On Hypercomplex Extensions of Quantum Theory

Transcription

1 Journl of Modern Physcs, 5, 6, Publshed Onlne Aprl 5 n ScRes. On Hypercomplex Extensons of Quntum Theory Dnel Sepunru RQE Reserch enter for Quntum ommuncton, Holon Acdemc Insttute of Technology, Holon, Isrel Eml: dnelsepunru@wll.co.l Receved 3 Jnury 5; ccepted 6 Aprl 5; publshed 9 Aprl 5 opyrght 5 by uthor nd Scentfc Reserch Publshng Inc. Ths work s lcensed under the retve ommons Attrbuton Interntonl Lcense ( BY). Abstrct Ths pper dscusses quntum mechncl schems for descrbng wves wth non-beln phses, Fock spces of nnhlton-creton opertors for these structures, nd the Feynmn recpe for obtnng descrptons of prtcle nterctons wth externl felds. Keywords omposton Algebrs, Hlbert Spces, Fock Spces, Non-Abeln Guge Felds. Introducton Stndrd Hlbert spce formulton of quntum theory provdes smple nd convenent schem for descrbng phenomen nvolvng electromgnetc nterctons. Expermentl evdence provng the exstence of non-beln guge felds mples tht proper extenson of ths theory must exst. In ths pper, we consder the constructon of consstent quntum mechncl frmeworks (quntum mechncl descrptons of wves wth non-beln phses s frst step nd second quntzton procedure s the next step) for descrbng non-beln guge felds. We llustrte the emergng structures employng the propertes of one- nd two-body sttes. Generlztons of Lorentz force nd the dervton of correspondng non-beln guge felds ccordng to the Feynmn-Dyson schem re treted n the one-, three-, nd seven-dmensonl spces of the nternl prmeters wth specl emphss beng plced upon the role of normed dvson (composton) lgebrs.. The Hlbert Spces Genelogcl Tree In order to generte the extensons of functonl nlytcl structures we use sequence of composton lgebrs s the mthemtcl foundton of the theory, thereby obtnng herrchy whch seems rch enough to ncorporte exstng expermentl nformton bout the known fundmentl nterctons. How to cte ths pper: Sepunru, D. (5) On Hypercomplex Extensons of Quntum Theory. Journl of Modern Physcs, 6,

2 D. Sepunru Before strtng t should be noted tht use of composton lgebrs leds to grve restrctons: stndrd vector product multplcton exsts only n vector spces of dmensons,, 3, nd 7 ccordng to the solutons of the followng relton (equton) [] []: n n n 3 n 7 () Begnnng wth the constructon of sngle prtcle sttes, the followng herrchy structures wth rel sclr products exst: ) Rel vlued stte functons wth rel sclr product trvl: ( f g) Tr( f g) fg b) omplex vlued stte functons wth rel sclr product:,, rel () R f fe fe complex ( f g) Tr( f g) fg fg c) Quternon vlued stte functons wth rel sclr product:,, rel (3) R f fe fe fe fe 3 3 quternon ( f g) Tr f g fg fg fg f3g3, (, ) rel, (4) R where f, f re functons over the rels wth ll the necessry propertes requred from the functonl nlyss; e e, { e, e } ;, ee ee e,,,, 3 ; e e, e3 ee. Tht structure my lso be generted by four dmenson vectors: { } f, g Tr f, g etr f, g e ;,, 3 (, ) (, ) {(, ) } {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e etr f g e (5) 3 3 (, ) (, ) {(, ) } {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e etr f g e 3 3 (, ) (, ) {(, ) } {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e etr f g e The sum of Equtons (5) gves us g ge ( f, g) Tr( f, g) ( f, g) e( f, g) e f, ef, e f, e R 3f 4 4 ge ge3 d) Octonon vlued stte functons over the rels wth rel sclr product: { } f fe octonon;,,,7 f, g Tr f, g f g rel (7) R e e, e, e ;, ee ee e,,,,,7; e e, ee f e ;,, k,,, 7 k k k f s completely ntsymmetrc seven-dmensonl nlog of the Lev-vt symbol wth the followng multplcton tble: (6) 699

