Journal of Theoretcs Vol.4- A Unfed Feld Theory Peter Hckman peter.hckman@ntlworld.com Abstract: In ths paper, the extenson of Remann geometry to nclude an asymmetrc metrc tensor s presented. A new co-varant dervatve s derved, and used n the commutator of two co-varant dervatves of a vector. Ths leads to two equatons whch descrbe spn bosons. The energy-stress tensor arses as a contracton of the curvature tensor, ts dvergence enables the number of dmensons of Space-Tme to be determned. A weak feld approxmaton gves potental equatons for both massless and massve bosons. The Top quark mass s calculated to be 73eV/c^. The cosmologcal term arses naturally from the feld equatons and s n agreement wth observaton. Keywords: Remann manfold, unfed feld theory. Symbols: umber of Dmensons reek ndces- Latn Indces 3 - h, Plancks Constant.x, Plancks constant h/π c, e Wavelength of a partcle ω A scalar feld ntroduced n secton 4 and ω has same dmensons as k t E 8 Speed of lght n vacuum.998x ms ravtatonal constant Electrc Charge.x k Ensten Constant 8π/c tme Cosmologcal Total energy of g ρ g ρ All unts are n SI ' Constant' a partcle The curvature scalar Unts.7x 4-9 Wavelength of a partcle / π m Relatvstc mass of a partcle -34 C Js - m Kg
. Affne Connecton The nvarant nterval between two ponts on a Remann manfold s ds g dx dx Let D be the co-varant dervatve, the nvarance of ds, requres D g. For the affne connecton to be determned by a metrc tensor only, two cases arse: Case I: The metrc and affne connecton are both symmetrc: Γ Γ g g.3 Wth the condtons gven by equaton., and.3, the symmetrc affne connecton are the Chrstoffel Symbols []. Γ g g + g.4 ( ) Case : The metrc and affne connectons are both asymmetrc: Γ Γ ĝ ĝ. Wth the condtons gven by equaton., and., the asymmetrc affne connecton s Γ ( ĝ + ĝ + ĝ ). A general affne connecton can be formed from equatons.4 and. A Γ + Γ h.7 ( ) + Where the magnary part of the connecton s asymmetrc n m and n; t can be shown that usng condton. wth the connecton.7, that the affne connecton s A Γ + Γ.8 where the asymmetrc affne connecton s completely asymmetrc. Γ. ( ĝ + ĝ + ĝ ).9. The Feld Equatons The commutator of the co-varant dervatves of a complex vector usng.8 for the affne connecton gves where By addng ρ H A ρ [, D ρ ] Ψ ρψ + H ρψ + Γ ρ Ψ D. Γ Γ + Γ Γ Γ Γ Γ Γ + Γ Γ + Γ Γ ρ ρ ρ ρ lowerng a, two equatons arse: ρ ρ ρ ρ R ρ + R ρ ρ Γ Γ + Γ Γ + Γ Γ Γ Γ Γ Γ + Γ Γ Ψ ρ ρ to both sdes of equaton. and equatng the LHS to zero, and Ψ + Γ Ψ ρ ρ ρ. Γ ρ ( + ) Ψ + H ρ Ψ.3 Contractng equaton.3 by settng nb gves [( + ) Ψ ΓΨ ].4 In a geodesc frame, ths reduces to ( + ) Ψ * * and snce ( Ψ Ψ ) ( Ψ ) Ψ D, t can be shown that s magnary. Contractng equaton. by lettng bs and nr gves ρ ρ ρ ρ
[ Ψ + Γ ψ ]. The vector can be elmnated from.4 and. by usng the relatons : g where, are x matrces where g g to gve Ψ Γ Ψ. + Γ Ψ.7 For g η, the eucldean metrc, the matrces are found to satsfy the followng, (see secton 3 for the calculaton of ) I I + where I s the x unt matrx. A soluton for the,,3 matrces s x y 3 z x y z I3 I 3 where the matrces, are gven by [], x y, x z y These are the matrces for Spn partcles, thus the vector bosons. z Ψ s the wavefuncton for spn 3. Calculaton of, dmensons of Space-Tme The symmetrc connecton n equatons. and.7 can be elmnated to yeld ( + ) Ψ 3. Multply the expresson n parenthess by c gves the total energy E, c E + E where E c from whch t follows that E Sg S 3. c where s the number of dmensons. Contractng equaton 3. gves S, so equaton 3. can be wrtten as whch can be wrtten as g 3.3 R g R + R g R + g R 3.4 compare wth Enstens feld equatons of gravtaton wth a cosmologcal term [], R g R + kt g gves + R 3. Wth the ad of the followng denttes: Γ + Γ + Γ + Γ f f Γ Γ Γf Γ Γ Γ Γf Γ 3
