5. INTERACTION BETWEEN PHOTON FIELD AND MATTER In th prsnc o lctric currnt nd chrg, ccording to Eqs. (7) nd (4), w hv th ollowing qution or th photon ild: i i = i w β i J, x c α (67) whr th spinor currnt dnsity J is givn by + nd j stisis th rltion J + j =, + + *. (68) j x + j x = j x (69) It would b nturl to rqust tht th spinor currnt dnsity J () x to hv only + positiv rquncy Fourir componnts. But th spinor currnt dnsity j my contin Fourir componnts k > k, nd or ths Fourir componnts this sprtion dos hold or ll rms o rrnc. In othr words, th positiv rquncy condition is not covrint. A covrint orm o this condition cn b writtn s: kp >, whr p is wll-dind 4-vctor. For ch physicl syst, thr lwys is wll-dind 4-vctor, nmly th totl nrgy-momntum 4-vctor o th syst. Thror, w hv + j 4 4 x ikx ikx d x J =θ( kp) d x, (7) i z > θ ( z) = / i z =, i z < (7)
12 Ruo Png Wng nd p ( p, ) = p th totl nrgy-momntum 4-vctor o th photon ild nd th chrgd mttr ild undr considrtion. On my obsrv tht, in th cntr o mss rm in which th totl momntum p o th whol syst in intrction is zro, J ( x ) hv only positiv rquncy Fourir componnts. It is sy to vriy tht th qution (67) cn b drivd rom th ollowing Lgrngin dnsity L = L + L, (7) int th Lgrngin dnsity o intrction givn by L int = i c τ J J τ. (7) According to th dinition o J ( x ), w hv dx d x V * T x τ J x J τ =. (74) Thror it is not ncssry to sprt J rom J * () x in th Lgrngin dnsity o intrction, nd th qution (67) cn lso b drivd rom th ollowing Lgrngin dnsity L = L + L, (75) int whr th Lgrngin dnsity o intrction is givn by L int = i c τ Js Js τ, (76) J x = J x + J x (77) * s. According to th rltion btwn th photon ild nd th our-vctor potntil A ( x ), th Lgrngin dnsity o intrction L int cn lso b writtn s L, int x = A xjx jxa x (78) whr A ( x ) is th positiv rquncy prt o Ax : 4 ikx 4 ikx d x A ( x ) =θ kp d x A ( x ). (79) On my obsrv tht i jx commuts A ( x ) nd A () x, th Lgrngin dnsity o intrction L int would bcom j ( x ) A ( x ), s in th clssicl lctrodynmics.
13 Th spinor ild thory o th photon Th qution (67) cn b solvd. W hv 4 = + d xg ( x x ) J ( x ), (8) whr () x is th r photon ild givn by xprssions (57), nd G Grn unction or th photon ild: ( ) x is th ixp ikx 4 iβ G d k k = 4 + w. k i α k + ε c (8) π 6. CONCLUSION I introducd spinor ild thory or th photon. Th spinor qution or th photon ild is quivlnt to Mxwll s qutions togthr th rltions btwn th our-vctor potntil nd lctric nd mgntic ilds, nd th Lorntz gug condition or th 4-vctor potntil. Th quntiztion o r photon ild is don, nd only trnsvrs photons r obtind. Th vcuum nrgy divrgnc dos not occur in this thory. Th solution or th photon ild in th prsnc o th lctric currnt nd chrg is ound, nd covrint positiv rquncy condition is introducd or sprting th photon ild rom its complx conjugt. APPENDIX A: INVARIANCE OF SPINOR EQUATIONS FOR ELECTROMAGNETIC FIELD AND POTENTIAL UNDER LORENTZ TRANSFORMATIONS By dirct vriiction, on my ind th ollowing rltions or mtrics s nd l : nd n nmp p p= [ s, α ] = i ε α, n, m=,,, (A) α l α = l δ α, n, m=,,. (A) n m n m nm Lt s considr Lorntz trnsormtion x cosh ϕ sinh ϕ x x x =. x x x sinh ϕ cosh ϕ x (A)
