A Wave Equation including Leptons and Quarks for the Standard Model of Quantum Physics in Clifford Algebra

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1 Jouna of Modn Physs, 4, 5, 49-7 Pushd Onn Dm 4 n SRs A Wav Equaton nudn Lptons and Quaks fo th Standad Mod of Quantum Physs n Cffod Aa Caud Davau, Jaqus Btand L Moun d a Land, Poué-s-Cotaux, Fan Ema: CaudDavau@nodntf, tandaqus-m@oanf Rvd Oto 4; vsd 7 Novm 4; aptd Dm 4 Copyht 4 y authos and Sntf Rsah Pushn In Ths wok s nsd und th Catv Commons Attuton Intnatona Lns (CC BY) Astat A wav quaton wth mass tm s studd fo a fmon pats and antpats of th fst naton: ton and ts nutno, poston and antnutno, quaks u and d wth th stats of oo and antquaks u and d Ths wav quaton s fom nvaant und th C oup nazn th atvst nvaan It s au nvaant und th U ( ) SU SU oup of th standad mod of quantum physs Th wav s a funton of spa and tm wth vau n th Cffod aa C,5 Thn many fatus of th standad mod, ha onuaton, oo, ft wavs, and Laanan fomasm, a otand n th fam of th fst quantzaton Kywods Invaan Goup, Da Equaton, Etomantsm, Wak Intatons, Ston Intatons, Cffod Aas Intoduton W us h a notatons of nw nshts n th standad mod of quantum physs n Cffod aa [] Th wav quaton fo a pats of th fst naton s a nazaton of th wav quaton otand n 67 fo th ton and ts nutno Ths wav quaton has otand a pop mass tm ompat wth th au nvaan n [] It s a nazaton of th homonous nonna Da quaton fo th ton aon []-[9] How to t ths pap: Davau, C and Btand, J (4) A Wav Equaton nudn Lptons and Quaks fo th Standad Mod of Quantum Physs n Cffod Aa Jouna of Modn Physs, 5,

2 C Davau, J Btand wth ˆ ˆ β m φ + qaφσ + m φσ, q, m ξ η φ ξ ση ξ η ( ) φ η ξ η σξ ˆ ( ) η ξ H ξ and η a sptvy th ht and ft Wy spnos of th ton Th β an s th Yvon- Takaayas an satsfyn ( φ) () () () β dt Ω+Ω ρ (4) Th nk wth th usua psntaton of th standad mod s mad y th ft and ht Wy spnos usd fo wavs of ah pat Ths ht and ft wavs a pats of th wav wth vau n C,5 W usd pvousy th sam aa C5, C,5 It s th sam aa, and ths xpans vy w why su-aas C, and C, hav n quay usd to ds atvst physs [] [] But th snatu of th saa podut annot f, ths saa podut n nkd to th avtaton n th na atvty It happns that vtos of C,5 a psudo-vtos of C 5, and mo nay that n -vtos of C,5 a ( 6 n) -vtos of C 5, Th nazaton of th wav quaton fo ton-nutno s smp f w us C,5 Ths s th fst ndaton that th snatu + s th tu on W xpan n Appndx A how th vs n C,5 s nkd to th vs n C,, a nssay ondton to t th wav quaton of a pats of th fst naton W hav notd, fo th ton aon fsty (s [8] 4), nxt fo ton + nutno [] th dou nk xstn twn th wav quaton and th Laanan dnsty: It s w known that th wav quaton may otand fom th Laanan dnsty y th vaatona auus Th nw nk s that th a pat of th nvaant wav quaton s smpy Th Laanan fomasm s thn nssay, n a onsqun of th wav quaton Nxt w hav xtndd th dou nk to to-wak ntatons n th pton as (ton + nutno) Now w a xtndn th dou nk to th au oup of th standad mod Th Laanan dnsty must thn th a pat of th nvaant wav quaton Moov w nazd th non-na homonous wav quaton of th ton, and w ot a wav quaton wth mass tm [], fom nvaant und th C GL(, ) oup and au nvaant und th U ( ) SU au oup of to-wak ntatons Ou am s to xpan how ths may xtndd to a wav quaton wth mass tm, oth fom nvaant und C and au nvaant und th U SU SU au oup of th standad mod, nudn oth to-wak and ston ntatons Fom th Lpton Cas to th Fu Wav Th standad mod adds to th ptons (ton and ts nutno n ) n th fst naton two quaks u and d wth th oou stats ah Wak ntatons atn ony on ft wavs of quaks (and ht wavs of antquaks) w wt th wav of a fmons of th fst naton as foows: φ φn φ φn, ˆ ˆ ˆ ˆ φn φ φσ n φσ φ φ φ φ φ φ φ φ,, d u d u d u d u ˆ ˆ ˆ ˆ φ ˆ ˆ ˆ ˆ u φ d φuσ φ σ d φu φ d φuσ φ σ d φd φu φd φu ˆ φ ˆ φ ˆ φ σ ˆ φ σ u d u d Th to-wak thoy [] nds th spnoa wavs n th ton-nutno as: th ht ξ and th ft η of th ton and th ft spno η of th ton nutno Th fom nvaan of th Da n () () 5

