Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.

Transcription

1 Algebrac Quantum Mechancs, Algebrac Spnors and Hlbert Space. B. J. Hley. Theoretcal Physcs Research Unt, Brkbeck, Malet Street, London WCE 7HX. Abstract. The orthogonal Clfford algebra and the generalsed Clfford algebra, C n, (dscrete Weyl algebra) s re-examned and t s shown that the quantum mechancal wave functon (element of left deal), densty operator (element of a two sded deal) and mean values (algebrac trace) can be constructed from entrely wthn the algebra. No appeal to Hlbert space s necessary. We show how the GNS constructon can be obtaned from wthn both algebras. The lmt of C n as n s shown to be the extended Hesenberg algebra. Fnally the relatonshp to the usual Hlbert space approach s dscussed.. Introducton. In my study of the structure of Clfford algebras partcularly n regard to the role they play n quantum mechancs through Paul spnors, the Drac spnors and Penrose's twstors, I was struck by how all the relevant results could be accounted for from wthn the algebra. There was no need to leave the algebra and to construct an external Hlbert space upon whch the elements of the algebra were deemed to act when they play the role of observables. Of course all of these algebras are fnte and non-nlpotent so that they are easly treated by standard algebrac analyss. My orgnal thoughts on these topcs can be found n Frescura and Hley (980, 984 and 987). However when t comes to the Hesenberg algebra these technques appear not to be applcable because ths algebra s nlpotent. However the earler work of Schönberg (957) and the later deas of Frescura and Hley (989) and Hley (99) suggested there was such a possblty provded one extended the algebra (Hley 00). Consderable

2 clarfcaton of the meanng of ths specfc extenson has been acheved by explorng the fnte dscrete Weyl algebra C n. (Morrs 967 and Hley and Monk 993). The mportance of these algebras for topc of dscusson n ths paper s that n the lmt as n the algebra contans the Hesenberg algebra. These algebras are fnte so that the same technques used n Clfford algebras can be drectly appled. Thus we have a unfed way of dealng wth the basc algebras that le at the heart of quantum mechancs. These results have only ntensfed my curosty as to why most f not all of the results can be obtaned wthout seemngly the need to resort to Hlbert space. Ths goes aganst the prevalng orthodoxy that appears to nsst that quantum mechancs cannot be done except n the context of a Hlbert space. Yet there have been other voces rased aganst the necessty of Hlbert space. Von Neumann hmself wrote to Brkoff (966) wrtng "I would lke to make a confesson whch may seem mmoral: I do not beleve absolutely n Hlbert space any more." (A detaled dscusson of why von Neumann made ths comment can be found n Réde 996). But there are more mportant reasons why an algebrac approach has advantages. As Drac (965) has stressed, when algebrac methods are used for systems wth an nfnte number of degrees of freedom (e.g., feld theory), we can obtan solutons to some physcal problems that gve no soluton n the usual Schrödnger pcture. Ths possblty arses because n these systems the equvalence between the Schrödnger representaton and the Hesenberg representaton no longer holds. Indeed n feld theory there s no longer one unque representaton. There exst many nequvalent representatons and these gve rse to more general structures as has been ponted out by Emch (97) and more recently n Haag (99). These works are of a more mathematcal nature but from the pont of vew of physcs what emerges s that the densty operator,, rather than the wave functon, plays a fundamental role. Ths means we have a natural way of descrbng more general quantum processes that nvolve, for example, thermal systems. Systems descrbed by wave functons then correspond to the specal case when = whch means the system s effectvely at zero temperature. But t s not merely a queston of the appearance of non-zero temperatures. As Prgogne (994) has already ponted out, nonntegrable systems gve rse to dffusve terms that correspond to the symmetry breakng n tme. Such systems cannot be descrbed n terms of wave functons. The only possble way of descrbng such systems s through the densty operator. These algebras are called generalsed Clfford algebras by Morrs (967)

3 Whle all the mportant new features that arse n the algebrac approach nvolve an nfnte number of degrees of freedom even n the case of systems wth a fnte number of degrees of freedom we fnd new features that are rarely dscussed n the physcs lterature. One such example, whch we shall dscuss later, s the appearance of algebrac spnors. We show that these spnors are a generalsaton of the ordnary spnor, whch have new mathematcal mplcatons that are gnored n the usual approach. Ths generalsaton s not only an enrchment of the mathematcs. Followng on from the work of Frescura and Hley (980a, 980b), Monk and Hley (998) have ponted out that the algebra provdes a very dfferent perspectve on the nature of quantum processes. In fact by emphassng the purely algebrac aspect of the approach we remove the dstncton between operator and operand and ths allows us to provde an nterpretaton n terms of the noton of process rather than n terms of partcles and/or felds n nteracton. These deas were developed to explore connectons wth non-commutatve geometry but we wll not develop these deas here (See Hley 99). Indeed t s not the purpose of ths paper to attempt to justfy ths more general and speculatve poston here. Rather we want to explore the mathematcal consequences of adoptng ths more general poston by brngng out ts relatonshp to the usual approach, a feature that s generally lost n the many detals provded, for example, n excellent two volume treatse by Brattel and Robnson (979). In partcular we want to present a smpler approach through whch we can see what results depend only on the structure of the abstract algebra, what features depend on a specfc representaton, usually a matrx representaton, and what features requre the constructon of a Hlbert space. To mantan generalty we start by recallng the man propertes of an abstract * algebra. We then show a how a measure, ω, called a weght s ntroduced, whch plays the role of the state. ω s a functonal, mappng elements of the algebra onto the real feld. It s not dffcult to show that ths s equvalent to ntroducng the densty operator n the usual approach To motvate the approach we frst explore the orthogonal Clfford algebra by showng how prmtve dempotents play a crucal role n ths approach. We use these dempotents can be used to construct mnmal left and rght deals. These deals are spanned by algebrac spnors and these elements play the role of 'wave functons' n the Hlbert space formalsm. But t must be emphassed these elements are contaned entrely wthn the algebra and no external vector space s needed. We have already stressed ths pont n Frescura and Hley (980a). In ths paper we show how these same deals are generated n the standard algebrac approaches descrbed n Haag (99), and n Emch 3

