Sierpinski (<1K) Multivector Quantum Mechanical Algebras

    "Stylists, instead of looking directly into the fact, cling to forms (theories) and go on entangling themselves further and further, finally putting themselves into an inextricable snare."     Bruce Lee.

    We will begin with a synopsis of conventional "ket based" QM due largely to Dirac which is usually formulated via algebras of complex matrices. This traditionally requires an "imaginary" i=Ö-1 which commutes with everything and has been geometrically interpreted in a number of ways.

    We saw in our discussion of spacetime flows how 1-vector fields are physically inadequate for describing particles. We will be interested here in <0;3;4>-vector "spinor" fields in Â4,1 @ C4 associating 3-vectors (dual to 2-vectors) with "spin" and 4-vectors (dual to 1-vectors) with "velocity".
    In this chapter we will justify associating the traditional QM ïñ with ½(1+e345) and ïñ with ½e13(1+e345) and discuss the consequent multipartcle algebras. In the following chapter we will establish a 5D form of the Dirac-Hestenes equation for a charged particle.

Quantum States
    We will ultimately represent basic quantum states by particular multivector fields yp in Â4,1 defined over particular 1-vector pointspaces and satisfying frame-invariant condition yp§yp = 0 where § is the geometric reverse conjugation. The central unit pseudoscalar i acts as our quantum i with i2=-1 ; i#=i#§=-i ; and i§=i. We thus associate "complex numbers" with central multivectors in Â4,1. Pensity ypyp§# will be self-scaling with complex frane-independant scaling facotr rp whose real positive modulus (rprp#)½ provides yp = rpyp~ for a "normalised" yp~ having idempotent pensity yp~yp~#§ .
    A composite state is then represented as a complex weighted superposition yp = åkak ypk over some possibly infinite functional basis where the ak are independant of p
    A composite state can then be represented as a event-dependant complex-valued function of a ket-type multivector, rp(a) reurning the complex ammount of "matter in configuration" a at event p.   
    In this chapter we will be frequently unconcerned with the p dependance, considering instead the geometric nature of yp at a particular event p. We will also be unconcerned with any "propagation equation" like Ñpyp = Fp(yp) to which permitted states are "solutions". How the system ecolves while unobserved is implicit in our defining yp over a spacetime p. Such Hamilton-Jacobi equations Ðe4A = -½m-1 (Ñp[e4*]A)2 + f(p) or Dirac Equation Ñpyp = (m-qap*)yp or Hamiltonian form Ðe4 yp = g(Ñp[e123], yp) as may be solved by yp evolving unobserved will not be of interest until later chapters. We are here concerned with what happens when we poke our clumsy "observer fingers" into the mechanism.
    We will assume an orthonormal basis {e1,e2,e3,e4,e5} with e12=e22=e32=e52=1 ; e42=-1. Any multivector can then be expressed in the form a+ib where a and b are even Â4,1 +  multivectors, and also as c+e5d where c and d are in Â3,1 space e5*.

Kets and Ketvectors
       Dirac's approach can be characterised somewhat uncharitably as "the quest for the 1-vector representation". Dirac represents quantum states by means of a ketvector yp representing the "state" of a "system" at a given spacetime event p.
     We will refer to a ketvector-valued function y defined over some eventspace BaseÌÂ3,1 as a ket y representing the state of the "entire" or "composite" system "across" Base . We denote the local state "at" event p by yp and the "composite" state across Base by y which must acordingly be regarded as a ket-field . In the context of traditional QM, kets and ketvectors are geometrically more akin to a 1-vector in a complex-coordinate vector space than a general multivector in a real-scalar geometric space. We here regard ketvectors as particular multivectors (ideals of a primary idempotent), initally from a 5D Â4,1 spacetime multivector algebra.
    A key Dirac hypothesis is that ketvector  r efi yp represents the same local state as does ketvector yp for any (potentially p-dependant) r>0, f Î Â . Consequently ket r efi y represents the same composite state as does ket y for any p-independant r>0, f Î Â . Thus, if you "double" a quantum state ket indicator y you get 2y indicating the same state as that indicated by y. Classically we expect "states" to be "doublable" in the sense of doubling the "amplitudes" of "oscillations" or similar "effects", but quantum states are "impervious to amplification" or "unscalable".
    The only caveat here is the particular case of a state which is capable of "cancelling itself out". The superposition y + zy always represents the same state at does y except for the particular case z=-1 in which case we obtain the zero state.

Dirac Conjugation

    The essence of Dirac's approach is that for any two states y and c we have a "number"-valued "inner-product" which we will call the Dirac product y»c traditionally denoted áyïïcñ. Dirac considers complex "numbers" but we will be more general and consider "number" to be something that commutes with other numbers. When ket y is a complex matrix with just one nonzero column, the correseponding bra is the conjugate transpose yT^ and the Dirac product is value of the sole non-zero element of the complex matrix psiT^c. Since this non-zero element lies somwehere on the lead diagonal, it is given by the matrix trace of the product matrix and corresponds (with a 2N factor) to the <0;N>-grade part of the geoemetric product y»c, ie. y»c º y»*c where multivector conjugation » is a Dirac conjugation corresponding to conjugate transpose of the matrix representation. We insist on odd N to ensure i central an pick » to be whichever of § and §# negates i=i so as to ensure that y»y is real nonnegative and that y»c = (c»y)^ = (c»y)» where ^ denotes complex conjugation.
    We then have the following key properties of Dirac conjugation:

    We say a multivector is Dirac real if y»=y and Dirac imaginary if y»=-y. .

Ideal Kets

    Suppose h1,h2,...,hk are commuting plussquare unit Dirac real multivectors so that hj2=1, hj»=hj and hihj=hjhi . Suppose further that each h either commutes or anticommutes with every extended basis element. Let uk(1±h1)(1±h2)..(1±hk) be one of the 2k disinct annihilating idempotents in Algebra{h1,h2,..,hk} .
    Anything that anticommutes with an hj is anihilated by u= so we can decompose a = a + a' where a Î Central()(h1,h2,..,hk)= Central(u) commutes with u and a' is has ua'u=0  where complex a = a(10) + a1h1 + ... akhk exploit i=i.
    Consider kets of the form au where a has nonzero scalar part. We can express au = (a+a')u where  a Î Central(u) and a' satisfies ua'u=0.
    Ket products simplify as = aubc..g and so all but the first (leftmost) factor can be aribitarily reordered without changing the product. and braket product (au)»bu = u(a»b + ¯u( a'»b' )) can be simplified to u(a»b) whenever it appears on the right of a ket .
    (au)»au = u(a»a + ¯u( a'»a' )) .
    Any ideal ket yp = yp_mvu has yp» = _mvuyp».
    Any bra-ket product appearing in a product of kets and bras based on u can be replaced with _mvprc1[u](a»b) u . Unless the ket is the rightmost term in the product the bra-ket product can be further reduced to (a»b + _mvprc1[u](a'»b;) u
    (au)»(bu)(cu) = = u(a»b)u cu = _prc_mv(u)(a»b)u cu
    (aua»)k = au(a»a)k-1ua» and if we insist a»a be central we have (aua»)k = (a»a)k-1aua» .

    Dirac conjugation provides a complex-valued inner product for composite states
    c»y  º  òCMdMp cp»yp     where CM is a particular M-curve of interest. Typically an e4 cotemporal 3-plane in nonrelativistic QM.

    It is frequently the case that statements involving a local ket yp remain true of "field" y provided products are "widened" into integrals and/or summations over appropriate domains. Thus an expression such as y»c might "hide" or embody and extremely elaborate and computationally intensive operation involving convolved integrations and infinite summations. Fortunately, we can often ignore such "under the hood details" and simply manipulate our "symbols" in accordance with geometric algebra.
    A ketvalued function of a single (classical time) variable y(t) can be regarding as representing the variable state of system "at" a single spacial location.

