THE A-TEMPORAL COSMIC SPACE AND A GENERALIZATION OF THE DIRAC EQUATION

Transcription

1 THE -TEMPORL COMC PCE N GENERLZTON OF THE RC EQUTON avide Fisaletti aelife nstitute an Loenzo in Camo PU taly fisalettidavide@libeo.it bstat model desibing an a-temoal sae-gavity endowed with a quantum wave stutue is oosed. n this model the most fundamental hysial quantities ae the density of osmi sae hysial quantity whih deends on the amount of matte esent in the egion into onsideation the eleti density of sae hysial quantity whih indiates the amount of hage esent in the egion into onsideation and a quantum numbe indiating a sot of otation-oientation of eah oint of sae. mathematial fomalism egading the wave funtion of quantum-gavity sae is develoed whih is based on a genealized Fisaletti-ia equation fo the density of osmi sae. genealized Fisaletti-ia equation fo the density of osmi sae ouled with eletomagneti inteation is also intodued.. ntodution t the beginning of the new millennium a etain onetual onfusion is evident in diffeent hysial theoies models and ideas fo eamle elativity theoies standad model quantum gavity natue of sae time and mass and sense of highe sae dimensions to name a few. t is tue that quantum mehanis and geneal elativity the two fundamental XX entuy hysial theoies have obtained huge suess fom the editive oint of view but it is also tue that they ovide a itue of the wold whih is somewhat inomlete and fagmented. The most imotant hallenge in today s fundamental hysis is to find a oheent and satisfatoy unifiation of these theoies and thus to inooate onsistently gavity in the quantum mehanial sheme. s egads the hallenge of quantum gavity in ode to omlete the evolution oened by geneal elativity and quantum mehanis loo quantum gavity edits that at the most fundamental level hysial sae has a ganula stutue namely is omosed by elementay gains a net of inteseting loos also defined sin elements o sin netwos beause thei quantum numbes and thei algeba loo lie the sin angula momentum numbes of elementay atiles having the Plan size. On the gound of the esults of loo quantum gavity nodes of sin netwos eesent the elementay gains of sae and thei volume is given by a quantum numbe that is assoiated with the node in units of the elementay Plan volume / V G / whee is Plan s edued-onstant G is the univesal gavitation onstant and is the seed of light. n this atile following the hilosohy of loo quantum gavity we want to suggest a new view a new hysial model whih stats fom the idea that gavity-sae is a- temoal in the sense that time eists only as a steam of mateial movements and hanges and is haateized by a quantum wave and ganula stutue. The ental idea suggested by this model is that hysial univese an be desibed by intoduing a fundamental level of desition whih an be alled quantum-gavity sae whih is a

