Photon polarization and Wigner s little group

Transcription

1 Photon polarization and Wignr s littl group Paw l Caban and Jaub Rmbilińsi Dpartmnt of Thortical Physics, Univrsity of Lódź Pomorsa 149/153, 9-36 Lódź, Poland Datd: January 15, 14 To discuss on-photon polarization stats w find th xplicit form of th Wignr s littl group lmnt in th masslss cas for arbitrary Lorntz transformation. As is wll nown, whn analyzing th transformation proprtis of th physical stats, only th valu of th phas factor is rlvant. W show that this phas factor dpnds only on th dirction of th momntum / and dos not dpnd on th frquncy. Finally, w us this obsrvation to discuss th transformation proprtis of th linarly polarizd photons and th corrsponding rducd dnsity matrix. W find that thy transform proprly undr Lorntz group. arxiv:quant-ph/341v 3 Oct 3 PACS numbrs: 3.65.Fd, 3.65.Ud, 3.3.+p I. INTRODUCTION In rcnt yars a lot of intrst has bn dvotd to th study of th quantum ntanglmnt and Einstin- Podolsy-Rosn corrlation function undr th Lorntz transformations for massiv particls [1,, 3, 4, 5, 6, 7]. In rcnt paprs [8, 9] also th masslss particl cas was discussd. On of th y ingrdints of ths paprs is th calculation of th xplicit form of th littl group lmnt for masslss particl in som spcial cass to analyz transformation proprtis of ntangld stats and rducd dnsity matrix. In this papr w driv th xplicit form of th Wignr s littl group lmnt in th masslss cas for an arbitrary Lorntz transformation and discuss th transformation proprtis of th linarly polarizd photons and th corrsponding rducd dnsity matrix obtaind by tracing out inmatical dgrs of frdom. As is wll nown in th Hilbrt spac of masslss particls th onparticl momntum ignvctors undr Lorntz transformation Λ ar multiplid by a phas factor dpnding on Λ and th particl four-momntum µ. W show that this phas factor dpnds only on Λ and th dirction of th momntum / but dos not dpnd on th frquncy. In contrast to othr paprs [8] this obsrvation nabls us to giv th dscription of th transformation ruls of th linarly polarizd photons which ar not ncssarily monochromatic. II. WIGNER S LITTLE GROUP FOR MASSLESS PARTICLES As is wll nown, th pur quantum stats ar idntifid with rays in th Hilbrt spac. For this rason, on th quantum lvl, w should us ray rprsntations of th classical symmtry groups. In our cas of th propr Elctronic addrss: P.Caban@mrlin.fic.uni.lodz.pl Elctronic addrss: J.Rmbilinsi@mrlin.fic.uni.lodz.pl ortochronous Poincaré group P+, which is th smidirct product of th propr ortochronous Lorntz group L + and th translations group T 4, its ray rprsntations so calld doubl-valud rprsntations ar faithful rprsntations of th univrsal covring of P+, i.., th smidirct product of SL, C and T 4. Morovr, th faithful rprsntations of P + ar homomorphic rprsntations of its univrsal covring group. W us th canonical homomorphism btwn th group SL, C univrsal covring of th propr ortochronous Lorntz group L + and th Lorntz group L + SO1, 3 [1]. This homomorphism is dfind as follows: With vry four-vctor µ w associat a twodimnsional Hrmitian matrix such that = µ σ µ, 1 whr σ i, i = 1,, 3 ar th standard Pauli matrics and σ = I. In th spac of two-dimnsional Hrmitian matrics 1 th Lorntz group action is givn by = AA, whr A dnots lmnt of th SL, C group corrsponding to th Lorntz transformation ΛA which convrts th four-vctor to i.., µ = Λ µ ν ν and = µ σ µ. Th rnl of this homomorphism is isomorphic to Z th cntr of th SL, C. Now, lt us focus on th cas of masslss particls. An xplicit matrix rprsntation 1 of th null light-con four-vctor can b writtn as = 1 + n 3 n n + 1 n 3, 3 whr n ± = n 1 ± in, n = /, = and dt = µ µ =. In this cas w choos th standard four-vctor as = 1,,, 1. In th matrix rprsntation 3 th following matrix is associatd with : =. 4