    Suppose now that u' is another dirac real idempotent with u'»u=0. If cp=cpu' and yp=ypu are kets based on the distrint idempotents then yp»cp = cp»yp = 0 and the two kets trivially satisfy the ket rules.
    ypcp» need not vanish but is "null" in that (ypcp»)2 = 0. Since (ypcp»)yp = 0 while (ypcp»)cp = yp |cp|2 we have (aypcp») = 1 + a(ypcp») so (aypcp») cp = cp + a|cp|2 yp and (aypcp») yp = yp and we can regard (aypcp) as introducing yp linearly.
    For kets based on the same idempotent we have yp»cp = uyp»cpu (yp»cp) u .
    ypcp»  = ypucp» .
    (yp»cp)2 = uyp»cpu yp»cpu = u (yp»cp)(yp»cp)) u = u (¯u(yp»cp) + ^u(yp»cp)) (¯u(yp»cp) - ^u(yp»cp)) u = u (¯u(yp»cp)2 + ^u(yp»cp)) + 2(^u(yp»cp)×^u(yp»cp) ) u
    Thus a more general ket can be regarded as y1pu1 + y2pu2 + ... + + ykpuk where u1,u2,...,uk are k mutually annihilating Dirac real idempotents.
    Transformation u1» = u1= has the effect of annihilating products like aui and uia for any i¹1. Its effect on au1 is to negate anything in a that anticommutes with u1.

Normalised Kets

    Kets and ketvectors do not normalise uniquely. Dirac conjugation provides a postive real Dirac Magnitude |y|» º (y»y)½ for (nonanti) kets (involving summations and integrations over particular subsets of Base), and dividing a ket by this magnitude does indeed provide a normalised ket y~ which also represents y. [  To accomodate antikets we require |y|» º |y»y|½ = ((y»y)2)¼ ]
    But efi y~ is another normalised ket representing the same state for any phase factor fÎÂ which we can even allow to be p-dependant. We will refer to the geometric multiplication of a ket by spinor efi as a phase rotation.
    Note therefore the important distinction between local normalisation yp~ º yp|yp»yp| so that yp~»yp~ = ±u     " p ; and CMp-normalisation
    y~p º yp |òCMpdMq yq»yq) |     where CMp is an understood possibly p-dependant M-curve in Base over which we wish y~q»y~q to integrate to ±1 .
    Even when Base is an unbounded space,  traditional QM insists that such integrals be finite as a condition on y.
    Local normalisation discards the potentially probabilistically relevant relative magnitudes of yp and yp+d .    

    If ï1ñ , ï2ñ, ... ïMñ are M kets then a1ï1ñ+a2ï2ñ+...+aMïMñ is also a ket for any complex a1,a2,... not all 0.
    Dirac's ket product ïfñïcñ = ïfcñ = ïcfñ is defined by Dirac only with regard to commuting kets. We here regard yc and yc as geometrically distinct multivectors representing potentially nonequivalent states.

    The Dirac conjuagte of a ket is known as a bra. Dirac coined the terms "bra" and "ket" because he wrote  y»  c as a "bra(c)ket ed pair" áyïïcñ. We have B = K» º { y»  : y Î K } .


    We here regard multivector local pensity y!p  º  yp(yp»)   =   ypyp»   =   ypuyp§   =   yp§(u)     as being more fundamental than yp.
    Because yp expresses itself as yp§ , ayp acts with ayp§ "Pensity" is an abbreviation for pure probability density but can alternatively be thought of as short for "propensity" or even "pointless misspelling of density".
    The term probability density or just density will be used here for the more general " ketbra" construct yc»   =   y'uc'» for possibly distinct kets y and c. Such a density has (ypcp»)2 =  (cp»yp)(ypcp») and hence has complex selfscale (cp»yp) which vanishes ((ypcp»)2 = 0) if cp and yp are "orthogonal".
     We will initially represent pensities with selfscaling (aka. "selfeigen") (y!p2=lpy!p) multivectors that contain only blades invariant under ». For »   =   #§ and N<7 this corresponds to <0;3;4>-vectors and an example pensity is ½(1+w) where w is a unit plussquare 3-vector. We refer to scalar lp = |y!p|s as the selfscale of the pensity at p, negative when yp is an antiket. y!p2=lpy!p implies y!p is either singular (ie. noninvertible) or the "scalar system" y!p=lp.

    Pensities combine symmetrically as y!~c! º ½ (y!c!+c!y!)   =   ½(y»c)yc» + (c»y)cy» ) ,   =   ((y»c)yc»)[#§+]   =   (y!c!)[#§+] ie. the Dirac-real component of y!c! .

    All nonzero pensities have nonzero scalar part. [ Proof :   y!<0>=0 Þ y»y=0 Þ y=0 Þ y! = 0  .]
    Given odd N, the multivector cyclic scalar-psuedoscalar rule provides c»y   =   u0-1 (yc»)<0,N>     and in particular y»y   =   u0-1 (yy»)<0> so though we can recover the complex inner product y»c from density yc» we can also recover its modulus |c»y|+2 from the pensities c! and y! as |y»c|+2   =   u0-1 y!*c!
[ Proof :  y!*c!   =   (y(y»c)c»)<0>   =   ((y»c)yc»)<0>   =   ((y»c)c»y)<0>   =   ((y»c)(c»y)u)<0>   =   (|y»c|+2y)u)<0>   =   |y»c|+2 u<0>  .]
    For example, ket ½(1+w) has pensity ½(1+w) with ½(1+w) *½(1+w) = ¼(1+w¿w) = ½ while |½(1+w)»½(1+w)|+=1 so _u0i=2 and |y1»y2|+2   =   ½(1+w1¿w2) .

    We can determine the "effect" of y» from   eijk*y! = y»(eijk)

    It is not possible to "retrieve" y=y'u from y! since ya would generate the same pensity for any a with aa»=1 as would y'bu for any b with b»(u)=u. y! accordingly contains less information than y but not being able retrieve the ket from the pensity is not that serious a problem since yc» = (y»c)-1 y!c! enables us to retrieve yc» from y! and c! apart from an arbitary central ("complex") phase factor.

    The Clifford kinematic rule Ñp*(bab§#)   =   2((Ñpb)a<-§#>b§#)<0;N>     with » = §# provides
    Ñp* y»(a) = 0 for all constant Dirac-real a   and Ñp* y»(a)   =   2((Ñpy)ay»)<0;N> for all constant imaginary a .
    In particular Ñp*y!p = 0       for any pensity (since 1»=1) and we have the ket kinematic rule
    Ñp*(yp»ayp)   =   2((Ñpyp»)ayp)<0,N>     for any 1-vector a.
    The most natural way to "normalise" a pensity is by normalising its contructive ket as y!~ º (y~)! º (y~)(y~)» so that y!~2 = ± y!~ . Since y! has nonzero scalar part we can efficiently normalise by enforcing the weaker y!~*y!~ = ± y!~<0> ;  rescaling so that (y!~)<0> = u<0> with
    y!~ = (y!<0>(y!*y!)-1)½ y!

    Note that y!×c!   =   (y!c!)<-»>   =   (y!c!)<1,2,5,6,9,10,...>     for » = #§.
[ Proof : y!×c! = ½(yy» cc» - cc» yy») = ½((y»c)yc» - (c»y) cy») = ½((y»c)yc» - (y»c)yc»)»)  .]
    y»y = (y»y)<0>u = (yy»)<0>u so the Dirac magnitude of the ket is the scalar part of the pensity.

    The idempotent and so noninvertible multivector operator y!~=   =   y!~» has y!~=(c!~)   =   |y~»c~|+2 y!~     so maps any pensity c!~ to y!~ scaled by the real nonnegative classsical probability for c ® y . It accordingly annihilates all pensities orthogonal to y!. In particular y!i~=(yS)   =   liy!i~ .