3 C Davau, J Btand thoy mposs to us φ fo th ton and φ n fo ts nutno satsfyn * ξ η η φ * ( ), n ξ ση φn ξ η η n ˆ φ η ξ η σξ φ η η ( ) ˆ n, n ( n ) η η n ξ Wavs φ and φ n a funtons of spa and tm wth vau nto th Cffod aa C of th physa spa Th standad mod uss ony a ft η n wav fo th nutno W aways us th matx psntaton (A) whh aows to s th Cffod aa C, as a su-aa of M 4 ( ) Und th daton R wth GL, w hav (fo mo dtas, s [6]): ato ndud y any M n () (4) x MxM, dt M θ, x x µ σ, x x µ σ (5) µ µ ξ Mξ, η Mˆ η, η Mˆ η, φ Mφ, φ Mφ (6) n n n n φ φ n M φ φn N ˆ φ ˆ ˆ ˆ ˆ n φ M φn φ Th fom () of th wav s ompat oth wth th fom nvaan of th Da thoy and wth th ha onuaton usd n th standad mod: th wav ψ of th poston satsfs (7) ψ γψ ˆ φ ˆ φσ (8) W an thn thnk th wav as ontann th ton wav φ, th nutno wav φ n and aso th poston wav φ and th antnutno wav φ : n φ φ n ξ η ξ,, n φ φn ˆ ˆ φσ ξ n n φσ ξ η And th antnutno has ony a ht wav Th mutvto ( x) spa-tm aa aus (s [] (65)) wth: w ot ( ) ( n n) ( n n) (9) s usuay an nvt mnt of th a dt φ φφ ξ η + ξ η () a ξη + ξ η ηη ηη () a ξη + ξη () n n ( ) aa aa dt + () Most of th pdn psntaton s asy xtndd to quaks Fo ah oo,, th towak thoy nds ony ft wavs: φd φu ˆ ηd ˆ ηu, φd, φu ˆ φ ˆ ηd ηu u φ d (4) Th wav s now a funton of spa and tm wth vau nto C,5 C5, whh s a su-aa (on C5, M8 :, (5) th a fd) of Th nk twn th vs n C,5 and th vs n C, s not tva and s dtad n Appndx A Th wav quaton fo a ots of th fst naton ads ( D ) L +M (6) 5

4 C Davau, J Btand Th mass tm ads wh w us th saa dnsts mρχ mρχ M (7) mρχ mρχ s and χ tms of Appndx B, wth 5 aa aa aa ρ ss ρ + +, (8) Th ovaant dvatv D uss th matx psntaton (A) and ads W us two potos satsfyn Γk (9) k D BP W P G µ, µ, µ µ µ µ µ µ µ () D LD L B LB µ W LWµ,,, () µ k µ k G LGµ, k,,, 8 () µ Th opatos at on quaks k on ptons: P ( ) ± ( L ), L ± () P P L (4) + 5 P P L (5) + 5 P P + (6) Th fouth opato ats dffnty on th pton and on th quak sto Usn potos: w an spaat th pton pat and w t (s [] (B4) wth a ) + I4 P ( I8 + L45 ), P ( I8 L45 ) I4 and th quak pat of th wav: + +, P P (7) (8) P ( ) L + L (9) P L () Ths ast aton oms fom th non-xstn of th ht pat of th Chomodynams W stat fom natos λ of th k SU au oup of homodynams wavs 5

5 C Davau, J Btand λ, λ, λ, λ4, λ5, λ6, λ7, λ8 To smpfy h notatons w us now,,, nstad Thn () vs W nam,,, λ, λ, λ, λ4, λ5, λ6, λ7, λ8 Γ k opatos ospondn to ( L L L L ) So w hav () () λ k atn on W t wth potos P + and P n (7): () Γ ( L L L L ) Γ P P P P Γ Γ4 ( ) L5 P, Γ5 ( ) L4 P Γ6 ( ) P L5, Γ7 ( ) P L4 ( P L L P ) Γ Evywh th ft up tm s, so a Γ k pot th wav on ts quak sto W an xtnd th ovaant dvatv of to-wak ntatons n th ton-nutno as: D + BP ( ) + W P ( ) (9) to t th ovaant dvatv of th standad mod (4) (5) (6) (7) (8) k D( ) ( ) + BP ( ) + W P ( ) + G Γk ( ) () k wh s anoth onstant and G a ht tms ad uons Sn I 4 ommut wth any mnt of C, and sn Pµ ( nd ) Pµ ( nd ) fo µ,,, and nd,,, ah opato Γk ommuts wth a opatos P k Now w us a nums a, a,,,,, k,,, 8, w t k 8 k,, Γ k, + + k () S ap S ap S S S S S 5

