The quantal algebra and abstract equations of motion. Samir Lipovaca

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1 Keywords: quantal, alebra, abstrat Te quantal alebra and abstrat equations o motion Samir Lipovaa slipovaa@aolom Abstrat: Te quantal alebra ombines lassial and quantum meanis into an abstrat struturally uniied struture Te struture uses two produts: one symmetri and one anti-symmetri Te loal struture o spaetime is ontained in te quantal alebra witout avin been postulated We will introdue an abstrat derivation onept and eneralize lassial and quantum meanis equations o motion to abstrat equations o motion in wi te anti-symmetri produt o te quantal alebra plays a ey role We will express te deinin identities o te quantal alebra in terms o te abstrat derivation In tis orm te irst deinin identity (te Jaobi identity) is analoous in orm to te Biani identity in eneral relativity wi is one set o ravitational ield equations or te urvature tensor Te identity or te unit element is equivalent to te important property tat te metri is ovariantly onstant Similarly, te Jaobi identity is analoous in orm to te omoeneous Maxwell equations Te anti-symmetri produt o te quantal alebra is releted in te antisymmetry o te eletromaneti tensor Introdution Te root ause o te onlit between quantum meanis and relativity oriinates rom te dierene in te underlyin Lie roups and Lie alebras In quantum meanis we deal wit unitary roups In relativity we deal wit ortoonal roup Te quantal alebra [1] is a natural abstrat struture o quantum teory Te alebra o quantions is onrete realization o te quantal alebra Tis unique matematial struture struturally extends non relativisti quantum meanis to a relativisti teory by eneralizin its underlyin number system (te ield o omplex numbers) It ontains bot quantum meanis and relativity in exatly our dimensions Sine te translation roup does not appear in te alebra o quantions, te spae is intrinsi Reimannian Tereore, te alebra o quantions struturally uniies quantum meanis and eneral relativity Te unneessary division property o omplex numbers was te main road blo in unoverin te relativity struture Null spae-time intervals do not ave an inverse in te alebra o quantions Imposin te unneessary division property removes relativity rom te alebra o quantions, orin us ba at usin omplex numbers Tis alebra is te only alebra tat is able to lit a deeneray o omplex numbers and as dierent alebrai and eometri norms Te alebrai norm o omplex numbers is deined as A ( z ) zz * were * is omplex onjuation We an expand A( z) in terms o te omponents o 1

2 z x iy and obtain A( z) x 2 y 2 M ( z) were M ( z) is a eometri onept (te trivial Eulidian metri in two dimensions) For omplex numbers te alebrai and eometri norms are equal I tis deeneray is lited, it will lead uniquely to te alebra o quantions Te two norms o quantions ave a remarable pysis interpretation Te alebrai property o quantions is related to standard quantum meanis Te eometri property is related to relativity In oter words, maps quantions to omplex numbers and non-relativisti quantum meanis, wile maps quantions into Minowsi our vetors and relativity Tus quantions are mixed relativity and quantum meanis objets In quantioni * pysis, Born interpretation o te untion as a probability density is naturally eneralized and replaed by Zovo s interpretation wi leads to Dira and Srodiner equations Ater tese introdutory remars about quantions as a onrete realization o te quantal alebra, let us outline te paper First, we ive a brie overview o te quantal alebra Ten te Hamiltonian as a ruial onept o bot lassial and quantum meanis is presented Based on tis undamental importane o te Hamiltonian in bot meanis, we introdue an abstrat derivation onept and eneralize lassial and quantum meanis equations o motion to abstrat equations o motion in wi te anti-symmetri produt o te quantal alebra plays a ey role We express te deinin identities o te quantal alebra in terms o te abstrat derivation In tis orm ertain onepts o te aue teories (te alebrai identities and te omoeneous dierential equations) are readily apparent We empasize tese onepts or te ravitational and eletromaneti ields Last, we inis wit Disussion and Conlusions Quantal alebra A quantal alebra is a real abstrat two-produt alebra wit a unit Denotin te elements o te alebra by small letters,, were is te spae o observables, te two produts, denoted by (symmetri) and (anti-symmetri) are Jordan and Lie produts respetively Followin are te deinin identities o te quantal alebra: te Jaobi identity, ( ) ( ) ( ) 0, te Leibnitz identity, ( ) ( ) ( ), and te Petersen identity, ( ) ( ) a ( ) 2