    The invertible (ay!~) = 1 + ((±a)-1)y!~ according as y!~2=±y!~ .
    More generally (ayc») = 1 + (abc»y)-1) bc»y-1yc» y!~ .

    We sometimes interpret a pensity field y!p as representing a "diffused localised entity". The probaility of the entity being at p is the real amplitude |y!p|+ = (yp»yp)½ divided by   òCkdkp |y!p|+ over some k-curve Ck of interest. The locally normalised y!p~ satisfying (y!p~)2 = ±y!p~ embodies the "orientation" and any other "parameters" of the entity if it is at p.
    We then have (1+ly!p~)2 = (1+(2l ± l2)y!p~) so that 1-y!p~ is idempotent for pensity y!p while 1+y!p~ is idempotent for antipensity y!p.

    If y!~2= y!~ then (ly!p~) = 1 + (l - 1)y!p~ while if y!~2=-y!~ then (ly!p~) = 1 - ((-l) - 1)y!p~ .

Pensity Superpositions

    If ket y = a1ï1ñ+a2ï2ñ+...+aMïMñ) for M orthonormal kets ï1ñ , ï2ñ, ... ïMñ and central a1,a2,... then
    y»y   =   åi |ai|+2 áiïïiñ   =   åi |ai|+2 u   so positive real scalar y»y   =   åi |ai|+2 ; and
    y! º yy»   =   (a1ï1ñ+a2ï2ñ+...+aMïMñ) (a1ï1ñ+a2ï2ñ+...+aMïMñ)»   =   åi|ai|+2ïiñáiï + åi¹j (aiaj^ ïiñájï +ajai^ ïjñáiï)
    But y!2 = y(y»y)y» = y»yy! so the normalisation condition for both y and y! is åi |ai|+2 = 1 .


    Traditional QM can be informally sumarised by the statement that "states collapse globally to eigenstates when observed locally". Just how, why, and indeed whether such collapses actually occur in nature are matters of extensive speculation. Does a "waveform" collapse "everywhere" instantaneously or do changes "radiate" outwards at finite speed? Is an observation a discontinuous all-or-nothing affair, or can one only partially collapse the wavefunction? Do cats qualify as observers? Does observation "drive" reality? And so forth. We will initially ignore these issues and formalise the mathematics of the idealised instantaneous local collapse.

Linear Operators
    We call any point dependant linear function ¦p: K ® K a linear ket operator. If ¦p=¦ is the same function at every point we will call it a universal operator. Most of the operators we are interested in are universal and we will often drop the p suffix. Statements involving ¦ should henceforth be regarded as applying either to a universal operator or at a particular point of interest.

    ¦ induces a natural linear bra operator ¦p:  B ® B defined by ¦p(c») y = c»¦p(y)       " y     [ ácï(¦ïyñ) = (ácï¦)ïyñ = ácï¦ïyñ = in Dirac's notation ] Linear operators can thus act like associative "multipliers" if we write them to the left of kets and to the right of bras .

    ¦ also induces a linear conjugate ket operator ¦» defined by     c» ¦»(y) = ¦(c)» y     or, equivalently, ¦»(y) = (¦(y»))»     ¦» is also traditionally known as the adjoint of ¦ though note this is an "adjoint" with regard to » rather than ¿ . We say ¦ is observable or real or self-adjoint if ¦» = ¦ . If ¦» = -¦ we say ¦ is imaginary. It is easy to show that ¦»»   =   ¦     and     (¦g)»   =   g»¦» .

    The general geometric linear operator ¦(y)=ayb has conjugate ¦»(y)=a»yb». Thus pensity y!p   =   ypyp» is real when regarded as a linear geometric ket operator y!p(cp) º y!pcp = yp(yp»cp) = (yp»cp)yp .
[ Proof : (acb)»y   =   u0-1((acb)»y)<0,N>   =   u0-1(b»c»a»y)<0,N>   =   u0-1(c»a»yb»)<0,N>   =   c»(a»yb»)  .]

    Any linear ket operator ¦ induces a linear pensity operator mapping pensities to pensities defined by
    ¦»(y!)  º  ¦y!¦»   =   ¦(yy»)¦»   =   (¦y)(y»¦»)   =   ¦(y)(¦(y))»   =   ¦(y)! . In particular
    y!»(c!)   =   y!=(c!)  º  y!c!y!   =   y(y»c)(c»y)y»   =   |y»c|+2 y! sends pensity c! to y! scaled by |y»c|+2 .

Eigenkets and Eigenpensities

    We say yp is a eigenket of linear ket operator ¦p if ¦p(yp)=apyp     " p for some "complex" eigenvalue scalar field ap. If ap=a is p-independant we will say the eigenvalue is universal It can be shown that if ¦ is real (self-adjoint), all its eigenvalues are real scalars. Further, eigenkets corresponding to distinct eigenvalues are orthogonal.
    An operator ¦p may have discrete eigenvalues at a given p, or a continuous range, or a mixture of the two. We will denote eigenvalues of ¦ by li where the subscript i can range discretely or continuously or both. If ¦ has just m distinct eigenvalues we will say ¦ has integer eigenrank m.
    Viewed as a ket operator, pensity yy» has eigenket y with associated real eigenvalue y»y.

    We say y! is a eigenpensity of linear pensity operator ¦p if ¦(y!)=ay! for some "complex" eigenvalue a.
    Since y!2 = y(y»y)y = (y»y)y! any pensity is an eigenpensity of itself having real scalar eigenvalue y»y. If y is normalised, this eigenvalue is unity. Pensity operator y!= has eigenpensity y! with associated real eigenvalue (y»y)2 which is 1 if y! is normalised .
[ Proof : y!=(y!) = y!3 = y(y» y)(y» y)y» = (y»y)2 y!  .]
    If y is an eigenket of ¦ with complex eigenvalue l then y! is an eigenpensity of ¦» with real eigenvalue ll^ .

    A celebrated mathematical result that we will simply state here is that if a real linear ket operator ¦ satisfies an algebraic equation
    ¦m + zm-1¦m-1 + ... +z1¦ + z01 = 0
for some complex valued z0,z1, but does not satisfy any "simpler" such equation, then ¦ has m distinct real eigenvalues corresponding to distinct orthogonal eigenkets that generate K.
    Thus, for example, ¦2=1 provides the decomposition y = ½(1-¦)y + ½(1+¦)y of a given ket y into two eienkets for ¦, with associated eigenvalue measures -1 and +1. These states are orthogonal in that
(½(1-¦)y)» ½(1+¦)y   =   ¼y»(1-¦)»(1+¦)y   =   ¼y»(1-f)(1+¦)y   =   0 .
    Let l1,...lm be normalised eigenkets associated with m discrete eigenvalues l1, ..lm. Linear operator  åj=1m ljlj» sends lk to lk  " k so if the m eigenkets are a complete set for K , åj=1m ljlj» can be regarded as the identity operation (scalar multiplication by unity).
    In the case of a continuous ranges of eigenvalues we must introduce ranged integrals of the form ò ll» dl to the discrete summation.

    Penrose attributes to Dirac [ "Emporer's New Mind" Ch.6 Nt.6 ] a key interpretation of the Dirac inner product: that ácïïyñ = c»y is the complex probability amplitude of (normalised) state ïyñ "jumping" to (normalised) eigenket ïcñ on observation. [  as opposed to to another unspecified (composite) state orthogonal to ïcñ ]
    If c and y are not normalised, the probability amplitude is given by = (c»y)(y»c) ( (c»c)(y»y) )-1 where real scalar (c»c)(y»y) > 0.  
    Under this assumption, the positive real scalar classical probability of y "collapsing to" eigenket c (both assumed normalised) is the squared modulus of the complex probability amplitude, and is accordingly given by the scalar product of their pensities.
    Probabilty(¦?(y)=c)     =   (cc»)*(yy») .
[ Proof : |c»y|+2   =   (c»y)(c»y)^   =   (c»y)(c»y)»   =   u0-1(c»y)(y»c)   =   u0-1(c»yy»c)<0>   =   u0-1(yy»cc»)<0>  .]
    c ® y and y ® c are hence classically equiprobable , but their respective complex probablity amplitudes are conjugate.