6 C Davau, J Btand and w t, usn xponntaton ( S) ( S ) ( S ) ( S ) ( S ) ( S ) ( S ) ( S ) ( S ) ( S ) xp xp xp xp xp xp xp xp xp xp () n any od Th st of ths opatos xp( S ) s a U ( ) SU SU L oup Ony dffn wth th standad mod: th stutu of ths oup s not postuatd ut auatd Wth th au tansfomaton ads µ µ xp S, D LDµ, D LD µ () xp ( ) D µ S Dµ (4) B B a (5) µ µ µ W P xp S W P xp S xp S [ ] µ µ µ (6) k k G µ Γ k xp( S) Gµ Γk µ xp( S) xp ( S) (7) Th SU oup natd y opatos Γ k ats ony on th quak sto of th wav: k k + P xp Γ P P P (8) Th physa tansaton s: Lptons do not at y ston ntatons Ths oms fom th stutu of th wav tsf It s fuy satsfd n xpmnts W t thn a U ( ) SU SU au oup fo a wav nudn a fmons of th fst naton Ths oup ats on th pton sto ony y ts U ( ) SU pat Consqunty th wav quaton s omposd of a pton wav quaton and a quak wav quaton: ( D ) L + m ρ χ χ χ ( D ) L + mρχ, χ χ Th wav quaton (9) s quvant to th wav quaton studd n [] [], wh (9) () D γ + m ρχ, γ γγγ () aφ + aφnσ+ aφn aφlσ+ aφ R χ ρ ˆ ˆ ˆ ˆ ˆ aφlσ aφr aφ aφnσ aφ + + n () + σ σ φr φ, φl φ () Ths wav quaton s quvant to th nvaant quaton: φ ϕ n ( D ) γ + mρ χ, (4) φn ϕ Ths wav quaton s fom nvaant und th Lontz daton R ndud y any nvt matx M satsfyn (5), (6), (7) [] It s au nvaant und th U ( ) SU oup [] natd y opatos P µ whh a potons on th pton sto of th opatos dfnd n () to (9) Thfo w nd to study ony th quak sto and ts wav quaton () W n y th dou nk twn wav quaton and Laanan dnsty that w hav makd fsty n th Da quaton [8], nxt n th pton as ton + nutno [] 54

7 C Davau, J Btand 4 Dou Lnk twn Wav Equaton and Laanan Dnsty Th xstn of a Laanan mhansm n opts and mhans s known sn Fmat and Mauptus Ths pnp of mnmum s vywh n quantum mhans fom ts nnn, t s th man ason of th hypothss of a wav nkd to th mov of any mata pat mad y L d Bo [4] By th auus of vaatons t s aways poss to t th wav quaton fom th Laanan dnsty But anoth nk xsts: th Laanan dnsty s th a saa pat of th nvaant wav quaton Ths was otand fsty fo th ton aon [8], nxt fo th pa ton-nutno [] wh th Laanan dnsty ads mρ (4) µ µ µ ( ησ µ η ξσ µ ξ ησ n µ ηn) ˆ R + + (4) µ µ µ ˆ Bµ ησ η + ξσ ξ + ησ n ηn (4) W µ µ µ µ ( Wµ Wµ ) ησ η R + n + ( ησ η ησ n ηn) (44) W sha stash th dou nk now fo th wav quaton (6) It s suffnt to add th popty fo () Ths quaton s quvant to th nvaant quaton: D L + m ρ χ (45) χ χ, χ (46) χ W t fom th ovaant dvatv (9) wth th opatos P n (4), (5), (6) and () and Γ k n () to (8) and wth n (8) A A D (47) A A B γ + W γ + W γ W G G G + G + G + G A B γ + W γ+ W γ W 6 G G G G G A B γ + ( W γ+ W γ W ) 6 (4) G + G + G + G + G + G Nxt w t ( Aγ + mρχ ) + ( Aγ + mρχ ) ( A γ + mρχ ) (4) ( Aγ + mρχ ) ( Aγ + mρχ ) Th auaton of th Laanan dnsty n th na as s sma to th pton as W t + (4) D L + mρ χ,, (48) (49) 55

8 C Davau, J Btand m ρ (4),,,,,, Th auaton of,,, pas th pa -n y th pa d-u and suppss th ξ tms, thn (4) (4) (44) om µ µ ( η σ µ η η σ µ η ) R d d + u u (44) B µ µ µ d d + u u ( η σ η η σ η ) (45) 6 W µ µ µ µ ( Wµ Wµ ) ηdσ η R + u + ( ηdσ ηd ηuσ η u ) (46) Sn th SU oup a nudd n auaton of and w t SU th auaton of has smats wth th µ µ 4 5 µ µ ( Gµ Gµ )( ηdσ ηd ηuσ ηu ) ( Gµ Gµ )( ηdσ ηd ηuσ ηu ) R + + R + + G R Gµ µ µ µ + ( ηdσ ηd ηuσ ηu + ηdσ ηd + η uσ µ ηu η µ µ dσ ηd η uσ ηu ) 6 7 µ µ µ µ µ µ µ ( Gµ Gµ )( ηdσ ηd ηuσ ηu ) ( ηdσ ηd ηuσ ηu ηdσ ηd ηuσ ηu ) (47) Ths nw nk twn th wav quaton and th Laanan dnsty s muh ston than th od on, aus t oms fom a smp spaaton of th dffnt pats of a mutvto n Cffod aa Th od nk, on fom th Laanan dnsty to th wav quaton, supposs a ondton of anaton at nfnty whh s duous n th as of a popaatn wav On th physa pont of vw, th a no dffuts n th as of a statonay wav Dffuts n whn popaatn wavs a studd Ou wav quatons, sn thy a ompat wth an ontd tm and an ontd spa, appa as mo na, mo physa, than Laanans Ths a ony patua onsquns of th wav quatons On th mathmata pont of vw th od nk s aways avaa It s fom th Laanan dnsty (4) and usn Laan quatons that w hav otand th wav quaton (6) 5 Invaans 5 Fom Invaan of th Wav Equaton Und th Lontz daton R ndud y an nvt M matx satsfyn W thn t whh mps Thn w t x MxM, dt M θ, x x µ σ, x x µ σ (5) µ µ η Mˆ η, η Mˆ η, φ Mϕ, φ Mϕ (5) u u d d d d u u φ d φ u M φd φu N,,, ˆ φ ˆ ˆ ˆ ˆ u φ d M φu φ d N µ N, L µ N N N, N, N, D ND N N ( ) (5) (54) (55) D L ND N L D L (56) 56