2 gavitational sae that ehibits wave stutue and ganula stutue. n atiula in this model eah egion of sae an be haateized by a wave funtion whih deends on two fundamental quantities: the density of osmi sae whih is lined to the amount of matte esent in the egion unde onsideation and a quantum numbe indiating a sot of otation-oientation of eah oint of sae. e will show that this wave funtion satisfy a fundamental genealization of the ia equation fo the density of osmi sae and that the standad ia equation of quantum theoy an be obtained fom this genealized ia equation in a atiula ase. e will analyze also the situation in whih the density of osmi sae is ouled with an eletomagneti inteation by intoduing a genealized ia equation fo the density of osmi sae with eletomagneti inteation.. The density of osmi sae and the gavitational inteation the eleti density of sae and the eletostati inteation On the basis of elementay eetion the assing of the time annot be eeived dietly as matte and sae; we an eeive only the ievesible hanges of matte in sae. Time annot be onsideed a eal hysial entity: it has not an autonomous eistene but eists only as a steam of ievesible mateial hanges haening in an a- temoal sae. t is emissible to assume on the gound of ou elementay eetion that hange does not un in time but hange itself is time. Time means hange movement and theefoe when thee is no hange thee is no time. Clos un in a- temoal sae. ith los we measue the duation and the numeial ode of all hanges. Phenomena un in sae-time only in the mathematial models of eality whih sometimes beome moe eal than eality itself whih instead on the gound of ou elementay eetion tuns out to be a-temoal. The stage in whih natual henomena haen is not sae-time but is an a-temoal sae. This is an altenative diffeent oint of view fom that onventionally adoted in hysis but is ehas moe oet and aoiate beause is moe oheent with eeimental fats i.e. with the fat that thee is no evidene that mateial objets move in time 4. tating fom the idea that sae is a-temoal inteesting esetives ae oened in theoetial hysis. n atiula in a-temoal hysial sae the fundamental inteations and hysial fields an be inteeted as seial states of sae unde etain iumstanes an be seen as entities whih deive fom eal oeties of sae 5. f time eists only as a steam of ievesible mateial hanges haening in sae we ae esented with the ossibility to ovide a new a-temoal inteetation of gavitational inteation. oding to this inteetation gavitational inteation is immediate in the sense that ats instantly though the density of osmi sae. Moeove gavitational foe is a-temoal: no movement of atile-wave is needed fo its ating. Gavity is tansmitted by the density of osmi sae eisting in eah oint of sae. The density of osmi sae an be onsideed the fundamental hysial oety whih haateizes univesal sae. t deends on the amount of matte esent in the egion into onsideation. The density of osmi sae assoiated with a mateial objet of mass m in the oints situated at distane fom the ente of this objet is defined by the Gm elation whee G is gavitational onstant. hile in the Newtonian view the soue of the gavitational field is mass in this model we suggest the idea that the gavitational field in eah oint of sae is a oety

3 that deends on the density of osmi sae eisting in that oint. t is the density of osmi sae that detemines the aeaane of a gavitational field in eah oint of sae. ntoduing the onet of the density of osmi sae assoiated with a given Gm mass m on the basis of elation the gavitational field assoiated with that density of osmi sae in the oints situated at distane fom its ente assumes the fom g ˆ. Equation shows that the gavitational field deives dietly fom the density of osmi sae haateizing a given oint. The fundamental ideas of the aoah hee suggested an be also eessed in the following way. e an say that if a egion of sae is haateized by a density of osmi sae given by in the oints situated at distane fom a given oint P this means that in the egion thee is a mateial objet of mass m and that the oint P is the ente of this mateial objet. t is the density of osmi sae that detemines the aeaane of a mateial objet in a given egion of sae. n othe wods onsideing a etain egion of sae the mass of a atile an be seen as the otion of that egion whee the density of osmi sae is bigge. This view allows us to inteet mass as a quantity whih is not muh diffeent fom sae as a onsequene of a oety of osmi sae namely its density. Newton s gavitational attation between two masses m and m situated at distane an be seen as a onsequene of a moe fundamental attation of two oints Gm of sae haateized by a diffeent density of osmi sae. n fat if is the density of osmi sae assoiated with a mateial objet of mass m in a given oint Gm of sae situated at distane fom its ente 4 is the density of osmi sae assoiated with a mateial objet of mass m in a given oint of sae situated at distane fom its ente is the distane between these two atiula oints we an wite: F g ˆ 5 whih eesents the geneal law of inteation G between these two densities of osmi sae the one assoiated with the mass m the othe assoiated with the mass m. Taing into aount elations and 4 equation 5 m m tuns out to be omletely omatible with equation F g G ˆ 6 whih is Newton s law of gavitational inteation witten in the standad fom. e an say that intoduing the density of osmi sae Newton s law 6 an be onsideed as a atiula ase of a moe geneal equation the equation 5 whih desibes the inteation between two densities of osmi sae the one assoiated with the mass m the othe assoiated with the mass m. The a-temoal model of gavitation inteation hee oosed oens theefoe this imotant esetive in theoetial hysis: two oints of sae haateized by a diffeent density of osmi sae attat eah othe and this onens all the oints of sae satisfying this ondition. nd the following inteetation of masses inteation deives: the mateial objets move in the dietion whee the density of osmi sae is ineasing. n othe wods a-temoal hysial sae an be seen as an elasti medium whih has a tendeny to shin. The moe medium is dense the stonge the foe of shining. The foe of shining is the gavitational foe 6. oding to the model hee oosed aeas of lowe density have a tendeny to move towads aeas of highe density. This is