    However, we do not adopt this assumption here, favouring
     Probabilty(¦?(y)=lj)   =   (y»lj)(lj»y) ( åi(y»li)(li»y) )-1
      =   lj!*y! (åili!*y!)-1
    where the summations are over all eigenkets of ¦ (and may include integrations for continuous eigenspaces). The li are here assumed normalised but y need not be. This assumption ensures that the total probability of collapsing to an eigenket of ¦ is unity.
    We say kets y and c are orthogonal if y»c=0 , corresponding to zero classical probabilities for y®c and c®y under any ¦.     


    We will initially regard a 1-observable as a p-dependant Dirac-real linear ket operator ¦p : K ® K taking yp to ¦p(yp). By a Dirac-real operator we mean   a self Dirac-adjoint one, ie. ¦(c)»y = c»¦(y) " kets c, y . We will initially consider observables that can be represented via geometric products such as ¦(y)=F^(y) º FyF^ for a multivector F and a general conjugation ^, or merely as ¦(y)=Fy. Later we will discuss "differentiating" operators such as momentum.
    In traditional QM, a 1-observation ¦?(y) is the non-deterministic "effect" of ¦ on a state y. Rather than taking y to well-defined ¦(y), ¦? can "collapse" y to any eigenket li of ¦ satisfying l-j»y ¹ 0 , returning as measure ¦?(y) the associated real scalar eigenvalue lj .
    If y = åi=1m zili for orthonormal real eigenvalued eigenkets li of ¦, then ¦?(y)=lj and ¦?(y)=lj with probability
    |lj»y|+2(åi=1m |li»y|+2)-1 = |y»lj|+2(åi=1m |y»li|+2)-1
    We can theoretically retrieve the eigenvalue measure ¦?(y) from the eigenket li¦?(y) as (eij..l* ¦(li)) (eij..l* li)-1 where eij..l is any blade present in li, and we will typically use the scalar part via ¦?(y) = (¦(li))<0> li<0>-1  . Physically, however, we have only the observed real scalar ¦?(y) (or some function of it) from which to infer the eigenket.

    If there are multiple potential eigenkets sharing a common eigenvalue we are unable to deduce the post measurement state ¦?(y) without recourse to the Neumann-Luders postulate that ¦?(y) be the orthogonal projection of y into the space of possible eigenkets. One consequence of this is that if y is already an eigenket of ¦ then ¦?(y)=y and the state is unchanged by the observation.

    Since ¦ is real (ie. ¦»=¦), the inner product ¦y º òCMdMp yp»(¦p(yp)) º y» ¦(y)   =   (¦(y))»y   =   u0-1 y»*¦(y) is a real scalar value for any ket y known as the averaged scalar measure of ¦ in y. It is often denoted by Ey(¦) or <¦> in the QM literature. If we attempt to observe ¦? in reproduceable state y many times and average the measures obtained (sum the measures obtained and divide by number of measures obtained), the averaged value will approach ¦y for large numbers of measurements.
    We will refer to multivector y»¦(y) = ¦yu as the average unstripped measure.
[ Proof : (y» ¦y)^ = (y» ¦y)» = y» ¦»y = y» ¦y    .]

    The averaged scalar measure of geometric 1-observable ¦(y) = fy for a given ket y is available from the pensity as
      fy º y»(fy)   =   u0-1 (y»fy)<0>   =   u0-1 y!*f   =   u0-1 y'»(u)*f   =   u0-1(u=y'»»(f))<0>
  =   ½ u0-1 (y!<0> + b*y!) when f=½(1+b).
    Note that (y!fy!)<0>   =   (y!2f)<0>   =   (y»y)(y!f)<0>   =   (y»y)y!*f .

    The averaged unstripped measure of ket operator y! in state with ket representor  c is c»(yy»)c = (c»y)(y»c) = |c»y|+2u which is u scaled by the classical probability of collapse c®y on making a y! observation.
    The averaged unstripped measure of pensity operator y!= in state with ket representor c is c»y!cy! = (c»y)(y»c)y! = |c»y|2y! , ie. y! scaled by the classical probability of collapse c!®y! on making a single y!= observation. Such "carrying" of the unobserved geometry into the measures space is a significant advantage in working with pensities rather than kets.

Uncertainty Principle
      ¦? can be regarded as a scalar-valued random variable having expected value ¦y in a given state y and we can consequently compute its standard deviation.

    For a given y and ¦, we can define a 0-centred 1-observable ¦-¦y by (¦-¦y)(c) º ¦(c) - ¦yc    , having expected measure 0 in y.
    We call (¦-¦y)2y º ((¦-¦y)2)y = (¦2)y - (¦y)2 the variance of ¦ in y and its positive square root provides a real postive scalar
    Dy(¦) º (¦-¦y)2y½     called the standard deviation or dispersion of ¦ in y. It measures the "variability" or "probabilistic spread" of observations of ¦ from the mean value. We say an observation ¦ is certain in state y if Dy(¦) = 0 and uncertain if Dy(¦) > 0.

    For general possibly noncommuting 1-observables ¦ and g we have the uncertainty principle      Dy(¦)Dy(g) ³ |(i¦×g)y|     .
[ Proof : If operators ¦ and g are real so is i¦×g . For real scalar t note that (¦-itg)»(¦-itg)   =   (¦+itg)(¦-itg)   =   ¦2+t2g2-it(¦g-g¦) . Whence
|(¦-itg)(y)|»2   =   ((¦-itg)(y))»(¦-itg)(y))   =   y»((¦-itg)»(¦-itg)(y))   =   y»(¦2+t2g2-2it(¦×g))(y)   =   ¦2y+t2g2y-2t(i¦×g)y
    Since |(¦-itg)(y)|» ³ 0 the quadratic discriminant 4(i¦×g)y2 - 4¦2yg2y £ 0 so ¦2yg2y ³ (i¦×g)y2 with equality only when repeated root t occurs with (¦-itg)(y)=0, It is easily verified that (¦-¦y)×(g-gy) = ¦×g so substituing the the 0-centred observables for ¦ and g we obtain
(¦-¦y)2y (g-gy)2y ³ (i¦×g)y2 as required  .]

    States y for which noncommuting ¦ and g attain this lower bound are known as minimal uncertainty states for ¦ and g. [  The ½ appearing in conventional statements of the uncertainty principle is here embodied in the definition ¦×g º ½(¦°g - g°¦) where ° denotes composition ]
    If ¦ and g commute, we have the trivial Dy(¦)Dy(g) ³ 0 but unless ¦ commutes with all the 1-observables of a system we must have Dy(¦) > 0.  

    String theory prevents ¦ from being measured to aribitary precision, allowing an arbitary uncertainty in g, by replacing Dy(¦)Dy(g) ³ h with Dy(¦)Dy(g)  ³ h + CDy(g)2 for very small C [ Smolin p165]  .      

    If two linear ket operators commute, then there is a complete set of kets that are simultaneously eigenkets for both operators, albeit with differing associated eigenvalues.
[ Proof :  See Dirac  .]
    In general. given k commuting linear operators ¦1,¦2,.. there is a complete set of kets yi such that each yi is an eignket of all k operators, having associated eigenvalue li 1 for ¦1 , li 2 for ¦2 and so forth. If two such eigenkets differ in their eigenvalues with regard to any one or more of the operators, then the kets are orthogonal.