9 C Davau, J Btand and w sha now study th fom nvaan of th mass tm A mps Ths vs ( φ ) ( φ) ( φ) s a dtmnants of a φ matx, ths s dt dt M dt M dt θ s (57) θ s s, ρ ρ (58) χ χ χ χ M Mˆ θ ρχ ρ χ ρ θ χ M Mˆ θ χ χ N χ θ (59) (5) (5) χ NN χ χ (5) Thn th fom nvaan of th wav quaton s quvant to th ondton on th mass tm nkd to th xstn of th Pank fato [] m ρ m ρ (5) m m (54) 5 Gau Invaan of th Wav Equaton Sn w hav pvousy povd th au nvaan of th pton pat of th wav quaton, t s ason nouh to pov th au nvaan of th quak pat of th wav quaton 5 Gau Goup Gnatd y P W hav h P L θ xp( θ P ) ( ) xp L µ µ µ To t th au nvaan of th wav quaton w must t (55) (56) B B B (57) θ θ χ χ xp L, χ χxp γ,,, Ths s satsfd aus θ θ σ σ d d, u u (58) φ φ φ φ (59) θ θ d d u u η η, η η, θ θ d d u u η η, η η (5) s s,,,,5 (5) θ 57

10 C Davau, J Btand A up tms n th matx χ ontan s φdσ and s φuσ tms W t θ θ σ d d d φ φ φ (5) And w fnay t θ θ θ σ σ s φ d σ φ d σ φ d σ φ d σ (5) θ χ χxp γ θ χ χ xp L θ D L + m D L + mρχ xp L ( ) ρχ Th wav quaton wth mass tm s au nvaant und th oup natd y P (54) (55) (56) 5 Gau Goup Gnatd y P W hav h Sn P W t L w t 5 Thn (5) s quvant to th systm o to th systm W thn t P L (57) 5 ( θp) xp( θl ) xp 5 (58) (59) Wµ Wµ µθ ( θp) xp( θl ) xp 5 (5) θγ,,, (5) ( θ) S ( θ) C os, sn (5) ˆ φ C ˆ φ S ˆ φ σ (5) d d u ˆ φ C ˆ φ S ˆ φ σ (54) u u d d d u d d u η Cη Sη, η Cη + Sη (55) η Cη Sη, η Cη + Sη (56) d d u d d u η Cη Sη, η Cη + Sη (57) u u d u u d η Cη Sη, η Cη + Sη (58) u u d u u d s C s S s + CS s s (59) 4 4 s C s S s + CS s s (54) s C s + S s + CS s (54)

11 C Davau, J Btand Ths mps Smay, pmutn oos, w t Ths mps and aso Ths mps Moov w t W thn t Nxt w hav and w t s C s + S s CS s (54) s s s s s s s s s s (54) s C s S s + CS s s (544) 5 5 s C s S s + CS s s (545) s C s + S s + CS s (546) 5 5 s C s + S s CS s (547) s s s s s s s s s s (548) s C s S s + CS s s (549) 6 s C s S s + CS s s (55) 6 6 s C s + S s + CS s (55) 6 s C s + S s CS s (55) 6 s s s s s s s s s s (55) s s, s s, s s (554) ρ ρ (555) A B A B χ, χ Bˆ Aˆ Bˆ Aˆ ( d d u u u ) ( ˆ ˆ ˆ ˆ ˆ u u d d d ) (556) A ˆ s ˆ φ ˆ φ ˆ φ ˆ φ ˆ φ σ (557) Bˆ sφ φ s φ s φ s φ σ (558) 7 Aˆ CAˆ SBσ ˆ (559) Bˆ CBˆ SAσ ˆ (56) C Sσ χ χ χ Sσ C Sn w t th sam aton fo and oos w fnay t ( L ) χ χ xp θ, 5 θγ xp D L + m ρχ D θl L + m ρχ 5 ( D ) L mρχ ( θl5 ) + xp (56) (56) 59