wth f3 to, the followng expresson arses when calculatng the dvergence of R g R Γ f 3 f 3 f [( ) D Γ + D Γ ] whch vanshes f (-3/)3/, gves, Space-Tme s d. The dvergence of equaton 3.4 becomes 4. The Weak Feld Approxmaton D ( R g R ) Γ f J + g f 3. From equaton 3., the equaton for the current denstes s h Ĵ D Γ + Γ D Γ Γ 4. ( h ) f f Γ f h where Γ Γ Γh The weak feld approxmaton for the asymmetrc metrc tensor s ĝ η + ω 4. Φ where w s scalar feld and η To an approxmaton, equaton 4. s Ĵ Γ f f 4.3 Usng equaton 4. for the asymmetrc metrc, applyng the gauge condton ĝ, and dvdng by w, equaton 4.3 becomes J Φ Φ k Φ 4.4 f f f f f ω J Ĵ ω ω f ω ωk Wth wconstant, equaton 4.4 reduces to the classcal wave-equaton. In general equatons 4.4 s for nteractons medated by massve bosons. For w constant, equaton 4.3 can be wrtten n a more famlar form: J f F f + f Φ where F f Φ f Φ f 4. If Φ then J F Maxwells equatons of Electro-magnetsm are obtaned [3]. For the specal case of wconstant, equaton 3. s found to be, to an approxmaton: f f R g R ω ( F F g F F f ) + g 4. compare wth Ensten s feld equatons [] gves T f 4 ω k F F g 4 FF The partcular case for f, s the energy-stress tensor of the Electro-magnetc feld [3]. Equatons 4.3 consst of 4 sets of equatons, for the ndex f3 to. It can be shown that there are 4 'Electrc' felds, gven by the followng equatons E E 3 φ 4.7a φ4 + 4A A 4 E E 4.7b φ φ There are 4 'Magnetc' felds gven by 4 A B f curla f A A 4 4.7c
A 3 3 3 3 Φ A 4 4 4 Φ 43 A ( Φ, Φ, Φ 3, φ 4 Φ 4 ) A ( Φ, Φ, Φ 3, φ Φ 4, φ Φ ) ( φ Φ, φ Φ, φ ) ( Φ, Φ, ) where,,3 the followng dentfcaton can be made: φ are the colour potentals φ 4 s the electro-magnetc scalar potental and the scalar potentals φ are related to the weak nteracton, φ. Top Quark Mass - A Soluton of the Wave Equaton The rest-mass of a partcle depends on the scalar feld w, whch s determned by the followng equaton A general soluton n sphercal co-ordnates s ω ω A Br x 4 (+ ) + x (+ ) + 3x ( + ) ω e e. r where are seperaton constants, A s a constant of ntegraton and,, 3 / { ( + + 3 B )}. Applyng the followng condtons x 4 x 3x e e ω (,,,) k e wth l, m, n ntegers, gves the partcular soluton for w: ω ( r,,,) / π r / 4π ( l+ m+ ) ( ) e e e n k πr ω.3 Wth, dr/dtc t can be shown that a soluton for wave-functon s e r ( + r dr ) Ψ Ψ.4 The curvature scalar s approxmately ω ρc where ρ c s the energy densty and Rcc scalar R s neglgble. The wave-functon.4 peaks when the expresson n brackets vanshes. Let lmn n equaton.3, solvng for r and usng equaton 7. gves the energy to be c E( n) 9 where For n> equaton. smplfes to c 9 πn ( e 9 ) e. π n ( ) e. E n Snce then Ψ Ψ whch mples that the wavefuncton Ψ s 4d and the matrces reduce to the 4x4 Drac matrces, hence the wavefuncton s now for Fermons. For n, E73eV/c^ whch s near the rest-mass of the Top Quark.. The Cosmologcal Term Wth, equaton 3. for the cosmologcal term s R 3. Usng the Robertson-Walker metrc [], the current value for the cosmologcal term s where ( - q ) H + c k/r.
q H k,-or + R For k, and deceleraton parameter Hubbles constant Scale factor of the Unverse - H 7Kms Mpc whch agrees well wth observatons. - the present value of the cosmologcal term s -3.3x ( q ) 7. Modfed De Brogle Momentum Equaton For It can be shown that 4 + Γ 4 7. P4 + 7. 4,,,3, then 4 and n the weak feld approxmaton equaton 7. for the 4 th component of momentum reduces to 4 P4 h 7.3 44 m For h h h, x 4 ct, h η h 44 and solvng equaton 7.3 gves c r m P4 + 3 crt 7.4 For r/tc and P 4 mc, from equaton 7.4 mc Snce the LHS of equaton 7. s postve, then length. 3 cr 7. r > >.x 3 c 3 m e greater than the Planck 8. Concluson The result of extendng the geometry of space-tme to nclude an asymmetrc metrc can be summed up by equaton 3., whch gves the wavefuncton for spn bosons. In the Standard model of partcle nteractons the central theme s that quanttzaton of the gauge felds coupled to the Hggs feld yelds bosons. Here the boson masses are determned by the couplng to gravty so no Hggs mechansm s requred. Equaton 3. s smlar to the Drac equaton for non-nteractng spn / fermons when, whch suggests that the vanshng of the th and th components of the wave-functon transforms bosons nto fermons. Ths s confrmed wth the calculaton of 73eV/c^ for a fermon whch seems to be the Top Quark. Further work n ths area s beng undertaken n order to fnd a connecton wth the Standard model. References Kenyon I, eneral Relatvty, Oxford Unversty Press, ew York (99) Townsend J, A Modern Approach to Quantum Mechancs,Mcraw-Hll Co (99) Jackson J,Classcal Electodynamcs, John Wley & Sons Inc (97)
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