14 Ruo Png Wng 4 W hv nd = cosh ϕ sinh ϕ, = cosh ϕ sinh ϕ x x x x x x j = cosh ϕ j ( x ) + sinh ϕcρ ( x ), cρ = cosh ϕcρ ( x ) + sinh ϕj ( x ). (A4) (A5) Th lst rltions cn b writtn in th trms o j ( x ) nd j ( x ): ( ( ) ) j = xp ϕ l α j ( x ). (A6) By using th rltions (A4), th qution (7) cn thn b writtn s th ollowing: Bcus so But ( cosh ϕ sinh ϕα ) + x ( cosh ϕα sinh ϕ ) +α +α + j =. x x x w thn hv (A7) xp ϕα = cosh ϕ sinh ϕα, (A8) xp ( ϕα ) = x = xp ( ϕα ) α +α +α = x x x = xp ϕ l α j ( x ). ( ( ) ), l l (A9) α = α (A) xp ( ϕl) = α + xp ( ϕ( l α ) ) x x (A) α +α xp ( ϕl) xp ( ϕl) j ( x ) =. x x
15 5 Th spinor ild thory o th photon According to th rltion (A), w hv thror n n α l = l α α, m=,, (A) ( l) ( l ) α xp ϕ = xp ϕ α α, xp ϕl α =α xp ϕ l α, m=,, nd th qution (A) bcoms (A) x ( x ) = α ( x ) j ( x ), (A4) ( x ) = xp ϕl. (A5) Th qution (A4) in th nw rrnc rm hs xctly th sm orm s Eq. (7), this mns th spinor qution or th lctromgntic ild is invrint undr Lorntz trnsormtions. Th invrinc o Eq. (4) cn b shown in similr wy. W hv xp ( ϕα ) = α xp ( ϕα ) + x x +α +α xp ( ϕl ) ( x ). x x c (A6) But xp ϕl α = α xp ϕ α l, m=,, (A7) thus Eq. (A6) cn b rducd to x ( x ) = α ( x ) ( x ), (A8) c ( ( ) ) ( x ) = xp ϕ α l. (A9) This qution hs xctly th sm orm s Eq. (4).
16 4 Ruo Png Wng 6 APPENDIX B: INVARIANCE OF SPINOR EQUATIONS FOR ELECTROMAGNETIC FIELD AND POTENTIAL UNDER SPACE ROTATION W hv nd Lt s considr n ininitsiml spc rottion x = x, x = x ε δ x, n=,,. (B) n n nmp m p mp, = nmp m x n x n mp, = x p = + ε δ, n =,,, (B) n n nmp m p mp, = ρ =ρ ( x ), j = j ( x ) + ε δ j ( x ), n=,,. (B) Th lst rltion is quivlnt to Th qution (7) cn b writtn s But so ( iδ s) j = iδ s j ( x ). (B4) + = x = ( + iδ s) αn εnlmδ lα j ( x ). lm, = x n m= lm, = Thn Eq. (B5) bcoms (B5) ε nlmα = is lα n iα n s l, l, n =,,, (B6) ε nlmδ lα = iδ sα n iα nδ s, n=,,. (B7) ( + iδ s) = α ( + i ) j ( x ). x δ s (B8) n n= xn
17 7 Th spinor ild thory o th photon 5 (B8) s On th othr hnd, w hv ( δ s ) x ( x ) = + i, so w rwrit Eq. ( x ) = α ( x ) j ( x ), (B9) which hs xctly th sm orm s Eq. (7). So th spinor qution or th lctromgntic ild is invrint undr spc rottions. Th invrinc o Eq. (4) cn b donstrtd xctly in th sm wy. REFERENCES. L. D. Lndu nd E. M. Lishitz, Clssicl Thory o Filds, 4 th d., Buttrworth Hinnn, 98.. J. D. Jckson, Clssicl Elctrodynmics, rd d., John Wily nd Sons, Nw York, E. Nothr, Nchr. Gsllscht, Wiss. Göttingn, 7 (98). 4. F. Mndl nd G. Shw, Quntum ild thory, (Chp. ), John Wily nd Sons, Chichstr, 984.
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