The Source of the Quantum Vacuum

Transcription

1 Januay, 9 PROGRESS IN PHYSICS Volum Th Souc of th Quantum Vacuum William C. Daywitt National Institut fo Standads and Tchnology (tid), Bould, Coloado, USA wcdaywitt@athlin.nt Th quantum vacuum consists of vitual paticls andomly appaing and disappaing in f spac. Odinaily th wavnumb (o fquncy) spctum of th zo-point filds fo ths vitual paticls is assumd to b unboundd. Th unboundd natu of th spctum lads in tun to an infinit ngy dnsity fo th quantum vacuum and an infinit nomalization mass fo th f paticl. This pap agus that th is a mo fundamntal vacuum stat, th Planc vacuum, fom which th quantum vacuum mgs and that th gaininss of this mo fundamntal vacuum stat tuncats th wavnumb spctum and lads to a finit ngy dnsity and a finit nomalization mass. Intoduction Th quantum vacuum (QV) [] consists of vitual paticls which a catd alon (photons) o in massiv paticlantipaticl pais, both of which a jumping in and out of xistnc within th constaints of th Hisnbg unctainty pincipl ( E t ); i.., thy appa in f spac fo shot piods of tim ( t) dpnding upon thi tmpoal ngy contnt ( E) and thn disappa. So th QV is an vchanging collction of vitual paticls which disappa aft thi shot liftims t to b placd by nw vitual paticls that suff th sam fat, th pocss continuing ad infinitum. Th photon componnt of th QV is fd to h as th lctomagntic vacuum (EV) and th massiv-paticl componnt as th massiv paticl vacuum (MPV). Th quantum filds ascibd to th lmntay paticls a considd to b th ssntial ality [] bhind th physical univs; i.., a st of filds is th fundamntal building bloc out of which th visibl univs is constuctd. Fo xampl, th vcto potntial fo th quantizd lctomagntic fild can b xpssd as [, p. 45] X X c A(; t) V () s [a ;s (t) xp (i ) + h:c:] ;s ; wh th fist sum is ov th two polaizations of th fild, jj, V L 3 is th box-nomalization volum, a ;s (t) is th photon annihilation opato, h:c: stands fo th Hmitian conjugat of th fist tm within th bacts, and ;s is th unit polaization vcto. This is th quantizd vcto potntial fo th EV componnt of th QV. Th vcto potntial satisfis th piodicity conditions o quivalntly A(x + L; y + L; z + L; t) A(x; y; z; t) () ( x ; y ; z ) (L)(n x ; n y ; n z ) ; (3) wh th n i can assum any positiv o ngativ intg o zo. Sinc th Planc constant is considd to b a pimay constant, th fild in () is a fundamntal fild that is not divabl fom som oth souc (.g. a collction of chagd paticls). This pap agus that is not a pimay constant and thus that th is a mo fundamntal ality bhind th quantum filds. Th most glaing chaactistic of th EV (and similaly th MPV) is that its zo-point (P) ngy [, p. 49] X X s! c X ;s is infinit bcaus of th unboundd natu of th (j i j<) in (3). Th sum on th ight sid of th qual sign is an abbviation fo th doubl sum on th lft and! c. Using th wll-nown placmnt X ;s! X s L 3 d 3 V X 8 3 s (4) d 3 (5) in (4) lads to th EV ngy dnsity c X V c 3 d ; (6) ;s wh th infinit upp limit on th intgal is du to th unboundd in (3). Th psnt pap dos two things: it idntifis a chagd vacuum stat (th PV [3]) as th souc of th QV; and calculats a cutoff wavnumb (basd on an ali indpndnt calculation [4]) fo th intgal in (6). Th PV modl is psntd in th Sction. In a stochastic-lctodynamic (SED) calculation [4] Puthoff divs th paticl mass, th cutoff wavnumb (in tms of th spd of light, th Planc constant, and Nwton s gavitational constant), and th gavitational foc. Th Puthoff modl is viwd in Sction 3 and th sulting cutoff wavnumb changd into a fom mo usful to th psnt nds by substituting divd lations [3] fo th Planc and gavitational constants. William C. Daywitt. Th Souc of th Quantum Vacuum 7