2 Now, lt us find th stability group of, i.., A SL, C which lavs invariant. All such A form a subgroup of th SL, C group, i.. stability group = {A SL, C: = A A }. 5 As is wll nown [1] th stability group of th fourvctor is isomorphic to th E group of rigid motions of Euclidan plan. W can asily find th most gnral A by solving th quation = A A. W gt A = i ψ z, 6 i ψ whr z is an arbitrary complx numbr. Sinc th SL, C is th two-fold covring of th Lorntz group, w rstrictd th valus of ψ to th intrval, π. Our purpos is to find th Wignr s littl group lmnt WΛ, corrsponding to and th Lorntz transformation Λ, namly WΛ, = L 1 Λ ΛL, 7 whr L L + is dtrmind uniquly by th following conditions: = L, L = I. 8 In ordr to find th corrsponding lmnt SΛ, in SL, C such that WΛ, = ΛSΛ,, i.. SΛ, = A 1 Λ AA, 9 whr ΛA = L, w hav to first calculat th matrix A. W can do it by solving th matrix quation Aftr simpl calculation w gt whr U n = = A A. 1 A = U n B, n 3 n 1 + n3 n n 3 1 rprsnts rotation R n which convrts th spatial vctor,, 1 to n, whil B = 1 13 rprsnts boost along z-axis which convrts to. Thrfor 1 A = 1 + n 3 n 1 + n 3 n n Not that according to Eq. 8 A = I. Now, an arbitrary Lorntz transformation ΛA is rprsntd in SL, C by th corrsponding complx unimodular matrix α β A =, αδ βγ = γ δ To calculat A Λ w simply us th formulas,3 to find and thn idntify = Λ. W find whr = 1 a 16 n 3 = b a 1, 17 n + = c a, 18 n =n +, 19 a = α + γ 1 + n 3 + β + δ 1 n 3 + αβ + γδ n + α β + γ δn +, b = α 1 + n 3 + β 1 n 3 + αβ n + α βn +, 1 c =α γ1 + n 3 + β δ1 n 3 + β γn + α δn +, and n = /. Thrfor w can find th xplicit form of SΛ, by mans of Eqs. 9 and 14. W hav to calculat only th lmnts SΛ, 11 and SΛ, 1, sinc th gnral littl group lmnt 6 dpnds only on th phas factor i ψ and complx numbr z. A straightforward calculation yilds finally th following formulas: i ψλ, = α1 + n3 + βn + b + γ1 + n 3 + δn + c a b1 + n 3 3 zλ, = αn + β1 + n 3 b + γn + δ1 + n 3 c a b1 + n 3 4 whr a, b, and c ar givn by Eqs. -. Th unitary irrducibl rprsntations of th Poincaré group ar inducd from th unitary irrducibl rprsntations of th littl group of th four-momntum µ i.., th E group in th cas of th masslss particls [1, 11]. Now, w hav two classs of th unitary irrducibl rprsntations of E: th faithful infinit dimnsional rprsntations and th on-dimnsional homomorphic rprsntations of E, isomorphic to its compact subgroup SO E. Bcaus thr is no vidnc for xistnc of masslss particls with a continuous intrinsic dgrs of frdom th physical choic is th last on [11]. Thus by mans of th induction procdur [1] th four-momntum ignstats transform according to th formula UΛ, = iψλ, Λ,. 5

3 3 In th abov formula UΛ dnots unitary oprator rprsnting Λ in th unitary rprsntation of th Poincaré group whil th hlicity fixs irrducibl unitary rprsntation of th Poincaré group inducd from SO; tas intgr and half-intgr valus only [1, 11]. W us invariant normalization of th four-momntum ignstats,, i.., p, σ, = δ σ δ p. Thus, whn analyzing th transformation proprtis of physical stats only th valu of th phas ψλ, is rlvant Eq. 3. So it is vry important to strss that th valu of th phas ψ dpnds only on Λ and n and dos not dpnd on th frquncy : ψλ, = ψλ, = ψλ,n. 6 Not also that momnta of masslss particls which ar paralll in on inrtial fram ar paralll for vry inrtial obsrvr, i.., = p p = p p, 7 whr = Λ, p = Λp. Indd, for masslss particls, and p ar paralll iff th corrsponding four-momnta ar Lorntz orthogonal, i.., µ p µ =. Sinc µ p µ is a Lorntz invariant thn this holds in all inrtial frams. Equation 7 can b also vrifid xplicitly by using Eqs Th abov proprty holds good only in th masslss cas. Now, using Eq. 3 w can immdiatly obtain th valu of iψλ, in a numbr of spcial cass considrd lswhr. Rotations: In th cas Λ = R w hav R = ΛU whr U SU SL, C, thus w put δ = α, γ = β, α + β = 1 8 and from 3 w gt th following simpl formula: iψr, = α1 + n3 + βn + α 1 + n 3 + β n. 9 For th givn rotation R th xplicit form of α and β can b xprssd by,.g., Eulr angls s, for xampl, Rf. [1]. Also not that from Eq. 4, w gt zr, =. 