3 D. Sepunru f e;, k, 3, 47, 57, 65, 64, 543, 736 for exmple. (8) k Then { } f, g Tr f, g etr f, g e ;,, 7 (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e etr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e etr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e e Tr f g e so we obtn g ge f, g Tr f, g f g f, ef,, e R 7 f 3 ge7 Mtrx multplcton s performed here s wth the usul ssoctve lgebrs due to the vldty of the Moufng dentty [3] ( x)( y) ( xy). () Now consder the sequence of structures generted by the complex sclr products: e) omplex vlued stte functons wth complex sclr product the stndrd mthemtcl formlsm of non-reltvstc quntum mechncs: f fe fe complex The structure my lso be generted by two dmensonl vectors: or n the mtrx nottons ( f, g) f g () f g Tr f g etr f g e f g e f g e () (, ) (, ) {(, ) } (, ) (, ) g f g f ef ge (3) (, ), The group of trnsformtons whch leves tht complex sclr product nvrnt s f) Quternon vlued stte functons wth complex sclr product f fe fe fe fe 3 3 quternon U. Smlrly to the prevous procedure, the requred complex sclr product my be generted by four dmenson (9) 7

4 D. Sepunru vectors: { } f, g Tr f, g etr f, g e ;,, 3 (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e etr f g e The summton then gves us: In mtrx nottons f g Tr f g etr f g e f g e f g e (4) (, ) (, ) {(, ) } (, ) (, ) g f g f ef ge (5) (, ), The group of trnsformtons whch leves tht complex sclr product nvrnt s U : f qfz; q ; z ( f, g ) ( qfz, qgz) z( f, g) z e z( f, g) ze z( f, g) z ( f, g) g) Octonon vlued stte functons over the rels wth complex sclr product: gn we hve nd n mtrx nottons f fe octonon;,,,7 { } f, g Tr f, g etr f, g e ;,,, 7 (, ) (, ) {(, ) } {(, ) } e f g e Tr f g etr f g e etr f g e (, ) (, ) {(, ) } (, ) (, ) f g Tr f g etr f g e f g e f g e (6) g f g f ef ge (7) (, ), The group of trnsformtons tht leves tht complex sclr product nvrnt s ( 4) h) Quternon vlued stte functons wth quternon sclr product f fe fe fe fe 3 3 quternon ) Octonon vlued stte functons wth quternon sclr product: U. ( f, g) f g (8) Q f fe octonon;,,,7 f ψ ψ e ; g ψ ψ e ; ψ ϕe ϕe ϕe ϕ3e3 quternon ψ ϕ7e ϕ4e ϕ5e ϕ6e3 quternon ψ3 χe χe χe χ3e3 quternon 7