2 Volum PROGRESS IN PHYSICS Januay, 9 Sction 4 agus that th QV has its souc in th PV. It accomplishs this sult by compaing th PV and QV ngy dnsitis. Th ad is asd to xcus th cous natu of th compaisons usd to ma th agumnt. Sction 5 commnts on th pvious sctions and xpands th PV thoy somwhat. Th d Bogli adius is divd in Appndix A to assist in th calculations of Sction 4. Th divation is supficially simila to d Bogli s oiginal divation [5], but diffs ssntially in intptation: h th adius aiss fom th two-fold ptubation th f paticl xts on th PV. Planc vacuum Th PV [3] is an omnipsnt dgnat gas of ngativngy Planc paticls (PP) chaactizd by th tiad (; m; ), wh, m, and () a th PP chag, mass, and Compton adius spctivly. Th chag is th ba (tu) lctonic chag common to all chagd lmntay paticls and is latd to th obsvd lctonic chag though th fin stuctu constant which is a manifstation of th PV polaizability. Th PP mass and Compton adius a qual to th Planc mass and lngth spctivly. In addition to th fin stuctu constant, th paticl-pv intaction is th ultimat souc of th gavitational (G m ) and Planc ( c ) constants, and th sting of Compton lations lating th PV and its PPs to th obsvd lmntay paticls and thi ba chag mc c mc ; (7) wh th chagd lmntay paticls a chaactizd by th tiad (; m; c ), m and c bing th mass and Compton adius ( c ) of th paticl. Paticl spin is not yt includd in th thoy. Th P andom motion of th PP chags about thi quilibium positions within th PV, and th PV dynamics, a th souc of both th f paticls and th QV. Th Compton lations (7) hav thi oigin in th twofold ptubation of th PV by th f paticl which polaizs and cuvs (in a gnal lativistic sns) th PV. Th paticl-pv intaction is such that th polaization foc ( ) and th cuvatu foc (mc ) a qual at th Compton adius c [3]: mc! c mc ; (8) wh th scond quation can b xpssd in its usual fom c mc. Th quimnt that th foc quality in (8) hold in any Lontz fam lads to th momntum (bp i ) and ngy ( E b opatos and to th d Bogli adius (Appndix A). Th so-calld wav-paticl duality of th paticl follows fom th coupling of th f paticl to th (almost) continuous natu of th PV whos continuum suppots th wav associatd with th wav popty ascibd to th paticl. 3 Puthoff modl On of th chags in th poduct tminating th chain of Compton lations (7) blongs to th f paticl whil th oth psnts th magnitud of th PP chags maing up th PV. Th fact that th ba chag is common to all th chagd lmntay paticls dpictd by (7) suggsts that phaps is masslss, and that th mass m in th paticl tiad (; m; c ) sults fom som action of th chag to th P filds. In a sminal pap [4] Puthoff, in ffct, xploits th ida of a masslss chag to div th paticl mass, th wavnumb c tuncating th spctum of th P filds, and th Nwtonian gavitational foc. This sction viws Puthoff s SED calculations and casts thm into a fom convnint to th psnt nds. Som mino licns is tan by th psnt autho in th intptation bhind quations () and (3) concning th constant A. Th Puthoff modl stats with a paticl quation of motion (EoM) fo th mass m m E zp ; (9) wh m, considd to b som function of th actual paticl mass m, is liminatd fom (9) by substituting th damping constant 3c 3 () m and th lctic dipol momnt p, wh psnts th andom xcusions of th chag about its avag position at hi. Th foc diving th paticl chag is E zp, wh E zp is th P lctic fild (B5). Equation (9) thn bcoms p 3c3 E zp ; () which is an EoM fo th chag that, fom h on, is considd to b a nw quation in two unnowns, and th cutoff