3 Te identities or te unit element e O are e, e 0 Te onstant a is related to Plan s onstant For quantum meanis it may ave any positive value It may be normalized to unity by re-salin te produt For lassial meanis, we ave a 0 Hamiltonian in lassial and quantum meanis Bot lassial and quantum meanis speiy ow te state o a system evolves wit time In lassial meanis, lie any oter property o te system, its total enery is determined by its state It is a untion o te position and momentum oordinates o te partiles omprisin te system Tis untion is nown as te Hamiltonian untion H o te system Te Hamiltonian untion ditates ow te state o a lassial system evolves trou time Te undamental equations o lassial meanis or a system o derees o reedom, written in te so-alled anonial orm are, p H H, q, 1, 2,, q p (1) Wen H does not depend expliitly on te time, te enery equation, were te total enery, is a onstant, ollows at one Te anonial equations (1), i expressed in terms o te Poisson braets, beome p [ H p ], q [ H q ] (2) were te Poisson braet o x and y is deined as x y y x [ x y] ( ) q p q p 1 (3) Lie lassial meanis, quantum teory tells us ow te state o a system evolves wit time Te ey role in te equation overnin tis evolution is played by an operator rater tan by te Hamiltonian untion, aordin to te eneral priniple tat, in quantum meanis, operators represent pysial quantities As in te lassial ase, te quantity in question is te total enery o te system It is represented in quantum teory by a Hermitian operator H wi we all te 3

4 Hamiltonian operator or te system Dira [2] too te point o view tat te Hamiltonian is really very important or quantum teory In at, witout usin Hamiltonian metods one annot solve some o te simplest problems in quantum teory, or example te problem o ettin te Balmer ormula or ydroen, wi was te very beinnin o quantum meanis It as been ound by 2i 2 i Dira tat in te quantum teory te expression ( x y y x) [ x, y], were is te Plan onstant, is te analoue o te Poisson braet (3) in lassial meanis Te simplest assumption is to tae over te equations (2) ormally into te quantum teory, replain te Poisson braets by teir quantum analoues Tereore it is assumed te equations o motion in te quantum teory to be p q 2i [ H, p ], 2i [ H, q ] (4) We see in bot lassial and quantum meanis, te Hamiltonian is a ruial onept Abstrat equations o motion A quantal alebra wit a 0 represents te abstrat struture o lassial meanis Te pase spae untions are onrete observables and teir ordinary produt and Poisson braet are te aloritms or te abstrat produts and: de, de ( ) p q p q 1 (5) Similarly, a quantal alebra wit a 0 represents te abstrat struture o quantum meanis Hermitian matries are onrete quantum meanial observables and al te Hermitian ommutator and al te Hermitian anti-ommutator are te aloritms or te abstrat produts and : de 1 ( ), 2i de 1 ( ) 2 (6) 4

5 Based on te equations (2) and (4) it seems reasonable to assume te abstrat equations o motion to be (7) were is te abstrat derivation assoiated wit in a sense i represents a Hamiltonian ten a onrete realization o mae (7) plausible For a 0 and is te time derivative d dt te abstrat equations o motion beome te anonial equations (2): Here stands or p and d dt d Te ollowin observations seem to dt, usin te aloritm (5) or te abstrat produt d dt [ ], q and stands or te Hamiltonian untion H Similarly, or a 0, usin te aloritm (6) or, te abstrat equations o motion beome te equations o motion in te quantum teory or p, q and 4 H : d dt i [, H ] [, H ] [ H, ] 2i i Wen rom equation (7) ollows 0 sine is an anti-symmetri produt Tus is a onstant o motion wit respet to In bot meanis tis means Ḣ 0 I we express te deinin identities o te quantal alebra in terms o te abstrat derivation we obtain: te Jaobi identity, 0, te Leibnitz identity, ( ) ( ), (8) and te Petersen identity, ( ) ( ) a Te identity or te unit element is e e 0 Te unit element beaves lie a onstant 5