4 the eason why objets with lowe mass have a tendeny to move towads objets with bigge mass. f the density of osmi sae is the most univesal hysial oety of a-temoal sae thee is howeve also anothe imotant hysial oety whih an be onsideed fundamental namely the eleti density of sae. The eleti density of sae is the hysial quantity whih indiates the amount of hage haateizing the egion of sae into onsideation and theefoe whih indiates the eleti oeties of a given egion of hysial sae. f we have a haged atile still in a given oint of sae this hage detemines a modifiation in the oeties of a-temoal osmi sae onening in atiula the tajetoies of othe haged atiles situated in that egion. n atiula the eleti density of sae assoiated to a hage q in the oints situated at Kq distane fom its ente an be defined though the elation e 7 whee l K is the onstant indiating the stength of the eleti foe with ε being the 4πε dieleti onstant of the vauum is the distane fom the ental quantum of sae. f a egion of sae is haateized by an eleti density of sae given by elation 7 this means that in this egion thee is a atile of hage q 7. Taing into onsideation its hysial dimensions the eleti density of sae an be inteeted as a measue of the eleti field fo unit of volume namely fo Plan volume l. ntoduing the onet of the eleti density of sae we an say that eah oint of a given egion endowed with the eleti density of sae 7 esents an eletostati field as a onsequene of this eleti density of sae. n atiula the eletostati field due to an eleti density of sae in a given oint at distane fom the ente of this eleti density an be eessed in the following way E l ˆ 8. Equation 8 shows lealy that the eleti field is a oety of sae deending on the value of the eleti density of sae haateizing that oint. e an say theefoe that the soue of the eleti field is not oely the eleti hage but the eleti density of sae. f mass an be seen as a stutue of sae deiving fom the density of osmi sae in a simila way hage an be seen as a oety of sae deiving fom the eleti density of sae. t is the eleti density of sae 7 that detemines the aeaane of a haged atile in a given egion of sae. n othe wods onsideing a etain egion of sae a hage an be seen as the otion of that egion whee an ootune eleti density of sae assumes its maimum value. n the lassial domain the eletostati foe ating between two hages q and q situated at distane an be seen as a onsequene of a moe fundamental inteation between two oints of sae haateized by a diffeent eleti density of sae. n fat Kq if e is the density of osmi sae assoiated with a atile of hage q in a l Kq given oint of sae situated at distane fom its ente e is the eleti l density of sae assoiated with a atile of hage q in a given oint of sae situated at distane fom its ente is the distane between these two atiula oints of 6 e e l sae we an wite: Fe ˆ 9 whih eesents the geneal law K of inteation between two eleti densities of osmi sae the one assoiated with the e 4