4 (97). As the am of ths paper s manly pedagogc, we rely heavly on the theorems proved n those volumes, partcularly the latter and wll not reproduce them here. We then go on to llustrate how these methods can be to specfc example that use two of the smplest Clfford algebra R, and H. R, s the Clfford algebra of a relatvstc spacetme wth one space dmenson whle H s the quaternon algebra. We choose these not for physcal reasons but because they are the smplest examples n whch to demonstrate the prncples nvolved. We also show how the densty operator s constructed and the role t plays n the whole structure. We then wden the dscusson to the dscrete Weyl algebra C n (Weyl 93 and Monk and Hley 993) ultmately generalsng to the symplectc Clfford algebra (Crumeyrolle 990). Ths enables us to extend the dscusson to nclude the Hesenberg algebra. We brng out clearly what s nvolved n ths structure by frst explorng C 3. In the mathematcs lterature ths gves the structure of the nonons frst ntroduced as a generalsaton of the quaternon many years ago by Sylvester (884). As the generalsaton to hgher n s straght forward but tedous we fnally dscuss the lmt as n to nclude the Hesenberg algebra to make contact wth standard quantum mechancs. In dong ths we show how the algebrac approach s related to the Hlbert space approach. Fnally we show how the densty matrx can be wrtten as a vector n a hgher dmensonal space, whch can be mapped nto a Hlbert space. Ths s the so-called GNS constructon, whch arses n a very straghtforward manner n the algebrac approach. We are then mmedately able to connect up wth the thermal feld theory approach of Umezawa (993) that explots the rch propertes of balgebras. Physcally ths approach allows a descrpton of the thermal propertes of quantum processes.. The Algebra. Snce our man ntenton of the paper s pedagogcal we begn by outlnng the propertes of the general algebra wthn whch we wll work. Recall that a lnear algebra, A, s a vector space wth () addton A + B A, () multplcaton by a scalar, A A, s an element of the real or complex feld, () together wth a product AB A. We wll assume the product to be assocatve. Furthermore we wll ntally consder the algebra to be of fnte rank, n, wth a fnte bass {e }. 4

5 Snce we requre the analogue of a Hermtan conjugate we assume the algebra, A, s equpped wth an nvolutve ant-automorphsm defned by *: A A wth the followng propertes : For A, B A (A*)* = A, ( A)* = *A* C (A + B)* = A* + B* (AB)* = B*A* We wll further assume the followng two requrements are satsfed, () For each A A, B n A such that A*A = B () A*A = 0 A = 0. So as not worry too much about the abstract nature of ths algebra, t s useful to keep n mnd a matrx algebra wth Hermtan conjugaton defned. 3. The Densty Operator. 3. The Algebrac prelmnares. We need some way of obtanng numbers from ths algebra that wll allow us to dentfy wth expectaton values used n quantum mechancs. We use a analogous method to that used n set theory by ntroducng a functonal such that : A R or C A A such that (A) =, R or C s a postve lnear functonal (called the expectaton or state functonal) satsfyng, ( A + B) = (A) + ( ); ( A) = (A) and (A*A) 0 If the algebra has a unt,, then we choose () =. Ths means that the state s normalsed. The collecton of all lnear functonals over A forms the dual space A* of the algebra. Now for A A, we have a real or complex number (A) dependng lnearly on A wth (A*A) 0. Ths enables us to defne a Hermtan scalar product through 5

6 A B = ( A B) A, B A. () The collecton of all postve lnear functonals over A s the postve cone A* (+) dual space A* of the algebra. of the In A* s called 'extremal' f t cannot be decomposed nto a lnear combnaton of two others. These extremal functonals are called 'pure states'. Thus n general we can form r (A) = (A) wth = = where are a set of extremal states. As the algebras we consder n ths paper are non-nlpotent, they have a set of dempotents. We can relate the extremal states to the prmtve dempotents of the algebra n the followng way. If { } are a set of prmtve dempotents, we can wrte = where j = j j. Snce () = we have r r r r () = () = j = j = = ( ) j =, j = Ths defnes the condton ω (ε j ) = j so that the condton = s satsfed. In ths way we establsh a one-to-one correspondence between extremal states and prmtve dempotents n the algebra. r = We emphasse that through out ths dscusson we reman entrely wthn the algebra A even though the precse choce of the functonal s left open at ths stage. However f we are thnkng n terms of a matrx algebra, then we can thnk of the lnear functonal as defnng the trace. However t s not necessary to fnd a matrx representaton of an algebra to defne the nvarant concept of a trace. For a fnte algebra the trace can be evaluated by examnng the coeffcents of the characterstc polynomal of the algebra. A detaled dscusson of ths technque as appled to the algebras treated n ths paper wll be found n Frescura and Hley (98). An dempotent s prmtve ff there does not exst two dempotents, such that = +. An dempotent s prmtve ff A = where A A and Z the centre of A. 6

7 3. The Physcs. In quantum mechancs we are nterested n expectaton values. If the system s descrbed by a pure state wth wave functon (r, t) then the expectaton value s defned through the nner product A = A = (r,t)a (r,t)d 3 r where the assumpton of the contnuty of the wave functon (r,t) forces us nto an abstract Hlbert space. In what follows we wll not make ths assumpton and wll work completely n the algebra. Thus Hlbert space n not essental to deal wth spn systems such as the Paul spnor, the Drac spnor and the twstor. In the usual approach a system n a mxed state requres a densty matrx. In ths case the r expectaton value A = A can be wrtten as = A = Tr( A) () where s the densty operator wth the propertes that =, Tr( A * A) 0 and Tr( ) =. Now n the usual approach usng Hlbert space we always have to make a separaton between the algebra of observables and the vector space on whch the observables operate, but when the system s descrbed by a fnte algebra A, there s no need to ntroduce ths extra external space. Everythng can be done wthn the algebra as there s a sutable vector space already contaned completely wthn the algebra. Indeed there are a set of subspace of ths vector space and these correspond to the mnmal left deals of the algebra. There s also a correspondng dual vector subspace, namely, the mnmal rght deals. It s through these deals that one can descrbe the physcal propertes of the physcal system wthout leavng the algebra. (See Frescura and Hley 980a, 980b, Benn and Tucker 983 and Monk and Hley 998) Let us recall how ths can be done. Recall that a left deal, I L conssts of a set of elements K such that I L = {K A AK I L A A} (3) 7

