Journal of Modern Physcs, 0, 3, 709-73 htt://dxdoorg/0436/jm038096 Publshed Onlne August 0 (htt://wwwscrporg/journal/jm) Wave Functon as Geometrc Entty Bohdan I Lev Bogolyubov Insttute for Theoretcal Physcs NAS Ukrane, Kyv, Ukrane Emal: bohdanlev@gmalcom Receved May 6, 0; revsed June 3, 0; acceted July, 0 ABSTRACT A secal aroach to the geometrzaton the theory of the electron has been roosed The artcle wave functon s reresented by a geometrc entty, e, Clfford number, wth the translaton rules ossessng the structure of Drac equaton for any manfold A soluton of ths equaton s obtaned n terms of geometrc treatment New exerments concernng the geometrc nature wave functon of electrons are roosed Keywords: Wave Functon; Clfford Algebra; Theory of Electron Introducton The roblem of how to geometrcal resentaton of the theory of the electron and nclude t n the scheme of the general relatvty theory s far from beng solved The exresson for the co-varant dervatve obtaned n [] from ntutvely consderaton, wth some nterretaton correctons ntroduced n [], s the generally acceted formula now Cartan [3] has showed, however, that fntedmensonal reresentaton of a comlete lnear grou of coordnate transformatons does not exst Moreover, the set of Drac snors reserves the structure of the lnear vector sace, but does not reserve the rng structure snce defnng the comoston oeraton nvolves some comlcatons Thus allowed states are deleted nasmuch as wave functon behavor under the arallel translaton cannot be calculated and arorately nterreted, and, besdes that, the states of the artcle ensemble cannot be determned As early as 878 G Frobenus [4] to rove a theorem, that exst only one assocatve algebra wth artton on the feld of real number-real number, comlex number and Clfford number The algebra, whch use the Clfford number ossesses rng structure [5] snce t s a vector sace over the feld of real numbers and hence makes an addtve grou whose low of elements comoston s dstrbutve rather than commutatve wth resect to addton Ths rng has deals whch may be obtaned by multlyng the searated element on the rght or on the left by rng elements [5] The deals resultng from ths rocedure are just the Drac snors of the standard aroach Now, the alcaton of the conventonal aroch by snor reresentaton n a Hlbert sace s well-known However attemt to dscus the Drac theory from Clfford algebra tself have been judged to have acheved lmted success Thus the reresentaton of the Clfford algebra by the Clfford number contans more nformaton on artcle roertes than snor reresentatons We known, that the Clfford algebra can be extended to nclude relatvty and lays essental role n the Drac theory of relatvstc electron [6,7] As was shown early [8-] covered all the standard features of quantum mechancs Clfford algebra gves [3] an unfyng framework of hyscal knowledge here ncludng relatvty, electromagnetsm and other hyscal matter When we ntroduce a Clfford rough scheme of quantum mechancs, as note n [0], we cannot gnore the emergng salent feature of ths formulaton It s that n ths case we obtan a quantum mechancal theoretcal framework nvokng only an algebrac structure that does not contan any further secfc requrement It ossble to show [8-], that the Clfford algebrac formalsm are comletely equvalent to the conventonal aroach to quantum mechancs Ths oen u the ossblty of a dfferent nterretaton an exlanaton of quantum henomena n term of a non-commutatve geometry and redct the new exerment for determnaton geometrcal resentaton of wave functon for elementary artcle The mean dea ths artcle are n resentaton the artcle wave functon by a geometrc entty, e, Clfford number, wth the translaton rules ossessng the structure of Drac equaton for any manfold A soluton of ths equaton can obtan n terms of geometrc treatment The novel exerments of geometrcal nature of wave functon determnaton can be roosed Wave Functon as Geometrc Entty We emloy the mean dea of corresondence between the snor matrces and the elements of an exteror algebra and thus defne the sace of states n terms of a sace Coyrght 0 ScRes