5 hage q the othe assoiated with the hage q. Equation 9 establishes that eletostati foe ats between egions of sae endowed with diffeent eleti densities of sae. t is imotant to obseve that taing into aount the definition of the eleti q q density of sae 7 Coulomb s law F e K ˆ is efetly equivalent to equation 9. e an theefoe onlude that intoduing the onet of the eleti density of sae Coulomb s law an be onsideed as a atiula ase of a moe geneal equation the equation 9 whih desibes the inteation between two eleti densities of sae. e ae esented theefoe with this inteesting esetive: two oints of sae haateized by a diffeent eleti density of sae inteat eah othe and this onens all the ais of oints of sae satisfying this ondition. nd the following inteetation of eletostati inteation deives: as eah oint of sae is haateized by the oety of the eleti density of sae not only the entes of the haged atiles move but also all the othe oints move aoding to equation 9.. -temoal gavitodynamis: the wave funtion of quantum-gavity sae and the Fisaletti-ia equation fo the density of osmi sae eveal authos have taen into onsideation the ossibility that sae is endowed with gavito/magneti oeties. n atiula in some eent atiles 8 9 Tuanyanin oosed the idea of a gavity-sae henomenon haateized by a non linea wave dynamis in whih the wave funtion desibing the state of sae is eessed in tems of G M two fields the gavitostati/eleti field E g ˆ having hysial dimensions of a G J wave veto l and the gavitoineti/magneti field B g having hysial dimensions of a fequeny t whee J L s L Mv is the obital angula momentum of the soue M is the soue mass s is the sin angula momentum of the soue. awing a stating oint fom these esults we want to suggest now a new view of a hysial sae endowed with a wave stutue and in whih time eists only as an ievesible motion of matte. oding to this inteetation the most fundamental hysial quantities desibing sae ae the density of osmi sae and a quantum veto Gs j defined as j and theefoe having the dimensions of an angula momentum fo unit of volume and multilied with the gavitation univesal onstant. This quantum veto j an be theefoe onsideed as a sot of otation-oientation of eah oint of sae. The quantum numbe j multilied fo / G an assume intege o halfintege multile values of the Plan edued onstant in ode to assue onsisteny with the esults of standad quantum mehanis. e will say thus that eah oint of sae is haateized by a deteminate value of the density of osmi sae and is endowed with a atiula otation-oientation lined to a atiula value of the quantum numbe j whih G an assume intege of half intege multile values of. n analogy with the oosal of Tuanyanin in the theoy hee suggested sae is desibed in tems of two fields: the gavistostati/eleti field defined on the basis of the 5

6 elation E g ˆ 4 whih is omletely omatible with and the v senϑ gavitoineti/magneti field defined on the basis of the elation ˆ j B g b 5 whee ϑ is the angle between and v bˆ is an unitay veto identifying dietion and vesus of L mv m being the mass of the atile assoiated with the density of osmi sae. The modulus of B g is B g v sen ϑ j v senϑ j os 6 whee is the angle between bˆ and j. t is imotant to obseve that the gavitostati/eleti field and the gavitoineti/magneti field hee defined have the hysial dimensions of a wave veto and a fequeny esetively. t aeas thus legitimate to intodue the wave funtion of sae ψ ˆ e πi v sen ϑ j v senϑ j os t ϕ 7 whee the amlitude is a funtion of the oint yz of sae. ine the density of osmi sae is the univesal hysial oety of sae will deend in geneal on the density of osmi sae and the seed of the atile detemined by this density of osmi sae. The wave funtion 7 of quantum-gavity sae an be onsideed the statingoint of an a-temoal inteetation of the univese in whih time eists only as motion of matte. n fat fo v and j onditions whih mean no motion one an easily see that this wave funtion assumes the fom ψ ˆ e πi ϕ 8 namely tuns out to be indeendent fom time: this means just that when thee is no motion thee is no time. G Now we want to analyze the atiula ase j and develo a mathematial G fomalism fo this. ine j oesonds to the aeaane of a atile of sin ½ one an intodue in this ase the idea of a geneal wave funtion of quantum-gavity sae ψ satisfying a ia-tye equation fo the density of osmi sae iγ ψ 9 whee t and γ ae the well-nown i elativisti maties γ γ σ i and σ ae the Pauli maties whih ae lined to σ j though the elation j G. Equation 9 an be onsideed the equation whih desibes a egion of sae haateized by a density of osmi sae whih detemines the esene of a atile of sin ½ without eletomagneti inteation namely in whih the density of osmi sae is not ouled with an eletomagneti inteation. The geneal solution of equation 9 an be eessed as P P ψ ψ ψ whee ψ eesents the wave funtion of sae assoiated with the aeaane of atiles of sin ½ and ψ eesents the wave funtion of sae assoiated with the aeaane of the oesonding antiatiles. P These two set of wave funtions of sae an be eanded as ψ b u 6