12 C Davau, J Btand Th wav quaton wth mass tm s thn au nvaant und th oup natd y P 5 Gau Goup Gnatd y P W hav h Sn P W t L w t 5 Thn (567) s quvant to th systm o to th systm W thn t Ths mps Smay, pmutn oos, w t P L (56) 5 ( θp ) xp( θl ) xp 5 (564) (565) Wµ Wµ µθ ( θp ) xp( θl ) xp 5 (566) θγ,,, (567) ( θ) S ( θ) C os, sn (568) ˆ φ C ˆ φ + S ˆ φ (569) d d u ˆ φ C ˆ φ S ˆ φ (57) u u d η Cη + Sη, η Cη + Sη (57) d d u d d u η Cη + Sη, η Cη + Sη (57) d d u d d u η Cη Sη, η Cη Sη (57) u u d u u d η Cη Sη, η Cη Sη (574) u u d u u d s C s + S s CSs + CSs (575) 4 4 s C s + S s + CSs CSs (576) s C s + S s + CSs CSs (577) 4 4 s C s + S s CSs + CSs (578) s s s s s s s s s s (579) s C s + S s CSs + CSs (58) 5 5 s C s + S s + CSs CSs (58) s C s + S s + CSs CSs (58) 5 5 Ths mps s C s + S s CSs + CSs (58)

13 C Davau, J Btand and aso s s s s s s s s s s (584) s C s + S s CSs + CSs (585) 6 Ths mps Moov w t W thn t Nxt w t wth (556) s C s + S s + CSs CSs (586) 6 6 s C s + S s + CSs CSs (587) 6 s C s + S s CSs + CSs (588) 6 s s s s s s s s s s (589) s s, s s, s s (59) ρ ρ (59) Aˆ CAˆ SBσ ˆ (59) Bˆ CBˆ + SAσ ˆ (59) χ χ C Sσ χ θγ Sσ C Sn w t th sam aton fo and oos w fnay t ( L ) χ χ xp θ, 5 xp ( D ) L + mρχ xp( θl5 ) D L + m ρχ D θl L + m ρχ 5 Th wav quaton wth mass tm s thn au nvaant und th oup natd y P (594) (595) 54 Gau Goup Gnatd y P W hav h P L (596) ( θp ) xp( θl ) xp (597) Sn P L w t (598) Wµ Wµ µθ ( θp ) xp( θl ) xp (599) Thn (597) s quvant to th systm o to th systm θγ,,, (5) ˆ θ φ ˆ φ (5) d d ˆ θ φ ˆ φ (5) u u 6

14 C Davau, J Btand W thn t Ths mps Nxt w t wth (556) η η, η η (5) θ θ d d d d η η, η η (54) θ θ d d d d η η, η η (55) θ θ u u u u η η, η η (56) θ θ u u u u s s, s s, s s (57) θ θ θ s s, s s, s s (58) θ θ θ s s, s s, s s (59) s s, s s, s s (5) s s, s s, s s (5) ρ ρ (5) ˆ θ ˆ θ A A, A A (5) ˆ θ ˆ θ B B, B B (54) θ θ χ χ χ θ Sn w t th sam aton fo and oos w fnay t ( L ) χ χ xp θ, xp D L + m ρχ D θl L + m ρχ ( D ) L mρχ ( θl ) + xp Th wav quaton wth mass tm s thn au nvaant und th oup natd y P 55 Gau Goup Gnatd y Γ W us now th au tansfomaton W an thn fot h must quvant to th systm ( θ) ( θ) (55) (56) C + S, C os, S sn (57) C + S (58) (59) Th au nvaan snfs that th systm G + m ρχγ G + m ρχγ G + m ρχγ G + m ρχγ,, (5) (5) 6

15 C Davau, J Btand Usn atons (57) and (58) th systm (5) s quvant to (5) f and ony f W nam f th au tansfomaton f: whh mps wth C os( θ ) and S sn ( θ ) G G θ (5) Γ (5) C + S xp ( θ f ) ( ) C + S Th quaty (57) s quvant to th systm Th quaty (58) s quvant to th systm Ths vs fo th nvaant saas W thn t (54) C + S (55) C + S (56) (57) η Cη + Sη, η Cη + Sη (58) d d d u u u η Cη + Sη, η Cη + Sη (59) d d d u u u η Cη + Sη, η Cη + Sη (5) d d d u u u η Cη + Sη, η Cη + Sη (5) d d d u u u s s s, s s, s s (5) s Cs Ss, s Cs Ss (5) s Cs Ss, s Cs Ss (54) s Cs + Ss, s Cs + Ss (55) s Cs + Ss, s Cs + Ss (56) s C s S s + CSs + CSs (57) s C s S s + CSs + CSs (58) s C s S s + CSs + CSs (59) s C s S s + CSs + CSs (54) ss s ss s (54) ss s ss s (54) s s s s s s (54) s s s s s s (544) s s s s s s s s s s (545)