70 B I LEV of reresentatons of a sace-tme Clfford algebra The constructon of such algebra [5,7] requres mosng a n vector sace R over the feld of real numbers, and the quadratc form whch assocated wth scalar roduct Can ntroduce the noton of -vectors e the roduct of n vectors from R, whch geometrcally sets the orented volume constructed uon these vectors Each -vector has comonents and s an element of sace of Cn Cn dmensons and each element, made u by a roduct of an arbtrary set of -vectors s an element of vector sace n n wth the dmensons C = n whch n tself s a drect sum of ts subsaces Clfford algebra can be constructed for the elements of such sace, whch makes ossble a sngle aroach to the examnaton of the nternal roduct of any vector assocated wth quadratc form, and the external roduct that the rng structure An arbtrary bass, smlar to the vector bass, can be ntroduce n any ont of the sace, allowng us to construct an nduced vector sace wth geometrcal characterstcs of dfferent tensor subsaces In the case fnte dmensonalty, the sace of functon s somorhc to the sace tself Thus, we can defne a sngle general characterstc n each ont of the manfold, regardng t as the drect sum of all ossble forms as elements of vector sace The drect sum of such tensor reresentatons can be attrbuted wth the Clfford algebra structure by means of the drect roduct [5] The fnal dmensonalty of the algebra s determned by the number of bass vectors, rovdes the rng structure and s resonsble for the exstence of an exact matrx reresentaton Moreover, the sace of functonal s somorhc to ths very lnear sace, and the algebra of outer roducts s somorhc to the algebra of the outer roduct of these very vectors In the case of sace-tme n the secal theory of relatvty, Clfford algebra s algebra constructed by the Drac matrx whch s assocated wth unt vectors The lnear combnaton roduct of ths matrx has all the roertes of the structure of Clfford algebra wth three comlex unts because one tme matrx 0 = and three sace matrx = Therefore, we can reroduce any element belongng to the nduced vector sace n the form of drect sum all ossble tensor resentaton The exstence of a unque set of lnear ndeendent forms defned at an arbtrary ont of the sace suggests that the nature of the forms translated over the manfold s smlar to the nature of forms whch characterze t [3] Ths may be also determned by the smlar form of the geometrc enttes as functons of elements of the nduced sace Makng use of ths bass, we consder the realzaton of the dfferent functon n the ordnary Eucldean sace In ths case the dfferent functon may be wrtten n terms of a drect sum of a scalar, a vector, a b-vector, a three-vector, and a seudo-scalar, = s v b t, that s gven by = 0 wth the reverse order of comoston, we have = s v b t and havng changed the drecton of each bass vector, we obtan = s v b t It s symmetry element for Clfford number If ntroduce the notce 5 0 3 (comlex number we wll note as ), can ntroduce the once one symmetry element as multlcaton by that resent as = After ntroducng symmetry element should be resent mathematcal oeraton on the feld of Clfford number The drect sum of tensor subsaces can be gven a rng structure wth the hel of drect tensor roduct n the followng symbolc notaton: = () where s an nner roduct or convoluton that decreases the number of bass vectors and s external roduct that ncrease number of bass vectors If every Clfford number multle on secal fxed matrx, whch have one own column wth elements own and other zero we can obtan the Drac snor wth four element Wth the hel ths column can reroduce snor resentaton every Clfford number Exst full corresondence between the snor column and the elements of an exteror algebra Next we would lke determne the rule of comarng two Clfford number n dfferent ont of manfold For ths goal should be determne the deformaton of coordnate system and rule of arallel dslacement on dfferent manfold An arbtrary deformaton of the coordnate system can be set n terms of bass deformatons e =, where s the Clfford number that descrbes arbtrary changes of the bass (ncludng arbtrary dslacements and rotatons) whch do not volate ts normalzaton, e, rovded = It s not dffcult to verfy that e = = = I and ths does not volate the normalzaton of the bass [5] Now, for an arbtrary bass, we can set, at each ont of the sace, a unque comlete lnearly ndeendent form as a geometrc entty that characterzes ths ont of the manfold For a four-dmensonal sace, such geometrc entty may be gven by = 0 e ee eee (3) eeee If ths ont of the manfold s occued by an elementary artcle, then ts geometrc characterstcs may be () Coyrght 0 ScRes
