Dispersive Property of the Quantum Vacuum and the Speed of Light

Transcription

1 Dispsiv Popty of th Quatum Vacuum ad th Spd of Light H. Razmi (, N. Baamzadh ( ad H. Baamzadh ( Dpatmt of Physics, Th Uivsity of Qom, 76466, Qom, I. R. Ia. ( azmi@qom.ac.i, azmiha@hotmail.com; ( baamzadh@yahoo.com; ( hbaamzadh@yahoo.com Abstact W wat to study th ifluc of th quatum vacuum o th light popagatio. At fist, by wokig i th stadad lia quatum thoy of th lctomagtic filds, it is show that th lctic pmittivity ad th magtic pmability of th vacuum mdium a chagd; but, th sultig spd of light is't modifid. Th, takig ito accout olia ffcts by cosidig th Eul-Hisbg Lagagia, th cospodig zo poit (vacuum gy ad th sultig modificatio of th spd of light a foud up to th fist o-vaishig coctio. Kywods: Quatum vacuum, Vacuum polaizatio, Spd of light, th Eul-Hisbg Lagagia

2 . Itoductio I classical physics, th vacuum, as a mpty spac f of ay gy ad matt, has som physical poptis as th lctic pmittivity ad th magtic pmability wh th fiit valu of th spd of light i vacuum is witt i tms of thm as c /. If th vacuum was "othig", th it could't hav ozo lctomagtic poptis with a impdac ( Z valu of about 77 Ohms wll- kow i lctical ad commuicatio giig []. I quatum physics, th zo poit (vacuum stat has otivial poptis. Th spotaous missio of adiatio [], th Lamb shift [], th Casimi ffct [4], ad th adiatio coctios [5] a xplaid basd o th quatum vacuum fluctuatios wll-kow i th stadad quatum fild thoy. Th quatum vacuum gy is cosidd i th ivstigatio of som oth famous subjcts as th cosmological costat (th dak gy poblm [6]. I ct yas, som sachs hav tid to show that th xpimtally vifid valus of th lctomagtic costats ad may b fudamtally foud basd o th poptis of th quatum vacuum as a polaizd mdium [7-9]. Th scaio cosidd i ths sachs is basd o this ida that th quatum vacuum is a mdium which is full of cotiuously catd ad aihilatd vitual paticls. Ths paticls a amd "vitual" ot bcaus thy a't al, but bcaus thy a't obsvabl du to th uctaity picipl costait o thi liftim ad gy valus. I this pap, w wat to study th popagatio of light i th vacuum basd o a dsciptio of th quatum vacuum as a ffctiv polaizd mdium. At fist, by wokig i th stadad lia quatum thoy of th lctomagtic filds, it is show that th lctic pmittivity

3 ad th magtic pmability a chagd; but, th p-dfid valu of th spd of light c / mais umodifid. Th, by wokig with th Eul-Hisbg Lagagia [] ad cosidig its olia ffcts up to th fist o-vaishig coctio, it is show that th spd of light is modifid du to th dispsiv popty of th quatum vacuum mdium.. Th Elctomagtic Poptis of th Quatum Vacuum ad th Spd of Light I this sctio w wat to s how th popagatio of a al photo i th quatum vacuum iflucs th vacuum lctic pmittivity ad magtic pmability ad th chck if th spd of light chags... A Fudamtal Quatum Mchaical Aalysis Lt us cosid a f photo popagatig with th fqucy i a volum box of th scal lgth of th od of a lcto Compto wavlgth i which a vitual pai of lctoposito polaiz th vacuum. I atual uit, th cospodig lctic ad magtic filds E ad B a simply witt as : E B (. m c Th cospodig lctic polaizatio vcto, by dfiitio, is qual to th umb of th lctic dipol momts p uit volum : P dp c m dv c (. Thfo, th vacuum lctic pmittivity is foud as :

4 ~ P m E (. Simila cosidatio fo th magtic pmability of th quatum vacuum du to th magtic cut loop of th vitual lcto-posito pai lads us to th followig xpssios : M dm IA c m dv c c (4. ~ B (5. M m wh A is th aa of th cut loop of th adius of th od of a fudamtal lcto Compto wavlgth, m ad M a th wll- kow magtic dipol momt ad th magtizatio vcto. Usig ( ad (5, it is foud that th sultig spd of light mais umodifid : v ~ ~ ( c (6.. Th Vacuum Polaizatio ad th Modifid Popagato As a simpl quatum fild thotical dvlopmt to th appoach i [8], usig th wll-kow modifid photo popagato du to th vacuum polaizatio D ( k ZD ( k i which th cofficit Z is th wll-kow omalizatio costat [], ad kowig that th popagato of ay fild is a fuctio of th squa fom of it, th lctomagtic fild pottial is omalizd as: F F A R ( k Z A ( k (7.