    If the operators associated with k 1-observables all commute, then the order in which observations are "made" becomes irrelevant and we can meaningfully speak of a simultaneous k-observation of k real scalar variables. After such an observation, y(t) is an eigenket for all k observation operators and the measure values can be considered as providing "current values" of real dynamic state variables. Let integer K be the products of the eigenranks of the k operators, correseponding to the number of distinctly observable states. If eigenvalues exist in continuous ranges, K is infinite, but can then informally be thought of as a geometric structure "ennumerating" the eigenkets.   The K eigenkets for the k-observable are indexed by the K eigenvalue combinations. Let lj denote a particular eigenvalue combination. and lj the corresponding simultaneous eigenket with j Î {1,2,..,K} .

    A k-observation ¦? is the measure resulting from the application of a k-observable. We can regard a k-observation as providing "currently pertaining values" for k eigenvalues characterising the "post observational mode" of the state. In the case of discrete eigenvalues, there are only finite K possible discrete "outcomes" for the observation. But in general, some or all of the eigenvalues may come from continuous ranges. Thus a k-observation ¦? is like a k-dimensional real 1-vector-valued function of kets, whose "coordinates" may be discrete or continuous values.
    k-observables are considered as both linear operators of kets, and real-valued functions of kets according as to whether ¦?(y) or ¦?(y) is denoted.

    A k-urbservable abbreviating k-urobservable is a set of k noncommuting scalar 1-observables (aka. complementary observables) associated with  a k-dimensional "property" of the unobserved system. We can only "observe" an k-urbservable by taking k successive "readings" along k seperate "axies of measurement" and the order in which we take the measurements will effect their final result. Indeed, only the final scalar reading obtained can be considered truly meaningful with regard to the system post urbservation.

Position N-observable
    Define pd,p(yp) º pd(yp) º (d-1¿p)yp. This is an origin-specific non-universal linear scalar operator corresponding to scalar multiplication by a particular point-dependant scalar coordinate. [  It almost ubiquitous in the physics literature to denote the momentum operator by p with the ith coordinate denoted pi but we will fly in the teeth of this and favour m or m for momentum here, retaining p to indicate a primary "point" or "position" parameter and p for the position operattor ]

    Traditional QM faulters at the first hurdle since the position obervable has no universal eignvalues.
[ Proof :  A universal eigenket ld with eigenvalue ld for pd would perforce satisfy ((d-1¿p)-ld)ld(p) = 0     " pÎBase     forcing ld(p)=0 outside the hyperplane      (d-1¿p)=ld) which would typically restrict a spacial 3-plane volume integral expression for ld»ld to a 2-plane and so cause it to vanish.  .]
    However pd has a single local eigenvalue (d-1¿p) at p so pd?(yp) = (d-1¿p) is certain for all y.

    If pd returns measure l when acting on y the collapsed state theoretically satisfies pd(cp) = lcp " p which can only hold if cp=0 whenever d-1¿p ¹ l . Within the hyperplane all states share the pd eigenvalue, so the Neumann-Luders postulate suggests that pd?(yp) collpases yp   to zero outide the hyperplane and preserves it within the hyperplane.
    Obviously such a discontinuos c might violate our criteria for acceptable ket fields so we must think of imprecision in the measure l "smearing out" the peak so that the collapsed ket is nonzero close to as well as within the hyeprplane.

    It is natural to combine the N position operators into a single 1-vector operater p º åj=1N ej pej so that pd = (d¿p) so that pyp = pyp . Rather than eigenvalues, geometric observable p has 1-vector eigenvalues corresponding to the measured position.

    Suppose yp (normalised over CM) represents a "particle event" in that yp»yp is nontiny  only in a Hermitian neighbourhood of an event c (ie. tiny whenever (p-c)(p-c) = (p-c)¿+(p-c) > e2) . Then òCM |dMp| yp»((d-1¿p)yp)   =   òCM |dMp| (yp»yp)(d-1¿p) will be very close to d-1¿c and careful consideration should convince the reader that averaged scalar measure pdy does indeed represent the "average likely d coordinate" of real probability distribution yp»yp ; and that 1-vector py is the "averaged likely position", even if constructing some apparatus to actually observe this measure over CM might be problematic.

    Clearly pei and pej commute so event p is an N-observable. T º Te4 º pe4 = (e4¿p) is known as the time observable.

    So what happens when we "measure" pd?  We will have some kind of apparatus capable of returning an event coordinate. A K-curve screen S Ì CM Ì Base perhaps. Assuming y»y òCM |dMp| yp»( pd(yp)) = 1, the expected measure will be òS |dKp| yp»( pd(yp))   =   òS |dKp| (d¿p) |yp|»2

Spin Observable

    Another simpler example of a geometric observation is provided by taking fp = f = ½(1+w) where w is a plussquare unit 3-blade such as e124 or e345 (ie. the "simplest" nonscalar multivector satisfying f2=f and f»=f when » = §#) Any ket y here assumed normalised over CM decomposes as ½(1+w)y + ½(1-w)y   =   fy + f§y correseponding to an equal superposition of eigenkets for f having eigenvalues 1 and -1 respectively which we might regard as being the only two possible measurements of the "spin" in "direction" w of a system .
    The observation f collapses y'u to a complex multiple of ½(1+w)y'u returning measure 1 with unnormalised complex probablity amplitude y»(½(1+w)y = ½ + ½y»(w(y)) ; or to a complex multiple of ½(1-w)y'u returning measure -1 with unnormalised complex probability amplitude ½ - ½y»(w(y)) .
    Because w»=w, y»(wy) is real and so the classical probablities of collapse are ½ + ½y»(w(y)) for measure +1 and ½ - ½y»(wy)) for measure -1 . The average expected measure is thus y»(wy) .

    Physicists typically characterise spin as a sort of 3D 1-vector built from Pauli matrices. They typically form s expressed as a linear combination of the three C4×4 matrices obtained by putting the same 2×2 Pauli matrix in to the two lead diagonal 2×2 blocks, and zeros elsewhere. These  matrices correspond to e145,e245, and e345 in Â4,1,.

Displacement Operators
                                         Thus far we have considered linear operators ¦(y)=c . Now we turn to "differentiating" operators and for these we must first consider displacement operators. Suppose we  can associate ket yp with the state of a system at a given spacetime event p. yp is then a ket-valued function of Â3,1 .
    We make the assumption of universal superpositions, postulating that if yp   = z1cp + z2xip at a given p with complex z1,z2 independant of p then yp+d   = z1cp+d + z2xip+d for any spacial displacement 1-vector d sufficiently small for p+d to remain within the eventspace over which our model y applies (hereafter refered to as the lab-space) which we assume to contain the "origin" event 0.

    The ket displacement operator Dp,dyp º yp+d is then linear .
    Assuming that yp+dd » yp+dd = yp»yp     provides the ket displacement operator normalisation condition Dp,d»Dp,d = 1. Assuming the preservation of both linear combinations and magnitudes, by no means mathematically inevitable, constitutes a physical assumption refered to by Dirac as a "kind of sharpening of the principle of supposition" [   Dirac p109 ] with regard to temporal displacements of a (non-relativistic) quantum system.

    Let ¦p be a 1-observable with ¦pyp = cp . 1-vector d induces a displaced observable ¦p,d defined by
    ¦p,dyp+d º cp+d = Dp,dcp = Dp,d¦pyp .
    From ¦p,dyp+d = ¦p,dDp,dyp we can thus derive     ¦p,d = Dp,d¦pDp,d-1     and so     ¦p,dDp,d = Dp,d¦p .