B I LEV 7 descrbed by the coeffcents of ths reresentaton A roduct of arbtrary forms of ths tye s gven by a smlar form wth new coeffcents, thus rovdng the rng structure Ths aroach makes t ossble to consder the mutual relaton of felds of dfferent hyscal nature [3,4] However, n what follows we consder new concet to descrton of a artcle and characterstc of manfold as a geometrc entty Defnng a sngle characterstc of manfold as a ont functon means assocatng each ont of manfold wth a ral number, determned by the functon value If the functon s dfferentable wth resect to ts argument, then we have dfferentaton of resent form In order to determne the oeraton of form translaton over an arbtrary manfold we have to defne the dervatve oeraton It may wrtten as a lnear form d = where x x that forms a bass of the vector sace of all changes along the curves assng through a gven ndvdual ont of the sace The acton of such an oerator on an arbtrary form may be resented as d = d d (4) where d and d may be called the dvergence and the curl of the relevant form, accordng to the defnton of the dfferentable varety, t s nsuffcent to have one non-sngular coordnate system to cover a manfold the toology of whch dffers from the toology of an oen set n the Eucldan sace The structure of such geometrc constructon should be comlemented wth a correlaton between the values of the transferred forms n dfferent onts of the manfold When assgnng ntrnsc values to the manfold characterstcs, t s necessary to ntroduce the low of form transformaton wth the change of the coordnate system Manfold mang s defned through mang of corresondng system of the form whch a certan grou L causes to transform A ossble transformaton of any geometrc characterstc whch caused both by the turn of a corresondng coordnate system as well as by the transformaton of the geometrc entty themselves, requres the use of Clfford algebra uon the elements of the nner vector sace grou (smlectc grou Sn ) Y = fz (5) where, Y, Z have reresentaton, constructed wth the elements of the smlectc grou and analogous to those of orgnal form, whle f are structural constant, deend on the manfold ont, they are reresented by tensor values of all dmensonal determned by the ntal vector sace A certan transformaton grou transforms each form accordng to the law = (6) where determnes the mang elements, of Clfford algebra n our case, and satsfes the condton = For ths algebra, we can wrte the frst structure equaton that defnes the co-varant dervatve as gven by [3]: = d (7) wth the gauge transformaton law for the constrant beng gven by = d (8) for the conservaton co-varant transformaton accordng to the same law = Here the tensor reresentaton of the constrant s smlar to that of an arbtrary form of the Clfford algebra The resent equaton called the frst structural equaton but now the form wll assume the value n Clfford algebra In ths case the arbtrary Clfford number can always be reduced to the canoncal form, but local deformatons of the roer bass become, however, s not observable snce the Tetroude form d corresonds to the second term of the gauge transformaton Then the second structure equaton that defnes the curvature form may be wrtten as F = d (9) wth the law of transformaton under the algebra beng gven by F = F The transfer equaton for the curvature tensor wth the resent transformaton law can wrtten n the followng form df F F = J (0) where J s the source form wth the analogous general reresentaton whch comles the transformaton J = J The obtaned equaton can be regarded as feld equaton, ts form externally smlar to analogous equaton for the connectvty form obtaned n L algebra Those equatons have a more general character as ther structure contans nterrelaton of geometrc characterstc whose tensor nature s dfferent In ths resentaton can be wrtten the fourth structural equaton whch demonstrates the deendence between covarant dervaton and form curvature: d F =0 () We can assume that the every elementary artcle n every ont of manfold can descrbe n the term Clfford number Then the artcle wave functon s reresented by a comlete geometrc entty A sum of robable drect forms of an nduced sace of the Clfford algebra Moreover, havng attrbuted the wave functon wth geometrcal sense we can obtan correct translaton rules for an arbtrary manfold [3] and come to some new quantum results assocated wth the geometrc nature of the wave functon For wave functon as geometrcal entty can wrte the frst structural equaton n the standard form: d = m () Coyrght 0 ScRes