wavnumb c. Th mass m of th paticl is thn dfind via th stochastic intic ngy of th chag whatv that may b. A asonabl guss is th intic ngy of th discadd mass m mc m p 3c 3 () alizing that, at bst, this choic is only a guid to pdicting what paamts to includ in th mass dfinition. Th dipol vaiation _p is xplaind blow. Th simplst dfinition fo th mass is thn m A _p c 3c 3 ; (3) wh A is a constant to b dtmind, along with and c, fom a st of th xpimntal constaints. Th th constaints usd to dtmin th th constants, c, and A a: ) th obsvd mass m of th paticl; ) th ptubd spctal ngy dnsity of th EV causd by adiation du to th andom acclations xpincd by th paticl chag as it is divn by th andom foc 8 William C. Daywitt. Th Souc of th Quantum Vacuum

3 Januay, 9 PROGRESS IN PHYSICS Volum E zp ; and 3) Nwton s gavitational attaction btwn two paticls of mass m. Th dipol momnt p in () can b adily dtmind using th Foui xpansions [6] p(t) p() xp ( it) d() (4) and E zp (; t) E zp () xp ( it) d() ; (5) wh p() and Ezp () a th Foui tansfoms of th dipol momnt vcto p and th fild E zp spctivly. Th mass of th paticl is dfind via th plana motion of th chag nomal to th instantanous popagation vcto in (B5) and sults in (Appndix B) _p (bx _p) 3 c5 c ; (6) wh bx is a unit vcto in som abitay x-diction and th facto accounts fo th -dimnsional plana motion. Whn th avag (6) is instd into (3), th constant m A c (7) mgs in tms of th two as yt unnown constants A and c. Acclation of th f ba chag by E zp gnats lctic and magntic filds that ptub th spctal ngy dnsity of th EV with which E zp is associatd. Th cosponding avag dnsity ptubation is [4] () c3 R 4 m c 3 A cr 4 4 ; (8) wh (7) is usd to obtain th final xpssion, and wh R is th adius fom th avag position of th chag to th fild point of intst. An altnativ xpssion fo th spctal ngy ptubation () mg c 3 R (9) is calculatd [4] fom th spactim poptis of an acclatd fnc fam undgoing hypbolic motion, and th quivalnc pincipl fom Gnal Rlativity. Sinc th two ptubations (8) and (9) must hav th sam magnitud, quating th two lads to th cutoff wavnumb c 3 c ; () A G wh G is Nwton s gavitational constant. Th final unnown constant A in () is dtmind fom th gavitational attaction btwn two paticls of mass m calculatd [4] using thi dipol filds and coupld EoMs, sulting in Nwton s gavitational quation F c3 c R m G AR ; () wh (7) and () a usd to obtain th final xpssion. Claly A fo th coct gavitational attaction, yilding fom () and (7) c 3 c () G and m c mg c 3 c c (3) fo th oth two constants. Th xpssions in th bacts of () and (3) a obtaind by substituting th PV xpssions fo th gavitational constant (G m ), th Planc constant ( c), and th Compton lation in (8). Th bact in () shows, as xpctd, that th cutoff wavnumb in (B5) is popotional to th cipocal of th Planc lngth (oughly th distanc btwn th PPs maing up th PV). Th bact in (3) shows th damping constant to b vy small, ods of magnitud small than th Planc tim c. Th smallnss of this constant is du to th almost infinit numb ( 99 p cm 3 ) of agitatd PPs in th PV contibuting simultanously to th P fild fluctuations. An asid: zittbwgung SED associats th zittbwgung with th EV [7, p. 396], i.. with th P lctic and magntic filds. In ffct thn SED tats th EV and th MPV as th sam vacuum whil th PV modl distinguishs btwn ths two vacuum stats. Taing plac within th Compton adius c of th paticl, th paticl zittbwgung can b viwd [, p. 33] as an xchang scatting btwn th f paticl and th MPV on a tim scal of about c c, o a fquncy aound c c. Th qustion of how th paticl mass divd fom th avaging pocss in (3) can b ffctd with th chag appaing and disappaing fom th MPV at such a high fquncy natually aiss. Fo this avaging pocss to wo, th fquncy of th avaging must b significantly high than th zittbwgung fquncy. This quimnt is asily fulfilld sinc c c c c. To s that th avaging fquncy is appoximatly qual to th cutoff fquncy c c on nds only consid th dtails of th avag (bx _p) R in (3) which involvs th intgal c d R 33 d. Ninty-nin pcnt of th avaging tas plac within th last dcad of th intgal fom 3 to 33 (th cosponding fquncy c in this ang bing wll byond th Compton fquncy c c of any of th obsvd lmntay paticls), showing that th ffctiv avaging fquncy is clos to c c. William C. Daywitt. Th Souc of th Quantum Vacuum 9

4 Volum PROGRESS IN PHYSICS Januay, 9 4 EV and MPV with tuncatd spcta Th non-lativistic slf foc acting on th f chag discussd in th pvious sction can b xpssd as [, p. 487] E slf d 3c 3 m (4) dt wh th adiation action foc is th fist tm and th nomalization mass is m 4 c 3c d (5) assumd h to hav its wavnumb spctum tuncatd at c. An infinit upp limit to th intgal cosponds to th box nomalization applid in Sction to quation (3) wh jn i j < is unboundd. Howv, if th nomal mod functions of th P quantum fild a assumd to b al wavs gnatd by th collction of PPs within th PV, thn th numb of mods n i along th sid of th box of lngth L is boundd and obys th inquality jn i j L p, wh is oughly th spaation of th PPs within th PV. Thus th cutoff wavnumb fom th pvious sction ( c p ) that cosponds to this n i placs th infinit upp limit odinaily assumd fo (5). So it is th gaininss ( ) associatd with th minimum spaation of th PPs in th PV that lads to a boundd i and n i fo (3), and which is thus sponsibl fo th finit nomalization mass (5) and th finit ngy dnsitis calculatd blow. Elctomagntic vacuum Combining (4) and (5) with a spctum tuncatd at c lads to th EV ngy dnsity [, p. 49] c X V c 8 3 d 3 c c 4 3 d d ;s 4c 4 3 c d c 4 c 4 c ; (6) wh th in font of th tipl intgal coms fom th sum ov s ; ; and wh c p and c a usd to obtain th final two xpssions. If th ngy dnsity of th PV (xcluding th stochastic intic ngy of its PPs) is assumd to b oughly half lctomagntic ngy ( ) and half mass ngy ( mc ), thn + mc 3 3 (7) is a ough stimat of this ngy dnsity. Thus th ngy dnsity (6) of th EV (th vitual-photon componnt of th QV) is at most on sixtnth (6) th ngy dnsity (7) of th PV. Although this stimat lavs much to b dsid, it at last shows th EV ngy dnsity to b lss than th PV ngy dnsity which must b th cas if th PV is th souc of th EV. Massiv paticl vacuum Th ngy dnsity of th P Klin-Godon fild is [, p. 34] hjhji V 3 () V V d 3 E 3 () c d d E c 4 E d c 4 c + d ; (8) wh 3 () V8 3 is usd to liminat 3 () and E p c + coms fom (A5). Equation (8) lads to hjhji V c 4 c 4 3 c c 4 c c c 6 3 c + c d h + ( c ) i d + c x ( + x ) dx ; (9) wh c c is usd in th fist lin. Th final intgal is asily intgatd [8] and lads to th xpansion in th scondto-last xpssion. Th final xpssion follows fom th fact that th scond ( c 4 ) and high-od tms in th xpansion a vanishingly small (th atio c is usd as a ough avag fo th atio of th Compton adii of th PP and th obsvd lmntay paticls). So th ngy dnsity in (9) is on thity-scond (3) of th PV ngy dnsity in (7). Th tm und th adical sign in (8) cosponds to th squad momntum of th massiv vitual paticls contibuting to th avag vacuum dnsity dscibd by (8). Th scond tm in th lag panthsis of (9) is appoximatly th lativ contibution of th vitual-paticl mass to th ovall ngy dnsity as compad to th cofficint in font of th panthsis which psnts th ngy dnsity of th vitual-paticl intic ngy. Thus th intic ngy of th vitual paticls in th MPV dominats thi mass ngy by a facto of about 4. 