8 These are just the algebrac spnors dscussed n Frescura and Hley (980a). They are generated by the prmtve dempotents ntroduced above. For example, f s a prmtve dempotent, then a mnmal left deal I L s generated as follows; K = A A A and K I L Now a rght deal I R s defned through I R = {K A KA I R A A} (4) So that a rght deal I R can also be generated by gvng K ' = A A A and K ' I R Snce ( )* = we can generate a rght deal that s dual to the left deal and s defned through K* = A* A* A Thus f A ε I L and B* ε I R then equaton () becomes ( B A) = B A (5) where B A C. In order to make contact wth the expectaton values used n physcs, the pure state 'wave functon' s thus replaced by an element of the mnmal left deal and ts Hermtan conjugate s replaced by an approprate element of the rght deal. Then f B I L and B * I R, A = ( B * AB ) A = Tr( A) (6) where s the densty operator for the pure state. We can now construct a densty operator as a element n the algebra as follows: A = ( B AB ) = ( AB B ) = ( A ) = ( A) Here the densty operator s 8

9 = B B * (7) so that the densty operator can be wrtten as an element of the algebra, A. We can also regard B as the 'square root' of the densty operator, B =. Let us make explct the role of the prmtve dempotent by wrtng = A A* A A Thus we see that the densty operator s an element of a two-sded deal subject to the condtons that s a postve and ( ) =. Ths puts a normalsaton constrant on A. We can check that ths densty operator actually does correspond to a pure state by showng drectly that tself s dempotent. In fact = (A A*)(A A*) = A( A*A )A = A A* = where R or C. can always be re-defned to absorb. In the case of the mxed state we have r A = (B AB ) = Tr A = where the densty operator takes the form r r = = B B * = B = r = ( ) Tr( A) (8) ( ) B * ( ) (9) If we are able to wrte the densty operator n the form ths can be done later, then we can wrte = DD * and we wll show how A = ( ADD * ) = ( D * AD) Ψ A Ψ (0) Thus even n a mxed state we can descrbe the system by a sngle vector D. Ths s equvalent to replacng the densty matrx as a sngle 'wave functon' Ψ n the usual Hlbert space formalsm. Thus we have the possblty of descrbng a thermal system by a wave functon mplyng that ths wave functon s a functon of temperature. In other 9

10 words the wave functon becomes a functon of temperature. Ths s what s called the GNS constructon (see for example Emch 97). From these results we see that the algebrac approach gves prmary sgnfcance to the densty operator and that ths operator les entrely wthn the algebrac structure tself. The 'wave functon' s replaced by an element of the left deal, whch, as we have seen, s an element of the algebra. Thus at least for fnte algebras the Hlbert space s an external auxlary feature that s not essental to descrbe quantum systems. Thus we can regard the wave functon smply as a devce to enablng us to calculate expectaton values usng famlar mathematcs and not as some feature descrbng the state of the system. Ths whole approach allows us to start from the more general mxed state, seeng the pure state as a partcular rather smple state. Methodologcally ths s much more satsfactory than the usual approach because we are not faced wth the need to generalse the formalsm when we fnd we have to deal wth more general systems that requre 'outsde' factors to somehow destroy coherence and form mxed states. Indeed the physcst's emphass on the wave functon as beng a descrpton of the state of a quantum system and therefore a prmtve element of the descrpton may be too restrctve (See Prgogne 994). A more general approach based on the algebra as suggested by Monk and Hley (998) has the possblty of avodng some of the nterpretaton problems n quantum mechancs. However ths comment should not be taken to mply that ths new approach wll enable us to solve all the problems of nterpretaton. It does not. What t does s to open up another way of lookng at quantum phenomena n whch the algebra tself s playng a prmary role. In other words snglng out the densty operator as the prmary feature of the mathematcs opens up the possblty of gvng a dfferent meanng to the formalsm (See Fernandes and Hley 997 and Brown and Hley 000). As I have argued elsewhere, ths approach suggests that a noton of process rather than partcles/felds-n-nteracton should be taken as basc (Hley 995). In a process based phlosophy there are no separately exstng objects. They are nvarants of the total unfoldng process. Here the wave functon s regarded as a transton probablty ampltude rather than a functon of state of some system. As Bohr nssted, there s no sharp separaton between subject and object so we should have no sharp separaton between operator and operand. The mathematcs of the algebrac approach reflects these notons and makes no such dstncton. Thus we do not have 'operators' playng the role of 'observables' whch are to 'act' on vectors n a dfferent abstract space. All aspects of the process are unted n the same structure descrbed by elements of the same algebrac 0

11 structure, elements that only have meanng n the context of the total algebra. Ths approach has some nterestng consequences that have already been dscussed n some detal by Hley and Monk (993) and wll not be dscussed further here. The emphass n ths paper wll be on those aspects of the algebrac structure that are relevant to ths general dscusson. 4. Mnmal Left Ideals, Algebrac Spnors. 4. Pure States. Let us now llustrate how these deas work by consderng a fnte, lnear *algebra on whch a state s defned. We requre to fnd a set of mnmal left deals. We wll not follow the technques based on the characterstc polynomal whch have already been outlned n Frescura and Hley (978). Rather we follow the method that used by Emch (97) and s essentally based on the fact that ( j ) = j as dscussed earler. They produce exactly the same results but ths method s the one conventonally used n algebrac quantum mechancs. (See, for example, Haag 99) We can construct left deals by solvng the equaton (B*K) = 0 for K and for all B* A. Here we are usng the defnton { ( B * K) = 0, B } () I Lω = K to construct the left deals. To show ths defnton does n fact produce a left deal take any A and B n A and any K Lω and use assocatvty to establsh ( A * ( BK )) = B * A (( ) * K) = 0 () showng that Lω s a left deal n A. The left deal generated n ths way s sometmes called the Gel'fand deal. Smlarly a rght deal can be generated by the sub-set Rω = { R ( RB * ) = 0, B } 4.. A Smple Example to llustrate these Ideas.

