Search For Gravity in Quantum Evolution

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Horace Hart
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1 Sech Fo Gviy in Qunum Evoluion Alok Pndy Depmen of Physics, Univesiy of Rjshn, Jipu 34, Indi. Xiv:qun-ph/939 5 Sep The possibiliies of cuvue of spce-ime in he meic of qunum ses e invesiged. The cuvue of he meic coesponding o wve funcion of Hydogen om is deemined. Also, Einsein enso is descibed fo given qunum se. PACS numbes:.4.+m, 3.65.Bz, 4.6.+n * E-mil: belgem@dinfosys.ne

invesiged. The cuvue of he meic coesponding o wve funcion of Hydogen om is deemined.

2 .Inoducion The possible pplicions of he meic of qunum ses in he configuion spce clculed by Alok e l [], e invesiged in his lee. Physiciss ofen emp o descibe he gviy of ss, glxies nd he Univese. Now, i is woh sking wh is he cuvue of he spce-ime coesponding o n objec like Hydogen om. How he qunum undesnding of Hydogen om ccommodes wih he genel heoy of eliviy? The pesen sudy is n emp o invesige his spec of gviion in he ligh of sudy of qunum evoluion []. In he consucion of clssicl heoy of gviion in Genel Reliviy, we ecognize cein significn sges viz., (i) definiion of mnifold (ii) fomulion of meic enso (iii) descipion of pllel nspo nd geodesics, nd lsly (iv) fomulion of cuvue enso nd Einsein s field equion []. In he sudies of qunum evoluion lso, ll hese ides peining o gviy hve been evisied. Alhough, hey e no idenified excly wih qunum gviy bu hey e coming up neveheless in he sme mnne in which genel heoy of eliviy ws developed. Howeve, i cnno be clled qunizion of gviy, bu i does led o bee undesnding of qunum nue of gviy. Duing ps few yes mny

The pesen sudy is n emp o invesige his spec of gviion in he ligh of sudy of qunum evoluion []. In he consucion of clssicl heoy of gviion in Genel Reliviy, we ecognize cein significn sges viz.

3 geomeic conceps elevn o undesnding of qunum nue of gviy hve been discoveed in he sudy of qunum evoluion, fo exmple conceps like lengh, disnce nd geomeic phses, hve been descibed in hese sudies [3-4]. The meic of qunum se spce hs been fomuled [3-5]. Also, he meic of qunum ses hs been descibed in he configuion spce []. Mnifold nd spheicl mnifold hs been descibed in he sudy of qunum evoluion [5-6] nd lso, pllel nspo nd geodesic fomulion hs been exploed [7]. Pesenly, in sech of qunum nue of gviy vious peubive nd nonpeubive ppoches e in pogess such s: nonpeubive ppoch o qunum gviy in sech of qunum sucue of spce-ime, iniied by Ashek nd ohes [8], heoy of supe gviy [9], nd Hwking s ph of bidging clssicl nd qunum heoies of gviion [-]. And mny divese nd non-sndd emps e going on. And ye hee is scope fo exploing he nue of geomeic nd gviionl popeies ssocied wih physicl objecs idenified by hei especive qunum ses. In his lee we nlyze nd exploe he possible pplicions of he meic enso [] in he pemise of Genel Theoy of Reliviy. We believe h hese sudies of qunum evoluion will led us o bee undesnding of qunum nue of gviy. 3

Mnifold nd spheicl mnifold hs been descibed in he sudy of qunum evoluion [5-6] nd lso, pllel nspo nd geodesic fomulion hs been exploed [7].

4 . Cuvue in qunum evoluion In he sudy of qunum evoluion he meic of qunum ses in he configuion spce hs been obined [] s: g Ψ Ψ Re. () xµ xν And he coesponding invin ds is given by []: ds Ψ Ψ µ x x ν µ ν Re dx dx. () If we choose o wie his line elemen in specific fshion, e.g. in he pol co-odines (,,,) wih signue (+, +, +, -), he foesid expession ppes s: ds Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Re d Re d Re d Re c d + + c c (3). We cn compe his meic wih ny ohe genelized meic defined in he sme coodines nd wih he sme signue. Thus, we exmine nd nlyze is elevnce by comping i wih Robeson-Wlke line elemen, which epesens meic evolving wih ime: ds d S k ( ) + ( d + d ) c d. (4) The compison of g fom (3) wih (4) implies: 4

We cn compe his meic wih ny ohe genelized meic defined in he sme coodines nd wih he sme signue.

5 5 Ψ Ψ c c Re -c, (5) nd Ψ Ψ k S Re ) ( ; (6) which implies Ψ Ψ S k Re ) (. (7) Thus we cn clcule cuvue k, which obviously depends upon he nue of he wve funcion Ψ nd he scle fco S(). Also we find h Ψ Ψ Re ) ( S, (8) nd Ψ Ψ Re ) ( S. (9) To illuse n exmple, we clcule cuvue of he meic coesponding o wve funcion Ψ of Hydogen om, s descibed in [] by: d e C d e C d e C d e C ds + + () The compison of he equion (4) wih equion () gives ( ) e C k S ) ( ;

Also we find h Ψ Ψ Re ) ( S, (8) nd Ψ Ψ Re ) ( S.