    We have ¦p,d = Dp,d¦pDp,d-1 = ¦p + 2e( Ðdצp) + O(e2)     where Ðdצp º ½( Ðd¦p-¦p Ðd) .
    Thus the observed displacement-directed gradient of an observable is (twice) the commutative product of the observable and the displacement.

Momentum N-urbservable
    We define directed ket derivative Ðd to be the ket operator Ðdyp º Lime ® 0 (yp+ed - yp)e-1   =   Lime ® 0 (Dp,dyp - yp)e-1 . Since we can multiply Dp,d by   eap,di for arbitary real ap,d we obtain an arbitary imaginary additive term ai  where a = ( Lime ® 0ap,ed)i in the derivative (somewhat akin to the arbitary "additive constants" pertaining in indefinite integrals).

    Differentiating ¦p,dDp,d = Dp,d¦p we obtain ( Ðd¦p.d)Dp,d + ¦p.d( ÐdDp,d) = ( ÐdDp,d)¦p

    For small scalar e we have Dp,d » 1 + e Ðd so normalisation condition Dp,d»Dp,d=1 Þ Ðd» = - Ðd Thus Ðd is an imaginary operator and so i Ðd is a real observable.
[ Proof : (1+e Ðd»)(1+e Ðd=1 Þ e( Ðd»+ Ðd + O(e2)=0 . Alternatively, differentiating normalisation condition y»y = u gives (Ðdy»)y +   y»(Ðdy) = 0 Þ Ðd» = -Ðd .  .]

    When acting as an observation, we thus expect i Ðd? to collapse a state yp to an eigenket of iÐd , ie. to a solution of iÐd yp = mdyp for a real scalar d-directed momentum md such as yp = (-mdi(p¿d)) y0.
    It is customary to define a real directed four-momentum operator by md º h Ðpd   =   h(d¿Ñp) where   h º hi = h(2p)-1i ( ie. i in natural units) will normally appear with "mass" or "charge" scalars which it can be thought of as "quantising". Thus Ðd =  h-1 md .
    [  Note that mej = h /xj = -h x/xj for j=1,2,3 in Â1,3 timespace hence i-1(2p)-1h /xj is common in the literature. ]

    We have undirected four-momentum operator m º åi=1N ei mei   =   åi=1N hei Ðei   =   hÑp . Physicists usually think of the momentum operator as a 1-vector, using "four" only in the sense of having four dimensions. In our Â4,1 geometric model four-momentum is a hyperblade and so actually is a 4-vector.
    If y is analytic in a flat spacetime so that the Ðei commute, momentum can be regarded as an N-observable. Otherwise its an N-urbservable.    Note carefully that there is no concept of "mass" in the momentum operator.

    ma× pb = ½h(b-1¿a) .
[ Proof :    ma pbyp   =   h Ða((b-1¿p))yp)   =   h(Ða(b-1¿p))yp + h(b-1¿p)(Ðayp)   =   h(b-1¿a)yp + pb mayp  .]
    Applying the uncertainty principle to ma and pb thus gives Heisenberg's uncertainty principle Dy( ma)Dy( pb) ³ ½h(2p)-1|b-1¿a|   with coordinate form Dy( mei)Dy( pej) ³ ½dij h(2p)-1 .
    This is often interpreted as meaning that one cannot measure the position of a particle withour effecting its momentum, and vice versa, but it is actually more profound even than this. It says the momentum and position of particle on a given axis/direction cannot meaningfully be regarded as having exact values even in the absence of observation.

    Hp º me4 º h(e4¿Ñ) is traditionally regarded as the "energy operator" or Hamiltonian at p. We say yp is e4-isolated if  Hp is independant of t=e4¿p , ie. if the system evolution operator is time invariant. An eigenket of the Hamiltonian is a solution to hÐe4yp = Epyp and if Hp is independant of t so to will be the scalar energy eigenvalue Ep.

    Thus as a linear function of kets, the Hamiltonian describes how yp=y(P,t) evolves over t, while as an observable it measures the "energy" of y .

    Applying the uncertainty principle to H= me4 and T=pe4 gives the time-energy uncertainity Dy(H)Dy(T) ³ ½h(2p)-1 indicating that the more precise the time at which we measure an energy, the greater the uncertainty of the result.

Momentum vs Velocity

     In classical mechanics, the difference between momentum and velocity of a particle is simply Mp=mVp where m is a positive scalar inertial mass. For a given m, they are essentially the same thing. In quantum mechanics the difference is profound. A particle's velocity - its "instantaneous direction and quantity of travel" is assumed to vary in a potentially chaotic and unpredictable manner, "zipping about" with collossal acelerations and perhaps being generally wierd in other ways, ceasing to exist or "bifurcating" for fleeting periods), but all in such a way that the net traversal over a nontiny time interval is consistant with a more smoothly varying frequently small "average drift" velocity. It is the multiplication of this "averaged out" velocity by mass rather than the instantaneous velocity that gives the quantum mechnanical "momentum" of a particle.
    A path p(t)=e4t is considered to represent a particle "at rest with respect to e4" in that repeated observations of the instantaneous "velocity" of the particle average out to give e4 and we regard the partcle as having momentum me4.

    It may be helpful at this point to consider how we might seek to "measure" the momentum (rather than the velocity) of a particle. To estimate the momentum we might arrange for the particle to collide with a "better behaved" "less quantum wierdy" test particle whose consequent change in momentum we are easily able to record. But the change in momentum of our impacted test particle will not be instantaneous if we assume it to arise from an interplay of predominantly repulsive forces rather than the instantaneous impact of "crisp outter shells" of two small "solids" so cannot be said to represent an instantaneous property of the particle.
     We might contrive to measure the momentum of a particle to a reasonable accuracy without greatly effecting its momentum, and expect subsequent momentum measurements return substantially similar results.

    To measure the rapidly changing instantanous velocity of the particle, however, we would have to measure its position at two distinct but extremely temporally close events. But if we are going to measure the position of two very close events and use their spacial seperation to compute a velocity, we will require extremely accurate spacial positional readings for the particle, and it is impossibe to obtain a truly acurate measurement of the first position without "rerandomising" the momentum and so effecting the second position measurement. We conclude that it is impossible to observe the instantanoes velocity of a particle with any degree of accuracy, but it is far from clear that such a mesurement would be of physical significance anyway, given its intrinsic obsolescence.

Angular Momentum ½N(N-1)-urbservable
    In nonrelativistic 3D QM the angular momentum operator is defined as a 3D 1-vector operator L, = p×m = (pÙm)e123-1 for spacial position p and momentum m 1-vector operators within e123.
    Relativistically, it it more natural to consider the 4D 2-blade pÙm but recall that m is actually hyperblade operator hÑp so we have 3-blade
    L, = p.(hÑp) = (pÙÑp)h = h(pÙÑp) = 2h(p×Ñp) = 2p×(hÑp) = 2p×m ; and also the more exotic operator
    pÙm   =   (åj=1N ej pej)Ù (åk=1N ek mek)   =   åj;k=1N (ejÙek) pej mek
      =   åj<k=1N (ejÙek)( pej mek - pek mej)   =   h-1åj;k=1N ( ej(ej¿p)ekÐek - ekÐkej(ej¿p) )   =   ejek(ej¿p)Ðek - ekej((ej¿ek)+(ej¿p)Ðek)

    More generally, we have
    La,b º pa mb - pb ma   =   h ((a-1¿p)Ðb - (b-1¿p)Ða) .
    Even though La+lb,b = La,b we have not here indexed L, with a 2-blade as LaÙb, because the relative magnitudes of a and b effect the position operators. Traditional QM usually considers only orthonormal a and b within e123 , regarding La,b as the angular momentum about spacial 1-vector "axis" a×b = (aÙb)e123 .