5 Conclusion and commnts Th conclusion that th PV is th souc of th quantum filds is basd on th fact that ( c) is a sconday constant, wh on of th s in th poduct is th paticl chag and th oth is th chag on th PPs maing up th PV; and that th amplitud facto A in th P lctic fild (B5) is popotional to th chag on th PPs in th PV. Th ubiquitous natu of! in th quantum fild quations, 3 William C. Daywitt. Th Souc of th Quantum Vacuum

5 Januay, 9 PROGRESS IN PHYSICS Volum whth is an lctomagntic wavnumb o a d Bogli wavnumb, futh suppots th conclusion. Th Compton lations (7) and th Puthoff modl in Sction 3 both suggst that th paticl chag is masslss. To b slf-consistnt and consistnt with th Puthoff modl, th PV modl fo th Compton lations must assum that th Compton adius c c (m) mc is lag than th stuctual xtnt of th paticl and th andom xcusions of th chag lading to th mass (3). Th PV thoy has pogssd to this point without addssing paticl spin its succss without spin suggsting phaps that spin is an acquid, ath than an intinsic, popty of th paticl. A ciculaly polaizd P lctic fild may, in addition to gnating th mass in (3), gnat an ffctiv spin in th paticl. This conclusion follows fom a SED spin modl [7, p. 6] that uss a ciculaly polaizd P fild in th modling pocss in od to avoid too much spculation though, on qustion lft unxplod in this spin modl is how th P fild acquis th cicula polaization ndd to div th paticl s spin. Phaps th P fild acquis its cicula polaization whn th magntic fild pobing th paticl (a laboatoy fild o th fild of an atomic nuclus) inducs a ciculation within th othwis andom motion of th PP chags in th PV, ths chags thn fding a cicula polaization bac into th P lctic fild E zp of th EV, thus lading to th paticl spin. Appndix A d Bogli adius A chagd paticl xts two distoting focs on th collction of PPs constituting th PV [3], th polaization foc and th cuvatu foc mc. Th quality of th two foc magnituds at th Compton adius c in (8) is assumd to b a fundamntal popty of th paticl-pv intaction. Th vanishing of th foc diffnc c mc c at th Compton adius can b xpssd as a vanishing tnso 4-foc [9] diffnc. In th pimd st fam of th paticl wh ths static focs apply, this foc diffnc F is ( ; ; 3; 4) F ; i c mc [ ; ; ; i ] ; (A) c wh i p. Thus th vanishing of th 4-foc componnt F4 in (A) is th souc of th Compton lation in (8) which can b xpssd in th fom mc c ( c)(c c )! c, wh! c c c mc is th Compton fquncy cosponding to th Compton adius c. Th 4-foc diffnc in th laboatoy fam, that is F a F, follows fom its tnso natu and th Lontz tansfomation x a x [9], wh x (x; y; z; ict), a i i and ; ; ; 3; 4. Thus (A) bcoms C A (A) F ; ; ; ; d [ ; ; ; i ] c mc d mc c ; i c ; i L mc L mc c (A3) in th laboatoy fam. Th quation F 3 fom th final two bacts yilds th d Bogli lation p c d d d (A4) wh p mv is th lativistic paticl momntum, d c is th d Bogli adius, and d d is th d Bogli wavnumb. Using (8) and (A4), th lativistic paticl ngy can b xpssd as E d m c 4 + c p 4 c + c d c + d ; (A5) wh mc c, c c, and c a usd to obtain th final two xpssions. Th quation F 4 fom (A3) lads to th lation p L, wh L c is th lngth-contactd c in th ict diction. Th Syng pimitiv quantization of flat spactim [] is quivalnt to th foc-diffnc tansfomation in (A3): th ay tajctoy of th paticl in spactim is dividd (quantizd) into qual lngths of magnitud c c (this pojcts bac on th ict axis as L L); and th d Bogli wavlngth calculatd fom th cosponding