6 so h k C S ( ), () e wih S ( ) C e. () Theefoe k co. (3) We noice h he expession of k in equion (3) is gul ; ; nd. Bu he guliies nd e he coodine guliies nd he only el physicl guliy exiss i.e. he disnce wice of Boh s dius. 3. Einsein enso nd Einsein s equion Now we efomule Einsein enso nd Einsein s field equion wih he help of he meic of qunum ses. Hee we cuiously specify h his ppoch could be only one picul ype of emen wih Einsein s field equion in his new scenio. 6

Einsein enso nd Einsein s equion Now we efomule Einsein enso nd Einsein s field equion wih he help of he meic of

7 We cn clcule cuvue enso R nd hence Einsein s enso R Rg. Now we clcule he enegy- momenum enso T wih he specific fom: T ρ v vµ ν dxµ dxν ρc (4) ds ds whee ds cd. To chnge his ino qunum mechnicl expession we mke following nsfomion: Wih P µ i!, (5) x µ we ge! Ψ Ψ v µ, (6) m x x µ µ nd similly! Ψ Ψ v ν. (7) m x x ν ν Theefoe v µ v ν! Ψ Ψ Ψ Ψ. (8) m xµ xµ xν xν Alenively, wih he knowledge of meic enso in qunum mechnicl fom we cn expess enegy momenum enso s meic deivive of cion S, which cn be clculed fo given se s: T δ S (9) δh 7

To chnge his ino qunum mechnicl expession we mke following nsfomion: Wih P µ i!, (5) x µ we ge! Ψ Ψ v µ, (6) m x x µ µ nd similly!

8 Thus he qunum mechnicl expession of he efomuled Einsein s field equion is; 8πG ρ! Ψ Ψ Ψ Ψ R 4 Rg () c m xµ xµ xν xν Howeve, we e we h his is no n eigen-equion nd he ems occuing in his equion such s ρ nd m e no qunum mechnicl obsevble. Bu hee e sill wo dvnges of his fomlism viz. (i) We cn scein he Einsein enso in ems of univesl gviionl consn, by clculing he igh hnd side of equion () which yields numeic figue fo specific wve funcion unde consideion. (ii) Vious mologicl models descibed fo specific qunum ses cn be esed wih his efomuled equion. 4. The Meic nd he wve funcion of he Univese Lsly, wh we poin ou hee is eled wih he ide of he wve funcion of he Univese []. The wve funcion of he Univese s fomuled by Hley nd Hwking is meic dependen funcion: ~ h ij, Φ] d[ g ] d[ Φ]exp( I [ g ]) () Ψ ± [ c± The meic consideed in his wve funcion is clssicl one. We wn o poin ou hee h fo he new bon Univese (ely Univese) jus fe big- 8

(i) We cn scein he Einsein enso in ems of univesl gviionl consn, by clculing he igh hnd side of equion () which yields numeic figue fo specific wve funcion unde consideion.

9 bng, he vey iniil wve funcion ime, could be given s funcion of coodines wih espec o oigin. And heefe wih evoluion in ime, meic g cn be clculed wih he help of evolving wve funcion. Thus, he vey iniil wve funcion of he Univese hs o be coodine dependen nd heefe wve funcion of he Univese cn be expessed s funcion of meic g. Refeences [] Alok Pndy nd Ashok K. Ngw, qun-ph/584 v. [] S. Weinbeg, Gviion nd Cosmology: Pinciples nd Applicions of he Genel Reliviy (John Wiley & Sons, New Yok, 97). [3] J. Anndn nd Y. Ahonov, Phys. Rev. Le. 65 No. 4(99) 697-7; J. Anndn, Phys. Le. A47 (99) 3-8; J. Anndn, Foundion of Physics. No. (99) [4] A. K. Pi, Physics Lees A59 (99) 5-; A. K. Pi, Jounl of Physics: Mh. nd Gen. 5 (99) L. [5] J. P. Povos nd G. Vllee 76 (98) [6] Alok Pndy nd Ashok K. Ngw (communiced). [7] Joseph Smuel nd R. Bhndi, Phys. Rev. Le. 6 (988) [8] A. Ashek, Phys. Rev. Le. 57 (986) 44-47; A. Ashek, Phys. Rev. D 36 (987) ; A. Ashek, Non Peubive Cnonicl Gviy, IUCAA Pune Lecue Noes- (99). [9] Vn Nieuwenhuizen, Pee, SupeGviy, Physics Repos, 68 (98)

Weinbeg, Gviion nd Cosmology: Pinciples nd Applicions of he Genel Reliviy (John Wiley & Sons, New Yok, 97). [3] J. Anndn nd Y. Ahonov, Phys. Rev. Le. 65 No. 4(99) 697-7; J. Anndn, Phys. Le. A47 (99) 3-8; J.

10 [] S. W. Hwking, Com. Mh. Phys. 43 (975) 99-. [] J. B.Hley nd S.W. Hwking, Phys.Rev. D8, No. (983) 96; S. W. Hwking, Nucle Phys. B39 (984)

Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].

Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b]. Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,