    We have important commutation relationships [ IQT 8.1.1 ]:

    For brevity and compatability with the literature we define L1 º Le2,e3 ; L2 º Le3,e1 ; L3 º Le1,e2
    Though the three Lj do not commute with eachother, they all commute with L2 º L12+L22+L32 [ IQT 8.2.2 ] . This means that we can find a simultaneous eigenket of L3 and L2 . It can be shown that if yp is mutual eigenket of eigenvalue mh for L3 and lh2 for L2 then the (e3-specific) angular momentum ladder operators L± º L1 ± iL2 act as L3 eigenvalue changers in that L±(yp) remains a lh2 eigenvalued eigenket of L2 but is an (m±1)h eigenvalued eigenket for L3.
    The L± commute with L2 but L3×L± = ±½hL±.
    We also have |L±y|»2 = (l-m(m±1)) h2 |y|»2 which imposes l³m(m±1) with equality iff L±y=0 . [ IQT 8.2.4 ]

    The "spacial" "orbital" angular momentum we have discussed thus far is assciated with the anticommuting plussquare 3-blades e145,e245,e345 . We can extend it to include minussquare e125,e135, and e235 by allowing one or both of the a and b in La,b  to be timelike within e1234 but there is a complication. The commutation of La,b and Lc,d simplifies to the above result only if we have (c-1¿a)=(c¿a-1) and so forth, which essentially requires a2=b2=c2=d2

    L, does not commute with the Dirac Hamiltonian, however, and so is not conserved. What is conserved is the total angular momentum operator J º L + ½(2p)-1h s which is interpreted as a combination of "orbital" angular momentum and "intrinsic" angular momentum due to spin. J satisfies similar commutaion results to those of L and has similar ladder operators. Rather than an "angular four-momentum" we thus have an "angular six momentum". as we incorporate the Dirac-real plussquare 3-blades e145, e245, e345 (dual to e23,e13, and e12) within the angular momentum observable.

Electron Orbits
Y(x) = xl y(x~) = rl y(q,f) with spherical harmonic y(x~) = y(q,f) = ((4p)-1(2l+1)((l-m)!(l+m)!-1)½ Pml( cosq) (imf) solve Laplace equation Ñx2Y(x)=0 and it can be shown   [ GAfp 8.155 ] that in Â3
    y(x~) = y(q,f) = ( (l+m+1)Pml( cos(q)) + Pm+1l( cos(q))efe123-1 ) ((l-1)fe12)     solves (xÙÑx)y(x~) = -ly(x~) and hence Y(x) = |x|ly(x~) is monogenic (ÑxY = 0). [  where q is the polar angle within [0,p] ; f the longitudinal [0,2p] angle; Pml a Legendre Polynomial. ]

    l=m=0 is just the constant scalar idempotent y=1. l=0,m=-1 provides 2-vector solution y = efe123-1(-fe12) = ef(-fe12) e123-1 which we can regard as the free electron orbitting itself.

    Consider taking a directed derivative (d¿Ñx)(afe12) for unit d. It will depend exclusively on the change d¿ef in f on moving from x to x+d and have value ((a(d¿ef)e12)-1)(afe12) which for small d approaches (a(d¿ef)e12)(afe12) = a(d¿ef)(½p+afe12) providing
    Ñx (afe12)   =   Ñx¿(afe12)   =   a   efp+afe12)   =   (½p-afe12) ef.
    Now Ñef=(r sin(q))-1ef=R-1ef .

    Orbit states are typically characterised (enumerated) by four integers: a nonphysical nonzero positive principle quantum number n ennumeration index loosely associated with orbital energy and radius; a nonnegative orbital angular momentum azimuthal quantum number l < n associated with eigenvalue l(l+1)h2 for L2 ; an orbital magnetic moment quantum number  m with |m|£l associated with eigenvalue mh for L3; and s=±1 associated with intrinsic spin ±½h. We chose n such that for given l and m, the energy of the two n,l,m orbitals increases with n. The principle quantum number is thus a catalog number rather than a physical observable.
    Note that the square of the L3 = Le1,e2 observable m2h2 £ l2h2 < l(l+1)h2 so the L3 observable is always less that the "L magnitude" observable regardless of how "e12-aligned" L might be. This suggests that L is a 2-vector rather than a 2-blade.
    The shell associated with a given n comprises n subshells each associated with a given l than can hold up to 2(2l+1) electrons ecah having a distinct m and s value pair.
    The theoretical maximal capacity of the the n=1 shell is thus 2; that for n=2 is 2+6=8; for n=3 we have 2+6+10=18; for n=4 2+6+10+14=32; and so on with shell n having a theoretical maximum of 2n2 electrons.
    In practice the n=5 shell tends to "fill" at 32 rather than 50 with further electrons favouring n=6 and 7 orbits. The n=6 shell typically "fills" at 18 rather than 72 with the hypothesised noble gas Ununoctium having 118 electrons configured as 2+8+18+32+32+18+8. As Hotson oberseves, were we to "fill" n=7 up to 18 and add 8 electrons into a hypothesised n=8 shell we would have a noble gas (8 electrons in the outtermost valence shell) with 136 electrons.

Complex Matrix Representation
    This non-geometric represenation is so fundamental to the existing literature that we must address it here. However, we will ultimately have no use for it and move on to Âp,q multivector representations.
    A k-observable induces a basis of orthonormal simultaneous eigenkets, one for each of K possible combinations of readings (measures). It is thus somewhat like an orthonormal geometric K-frame (a set of K 1-vectors) where K is the product of the ranks of the operators associated with the observables. K is infinite if any of the k-obervable's eigenvalues are from a continuous range ragther than discretely valued. Kets act like a column 1-vector basis for CK which we can regard as K×K complex matrices having nonzero values in the first column only. We will refer to such a set of orthonormal eigenkets as an eigenbasis.
    The standard ket l-1 has real 1 ( or K if normalised) throughout the leftmost column, and 0 elsewhere. Any ket can be "generated" from the standard ket by multiplication by the matrix having the coordinates of the desired ket along the lead diagonal and zeroes elsewhere, though such a matrix is Hermitian only if the target ketvector representation has real coordinates.  
    Any linear operator g can then be regarded as a linear transformation of CK 1-vectors and is representable with regard to the eigenbasis by a CK×K matrix in the conventional way, with li»(g(lj)) providing the ith element of the jth column. Such a matrix is Hermitian (ie. its complex conjugate is equal to its transpose) if g is Dirac-real. If any of the operators of the k-observable generating the eigenbasis is represented in this way we obtain a real diagnonal matrix.
[ Proof : li»¦1lj = li» l1 jlj = l1 jdi j  .]
    Note that any unitary (U = U-1) matrix can be expressed as (iH) º eiH where H is Hermitian (H º H^T = H ) .
    Bras are then represented as K-D row-vectors containing the complex conjugate of the transpose of their associated ket. A bra is thus like a K-D row-vector which we can regard as a K×K complex matrice having nonzero values in the top row only.
    A bra-ket product matrix is nonzero only in the top left corner and is thus not so much a complex value as a complex multiple of the unit corner matrix, which is itself right-absorbed by kets and left-absorbed by bras. A ket-bra product matrix has {yc»}i j = {y}[1 i] {c»}[j 1] .

    y! is thus represented by a Hermitian CK×K matrix. Dirac conjugation » correspends to to the complex conjugate of the transpose »   =   T^   =   ^T.
    The classical probability of state y collapsing to eigenket corresponding to a particular combination of eigenvalues ïl1 l2 ..ñ is | ál1 l2 ..ïïyñ |2 , the real modulus of the complex coefficient of the particular eigenket in the eigenbasis formulation of y.
    Each further linear operator (of rank k) can be thought of as "splaying" out our eigenket basis by the introduction of k alternative eigenvalue "labels" into our ket namespace, requiring k eigenkets for every previous eigenket, multiplying the dimension of our 1-vectors by integer k. In the case of continuos ranges of eigenvalues, matters are complicated by the allowance of kets of infinite dimension.
    We say a k-observation is complete if there is only one simultaneous eigenket associated with each combination of k eigenvalues, so that a given k-measure uniquely specifies the resulting ket (up to a phase rotation).