5 D. Sepunru ψ4 χ7e χ4e χ5e χ6e3 quternon where ϕ, ϕ, χ, χ,,,, 7 re functons over the rels. 3 ( f, g) [ ψ, eψ ] ψ ψ ( eψ ) ( ψ e ) Q ψ ψ ψ e ψ ψ e ψ ψ ψ ψ ψ 4e ) Octonon vlued stte functons wth octonon sclr product: f fe octonon;,,,7 3. Fock Spce n Hypercomplex Quntum Mechncs (9) ( f, g) f g () O The next step s developng second quntzton procedure for our schem, for whch n del gs consstng of dentcl prtcles s consdered. Restrctng the dscusson to structures wth complex sclr product nd gvng the generl procedure for the reducton of tensor product lgebrs, sutble redefnton of the sclr products s obtned whch llows the proper extenson of the functon nlyss. Let us consder the tensor product of N Hlbert spces. The stte s defned by ( f f f ) f ( x ) f ( x ) f ( x ) Ψ,,, N N N () In generl Kronecker multplcton, n lgebrc operton dfferent from nner multplcton nd whch cnnot be reduced to t, s used. It s dstrbutve wth the followng propertes: ( f g ) ( f g ) ( f f ) ( g g ) () Tr ( f g ) Tr ( f ) Tr ( g ) (3) N( f g) N( f ) N( g) (4) ( f g) ( f ) ( g) dm dm dm. (5) Therefore, the product of N Hlbert spces hs the dmenson N. In the cse when quternons re used to descrbe sngle prtcle stte we obtn 4 N for the dmenson of both the system sttes nd the sclr products (nd 8 N for the octonons correspondngly). System sttes n quntum theory re not observble qunttes, therefore, we need not reduce ther dmenson. However, the sclr products re observble qunttes nd should be numbers belongng to the one of the composton lgebrs. Tensor products n stndrd quntum mechncl theory should stsfy the followng generl requrements of quntum mechncl system wthout ntercton: ) Ech component of tensor product s completely ndependent of the others. ) onstructon of tensor product spces do not spol the vldty of the superposton prncple n ech spce. In order to stsfy the bove condtons one needs two dfferent unts e (for exmple, for the quternon sttes) n the lgebrc bss of the theory: one whch does not commute wth some other unt e (these unts re used for the descrpton of the quntum mechncl stte n the sme spce) nd nother whch does commute wth tht sme unt e (ths e belongs to the second spce nd the quntum mechncl stte n tht spce should be completely ndependent of the quntum mechncl stte whch belongs to the frst spce). The use of the Kronecker multplcton leds to the vldty of the superposton prncple on the level of mny-body sttes. These sttes gn pper to be quntum mechncl sttes stsfyng the bsc prncples of quntum mechncl theory. Let us now consder n obvous exmple of wves wth non-beln phses: quternon quntum mechncs wth complex sclr product. On the level of one-body theory, the quntum mechncl stte s descrbed by the followng mtrx representton: 7

6 D. Sepunru Superpostons re defned by f Ψ ( f ) fe f fe fe fe fe 3 3 (6) fq gq fq gq Ψ ( ) Ψ Ψ fq gq fq gq ( fq gq) e fqe gqe Superpostons re lner only wth respect to complex numbers. Ths form genertes complex sclr product defned by (5). The followng form represents two-body stte: ( f, f ) Ψ f fe f f fe f fe fe The form of the three-body sttes nd so on s obvous. The quternonc unts re non-commutng nd t s cler tht only use of Kronecker products (drect product lgebrs) llows us to stsfy the condtons for constructon of the mny-body sttes. Usng (8) we hve: ( ( f, f), G ( g, g) ) ( f, g) ( f, g) (7) (8) Ψ { } (9) 4 Tht reduces the sclr product lgebr to ts sublgebr wth the bss, e e, e, e Further reducton s cheved through ntroducton of the proecton opertors It s esy to verfy tht Z ( e e) (3) Z e e Z Z, Z Z, Z Z Z Z Z. Fnlly, the requred redefnton s obtned ( ( f, f), G ( g, g) ) etr{ ( f, g) ( f, g) Z } etr ( f, g) ( f, g) Z { } Ψ (3) Usng (3) we hve Ψ { } (3) 4 ( ( f, f), G ( g, g) ) ( f, g) ( f, g) nd becuse of the fctorzed form of the sclr product, relzton of the second quntzton procedure my be crred out nlogously to the stndrd rules. There s no pror connecton between e nd e whch pper n front of the trces n the defnton of the sclr product nd the opertors Z, Z tht re nsde the sclr product. In generl, we ntroduce the followng lgebrc generlzton of the complex sclr product: 73

7 D. Sepunru ( N) ( N) f, f,, fn, G g, g,, gn etr{, G Z } etr (, G) Z (33) ( ) { } Ψ Ψ Ψ ( N ) Z form complex sublgebr of the drect product l- where e e; ee ee e; e e nd gebrs of the obtned construct. In three-body cse they re: ( N ) Z, Z e e e e e e 4 ( 3) Z e e e e e e 4 ( 3) nd so on. {, e } s the lbel for the genertors of the complex feld n ech spce. The complex lner opertors hve the followng form: Az where mtrx elements re c-number opertors over quternons nd n turn re ssumed to be t lest z-lner opertors. The quternon lner opertors hve the form Aq where s q-lner opertor over quternons. It hs the followng structure: e e e e 3 3 where re rel opertors. In occupton number representton the sttes of system of fermons (n spce, two-body cse) re gven by: ( ) Z ( e ) Z, ;,, 3 Z ( e ) Z Then nnhlton-creton opertors hve followng form: Z e Z e ;,, 3 ( ) ( ) Z e Z ( e ) (34) (35) ( e ) Z ( e ) Z ; ( e ) Z ( e ) Z ; ;,,, 3 74