6 wit respet to te abstrat derivation assoiated wit any element o te alebra Similarly, any element beaves lie a onstant wit respet to te abstrat derivation assoiated wit te unit element Remarably, in tis orm o te quantal alebra ertain onepts o te aue teories (te alebrai identities and te omoeneous dierential equations) are readily apparent Let us empasize tese onepts or te ravitational and eletromaneti ields Te ravitational ield From te Jaobi identity in (8) we an subtrat anoter Jaobi identity in wi and are interaned to obtain 0 ( ) ( ) ( ) 0 (9) I represents a ovariant vetor V and a ovariant derivative ten ( ) were R ives V V V V V R ; ; ; ; is te Riemann-Cristoel tensor or te urvature tensor Similarly, we an replae oter two terms in (9) wit ovariant vetors and ovariant derivatives to obtain V R V R V R 0 Te ator V ours trouout tis equation and may be anelled out We are let wit te irst Biani identity R R R 0 Let us suppose now ovariant vetor Ten,, and represent V, V, V respetively, werev a is a a a a ( ) ( ) V ( ) V a a ; V V V R V R a ; ; ; a ; ; ; ; a a ; 6

7 Similarly, or te seond and tird terms in (9) we obtain and ( ) ( ) V a ( ) V a ; V V V R V R a ; ; ; a ; ; ; ; a a ; ( ) ( ) V a ( ) V a ; V V V R V R a ; ; ; a ; ; ; ; a a ; respetively Te let-and side ives Tus te irst deinin identity (te Jaobi identity) in (8) is analoous in orm to te Biani identity in eneral relativity wi is one set o ravitational ield equations or te urvature 7 ( V V ) ( V V ) ( V V ) a ; ; ; a ; ; ; a ; ; ; a ; ; ; a ; ; ; a ; ; ; ( V V ) ( V V ) ( V V ) a ; ; a ; ; ; a ; ; a ; ; ; a ; ; a ; ; ; ( V R ) ( V R ) ( V R ) V ; R a a Te rit-and side ives ; a ; a ; V R V R V R V R V R a ; ; a a; ; a a ; V R V R V R V R V R V R ; a a ; ; a a ; ; a a ; V R V R V R V ( R R R ) ; a ; a ; a a ; V R V R V R ; a ; a ; a as te term in V a ; vanises due to te irst Biani identity Te remainin rit-and side term anels wit an idential term on te let-and side and we are let wit V ( R R R ) 0 a ; a; a ; Te ator V ours trouout tis equation and may be aneled out We are let wit te seond Biani identity R R R 0 a ; a; a;

8 tensor I e stands or te metri tensor ab and or te ovariant derivative, ten te identity or te unit element is analoous to te important property tat te metri is ovariantly onstant: e ab 0 (10) Te eletromaneti ield I in equation (9), ( ) is replaed by ( ) A were A is te vetor potential, yli permutations o indies produe analo expressions or te remainin terms o (9) and we obtain ( ) A ( ) A ( ) A 0 Tis equation an be rearraned as F F F 0 wi is te relativisti orm o te omoeneous Maxwell equations were F A A is te eletromaneti tensor Now, sine is an anti-symmetri produt, abstrat derivation, expressed as A A 0 or in terms o te 0 In terms o te vetor potential tis equation an be 0I we subtrat tis equation rom itsel and rearrane terms we obtain ( A A ) ( A A ) F F 0 Tus te anti-symmetri nature o te produt is releted in te antisymmetry o te eletromaneti tensor Disussion Based on undamental importane o te Hamiltonian in bot meanis, we did introdue an abstrat derivation onept to eneralize lassial and quantum meanis equations o motion to abstrat equations o motion Te anti-symmetri produt o te quantal alebra plays a ey role in tese abstrat equations Wen te deinin identities o te quantal alebra are expressed in terms o te abstrat derivation, ertain onepts o te aue teories (te alebrai identities and te omoeneous dierential equations) are readily apparent A eneral terminoloy or aue teories onsists o seven onepts: te aue roup, te aue ield, te interability onditions, te urvature, te alebrai identities, te omoeneous dierential equations, and te inomoeneous dierential equations Let us briely illustrate tese onepts or te ravitational and eletromaneti ields For te ravitational ield, te aue roup is te roup o eneral oordinate transormations in a real our-dimensional Riemannian maniold wit loal Minowsi metri Te aue ield is te Cristoel symbol 8