7 7 v d * ψ esetively. Hee u ae ositive-fequeny 4-sinos of sae while v ae negative fequeny 4-sinos of sae; they togethe fom a omlete set of othonomal solutions to 9. The label is an abbeviation fo the set j whee is the -momentum and Gs j with s is the label of the otationoientation of the genei oint of sae at distane fom the ente of the density of osmi sae namely whee assumes its maimum value. s egads the eessions of u and v equation 9 leads to the following esults: i u z e with z and i v z e 4 with z4 whee and E i E E 5 E E i E 6 E i E E 7 4 E E i E 8. whee E is the enegy of the atile assoiated with the density of osmi sae. Now by intoduing the quantities Ω P u u 9 Ω v v one an obtain P ψ and ψ fom ψ using P P d ψ ψ Ω and d ψ ψ Ω whee t t. One an also intodue the atile and antiatile uents assoiated esetively with

8 P the atile wave of sae ψ and the antiatile wave of sae ψ defined as ψ γ ψ P P P P J ψ γ ψ J whee ψ ψ γ. ine ψ and ψ seaately satisfies the ia-tye equation 9 the uents P J J P J and J ae seaately onseved. Theefoe one an intodue the idea of tajetoies of atiles P and antiatiles whih deive fom ψ and ψ on the basis of the equations: P P P d J t d J t P 4 P 5 esetively whee J J J J. dt J t dt J t t is also inteesting to obseve that the well-nown ia equation fo elativisti m atiles of sin ½ of standad quantum theoy i γ ψ 6 an be seen as a atiula ase of the equation 9. n fat equation 6 an be dietly obtained fom m m 9 in the ondition namely that is l and in oesondene the wave funtions and 4 beome the standad 4-sinos whih ae solutions of the standad ia equation of the standad quantum theoy fo atiles. This onsideation allows us to oen the following imotant esetive: equation 6 an be inteeted as the equation of the density of osmi sae fo the atiula oints of sae situated at distane l fom the ente of the density of osmi sae into onsideation. n othe wods the standad wave funtions of ia atiles an be seen as atiula ases of moe geneal wave funtions of quantum-gavity sae fo the atiula oints situated at distane l fom the ente of the density of osmi sae into onsideation. e ae theefoe esented with the ossibility that equation 9 an be onsideed as a moe fundamental equation than the standad ia equation 6: we an all equation 9 as the genealized ia equation fo the wave funtion of quantum-gavity sae without eletomagneti inteation o the Fisaletti-ia equation without eletomagneti inteation. Finally let us onside now a egion of sae haateized by a density of osmi sae whih detemines the esene of a atile of sin ½ whih is subjeted to an eletomagneti inteation namely a egion of sae in whih the density of osmi sae is ouled with eletomagneti inteation. n this ase we an intodue the idea of a wave funtion of quantum-gavity sae ψ satisfying a genealized ia-tye equation fo the density of osmi sae with eletomagneti inteation e l G iγ 4 ψ 7 whee e is the eleti density of sae K is K the eletostati onstant is the eletomagneti otential. The well nown ia equation fo elativisti atiles of sin ½ with eletomagneti inteation q m i γ ψ 8 an be seen as a atiula ase of the equation. n q e l G fat equation 8 an be dietly obtained fom 7 in the onditions 4 K Kq m m namely e and namely whih togethe oesond to Gl the ondition l. This onsideation allows us to oen the following imotant 8