12 As the man purpose of ths paper s pedagogc, I wll now demonstrate how these deas work n a smple real Clfford algebra R,. Ths example s not wthout physcal meanng as t forms the bass of relatvty n one spatal dmenson, and one tme dmenson. It s the background structure to spn networks and relatvty dscussed by Kauffman (99). The algebra s charactersed by the multplcaton table below e e e e e e e e e e e e wth e * = e ; e * = e ; e * = e. Defne the state by It s easy to show usng (B*A) = 0, that ( + e + e + e ) = (3) * (( )( a( + e ) + b( e + e ))) = 0 + e * + e * + e confrmng that produces an element n a left deal L defned by Ψ L = a + b (4) where = ( + e ) and = (e + e ). The element n the correspondng rght deal can be shown to be where = ( + e ) and = ( e e ) Ψ R = a * + b* (5) Then we can form

13 whch s unty for a normalsed state. ΨR* ( Ψ L ) = Ψ R Ψ L = a + b We now construct a densty operator from elements n these deals. Recallng the defnton of the densty operator defned n secton 4.0, we fnd that ths densty operator s [ ] = Ψ Ψ = L R ( a + b )+ ( ab + a b)e + ( a b ab )e + ( a b )e = a + ab* + a * b + b (6) We can generate a second set f elements n second left deal by usng the state so that ( + e + e + e ) = + (7) * (( )( c( e ) + d( e e ))) = 0 + e * + e * + e producng the element of the left deal, Ψ L defned by Ψ L = c + d. (8) where = ( - e ) and = (e - e ). The correspondng element n the rght deal s Ψ R = c * + d * (9) where = ( e ) and = ( e + e ). Then we fnd whle the densty operator s ΨR* ( Ψ L ) = Ψ R Ψ L = c + d 3

14 [ ] = Ψ Ψ = L R ( c + d ) + ( cd + c d)e + ( cd c d)e ( c d )e = d + c* d + cd * + c (0) We see mmedately that the two left deals are generated by the dempotents = (+ e ) and = ( e ) so that + = and. = 0. Snce these dempotents are prmtve we have generated two mnmal left deals. These are the results presented n Frescura and Hley (980a). A more lengthy dscusson of the Paul Clfford wll be found n Frescura and Hley (978). 4.3 A Matrx Representaton. These results can be made clearer f we consder the followng matrx representaton although these matrx representatons are not necessary n general; ( ) = 0 e ; e 0 ; e 0 ( ) = 0 () 0 ( ) = 0 In terms of matrces, the elements of the mnmal left deals become (Ψ L ) = a 0 and (Ψ b 0 L ) = 0 d () 0 c The appearance of two mnmal left deals may at frst sght seems surprsng snce n physcs there s only one column vector for spn 3. In secton 8 we wll show exactly how these two spnors are merged nto one. At ths stage we merely remark that the two spnors are equvalent under the nner automorphsm defned by 3 It should be noted that mnmal left deals are not always represented as sngle column matrces. For example ( Ψ L ) = a b a b s a mnmal left deal. a + b a + b 4

15 (e ) = 0 0 The correspondng elements of the mnmal rght deals become (Ψ R ) = a* b * 0 0 and (Ψ R ) = 0 0 (3) d * c * Fnally let us return to the defnton of the state and see how t fts n wth the matrx representaton. In ths case the densty operator takes the specal form shown above so that ( ) = Tr ( + e + e + e ) + e + e + e whereas for the densty operator takes the form so that + e + e + e ( ) = (4) ( ) = + (5) ( ) = Tr ( + e + e + e ) Thus we fnd consstency snce equatons (4) and (5) are dentcal to equatons (3) and (7). Ths means that the state defned by generates the left deal generated by, and vce versa. The reason for ths s as follows. Snce (B*K) = 0 for B A, we must have (K*K) = 0 for K L. Snce ( * ) 0, L. Thus the left deal generated by (B*K) = 0 s always the complement of the deal generated by. We wll dscuss ths pont agan n secton Mean Values. As we have seen above we can wrte Now f we wrte A = ( Ψ R AΨ L ) A = a 0 + a e + a e + a e Then usng the above expressons for L and R above we fnd ( ) = a + b Ψ R AΨ L [( )a 0 + ( a * b + b * a)a + ( b * a a * b)a + ( a b )a ] + e ( ) 5

16 whle ( ) = a + b A = Ψ R AΨ L We can also wrte [( )a 0 + ( a * b + b * a)a + ( b * a a * b)a + ( a b )a ] (6) A = ( Ψ R AΨ L ) = ( AΨ L Ψ R ) = ( A ) = Tr ( A) Indeed one can evaluate (A ) and show t s equal to the result calculated n the last equaton. 5. Mxed States and the GNS Constructon. In order to llustrate how to deal wth mxed states for the algebra R,, we start wth the matrx representaton defned n secton 4.3 and choose the specfc densty matrx ( ) = 0 0 It can quckly be checked that n fact corresponds to mxed state by showng. Rememberng that s equvalent to takng the trace, t s straght forward to show that ( A) = ( a 0 + a e + a e + a e ) = ( + )a 0 + ( )a (7) Let us now try to fnd the Gel'fand deal usng the technque outlned n secton 4. and 4.. We start by wrtng ths deal n the general form Ψ L = z + z e + z e + z e Then we need to solve the equaton wth (( a 0 + a e + a e + a e )( z + z e + z e + z e )) = 0 ( a 0 ) = a 0 ( + ); ( be ) = 0; ( ce ) = 0; ( de ) = d( ) We can then show that the equaton we have to solve s 6