16 C Davau, J Btand Nxt w t and w t wth (B7) and (B8) Ths vs th awatd sut ρ ρ (546) A B A B χ, χ Bˆ ˆ ˆ ˆ A B A A B A B χ, ˆ ˆ χ B ˆ ˆ A B A (547) (548) A CA SA, B CB SB (549) A CA SA, B CB SB (55) ρ ρ (55) χ Cχ S χ (55) χ Cχ S χ (55) Th han of sn of th phas twn (57) and (55) oms fom th antommutaton twn and 56 Gau Goups Gnatd y Γ k, k > W us wth k th au tansfomaton ( θ) ( θ) C + S, C os, S sn (554) C S (555) Th au nvaan snfs that th systm must quvant to th systm (556) G + m ρχγ G + m ρχγ G + mρχγ, G + mρχγ Usn atons (554) and (555) th systm (558) s quvant to (557) f and ony f aus w t, (557) (558) G G θ (559) ρ ρ (56) χ Cχ + Sχ (56) χ Cχ Sχ (56) Th as k s dtad n C and th as k 8 s dtad n C Cass k 4 and k 6 a 64

17 C Davau, J Btand sma to k and ass k 5 and k 7 a sma to k y pmutaton of ndxs of oo 6 Conudn Rmaks Fom xpmnta suts otand n th aatos physsts hav ut what s now known as th standad mod Ths mod s nay thouht to a pat of quantum fd thoy, tsf a pat of axomat quantum mhans On of ths axoms s that ah stat dsn a physa stuaton foows a Shödn wav quaton Sn ths wav quaton s not atvst and dos not aount fo th spn / whh s nssay to any fmon, th standad mod has vdnty not foowd th axom and has usd nstad a Da quaton to ds fmons Ou wok aso stats wth th Da quaton Ths wav quaton s th na appoxmaton of ou nonna homonous quaton of th ton Th wav quaton psntd h s a wav quaton fo a assa wav, a funton of spa and tm wth vau nto a Cffod aa It s not a quantzd wav wth vau nto a Htan spa of opatos Nvthss and onsqunty w t most of th aspts of th standad mod, fo nstan th fat that ptons a nsnstv to ston ntatons Th standad mod s muh ston than nay thouht Fo nstan w fsty dd not us th nk twn th wav of th pat and th wav of th antpat, ut thn w ndd a at Cffod aa and w oud not t th nssay nk twn vsons that w usd n ou wav quaton W aso ndd th xstn of th nvs to ud th wav of a systm of pats fom th wavs of ts omponnts And w ot two na dntts whh xstd ony f a pats of th na wav w ft wavs, ony th ton havn aso a ht wav Th most mpotant popty of th na wav s ts fom nvaan und a oup nudn th ovn oup of th sttd Lontz oup Ou oup dos not xpan why spa and tm a ontd, ut t spts ths ontatons Th physa tm s thn ompat wth thmodynams, and th physa spa s ompat wth th voaton of paty y wak ntatons Th wav aounts fo a pats and ant-pats of th fst naton W hav aso vn [] [8] [9] [] th ason of th xstn of th natons; t s smpy th dmnson of ou physa spa Sn th SU au oup of homodynams ats ndpndnty fom th ndx of natons, th physa quaks may omnatons of wavs of dffnt natons Quaks omposn potons and nutons a suh omnatons Ou wav quaton aows ony two masss at ah naton, on fo th pton pat of th wav, th oth on fo th two quaks Th mxn an v a dffnt mass fo th two quaks of ah naton Sn th wav quaton wth mass tm s au nvaant, th s no nssty to us th mhansm of spontanous symmty akn Th saa oson tany xsts, ut t dos not xpan th masss A wav quaton s ony a nnn It sha nssay to study aso th oson pat of th standad mod and th systms of fmons, fom ths wav quaton A onstuton of th wav of a systm of dnta pats s poss and ompat wth th Pau pnp [] [7] Rfns [] Davau, C and Btand, J (4) Nw Inshts n th Standad Mod of Quantum Physs n Cffod Aa J- Pu, Poué-s-Cotaux [] Davau, C and Btand, J (4) Jouna of Modn Physs, 5, - [] Davau, C (99) Equaton d Da non néa PhD Thss, Unvsté d Nants, Nants [4] Davau, C (997) Advans n Appd Cffod Aas, 7, [5] Davau, C (5) Annas d a Fondaton Lous d Bo,, [6] Davau, C () L spa-tmps dou JPu, Poué-s-otaux [7] Davau, C () Advans n Appd Cffod Aas,, [8] Davau, C () Dou Spa-Tm and Mo JPu, Poué-s-Cotaux [9] Davau, C () Nonna Da Equaton, Mant Monopos and Dou Spa-Tm CISP, Camd [] Dhuvs, R (99) Tnsus t spnus PUF, Pas Th vson s an ant-somophsm hann th od of any podut (s [] ) It s spf to ah Cffod aa Th Appndx A xpans th nk twn th vson n C, and th vson n C,5 65

18 C Davau, J Btand [] Hstns, D (986) A Unfd Lanua fo Mathmats and Physs and Cffod Aa and th Intptaton of Quantum Mhans In: Chshom, JSR and Common, AK, Eds, Cffod Aas and Th Appatons n Mathmats and Physs, Rd, Dodht, - [] Wn, S (967) Physa Rvw Ltts, 9, [] Davau, C (4) Gau Goup of th Standad Mod n C,5 ICCA, Tatu [4] d Bo, L (94) Annas d a Fondaton Lous d Bo, 7 66