7 B I LEV f suose that the covarant dervaton roortonal same wave functon as = m where m s mass coeffcent The form of ths equaton s analogous to the Drac equaton n the snor reresentaton Among these results we ndcate the observaton that Drac equaton n the geometrc reresentaton s nothng but the translaton equaton n the general relatvty sense, hence ts solutons may be nterreted geometrcally Moreover, the geometrc reresentaton of the wave functon yelds other results concernng the nterference of elementary artcles whch just may reveal the geometrc nature of the wave functon [5] Ths equaton s more nformatve for several reasons The frst one s that snors are only secal rojectons of Clfford numbers [5], Drac snors are reresented only by deals n ths algebra, and thus t s mossble to ntroduce the comoston oeraton on the snor set And the most mortant dfference s that comlete grou of lnear transformatons of the coordnate system does not exst for snors [3] As follows from the revous analyss, a comlete transformaton grou assocated wth the structure equaton exsts only n the Clffordnumber reresentaton of the wave functon The frst structure equaton for the wave functon reroduces the form of the Drac equaton and, as t has been shown n [5], ts solutons are smlar to those for the snor reresentaton Ths solves the roblem of fnte-dmensonal reresentaton of the wave functon under the comlete lnear grou of coordnate transformatons The Drac equaton for wave functon can be obtaned mnmze the acton, constructed from geometrcal nvarants S = m d The Lagrange multler m ensure the normalzaton condton for wave functon d = The resent acton non degeneracy on the soluton of Drac equaton n contrast wth standard aroach In our aroach the dynamc equaton for wave functon s resented as rule the arallel translaton for characterstc of elementary artcle on the arbtrary manfold As roof n [5] the each even number = for 0, the Clfford number n Eucldean sace may be reduced to the canoncal form, e, = x ex (3) where = descrbes all the coordnate transformatons assocated wth the translaton and rotaton of coordnates and wth the Lorentz transformaton n the Eucldean sace As smle to see d s scalar and n the hyscal nterretaton of ths geometrc entty s rather evdent snce x can be assocated wth the robablty densty of fndng a artcle n an arbtrary satal ont, and s the angle that determnes the egenvalue of a artcle wth ostve or negatve energy We can take =0 for an electron and = π for a ostron Thus t becomes ossble to descrbe the ntermedate states of the artcle snce the form of the wave functon of an arbtrary ensemble of artcles s analogous [7] As roof n the book [5] the odd art of general Clfford number can resent as even art whch multled on the solate element of ths algebra γ 0 and thus not roblem wth manulaton of full Clfford number The structure equaton thus obtaned s wrtten n the ntroduced terms s comletely equvalent to the Drac equaton, and has well known solutons both for the calculaton of the hydrogen atom sectrum and for the nterretaton of electron states [5] 3 Exermental Determnaton of Geometrc Entty General roblem consst to exermental observaton geometrcal resentaton of wave functon Usually, that n general relatvty the geometrcal resentaton of wave functon should be crucal role and wll be observed for many cases But we would lke suggest on ossblty to determne geometrcal character of wave functon whch relate to modern real condton We consder the case of artcle dffracton on two holes that mght be helful n revealng the geometrc character of the wave functon In our reresentaton, two ossble assng the one artcle can be descrbed by the wave functons reresented by geometrc enttes n the canoncal form, e, and = x ex = x ex For electrons we have = =0 The canoncal form of the nterference two ossble assng of the wave functon of electron should be smlar, e, = x ex = = = Now the ost-nterference result can be wrtten as x (4) In the case of even Clfford numbers, when corresonds to Lorentz rotatons, e, when can be wrtten as =ex B, where B = b s a double vector, and are constant numbers, and b s a vector whose modulus s equal to one, the result of nterference, s gven by the standard exresson, e, = x = cos (5) For lane monochromatc waves [5], the soluton of Coyrght 0 ScRes