17 [ ] + a [ + ] + [ ] + a [ + ] = 0. a 0 ( + )z + ( )z ( )z + ( )z a ( + )z + ( )z ( )z + ( )z Ths equaton has no non-trval soluton except for =, = 0; and = 0, = whch are the results we obtaned n secton 4.. Hence we cannot extend these methods to general values of and therefore we have to use an alternatve method, whch we wll dscuss below. However before gong on to explore a more general method let us frst explore how the mxed state can be handles wthn R,. In secton 3. equaton (9), we showed that the densty matrx could be wrtten n the form r = B B = whch n ths example can be wrtten as * = + = Ψ R Ψ L + Ψ R Ψ L where Ψ L = ( + e ) = wth Ψ = R ( + e ) = and Ψ L = ( e ) = wth Ψ R = ( e ) = so that = + = + (8) Then gves A = Tr( A) = A ( ) + ( A) A = ( + )a 0 + ( )a (9) 7

18 Ths then agrees wth the result calculated usng the matrx representaton above. Thus we see that we use both mnmal deals n constructng the densty operator. Ths result has already been used for the Drac Clfford algebra by Frescura and Hley (987) 5. GNS Constructon. As we ndcated n secton 3. we would lke to construct a D so that we can wrte the densty operator as = DD * whch n turn allows us to wrte A = ( D * AD) = ( DD * A) = ( A) = Ψ A Ψ so that D s the algebrac equvalent of Ψ. Ths would enables to fnd the algebrac background to the GNS constructon. From equaton (8) together wth the fact that, are the orthogonal prmtve dempotents satsfyng jj = j jj, we can mmedately wrte so that D = (30) = A = A r r = ( + )a 0 + ( )a whch s the result produced n equaton (9). Thus we have obtaned exactly the same result as that obtaned by matrx representaton used n secton 4.. It s now obvous how to generalse ths. We smply wrte whch mmedately gves A = r D = (3) = ( D * AD) = ( A) Ψ A Ψ 5. The Matrx Representaton 8

19 The GNS constructon s usually presented as a matrx representaton (See Emch 97). We can mmedately construct a matrx representaton f we frst notce that any matrx transformaton of the form C D can be wrtten n the form ( C D) ˆ where ˆ s the n n densty matrx wrtten as a column wth n entres. But we are nterested n formng where A = Tr( A) = Tr( DD * A) = Tr( D * AD) = Tr( ( D * ) ( A ) ( D) ) (3) ( D) s a column wth n entres. Ths can be wrtten as ( D) = j r ( ) jj = B j j = r j = ( ) (33) In the matrx representaton we can wrte That s where ( B j ) = 0. 0 j 0. 0 ( B j ) s the n column matrx wth j appearng n the s = [j(n + )+] element. Here j s the j th egenvalue of the densty matrx n the representaton used n secton 5. Ths enables us to ntroduce the column matrx ( D) wth n the j js sth element satsfyng s = [j(n + )+] as j runs from to r. Then we can wrte A = ( A) = Tr( A) = ( (D * )( (A) ()) (D)) (34) 5.3 Specfc example Let us llustrate the above dscusson by returnng to the example n secton 5.. Here we fnd that (D) becomes 9

20 ( D) = 0 0 (35) so that a 0 + a a a A = Ψ A ˆ a Ψ = ( 0 0 ) + a a 0 a a 0 + a a a a + a a 0 a 0 0 A = ( + )a 0 + ( )a whch s dentcal to equaton (7) showng that we can get exactly the same result as the standard method usng the densty matrx. The advantage of ths method les n the fact that when generalsed to a Hlbert space constructon, whch we wll dscuss later, we have 'wave functons' dependng upon temperature. Ths leads us to a more generalsed quantum mechancs dscussed by Umezawa (993), 5.4 A more Physcal Example. To ths pont we have been consderng mathematcal structures per se. Let us now turn to see how these technques n a physcal problem. Consder a thermal spn system n an external magnetc feld. A typcal densty operator can be wrtten n the form = exp[ H] Tr exp[ H] (36) where = /kt and H s the Hamltonan H = B 3. Snce we are usng Paul spn matrces we must use the standard multplcaton rules for Paul spn matrces,.e. the quaternon Clfford algebra, H and not R, as used above. Usng the standard representaton for the Paul spn matrces = 0 0 = 0 and 3 0 = 0 0 0

21 we fnd + tanh = 0 0 tanh (37) then t s straghtforward to show ( a+ b x + c y + d z ) = a + d tanh (38) Let us now try to fnd the Gel'fand deal usng the technque outlned above. We start by wrtng ths deal n the general form Ψ L = z + z + z + z 3 3 Then we need to solve the equaton wth (( a + a + a + a 3 3 )( z + z + z + z 3 3 )) = 0 ( a) = a; ( b ) = 0; ( c ) = 0; ( d 3 ) = d tanh It s straghtforward to show that the equaton we eventually have to solve s ( a a 3 tanh )z + ( a + a tanh )z + ( a a tanh )z + ( a tanh + a 3 )z 3 = 0. Once agan ths equaton has no non-trval soluton so we cannot used the methods outlned above n secton 4 and we are forced to use the GNS constructon. To do ths we need a par of prmtve dempotents from the algebra H. These are so that = ( + 3 ) and = ( 3 ) (39) D = ( + 3 ) + ( 3 ) (40)

22 If we wrte A = a + b + c + d 3 then A = ( A ) = ( D * AD) = a + d tanh whch s the value found n equaton (38). In the matrx representaton dscussed n 5. we fnd ( D) = Ψ = + tanh 0 0 tanh wth ( ) = 0 0 (4) Agan t s straght forward to check that ( ( A)) = Ψ ( A) Ψ = a + d tanh 6. Umezawa's Approach. In the above generalsaton to mxed states we need to consder a product representaton A. It s of nterest to compare the above approach wth that ntroduced by Umezawa (993). Suppose we start wth a general thermodynamc system wth the partton functon gven by Z( ) = Tr( exp[ H] ) where = /kt and H = H 0 N. Here H 0 s the partcle Hamltonan and s the chemcal potental. The ensemble average s gven by A = Tr ( exp[ H]A ) Z( ) We want to fnd a state Ω( ) such that