spactim gomty. Thus th dvlopmnt in th pvious paagaphs povids a physical xplanation fo Syng s spactim quantization in tms of th two ptubations and mc th f paticl xts on th PV. Appndix B Chag EoM with th slf foc Combining (4) and (5) lads to th chag s slf foc E slf d 3c 3 dt! (B) with! c p. Adding (B) to th ight sid of (9) thn yilds th x-componnt of th chag s acclation cosponding to (): dx x! dt x + 3c3 bx E zp (B) which can b solvd by th Foui xpansions and x(t) x() xp ( it) d() (B3) E x (; t) E x () xp ( it) d() (B4) wh E x bx E zp, and wh th P lctic fild E zp is assumd to hav an upp cutoff wavnumb c [4, 3]: E zp (; t) R X c d d b ()A xp i (!t + ()) ; (B5) William C. Daywitt. Th Souc of th Quantum Vacuum 3

6 Volum PROGRESS IN PHYSICS Januay, 9 wh R stands fo al pat of ; th sum is ov th two tansvs polaizations of th andom fild; th fist intgal is ov th solid p angl in -spac; p b is th unit polaization vcto; A! is th amplitud facto which is popotional to th ba chag of th PPs in th PV;! c; and is th andom phas that givs E zp its stochastic chaact. Th invs Foui tansfom of E x fom (B4) wos out to b X E x () (!) xp [ i ( + ())] + + ( +!) xp [ i ( + ())] c d d bx b ()A (B6) in a staightfowad mann, wh (!) and (+!) a Diac dlta functions. Equation (B6) is asily chcd by insting it into (B4) and compaing th sult with bx E zp fom (B5). Calculating x and dxdt fom (B3) and insting th sults, along with (B4), into (B) lads to th invs tansfom x() 3c 3 Ex() ( +! ) + i 3 (B7) fo x(t). Thn insting (B7) into (B3) yilds 3c 3 X c x(t) R d d bx b ()A (B8) xp [ i (!t + ())] ( +! )! + i! 3 fo th andom xcusions of th chag. Diffntiating (B8) with spct to tim whil discading th small tms in th dnominato lads to th appoximation 3c 3 X _x(t) R d 3 bx b () (B9) A i! xp [i (!t + ())]! fo th x-dictd vlocity, fom which th dipol avag (6) which diffs fom () only in dnominato on th ight sid of (B). Th last two tms in th dnominato a ods of magnitud small than on:! < c and c < c c p c. Thus th chag s slf foc is not a significant considation in th dfinition (3) of th paticl s mass. Submittd on Sptmb, 8 / Accptd on Sptmb 3, 8 Rfncs. Milonni P.W. Th quantum vacuum an intoduction to Quantum Elctodynamics. Acadmic Pss, Nw Yo, Winbg S. Th sach fo unity nots fo a histoy of Quantum Fild Thoy. Dadalus, v. 6, 977, Daywitt W.C. Th planc vacuum. Pogss in Physics, 9, Jan., v.,. 4. Puthoff H.E. Gavity as a zo-point-fluctuation foc. Phys. Rv. A, 989, v. 39, no. 5, d Bogli L. Un tntativ d intpétation causal t non linéai d la mécaniqu ondulatoi. Gauthi-Villas, Pais, 956. S Ch.. 6. Haisch B., Ruda A., Puthoff H.E. Intia as a zo-point-fild foc. Phys. Rv. A, v. 49, no., 994, Sction III and Appndix A of this fnc point out a small poblm with th calculations in Sction III of f. [4]. Th poblm is asily coctd by using Foui xpansions in Sction III of [4]. 7. d la Pña L., Ctto A.M. Th quantum dic an intoduction to Stochastic Elctodynamics. Kluw Acadmic Publishs, Boston, Dwight H.B. Tabls of intgals and oth mathmatical data. Th Macmillan Co., 3d d., NY, 947. S quation Jacson J.D. Classical Elctodynamics. John Wily & Sons, st d., nd pinting, NY, 96.. Syng J.L. Gomtical Mchanics and d Bogli Wavs. Cambidg Univsity Pss, 954. S pp.6 7. _p (bx _p) _x (t) 3 c5 follows, wh c is usd to liminat, and d 3 c d d c (6) (B) is usd to xpand th tipl intgal duing th calculation. Diffntiating (B8) twic with spct to th tim lads to th dipol acclation that includs th chag s slf foc: p 3 c c c R X c d d b () A xp [i (!t + ())] +! ; + i c (B) 3 William C. Daywitt. Th Souc of th Quantum Vacuum