3 Now lt us considr th spcial cas of th rotation Rχn around th dirction n. Th matrix U Rχn SU rprsnting Rχn can b writtn in th form [13] Thus, U Rχn = i χnj σ j = cos χ + inj σ j sin χ. 31 U Rχn = cos χ + in3 sin χ in sin χ in + sin χ cos χ in3 sin χ. 3 Thus insrting th corrsponding valus of α and β to Eq. 9, w find in this cas iψrχn,n = iχ. 33 As th nxt xampl, w considr th rotation Rχẑ around th z-axis. In this cas s Rf. [13] i χ U Rχẑ = i χ, 34 thrfor from Eq. 9, w gt th sam formula as prviously iψrχẑ,n = iχ. 35 Boosts: Pur Lorntz boost Λv in an arbitrary dirction = v v can b rprsntd by th following SL, C matrix [1]: Av = 1 ξj σ j = cosh ξ + 3 sinh ξ sinh ξ + sinh ξ cosh ξ 3 sinh ξ 36 whr th paramtr ξ is connctd with th vlocity of th boostd fram by th rlation tanhξ = v 37 and ± = 1 ± i w us natural units with th light vlocity qual to 1. Insrting th corrsponding valus of α and β into Eqs. -, w arriv at th rlations of th form a = coshξ + n sinhξ, 38 b = coshξ + n nsinhξ + 3 ncosh ξ 1, 39 c = n + + sinhξ + ncosh ξ Now, th corrsponding littl group lmnt can b obtaind from Eqs. 3,4. W considr hr two spcial cass: boost Avn along th n dirction and boost Λvẑ along th z dirction. In th first cas, w hav = n and from Eqs and 3,4, w find iψλvn, = 1, zλvn, =. 41 In th scond cas, w hav α = 1 1 v δ = 4, β = γ = v Insrting abov valus to Eqs. 3 and 4, w gt ψλvẑ, =, zλvẑ, = n 1 v n3. 43,

4 4 III. TRANSFORMATION LAW FOR LINEARLY POLARIZED LIGHT Now, w apply th abov rsults to discuss som transformation proprtis of th polarizd stats and rducd dnsity matrix for photons. Lt us considr first th classical lctromagntic fild. As is wll nown th monochromatic plan lctromagntic wav is in gnral polarizd liptically. In th spcial cas of th linar polarization w can dal also with th plan wav which is not ncssarily monochromatic. In this cas th lctromagntic fild tnsor can b writtn as F µν x = F µν fct n x 44 for th linary polarizd wav propagating in th n dirction, whr th tnsor F µν is x µ indpndnt. It is vidnt that th abov formula is covariant undr Lorntz transformations. It mans that Lorntz transformations prsrv linar polarization of an arbitrary plan wav not ncssarily monochromatic. W show that it is also th cas on th quantum lvl. As is wll nown s,.g., Rfs. [11,14] th onphoton rprsntation spac is spannd by th vctors {, 1,, 1 } bcaus th parity oprator changs th sign of th hlicity. Lt us considr first th monochromatic linarly polarizd plan wav. Th photon stat corrsponding to such a wav is of th form [14], φ, n, φ = 1 1 = 1 iφ,, 45 whr = and th momntum indpndnt angl φ dtrmins th dirction of th polarization in th plan prpndicular to th dirction of th propagation n. Th gnral linarly polarizd stat corrsponding to th wav 44 has th form g, φ,n = 1 iφ d g, n,, 46 whr n is fixd. Th stat 46 is a tnsor product of momntum dirction and polarization stats in ach Lorntz fram. Lt us not that stats blonging to th propr Hilbrt spac wav pacts cannot b xactly linarly polarizd stats. Howvr, linarly polarizd stats 46 can b approximatd as tmprd distributions with an arbitrary accuracy by squncs of wav pacts. It is intrsting to point out a paralllism btwn classical and quantum dscription of idal linarly polarizd stats. Namly, on th classical lvl thy hav infinit total lctromagntic nrgy whil on th quantum lvl thy li out of th propr Hilbrt spac of th wav pacts, i.., thy ar distributions. Now, w show that for vry inrtial obsrvr th linary polarizd stat 46 rmains linary polarizd. Indd, taing into account Eqs. 3-6 w find whr UΛ g, φ,n = 1 iφ+ψλ,n d g, n, = g, φ + ψ,n, 47 g = a g, 48 a a is givn by Eq. and by virtu of Eq. 7 th dirction n is fixd, too. Thrfor, th stat w rcivd is again linarly polarizd. Now, w discuss th transformation of th rducd dnsity matrix for linary polarizd plan wav. In gnral for th rducd dnsity matrix dscribing th hlicity proprtis of th stat f = dµf, 49 w obtain th following formula: dµfσ f ˆρ σ = dµ f, 5 whr w hav usd th Lorntz invariant masur dµ = θ δ d 4 d3. 