B I LEV 73 the Drac equaton s gven by x u x3 x = e, where 3 s the Paul matrx, u s artcle amltude, and s artcle momentum The second soluton s smlar excet for the hase shft, e, we have x u x 3x = e We see that now electron nterference s descrbed by the well known formula Next we assume that deformatons of the reference system can change the geometrcal entty and can gve the result, whch rncal are dfferent as standard aroach The exstence of ths effect can be verfed exermentally A coherent electron beam should be dvded nto two beams, the latter should be assed through searate regons wth varable basc geometrc characterstcs The change of the wave functon assng through dfferent regons can be wrtten as = a b where a and b descrbe the transformaton of artcle characterstcs n the regons a and b If the sequence order s changed, = b a ab ba then electron nterference should corresond to the last case of the revous analyss, e, the nterference attern should be dfferent from the standard case The varous regons can be nfnte solenods of the Aaronov-Bohm exerment wth dfferent drectons of the magnetc flux If change the drecton of the magnetc flux n two solenods we can observe the dfferent nterference cture, whch wll be dfferent as standard Another way to observe the dfference of the nterference atterns s to ass electrons along and across the solenods The dfference s gven rse to only by the geometrc reresentaton snce n the frst case the flux s not changed as dstnct to the ooste case We can assume that ths effect mght be also observed for neutron nterference, the regons of varaton of the wave functon geometrc comonents beng two nclusons wth dfferent mass numbers occurrng on the neutron roagaton ath A smlar exerment had been roosed n aer [5], however, t has not been erformed tll now 4 Conclusons We can assume that the Clfford algebrac formalsm s comletely equvalent to the conventonal aroach to quantum mechancs Quantum mechancs holds about the basc henomenon of quantum nterference The frst structure equaton s wrtten n the ntroduced geometrcal terms s comletely equvalent to the Drac equaton, and has well known solutons both for the calculaton of the hydrogen atom sectrum and for the nterretaton of electron states [5] We may realze t usng the basc elements, and the structure of the Clfford algebra Ths oen u the ossblty of a dfferent nterretaton an exlanaton of quantum henomena n term of a noncommutatve geometry and redct the new exerment for determnaton geometrcal resentaton of wave functon for elementary artcle We have to consder the basc foundatons of quantum mechancs as basc framework reresentng concetual enttes [8] I would lke to thank Prof J Klauder for very useful dscusson REFERENCES [] W A Fock, Geometrzerung der Drachschen Theore des Elektrons, Zetschrft fur Physk, Vol 57, 99, 6 [] V A Zhelnorovch, Theory of Snors and Alcaton n Mechancs and Physcs, Nauka, Moskow, 98 [3] E Cartan, Lecons sur la Theore des Sneurs, Actualtes Scentfques et Industres, Pars, 938 [4] G Frobenus, Uber Lneare Substtuton and Blnear Formen, Grelle, Vol 84, 878, -63 [5] G Kasanova, Vector Algebra, Presses Unverstares de France, 976 [6] B J Hley and R E Callaghan, The Clfford Algebra Aroach to Quantum Mechancs A: The Schrodnger and Paul Partcles, Unversty of London, London, 00 [7] C Doran and A N Lasenby, Geometrc Algebra for Physcst, Cambrdge Unversty Press, Cambrdge, 003 [8] E Conte, An Examle of Wave Packet Reducton Usng Bquaternons, Physcs Essays, Vol 6, No 4, 994, 53-535 [9] E Conte, Wave Functon Collase n Bquaternon Quantum Mechancs, Physcs Essays, Vol 7, No 4, 994, 4-0 do:04006/30960 [0] E Conte, On the Logcal Orgns of Quantum Mechancs Demonstrated By Usng Clfford Algebra: A Proof that Quantum Interference Arses n a Clfford Algebrac Formulaton of Quantum Mechancs, Electronc Journal of Theoretcal Physcs, Vol 8, No 5, 0, 09-6 [] A Khrennkov, Lnear Reresentatons of Probablstc Transformatons Induced by Context Transtons, Journal of Physcs A: Mathematcal and General, Vol 34, No 47, 00, 9965-998 do:0088/0305-4470/34/47/304 [] M Cn, Partcle Interference wthout Waves, Electronc Journal of Theoretcal Physcs, Vol 3, No 3, 006, -0 [3] B I Lev, Algebrac Aroach to the Geometrzaton of the Interacton, Modern Physcs Letters, Vol 3, No 0, 988, 05 [4] J M Benn and R W Tucker, The Dfferental Aroach to Snors and ther Symmetres, Nuovo Cmento A, Vol 88, 985, 73 [5] A G Klen, Schrödnger Involate: Neutron Otcal Searches for Volatons of Quantum Mechancs, Physcs B, Vol 5, 988, 44 Coyrght 0 ScRes