5 Applicatio of th appopiat Foui tasfom latios fo th gy-momtum to th spac-tim ot sd sd l th followig modifid foms of th lctic pottial, th magtic pottial, th lctic fild, th magtic fild, th lctic polaizatio vcto, ad th magtizatio vcto: ' Z A' Z A E' Z E B' P vacuum Z E M vacuum Z B Z B (8. Th dict sult of th abov latios is that th vacuum has b polaizd with th followig modifid foms of th lctic pmittivity ad th magtic pmability : vacuum Z vacuum (9. Z Of cous, th spd of light dos't chag : v vacuum vacuum c (.. Th Quatum Vacuum as a Nolia Dispsiv Mdium I od to cosid th olia ffcts, it is appopiat to us th Eul - Hisbg fomulatio of quatum lctodyamics. I what follows, aft a shot itoductio to th ffctiv Eul- Hisbg thoy, it is show that th vacuum bhavs as a dispsiv mdium ad modifis th spd of light.

6 .. Th Eul-Hisbg Lagagia ad Hamiltoia Dsitis As is wll-kow, th ffctiv low gy (up to th scod od of th fi stuctu costat Eul-Hisbg Lagagia dsity is [, ]: = ( E B ( E B 7( E. B 4 45m (. I tms of th fou pottial A ad by applyig th Lotz gaug coditio, th Eul- Hisbg Lagagia ad Hamitoia dsitis a foud as: 9m A A A. A 4 4 A A A. A 7 A A A. A A A A. A ; A A + A. A 4 A A + A. A A A A. A (. m Th Hamiltoia fuctio cosists two tms wh th fist tm is th stadad wll-kow fuctio: +. d x (, H AA A A ad th scod tm is th olia coctio: m +.. H A A A A A A A A d x 4 (4.

7 .. Th Dispsio Rlatio du to th Nolia Effcts By cosidig th stadad Foui xpasio i th scod quatizatio of th fild A ( x i tms of th catio ad aihilatio opatos i a volum box V [5], th gy xpctatio valus ( E ad ( E a simply calculatd i tms of th photos umb ad thi momtum k as i th followig: ( E H k k, ; ( E H m 4 4V k, k, ( ( ( k. k k k kk k k (5. Th fist xpssio ( E is th wll-kow zo poit (vacuum gy i th stadad lia quatum lctodyamics. Th scod coctio tm ( E is a mo complicatd xpssio which ca b simplifid by som cosidatios. Th distac scal cospodig to th gy gim ud cosidatio h is costaid up to th lcto Compto wavlgth; this costait ad this fact that th olia ptubatio tm is at th scod od of th fi stuctu costat (, lad us to apply a cutoff valu fo th momtum up to th lcto mass m. Cosidig this cutoff valu ad usig d k' V k ' ( sult i: ( E k ( ( k, (6. Usig (5 ad (6, th total coctd zo poit (vacuum gy is foud as: Why ca w us th stadad Foui xpasio whil w a cosidig olia ffcts? I spos to this qustio, w should say that th olia coctios h a vy small so that th lctomagtic fild vaiatio is ough smooth that w ca wok i th sam way as i a ptubativ lia gim thoy.

8 E ( k 6 k, (7. Th abov sult asos o this fact that th quatum vacuum is a dispsiv mdium with th followig dispsio latio : ( k ( k (8. 6. Modificatio of th Spd of Light Accodig to th dispsio latio (8, th spd of light should b modifid as (comig back fom th atual uit : v c( (9. 6 Th umical coctio foud i (9 is about o p millio which sms it ca b asily vifid i xpic; but, th mai challg is i th xpimtally achivmt of th quid valu fo th lctic o magtic filds big abl to polaiz th vacuum []. 4. Discussio Why th sult (9 is icompatibl with th Lotz ivaiac? Sic th quatum vacuum sposs to th motio of th light as a ffctiv mdium with itativ back actios, dalig with vaiabl valu of th spd of light is uavoidabl. Such a vaiabl valu fo th spd of light is cosidd i a umb of byod stadad modls (.g. difft appoachs to quatum gavity i thm th Lotz ivaiac picipl is violatd. Th swift shot gamma ay busts is

9 a alady kow xampl fo th gy-dpdt vlocity of light at small gis tha th quatum gavity high gy scals []. 5. Coclusio I this pap, cosidig th quatum vacuum as a ffctiv mdium, it has b discussd about th oigi of th lctomagtic poptis of th vacuum basd o th adiatio coctios du to th vacuum polaizatio. It has b show that although th lctic pmittivity ad th magtic pmability of th vacuum a chagd du to th polaizatio mad by th popagatio of a al photo, th spd of light mais uchagd whil o woks i th stadad lia thoy of lctodyamics. By cosidig th olia coctios i th Eul-Hisbg modl i a gy gim up to a valu lss tha th lcto mass, th dispsio latio (8 ad th cospodig modificatio of th spd of light (9 hav b foud. Such quatum vacuum iflucs hav b alady kow i a umb of itstig ffcts as light by light scattig [4-5] ad photo splittig i vacuum [6]. Rfcs [] P. C. Clmmow, A Itoductio to Elctomagtic Thoy, p. 8. Cambidg Uivsity Pss (97. [] P. A. M. Diac, Th Quatum Thoy of th Emissio ad Absoptio of Radiatio, Poc. Roy. Soc. Lod. A.4, 4 (97. [] H. A. Bth, Th Elctomagtic Shift of Egy Lvls, Phys. Rv. 7, 9 (947.

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