23 A = Tr ( exp[ H]A ) = Ω( ) (A) Ω( ) Z( ) Suppose H n = E n n, then A = Z ( ) n A n exp[ E n ] = Ω( ) (A) Ω( ) n Can we fnd Ω( )? To ths effect let us wrte Ω( ) = n n f n ( ) so that * Ω( ) (A) Ω( ) = f m ( ) f n ( ) m A n = Z ( ) n A n exp[ E n ] n,m n whch wll be satsfed only f f m * ( ) f n ( ) = Z ( )exp[ E n ] mn However t s mpossble to satsfy ths relaton f f n ( ) are smply complex numbers. Umezawa suggests we wrte f n ( ) = Z ( )exp[ E n ] n Here n s a copy of the orgnal set of kets. Then f m * ( ) f n ( ) = Z ( )exp[ ( E n + E m ) ] mn Ths means we can wrte Ω( ) = Z exp[ E n ]n n. (4) n We can show ths s exactly the same vector as s obtaned from the GNS constructon. We wll smply llustrate the smlarty usng the example of secton 5.4. It s straghtforward to show that and Z exp[ E ] = + tanh 3

24 Z exp[ E ] = tanh Snce and We fnd 0 = = 0 Ω( ) = + tanh 0 0 tanh whch s dentcal to equaton (4), the result we obtaned from the GNS constructon. What we see from ths approach to the GNS constructon s that we need to "double up" the algebra snce we need two copes of the vector space. To put t more formally we need to form the balgebra. Indeed ths s exactly what Umezawa (993) proposed n hs approach to thermal feld theory. Startng from the boson algebra generated by the set {, a, a } where a s the boson destructon operator and a the correspondng creaton operator, he ntroduced an addtonal par of annhlaton and creaton operators a and a whch he attrbutes to a "ghost" feld (Umezawa 993a). Of course the dea of a ghost feld s not very appealng from the physcs pont of vew, however ths approach opens up the possblty of usng ths doublng to account for quantum dsspaton (Celeghn, Rasett, and Vtello 99). Hley and Fernandes (997), followng from Bohm and Hley (98), have shown ths doublng of the number of degrees of freedom s also requred n gong from two pont functons n confguraton 4

25 space descrpton to an algebrac phase space descrpton. Ths gves an alternatve vew on dsspaton. These deas wll be developed further n a forthcomng paper. 7. Extenson to the Hesenberg Algebra. All of the prevous examples have used the orthogonal Clfford algebra n some form or other. What we would lke to do now s to show how the whole procedure can be extended to the Hesenberg algebra. As ponted out n a prevous paper (Hley 00), the man problem n followng the above approach arses from the fact that the Hesenberg algebra s a nlpotent algebra of degree three under the product [A, B]. As a consequence of a well-known theorem, nlpotent algebras do not contan any non-trval dempotents so t s not possble to construct any non-trval left deals. We want to approach Hesenberg algebra n two steps. Rather than go to the full algebra mmedately, we wll start by consderng the fnte algebra ntroduced by Weyl (930) whch contans the Hesenberg algebra n the contnuum lmt. The Weyl algebra has an extremely smple structure. It s a fnte polynomal algebra C n generated by the set of elements {, e, e } subject to e e = e e e n =, e n =, wth = exp[ /n] (43) where n s an nteger. Ths algebra has a long hstory beng frst explored by Sylvester n 884 as an example of a generalsaton of the quaternons. He called the elements nonons for n = 3 and n-ons more generally. What Weyl shows s that we can wrte e = exp[ P] and e = exp[ X] where = / pn and = / xn so that we begn to see the begnnngs of the Hesenberg algebra. Indeed n the lmt as n ths algebra does approaches the Hesenberg algebra wth X and P representng the poston and momentum operators.. The mportance of the algebra from our pont of vew s that t s not nlpotent. In order to fnd the dempotents usng the method descrbed n the secton 4, equaton () we need to generate the sub-set satsfyng = { K ( B * K) = 0, B } In order to see how ths works n ths example we agan llustrate the procedure wth an example. 5

26 7. The nonons. We now take the smplest example of the structure descrbed n the last sub-secton, namely, n = 3, the nonons, and wrte down the explct multplcaton table explctly. e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e wth e * = e ; e * = e ; e * = e ; e * = e, and (e *)* = e ; * =. We defne the state by ( a) = a; be ( ) = b; ce ( ) = c; Rest = 0. Now we need to solve the equaton (B*K) = 0, where B = + e + e + e + e + e + e + e +Ωe = + e + e + e + e + e + e + e +Ω e and K = z+ z e + z e + z e + z e + z e + z e + z e + z e. It s tedous but straght forward to show that the soluton of the equaton (B*K) = 0 s ( + + )( z + z + z ) + ( + + )( z + z + z ) + + +Ω whch gves soluton (A) as ( ) = 0 ( )z + z + z z = ; z = ; z = 6

27 z = ; z = ; z = z =; z = ; z = so that the left deal s 3Ψ L (A) = ( + e + e )a + ( e + e + e )b + ( e + e + e )c (44) Soluton (B) s z = ; z = z = ; z = z =; z = ; z = ; z = ; z = 3Ψ L ( B) = ( + e + e )a + ( e + e + e )b + ( e + e + e )c (45) We see from these two solutons we have two dempotents ( ) and = 3 + e + e = 3 + e + e from ( ). We can fnd the thrd = So that 00 = 3( + e + e ) from whch we can generate the thrd left deal by multplyng ths dempotent from the left by each element of the algebra. For completeness we fnd ths thrd deal s explctly 3Ψ L ( C) = ( + e + e )a + ( e + e + e )b + ( e + e + e )c (46) It s now straght forward to use these three left deals to construct densty operators and mean values for both pure and mxed states as descrbed n secton 4 and 5. It s also possble to construct the GNS representaton n a straght forward manner. We wll not go nto the detals here. 7. Generalsaton to arbtrary n. The generalsaton to arbtrary n s also very straght forward. It s easy to obtan a general expresson for the famly of dempotents defned n the prevous secton. They are 7