51 It should b notd that in gnral th stat f can b a tmprd distribution it dos not ncssarily blong to th Hilbrt spac but rathr to th Gl fand tripl, as for xampl four-momntum ignstats. In such a situation th formula 5 should b undrstood as a rsult of a propr rgularization procdur. Applying th abov considrations to th dnsity matrix dscribing th stat g, φ,n w gt th following rducd dnsity matrix: i.., ρ σ g, φ,n = 1 i σφ, 5 ρg, φ,n = 1 1 iφ iφ 1, 53 which in fact rprsnts a rducd pur stat bcaus ρ = ρ. Th abov dnsity matrix transforms proprly undr Lorntz transformations, namly ρ iψλ,n iψλ,n = iψλ,n ρg, φ,n iψλ,n = 1 1 iφ+ψ iφ+ψ 1 = ρg, φ + ψ,n. 54

5 5 W strss that th fact that th linarly polarizd stat admits a covariant rducd dnsity matrix dscription in trms of hlicity dgrs of frdom is rlatd to th proprty of th Lorntz transformation that it dos not gnrat ntanglmnt btwn momntum dirction and hlicity. Finally, lt us not that th von Numann ntropy corrsponding to th dnsity matrix 53 is qual to zro. Evidntly it is Lorntz-invariant in viw of Eq. 54. Our discussion can b asily rcast in trms of polarization vctors dfind according to Rf. [11], for diffrnt approach s also Rf. [15]. IV. CONCLUSIONS W hav found in this papr th xplicit form of th Wignr s littl group lmnt in th masslss cas for arbitrary Lorntz transformation. Using this rsult w hav shown that th light wav which is linarly polarizd but not ncssarily monochromatic for on inrtial obsrvr rmains linarly polarizd also for an arbitrary inrtial obsrvr. W hav also shown that th rducd dnsity matrix dscribing linarly polarizd photon, obtaind by tracing out inmatical dgrs of frdom, transforms proprly undr Lorntz group action. Morovr th corrsponding von Numann ntropy is a Lorntz scalar. Acnowldgmnts This wor was supportd by Univrsity of Lódź and th Laboratory of Physical Bass of Procssing of Information. [1] A. Prs, P. F. Scudo, and D. R. Trno, Phys. Rv. Ltt. 88, 34. [] M. Czachor, Phys. Rv. A 55, [3] M. Czachor, in Photonic Quantum Computing, ditd by S. P. Holating and A. R. Pirich SPIE - Th Intrnational Socity for Optical Enginring, Bllingham, WA, 1997, vol. 376 of Procdings of SPIE, pp [4] J. Rmbilińsi and K. A. Smolińsi, Phys. Rv. A 66, 5114, quant-ph/4155. [5] P. A. Alsing and G. J. Milburn, Quantum Inf. Comput., 487. [6] H. Trashima and M. Uda, Int. J. Quant. Inf. 1, [7] R. M. Gingrich and C. Adami, Phys. Rv. Ltt. 89, 74. [8] N. H. Lindnr, A. Prs, and D. R. Trno, J. Phys. A: Math. Gn. 36, L449 3, quant-ph/3417. [9] A. J. Brgou, M. Gingrich, and C. Adami, Phys. Rv. A 68, 41 3, quant-ph/395. [1] A. O. Barut and R. R acza, Thory of Group Rprsntations and Applications PWN, Warszawa, [11] S. Winbrg, in Lcturs on Particls and Fild Thory, ditd by S. Dsr and K. W. Ford Prntic-Hall, Inc., Englwood Clifs, N. J., 1964, vol. II of Lcturs dlivrd at Brandis Summr Institut in Thortical Physics, p. 45. [1] G. A. Korn and T. M. Korn, Mathmatical Handboo for Scintists and Enginrs McGraw-Hill Boo Company, Nw Yor Toronto London, [13] F. Gürsy, in Rlativity, Groups and Topology, ditd by C. DWitt and B. DWitt Gordon and Brach, Nw Yor London Paris, 1963, Lcturs dlivrd at Ls Houchs Summr School 1963, p. 91. [14] S. Winbrg, Th Quantum Thory of Filds, vol. I Cambridg Univrsity Prss, Cambridg, [15] A. Prs and D. R. Trno, J. Modrn Optics 5, , quant-ph/818.