8 D. Sepunru ; no summton ; (36) ; nd thus we hve lmost cnoncl fermon commutton reltons for the nnhlton-creton opertors: { }, no summton,, 3 (37) The lst exmple wth smlr structure s octononc quntum mechncs wth complex sclr product. Relzton of ths cse occurs through mtrx representton of the one body stte (Equton (6)): f Ψ ( f ) fe where f fe octonon ;,,, 7. The remnng constructon s dentcl to the prevous cses due to the Moufng dentty (Equton ()) ( x)( y) ( xy), x, y octonons Our cse correspondng to the choce e ; e s lbel for one of the octononc unts. The nnhlton-creton opertors for system of fermons hve the unusul propertes becuse of the followng multplcton rule for octonons: e 3e7 e,, 3 Let us consder, defned s n (34), where,, 7 nd, defned s n (35),,, 7. Then But nd Then nd,,, 7 (38) ; 3, > 3, < Z 3 Z 3,, 3 (39) ( e ) Z,, 7 (4) ( e ) Z,,, 7 ; 3; > 3; < (4) 75

9 D. Sepunru,, 7,,,,,,,,,, 3 3 ; ;,,3 ( 3 ) ( 3) (4) ( 3 ) ( 3),, All these quntum mechncl schems shres common fetures: sttes tht stsfy the z-lner superposton prncple, sclr products re z-lner nd the followng theorem s vld (here only the two-body cse s consdered s generlzton to other cses s obvous): Beckett Theorem Proof: In generl ( ( f, f), G ( gz, g) ) ( ( f, f), G ( g, gz) ) Ψ Ψ (43) ( Ψ ( f, f), G ( gzg, ) ) {(, ) (, ) } (, ) (, ) { } e Tr f g z f g Z e Tr f g z f g Z etr ( f, g) ( f, g) ( ) ( ) [ ] b e e e etr ( f, g) ( f, g) ( ) b( e ) [ e e] etr ( f, g) ( f, g) ( ) ( ( e) [ e e] etr ( f, g) ( f, g) ( ) b( e) [ e e] ( ( f, f), G ( g, gz )) Ψ ( Ψ ( f, f,, fn ), G ( g, g,, gz,, g,, gn )) ( Ψ ( f, f,, fn ), G ( g, g,, g,, gz,, gn )) expresses the sttement tht the observble qunttes re gven only n terms of ther reltve phses. 4. Interctons Now we re ble to study the prtcle nterctons. hoosng the Feynmn route to nvestgte the vlble optons, we begn [4] wth clsscl Newtonn equtons of moton for sngle, solted prtcles mx F x, x, t ;,, 3 (44) 76

10 D. Sepunru supplemented by Hesenberg (quntum) commutton reltons x, x k (45) m x, x k δ k (46) Then the chrge movng n the gven electromgnetc feld exerts the Lorentz force where E( xt, ) nd (, ) (,, ) (, ) ε (, ), ( ) F xxt ee xt e xh xt c (47) kl k l H xt re defned by the Mxwell equtons dvh (48) H curle. The prtcle nternl prmeters re descrbed by the chrge nd current denstes ρ,, whch re responsble for couplng wth the externl electromgnetc feld. They re defned by the ddtonl pr of the Mxwell equtons: nd (49) dve 4πρ (5) E curlh 4π ρ dv The presence of tht conservton lw tells us tht we hve del wth ddtonl nternl symmetry of the system. Now let us return to Equton (). If we use conventonl vector product multplcton, then the spce dmensons re fxed by the roots of tht equton n, n 3 or n 7. There re lwys only three dmensons n the rel (outer) world wth no expermentl evdence whtsoever contrdctng ths premse. Inner spce, however, hs dmenson of n. Vector products n the nner spce re dentclly zero snce ll vectors re prllel to ech other. ontnung wth the Yng-Mlls [5]-Shw [6] extenson of Mxwell electrodynmcs, whose soluton ws obtned by.r. Lee [7] nd S. K. Wong [8], we use ther nottons n our subsequent dscusson. Accordng to the Feynmn-Dyson schem we dd Wong s equtons for the prtcle crryng the sotopc spn I,,, 3. (n the tme xl guge A ). spn components,, 3. ommutton reltons re now gven by mx F x, x, t ;,, 3 (5) (5) b c I gε AI x ;,, 3;,, 3 (53) bc b A re the vector potentls wth spce components,, 3 nd sotopc x, x k m x, x δ k k 77