28 n = j e n j = 0 ( ) j, j = 0,,...,n. (47) Each one of these dempotents can be used to generate a set of n left deals so that the technques dscussed n secton 4 and 5 can also easly be generalsed to arbtrary n. 8. Hlbert Space and the GNS Constructon. Now we want compare our approach wth the approach through Hlbert space. Ths means showng how the mnmal left deals are related to vectors n a Hlbert space. In order to begn we must frst ntroduce the noton of -equvalence classes of elements of the algebra, A. To do ths consder two elements A and B of A. Whenever A B ω, we say A and B are -equvalent. Snce ω s a lnear sub-space of A t s clearly an equvalence relaton. For every A A we denote ts equvalent class by Ω(A). We now equp the set of equvalent classes wth a vector space structure I ω by defnng Ω(A) + Ω(B) = Ω( A + B) A, B A, λ, µ R or C We can equp I ω wth a pre-hlbert space structure by defnng ( Ω(A),Ω(B) ) =Ω(A * A) Wth ths we can defne a norm Ω(A) =Ω(A * A). If necessary we can use ths norm to complete I ω thus formng a Hlbert space. 8.. Examples. To llustrate the mplcatons of ths structure we agan examne the example R, gven above. We take the algebrac spnor defned above n equaton (8), namely, Ψ L = a ( - e ) + b (e - e ). We then see clearly that and e Ω(), whle e and e Ω(e ). The Hlbert space s thus two-dmensonal so we can wrte H ω = Ω() + Ω(e ) (48) 8

29 Where Ω() and Ω(e ) are orthogonal, provdng a bass for the two-dmensonal Hlbert space H ω. Notce that when we pass to the equvalence class we lose some of the mathematcal structure n that both L and L form equvalent Hlbert spaces. In standard quantum mechancs t does not seem necessary to dstngush between these two spaces. In the case of the nonons we can show that the three deals L, L and L3 belong to a sngle equvalence class. Thus wth the obvous generalsaton of equaton (48) we can construct a Hlbert space by wrtng H ω = () + (e ) + (e ). (49) Ths gves a three-dmensonal Hlbert space as expected. Notce that n ths case we are dealng wth a pure state so that the GNS constructon gves us an rreducble representaton. We could generalse to the mxed state by choosng a dfferent state ω'. For example we could start wth the densty matrx gven n equaton (36) but choosng for the Hamltonan H = B ( e ). The GNS constructon wll go through exactly as explaned n secton 5. As the example chosen s not motvated by any relevant physcal system we wll not dscuss t further here but leave t as an exercse for the nterested reader. It should also be stressed that these are not the only left deals n the algebra. Indeed there are an nfnte number of them obtaned from nner automorphsm. Thus we can always fnd another left deal by frst fndng a new set of dempotents usng = S S S A (50) It mght be of some nterest to note that a sub-set of these S generate a set of dscrete Fourer transformatons. 8. Hlbert Space contnued. We now defne for all A A 9

30 π ω (A) : H ω H ω, so that π ω (A)Ω(B) = Ω(AB). (5) We must ensure that ths defnton mples that Ω(AB') s ndependent of B' n Ω(B) so that t does not matter whch Ω-equvalent element B s used to label the equvalence class Ω(B). To ths effect Ω(AB) Ω(A B ) = Ω A B B ( ( )) =Ω AB ( B ) = Ω B B (( ) * ( A( B B ))) (( ) * A * A( B B )) A *. Ω( B B ) = 0 Ths s zero because B B' Lω. Thus t does not matter whch element n the equvalence class s used to label the class. Now we need to verfy that ω(a) s, ndeed, a representaton. ( A + B)Ω(C) = Ω ( A + B)C ( AB)Ω(C) =Ω( AB)C ( ) = Ω(AC) + Ω(BC) = ( ( A) + ( B) )Ω(C) ( ) =ΩABC ( ( )) = (A) (B)Ω(C) ( (A) * Ω(C),Ω(D) ) = ( Ω(C), ( A)Ω(D) ) = ( C * AD) ( ) = ( Ω( A * C),Ω(D) ) = (A*)Ω(C),Ω(D) = ( AC) * D ( ) Thus ω s a homorphsm of the nvolutve algebra A to the set of operators actng on the pre-hlbert space Lω. It s not dffcult to show that, f necessary, we can extend ω (A) to a bounded operator on H ω n a unque manner. Furthermore snce ω(a*) = ω(a)*, ω(a) s self-adjont for every A A Return to the Example. Returnng to R,, n the orthogonal bass (), (e ) we fnd () = 0 (e 0 ) =

31 (e ) = 0 (e 0 ) = 0 0 Thus we have produced an rreducble representaton and we can wrte ( ) = + e + e + e + + (5) whle Ω() = and Ω(e 0 ) = 0 (53) Thus n a pure state defned by (A) gven above, we fnd that the representaton for the famlar real Paul spnor. s In the case of the nonons equaton (53) s generalsed to Ω( ) = 0 0 ( ) = Ω e 0 and Ω e 0 ( ) = 0 0 Gvng us the bass for the three-dmenson representaton dscussed n secton 8.. It s not dffcult to show that the correspondng matrx representaton can be wrtten as ( e ) = ; e 0 0 ( ) = and e 0 0 ( ) = The rest of the matrxes can be found usng the multplcaton table above. Notce also that ths approach enables us to attrbute to each A A, a vector Ω(A) = (A)Ω(). 3