19 C Davau, J Btand Appndx A Cauaton of th Rvs n C 5, H ndxs µ, ν, ρ, hav vau,,, and ndxs a,,, d, hav vau,,,, 4, 5 W us th foown matx psntaton of C,5 : wh γ µ I4 I Lµ ; L4 ; L5 ; ; γ µ I4 I I σ ; ;,, σ γ γ γ γ I σ a Pau mats Ths vs W t aso Smay w t L γµ γ µν ν γ LL γµ γ γ ν µν µν µ ν γµν γρ γµνρ Lµνρ Lµν Lρ γµν γρ γµνρ L γ L L γ I L LL L L I4 45 I4 I 4 I4 L5 LL5 I L L 4 γ γ µ 4, L µ µ µ 5 γµ γµ γ γ µν 4, L µν µν µν 5 γµν γµν (A) (A) (A) (A4) (A5) (A6) (A7) (A8) (A9) Saa and psudo-saa tms ad L L L γ γµνρ γµνρ µνρ µνρ 4, Lµνρ 5 γµνρ γ γ µ 45, L µν µ µν 45 γµ γµν γ, µνρ µνρ 45 L4 γ µνρ (A) (A) (A) I, I 4, I 8 a unt mats Th dntfaton poss aown to nud n ah a Cffod aa aows to ad a nstad of ai n fo any ompx num a ant-ommuts wth any odd mnt n spa-tm aa and ommuts wth any vn mnt 67

20 C Davau, J Btand Fo th auaton of th -vto tm w t Ths vs Fo th auaton of th -vto tm w t Ths vs Fo th auaton of th -vto tm w t Ths vs wth (A) and (A9) Fo th auaton of th 4-vto tm ( α ω) + I4 αi8 + ωl45 ( α ω) I4 ( α ω) I4 αi8 ωl45 ( α + ω) I4 a 4 5 N La N L4 + N L5 + N µ Lµ 4 5 N, N, N µ µ (A) (A4) β δ a γ (A5) a N La βi4 + δ+ a βi4 + δ+ a a 45 µ 4 µ 5 µν N La N L45 + N Lµ 4 + N Lµ 5 + N Lµν 45 µ 4 µ 5 µν N, N γµ, N γµ, A N γµν (A6) (A7) a + + A N L a + A a N La N µ L 45 N µν L 4 N µν L 5 N µνρ µ + µν + µν + Lµνρ µ 45 µν 4 µν 5 µνρ N γµ, N γµν, N γµν, N γµνρ (A8) d B C (A9) a d B + C + N L a d + B + C + (A) w t ad N Lad N µν L 45 N µνρ L 4 N µνρ µν + µνρ + Lµνρ 5 + N L D N γ, f N γ, N γ, ζ N (A) µν 45 µνρ 4 µνρ 5 µν µνρ µνρ Ths vs wth (A4) and (A) ad N L ad Fo th auaton of th psudo-vto tm w t D + f + + ζ D f + + ζ ad N Lad N µνρ Lµνρ 45 + N L4 + N L5 (A) 68

21 C Davau, J Btand s Ths vs wth (A7) and (A) W thn t h N γ, η N, θ N (A) N ad µνρ µνρ L ad h η θi4 h+ η θi4 ( α ω) I4 ( ζ ) ( β θ) I4 ( δ η) ( β θ) I + ( a h) + ( B + C) + ( d + ) + ( δ + η) ( α ω) I + ( + ) + ( A + D) + ( f ) + ( ζ + ) (A4) A D + + f a + h + B + C + d (A5) Ths mps In C, th vs of α + ω A D + + f + ζ (A6) β + θ + a + h + B + C + d + + δ η (A7) β θ + a h + B + C + d + + δ + η (A8) ( ) ( A D) ( f ) α ω ζ + (A9) A A + A + A + A + A 4 A A + A A A + A 4 w must han th sn of vtos A, B, C, D, and tvtos, d,, f and w thn t ( α + ω) + ( + ) + ( A + D) + ( f ) + ( ζ ) (A) s β + θ + a + h + B C + d + δ η (A) β θ + a h B + C d + + δ + η (A) ( ) ( A D) ( f ) α ω ζ + (A) Th vs, n C,5 now, of A A + A + A + A + A + A + A A A + A A A + A + A A Ony tms whh han sn, wth (A), (A8) and (A), a saas and ω, vtos,, d, and vtos A, B, C Ths hans of sn a not th sam n C,5 as n C, Dffns a otd y th fat that th vson n C,5 aso xhans th pa of and tms W thn t fom (A5) ( α ω) I4 ( ζ ) ( β θ) I4 ( δ η) ( β θ) I + ( ) ( + ) ( + ) + ( δ + η) ( α + ω) I + ( + ) + ( + ) + ( ) + ( ζ ) A D + + f a + h + B C + d + 4 a h B C d 4 A D f (A4) 69