9 esetive: equation 8 an be inteeted as the equation of the density of osmi sae ouled with eleti density of sae and eletomagneti inteation fo the atiula oints of sae situated at distane l fom the ente of the density of osmi sae and the eleti density of sae into onsideation. This imlies also that the standad wave funtions of ia atiles with eletomagneti inteation an be seen as atiula ases of moe geneal wave funtions of quantum-gavity sae ouled with eletomagneti inteation fo the atiula oints situated at distane l fom the ente of the density of osmi sae and the eleti density of sae into onsideation. e ae theefoe esented with the ossibility that equation 8 an be onsideed as a moe fundamental equation than the standad ia equation 55: we an all equation 8 as the genealized ia equation fo the wave funtion of quantumgavity sae with eletomagneti inteation o the Fisaletti-ia equation with eletomagneti inteation. The ossibility is oened that the Fisaletti-ia equation with eletomagneti inteation 7 an be esolved analogously to the oesonding standad ia equation of quantum theoy by intoduing a Feynman oagato inside a etubative method with eset to the eleti density of sae 7. n absene of the eleti density of sae we have the wave funtions of quantum-gavity sae without eletomagneti inteation solutions to the Fisaletti-ia equation these solutions on the basis of this etubative teatment an be inteeted as aoimations at the zeoth ode of the solutions to the Fisaletti-ia equation with eletomagneti inteation 7. hen the eleti density of sae is esent and is lead to its hysial value we have the solutions to the Fisaletti-ia equation fo the density of osmi sae with eletomagneti inteation 7 at the fist ode. bout the solutions of the Fisaletti-ia equation with eletomagneti inteation 7 and thei onsequenes futhe eseah will give you moe infomation. 4. Conlusions tating fom the idea that hysial sae is a-temoal in the sense that time eists only as motion of matte and intoduing the onets of the density of osmi sae the eleti density of sae and of the otation-oientation of eah oint of sae a new view of quantum-gavity sae aises. n this view sae an be desibed in tems of two fields the gavitostati field and the gavitoineti field whih ae defined in tems of the density of osmi sae and of the otation-oientation of eah oint of sae. n vitue of the natue of these fields a wave funtion of gavity-sae and theefoe the idea of a G wave quantum-gavity sae an be intodued. n the atiula ase j whih oesonds to the aeaane of a atile of sin ½ genealized Fisaletti-ia equations fo the density of osmi sae both without eletomagneti inteation and theefoe without the ouling with an eleti density of sae and with eletomagneti inteation and thus with the ouling of an eleti density of sae have been intodued. The well nown ia equations without eletomagneti inteation and with eletomagneti inteation fo elementay atiles of standad quantum theoy an be seen as seial ases of the genealized Fisaletti-ia equations fo the density of osmi sae without eletomagneti inteation and with eletomagneti inteation esetively. The lin between the standad ia equations without eletomagneti inteation and with eletomagneti inteation fo elementay atiles and the genealized Fisaletti-ia equations fo the density of osmi sae without eletomagneti inteation and with eletomagneti inteation has the imotant 9

10 onsequene that hysial sae tuns out to be not indefinitely divisible but to have a minimal size equal to Plan length. n this way the model hee suggested ovides an imotant justifiation of the fat that in a etain sense the idea of a wave stutue of sae must not be onsideed inomatible with the idea of a ganula stutue of sae. n sum the inteetation of quantum-gavity sae oosed in this atile an be onsideed the stating-oint of a modifiation in the th entuy field-geomety aadigm towads a eal ganula-wave-dynami itue and deely holisti view of the univese. Refeenes. oli. and Fisaletti. 5. tive Galati Nuleus s a Renewing ystem Of the Univese. Eletoni Jounal of Theoetial Physis Rovelli C.. Loo Quantum Gavity. Physis old oli. and oli. 4. Mathematial Time nd Physial Time n The Theoy Of Relativity. Eletoni Jounal of Theoetial Physis htt:// 4. oli. and oli. 4. -temoal gavitation. Eletoni Jounal of Theoetial Physis Fisaletti. and oli. 5. Towad an a-temoal inteetation of quantum otential. Fontie Pesetives oli. and oli. 5. -temoal gavitation and hyothetial gavitational waves. Eletoni Jounal of Theoetial Physis Puell E.M. 97. La fisia di Beeley Vol. : Elettiità e magnetismo. Bologna: Zanihelli. 8. Tuanyanin. 5. ave Gavity the way towads Plan sale. Millennium Relativity Paes etion. 9. Tuanyanin. 6. ave dynamis of quantum gavity-sae. Geneal iene Jounal.. Fisaletti. 8. Pesetives towads the density theoy of eveything ; ientifi nquiy Nioli H. 4. Bohmian atile tajetoies in elativisti femioni quantum field theoy. axiv:quant-h/5 v 7 et 4.. Nioli H. 4. Found. Phys. Lett Bjoen J.. and ell Relativisti quantum mehanis. New Yo: MGaw-Hill.