32 When generalsng to nclude an nfnte algebra we requre the representaton space H ω to be defned n terms of a strong closure of these vectors. Ths means that () s cyclc for ω. A vector s cyclc relatve to the algebra f the set {π ω (A) A A} s dense n H ω. Ths means that for any vector H ω (A) can be made as small as desred. In concluson then our constructon assocates to every state representaton ω of A. We can then wrte the expectaton value as on A a cyclc (A) = Ω (A) Ω where we have wrtten Ω() = Ω. 8.4 The Lmt n The smplest way to proceed to generalse the n-ons s to go to the lmt n. Ths lmt has already been dscussed by Hley and Monk (993) and Hley (00), ther argument beng based on earler work by Weyl (93). We wll brefly outlne the argument agan here for the sake of completeness. We have wrtten e = exp[ P] and e = exp[ X] where = / pn and = / xn so that we can regard e as a translaton n space and e as a translaton n momentum space. Thus we can regard (e ) s as translatng from x k x k s. In the lmt ths corresponds to the transformaton (x) (x s). Smlarly (e ) t takes x k kt x k, whch n the lmt corresponds to (x) exp[tx]. (x). Ths shows how the Schrödnger representaton emerges from the dscrete Weyl algebra. In ths case the dscrete Fourer transformaton mentoned above n secton 7. becomes the usual Fourer transformaton that takes the x- representaton to the p-representaton. If we now examne the lmtng process on the prmtve dempotent 00 we fnd ( ) = n 00 n exp[ X ] d exp[ X] (x) (53) Thus we see that n the lmt the Weyl dempotent 0 (n) plays the role of the Drac delta functon n the contnuum algebra. If we start drectly from the Hesenberg algebra defned through the standard commutaton relatons [X, P] =, we mss the dempotent, 3

33 the Drac delta functon whch we must add later when we have constructed the Hlbert space. If we reman n the algebrac structure then we can formally wrte 00 (n) = then we fnd the relatons (54) X = 0, P = 0, and = If we dentfy E = ntroduced by Frescura and Hley (984) then the followng relaton result: (56) EX = 0, PE = 0, and E = E The set {, X, P, E} then generates what we have called the extended Hesenberg algebra. Further detals of ths structure can be found n Hley (00). 9. Conclusons. We have shown how an approach to quantum mechancs can be bult from the algebrac structure of the Clfford algebra and the dscrete Weyl algebra (or the generalsed Clfford algebra). These algebras can be treated by the same technques that do not requre Hlbert space yet enable us to calculatng mean values requred n quantum mechancs. In the appendx we compare the two approaches n table form. We have also shown how these technques are related to the Hlbert space, whch effectvely requres addtonal structure that nvolves the space-tme contnuum. Yet these algebras already contan geometrc feature not only of space-tme tself, but of a generalsed phase space as already ponted out n Bohm and Hley (98 and 983). The full mplcatons of ths geometrc structure wll be dscussed n another publcaton. 0. Appendx. In ths appendx we lst the correspondence between the algebrac approach to quantum mechancs whch we call "The Algebra of Process" and the usual approach whch we call "The Algebra of Observables". We also nclude the extra step whch nvolves a projecton of the ket vector space nto L (x). 33

34 Algebra of Process. *Algebra Left deal, L L Algebra of Observables. Algebra External vector space, V x = (x) Generated by dempotents j, j = j A j. Bass of deal j wth j jj Bass [ n, AND j n] [ n] Rght deal, R R Dual vector space, V* x = (x) Generated by dempotents j j = j j A. [*: L R, * = ] [ : ] Inner product. ω(rl) = R L Inner product : j j, j, ( ) * x ( x)d 3 x [Trace : coeff of r- of characterstc poly.] (L*L) = ( A ) = ( A ) = A R Outer product, L.R B C L.L* B B* Completeness relaton. Outer product, Projecton operator P = Completeness relaton. = = x x = x x ( ) 34

35 * Densty operator, = B B Densty operator,, + j j + j =, j ( < j) ( ) = + j j + j, j( < j ) ( ) [ρ =ρ, Tr(ρ) =, ρ+ve and lnear] For dagonal For dagonal = = ( x, x) = * x ( ) ( x) Expectaton values. Expectaton values. A = ( A) = B* AB A = Tr A A r = r ( ) = r r A r r A = * x ( ) ( x)d 3 x GNS constructon GNS constructon = D*D = D*D,.e. D = Ω a cyclc vector A = ( D * AD) A = Ω ( A) Ω. References. I. M. Benn and R. W. Tucker, (983), Fermons wthout Spnors, Comm. Math. Phys. 89, G. Brkhoff (966), Lattce Theory, AM. Math. Soc. Coll. Pub

36 D. Bohm and B. J. Hley, (98), On a Quantum Algebrac Approach to a Generalsed Phase Space, Found. Phys.,, D. Bohm and B. J. Hley, (983), Relatvstc Phase Space Arsng out of the Drac Algebra, n Old and New Questons n Physcs and Theoretcal Bology, Ed., A. van der Merwe, pp , Plenum, New York. O. Brattel and D. W. Robnson, (979), Operator Algebras and Quantum Statstcal Mechancs I and II, Sprnger, Berln. M. R. Brown, and B. J. Hley, (000), Schrödnger Revsted: an Algebrac Approach. quant-ph/ E. Celeghn, M. Rasett, and G. Vtello, (99), Quantum Dsspaton, Ann. Phys. 5, A. Crumeyrolle, (990), Orthogonal and Symplectc Clfford Algebras: Spnor Structures, Kluwer, Dordrecht. P. A. M. Drac, (965), Quantum Electrodynamcs wthout Dead Wood, Phys. Rev., 39B, G. G. Emch, (97), Algebrac Methods n Statstcal Mechancs and Quantum Feld Theory, Wley-Interscence, New York, F. A. M. Frescura and B. J. Hley, (978), Mnmum and Characterstc Polynomals and Non-relatvstc Spnors n the Paul Clfford Algebra, (unpublshed). F. A. M. Frescura and B. J. Hley, (980) The Implcate Order, Algebras, and the Spnor, Found. Phys., 0, 7-3. F. A. M. Frescura and B. J. Hley, (980) The Algebrazaton of Quantum Mechancs and the Implcate Order, Found. Phys., 0, F. A. M. Frescura and B. J. Hley, (984), Algebras, Quantum Theory and Pre-space, Rev. Brasl Fs., Volume Especal, Os 70 anos de Maro Schonberg,

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