9 Te interability ondition is iven by te relation R 0 were R is te Riemann urvature tensor Te urvature is deined exatly by te Riemann urvature tensor Te Riemann urvature tensor exibits symmetry properties wi an be read o diretly rom its deinition: were R R R R R R R R R 0 is te ovariant orm o te Riemann urvature tensor Tese symmetry properties are reerred to as alebrai identities Te seond identity is reerred to as te irst Biani identity Tain te ovariant derivative o te Riemann tensor one obtains te omoeneous dierential equations reerred to as te seond Biani identity: Te Einstein equation G R R R 0 ; ; ; 8T represents te inomoeneous dierential equations were G R 1 R 2 is te Einstein tensor and T is te stress-enery tensor R is te Rii tensor wi is te irst trae o te Riemann tensor Te seond trae is te urvature salar R Similarly, or te eletromaneti ield, te aue roup is te unitary roup o pase transormations U ( 1 ) Te aue ield is iven by te our-potential A ( x) Te interability onditions are A A 0 Te eletromaneti tensor F A A is te 9 urvature, wi tus appears as te analoue o Riemann s urvature tensor Te urvature tensor F is subjet to only one alebrai identity It is antisymmetry F F omoeneous dierential equations are represented by te ollowin dierential identity F F F 0,,,, 0 Te wi is analoous to te seond Biani identity in eneral relativity Te inomoeneous dierential equations are iven by te soure equation or te eletromaneti ield F 4 J In te ravitational ield setion we sowed ow te Jaobi identity o te quantal alebra is analoous in orm to te Biani identities Similarly, in te eletromaneti ield setion we sowed ow te Jaobi identity is analoous in orm to te omoeneous dierential equations In addition, te anti-symmetri nature o te produt is releted in te antisymmetry o te

10 eletromaneti tensor Peraps it is not surprisin wy te identity or te unit element e o te quantal alebra is analoous to te important property tat te metri is ovariantly onstant I O J is te entralizer o J, ie te set o all observables in O su tat J 0, were J e, ten O,,, J e is a quantal alebra [1] J plays a unique role in te alebra, and introdues relativity into te quantal ramewor Speiially, one J is seleted, quantions are deined into a subspae o SO ( 2, 4 ), te entralizer spae o O J ( 2, 4 ) Te entralizer redues itsel to a omplex Minowsi spae o dimensionality 8: M 0 ( C ) M 0 im 0 Usin te Leibnitz identity on ( J J ) we obtain te identity or te unit element ( e 0 ), tus, e, in essene, introdues relativity into te quantal ramewor e is a unique element in te quantal alebra Similarly, te metri tensor plays a unique role in eneral relativity Equation 10 suests tey are analoous to ea oter Would te Leibnitz and Petersen identities o te quantal alebra (8) be someow analoous to te inomoeneous dierential equations or te ravitational and eletromaneti ields? Tis is an open question Conlusions Wen te deinin identities o te quantal alebra are expressed in terms o te abstrat derivation, ertain onepts o te aue teories (te alebrai identities and te omoeneous dierential equations) are readily apparent or te ravitational and eletromaneti ields Anowledments Te results presented in tis paper are te outome o independent resear not supported by any institution or overnment rant Reerenes [1] E Grin, Te Alebra o Quantions (Autorouse, Indiana, 2005) [2] Paul A M Dira, Letures on Quantum Meanis (Dover Publiations, In, Mineola, New Yor, 2001) 10