11 D. Sepunru [ ] I, I ε I ; (54) b bc c x, I (55) Prtcle moton s ffected by the generlzed externl Lorentz force where (, ) E xt E ( xt, ) I nd (, ) (, ) (,, ) (, ) ε (, ) F xxt ge xt g xb xt (56) kl k l B xt B xt I sotopc nternl spce of the prtcle. They re the solutons of clsscl Yng-Mlls equtons re three-dmensonl vectors both n spce nd n bc b c B gε AB (57) B bc b c ( E gε AE ) ε k k k f the Weyl orderng prescrpton used. The other two Yng-Mlls equtons defne the chrge nd current denstes: (58) E gε AE ρ (59) bc b c E ε ( B gε AB) bc b c k k k Thus gn we hve to del wth two sets of vrbles tht descrbe the system dynmcs. Therefore the followng system of equtons suggests tself for further nvestgton: (6) [ ] x, x k m x, x δ k k I, I f I; bc,,,,,7; (6) b bc c The generlzed Lorentz force s expected to hve the form where (, ) E xt E ( xt, ) I nd (, ) (, ) x, I ;,,,7;,,3. (6) (,, ) (, ) ε (, ) F xxt ge xt g xb xt (63) kl k l B xt B xt I dmensonl vectors n the prtcle nner spce. They re the expected solutons of the followng clsscl Yng- Mlls equtons: re three-dmensonl vectors n outer spce nd seven- bc b c B gf A B (64) B bc b c ( E gf A E ) ε k k k Together wth the properly defned chrge nd current densty: (65) E gf A E ρ (66) bc b c E ε ( B gf A B ) bc b c k k k Although deservng of ttenton, ths opton hs not yet been treted n the lterture. However, reltvstc nd quntum verson of the proposed theory should be developed. (67) 78

12 D. Sepunru 5. oncluson The centrl pont of the present dscusson s connected to the possble role of composton lgebrs n current nd future pplctons n physcs. Here we consder only the closest neghborhoods to the stndrd complex Hlbert spce n detl. The common feture of the schems heren presented s tht they provde rch structures, potentlly contnng the requred symmetres for ncludng both strong nd grvtton nterctons nto the overll unfcton pcture whle use of composton lgebrs leds to severe lmttons upon the dmensons of the nner nd outer spces. They dctte tht the mthemtcl opertons llowed the form of the couplng of externl forces wthn the gven physcl system. However, n ths pper the ntercton felds were only treted clssclly. Much more work needs to be done n order to clrfy the physcl content of the suggested constructs. References [] Eckmnn, B. (943) ommentr Mthemtc Helvetc, 5, [] Rost, M. (996) Doc. Mth. J. DMV,, 9-4. [3] Moufng, R. (933) Abhndlungen us dem Mthemtschen Semnr der Unverstät Hmburg, 9, [4] Dyson, F.J. (99) Amercn Journl of Physcs, 58, [5] Yng,.N. nd Mlls, R.L. (954) Physcl Revew, 96, [6] Shw, R. (955) Ph.D. Thess, mbrdge Unversty, mbrdge, pt. II, ch. III. [7] Lee,.R. (99) Physcs Letters A, 48, [8] Wong, S.K. (97) Il Nuovo mento A, 65,