22 C Davau, J Btand Ths nk twn th vson n C, and th vson n C,5 s nssay to t an nvaant wav quaton It s not na, fo nstan th vson n C s not nkd to th vson n C, Appndx B Saa Dnsts and χ Tms Th a suh ompx saa dnsts: W usd n [] wth φ φ ( + σ ) and φ φ ( σ ) R ( u u u u ) ( u u u u ) ( u u u u ) ( u u u u ) ( u u u u ) ( u u u u ) ( d d d d ) ( d d d d ) ( d d d d ) ( d d d d ) ( ) d d d d ( d d d d ) ( u d u d ) ( u d u d ) ( u d u d ) ( u d u d ) s ξ η + ξ η η η η η (B) s ξ η + ξ η η η η η (B) s ξ η + ξ η η η η η (B) s ξ η + ξ η η η η η (B4) 4 s ξ η + ξ η η η η η (B5) 5 s ξ η + ξ η η η η η (B6) 6 s ξ η + ξ η η η η η (B7) 7 s ξ η + ξ η η η η η (B8) 8 ( u d u d ) ( u d u d ) s ξ η + ξ η η η η η (B9) 9 ( u d u d ) ( u d u d ) s ξ η + ξ η η η η η (B) ( u d u d ) ( u d u d ) s ξ η + ξ η η η η η (B) ( d u d u ) ( u d u d ) s ξ η + ξ η η η η η (B) ( u d u d ) ( u d u d ) s ξ η + ξ η η η η η (B) ( d u d u ) ( u d u d ) s ξ η + ξ η η η η η (B4) 4 ( d u d u ) ( u d u d ) s ξ η + ξ η η η η η (B5) L 5 aφ + aφnσ+ aφn aφlσ+ aφ R χ ρ ˆ ˆ ˆ ˆ ˆ aφlσ aφr aφ aφnσ aφ + + n, and w nd now ( s d s d s u s u s u ) ( s u s u d d d ) ( s ˆ ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ˆ u u s d s d s d s d d u u u ) 4φ 6φ 7φ φ 4φ σ φ φ 7φ φ φ σ ρ χ φ φ 7φ φ φ σ 4φ 6φ 7φ φ 4φ σ ( s d s d s u s u s u ) ( s u s u d d d ) ( s ˆ ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ˆ u u s d s d s d s d d u u u ) 5φ 4φ 8φ φ 5φ σ φ φ 8φ φ 4φ σ ρ χ φ φ 8φ φ 4φ σ 5φ 4φ 8φ φ 5φ σ ( s d s d s u s u s u ) ( s u s u d d d ) ( s ˆ ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ˆ u u s d s d s d s d d u u u ) 6φ 5φ 9φ φ φ σ φ φ 9φ φ 5φ σ ρ χ φ φ 9φ φ 5φ σ 6φ 5φ 9φ φ φ σ (B6) (B7) (B8) (B9) 7

23 C Davau, J Btand Appndx C Gau Invaan, Dtas C Gau Goup Gnatd y Γ W nam whh mps f th au tansfomaton f Th quaty (C) s quvant to Th quaty (C4) s quvant to W t : Γ (C) θ xp ( θ f ) ( ) θ θ θ (C) (C) (C4) (C5) θ φd φu φd φu ˆ ˆ θ φ ˆ ˆ u φ d φu φ d φ d φ θ u φd φu ˆ φ ˆ θ ˆ ˆ u φ d φu φ d θ θ d d, u u (C6) (C7) η η η η (C8) η η η η (C9) θ θ d d, u u η η η η (C) θ θ d d, u u Ths vs fom whh w t η η, η η (C) θ θ d d u u s s s s s s (C) θ θ,, s s s s s s (C) θ θ 4 4, 5 5, 6 6 s s s s s s (C4) θ θ 9 9, 8 8, 7 7 s s s s s s (C5) θ θ,, s s s s s s (C6) θ θ 4 4, 5 5, ss ss,,,,5 (C7) ρ ρ (C8) θ χ χ (C9) 7

24 C Davau, J Btand Ths atons a th awatd ons aus C Gau Goup Gnatd y Γ 8 W nam f 8 th au tansfomaton whh mps Ths vs W thn t θ χ χ (C) θ θ ( ) ( θ ) θ θ ( ) ( θ ) + (C) + (C) G G θ (C) f8 : Γ8( ) (C4) θ xp ( θ f8 ) ( ) θ θ (C5) θ xp θ xp θ xp xp θ θ φ d φd, φ u xp φu xp θ θ φ d φd, φ u xp φu θ θ φ d xp φd, φ u xp φu xp θ, xp θ, xp θ η η η η η η d d d d d d xp θ, xp θ, xp θ η η η η η η d d d d d d xp θ, xp θ, xp θ η η η η η η u u u u u u (C6) (C7) (C8) (C9) (C) (C) (C) (C) (C4) 7

25 C Davau, J Btand θ θ θ η u xp ηu, η u xp ηu, η u xp ηu (C5) Ths mps s xp θ s, s xp θ s, s xp θ s s xp θ s, s xp θ s, s xp θ s s xp θ s, s xp θ s, s xp θ s s xp θ s, s xp θ s, s xp θ s s xp θ s, s4 xp θ s4, s5 xp θ s5 W thn t th awatd suts (C6) (C7) (C8) (C9) (C4) ss ss,,,,5, ρ ρ (C4) θ θ θ χ xp χ, χ xp χ, χ xp χ (C4) 7

26