Scalar Pensity as boolean property

    The simplest possible nonzero pensity is a scalar field y!p deriving form a 1-D ketvector yp with a 1-D complex-cordinate 1-vector representation defined over pÎBase .
    As a 1-D complex 1-vector, yp can be regarded as a complex scalar field yp over Base, our primary idempotent u is simply the real scalar 1. Since, at a given p, yp is "impervious" to multiplication by "complex numbers", all nonzero yp represent the same state at p. Thus there are only two distinguishable states at a given p, characterised by yp=0 and yp¹0. The pensity y!=yy» = y^y = |y|+2 is simply the squared modulus of this scalar and is accordingly a positive real scalar field. We can interpret y!p as the probablity of an observation at p collapsing yp (or equivalently y!p) to 1 rather than to 0 , provided that y! is appropriately scaled (normalised) to integrate to 1 over an appropriate subspace of Base.
    We interpret y!p=1 as indicating some physical "boolean property" being "true" at p . One example is whether a "particle" is "present" or "absent" at event p.
    A 1-D ket yp is thus a "complex probability field" , often referred to as the wavefunction of a "particle", providing via y!p a "statistical template" for "appearance likelihood".
    If we observe the property to be "true at" p0 we collapse the yp waveform to a ket satisfying yp0 = 1.

    Geometrically, we can consider the "complex" ket wavefunction to be yp = rp(-hqp) = rp(iqp) in natural units , for scalars rp and qp, with i=e12345 taking the i role. ypyp» = rp2. The generalised spacial momentum is provided by mp = Ñ[e123]qp and qp is known as the phase. Continuity assumptions in yp mean that the phase qp must vary continuously except over nodal K-curves over which rp=0

    Such complex wavefunctions frequently satsify (to good approximation) Schrodinger's equation Ðe4ßyp = (½m-1hÑp[e123]2 + h-1f(p))yp [ ie. Ðe4ßyp = (-½m-1Ñp[e123]2 + f(p))*yp in natural units ]     for real scalar potential f(p) .

Bohm Quantum Potential
Bohm inserts complex yp =   rp(-qph-1)    into the nonrelativistic Schrodinger equation to obtain a "conservation equation"
    rp2/t + Ñ[e123]¿(rp2(Ñ[e123]q)m-1)=0 ; and a modified Hamilton-Jacobi equation
     qp/t + ½m-1(Ñ[e123]qp)2 + Vp + Qp = 0 [ Holland 3.2.17 ] where real scalar quantum 0-potential Qp = ½m-1h2 (Ñ[e123]2rp)rp-1 is dependant only on rp in a way independant of the magnitude of rp (ie. |Qp| can be large for small |rp|) and so is highly nonlocalised.

    The phase qp also provides the action, with orbit momentum 1-vector  mp = Ñp qp independant of probability wieghting rp.
     More generally we have a quantum 1-potential ap = ap(qp, Ñpqp, ...) that is "shaped by" the action and its derivatives, yielding different mechanics to those obtained by extending the Lagrangian to a function of higher temporal derviatives of position.

<0;3> pensity as qubit
    Our » = §# hypothesis leads us to expect pensities of grade <0;3> and since we want idempotents ( (y!p~)2=y!p~ ) we might expect y!p~=½(1+ap) where ap is a 3-vector with ap2=1 . Conventional QM delivers this via a remarkable degree of obsfucation which we summarise here.

    The <0;3> pensity provides the "internal" degrees of freedom of a particle and is usually considered without reference to spacial position. We will drop the p suffix in much of the below and the reader should consider y as being an evaluation over some region in which we "know" a particle will be.

      Consider first the 2×2 complex matrix representation of a 2D ketvector which we can consider as a general superpostion of two basis kets
= æ 10 ö ;= æ 00 ö ; y= æ z00 ö ; y»= æ z0^z1^ ö     for complex z0=r0eq0i , z1=r1eq1i
è 00 ø è 10 ø è z10 ø è 00 ø
↑ and ↓ can be regarded as eigenkets of eigenvalue 1 and -1 respectively for the linear operator s3º æ 10 ö .
è 0-1 ø

    Dirac C2×2 conjugation is traditionally provided by Hermitian matrix conjugation »   =   T^ and so the  normalisation condition is
y»y= æ z0^z0+z1^z10 ö = æ r02+r120 ö = æ 10 ö
è 00 ø è 00 ø è 00 ø

    The astute reader will recognise this as precisely matching our "abstract" specification for a single qubit.

Pauli Algebra
    QM traditionally represents C2×2 matrices with Pauli "spinor" algebra. This postulates four "spinors" 1, s1, s2, s3 having multiplication table
aba which we can summarise as sj sk= eijki si for distinct ijk ; si2=1     where "imaginary scalar" i commutes with the si and has i2 =-1. Thus the si anticommute with the product of any two being the dual of the third , signed cyclicly.
    Note that s1 s2 s3=i s32 = i 1.
    Physicists often write a.s = a1 s1+ a2 s2+ a3 s3  and with this dubious notation (a.s)(b.s) = (a.b)1 + i((a×b).s) where _corss denotes the traditional 3D vector product. In particular (a.s)2 = a21 and hence (i(a.s) = cos(|a|) + i(a.s) Sin()(|a|)
1 s1 s2 s3
b11 s1 s2 s3
s1 s1 1-i s3+i s2
s2 s2+i s3 1-i s1
s3 s3-i s2+i s1 1

    Such a basis is provided by Pauli matrices
1= æ 10 ö ; s1= æ 01 ö ; s2= æ 0-i ö ; s3= æ 10 ö
è 01 ø è 10 ø è i0 ø è 0-1 ø
with an arbitary complex 2×2 matrix being expressed via
æ a11a12 ö = ½(a11+a22)1 + ½(a12+a21) s1   + ½i(a12-a21) s2 + ½(a11-a22) s3
è a21a22 ø
    For a Hermitian (AT = A^) matrix, a11 and a22 are real and a21=a12^ making all four coefficients  ½(a11+a22),½(a12+a21),-i½(a12-a21), and ½(a11-a22) real. In particular, the unit corner matrix (aka. primary idempotent) given by a11=1, a12=a21=a22=0 is represented by ½(1+ s3).
    Note that Hermitian matrix conjugation negates i1 while preserving 1 and the sj.
    Hestenes proposed regarding the si as bivectors in Â1,3 + . We favour Â3,1 + here and observe  that s1=e14 ; s2=e24 ; s3=e34 satisfy the Pauli algebra with i=e1234 and si= si for (e4-specific) geometric Hermitian conjugation.
[ Proof : si sj = eie4eje4 = eij = eijkek    = -eijk4e4ek = eijki sk = eijki sk     i¹j .     si2=-e42ei2=1 .     i = s1 s2 s3= e1e4e2e4e3e4 = e1234  .]
Thus æ a11a12 ö = ½(a11+a22) + ½(a12+a21)e14   - ½(a12-a21)e13 + ½(a11-a22)e24
è a21a22 ø

[  e1,e2, and e3 also satisfy the Pauli algabra (with i=e123) but representing qubits by bivectors enables us to represent a k-quantum register as the geometric product of k commuting multivectors. We wish representors for distinct "independant" qubits to commute geometrically to  satisfy, for example, ï[1][2]ñ =ï[2][1]ñ = ï[1]ñ ï[2]ñ = ï[2]ñ ï[1]ñ ]

    The (non-normalised) pensity matrix yy» is given by singular Hermitian matrix
yy»= æ z0z0^ z0z1^ ö = æ r02 r0r1e(q0-q1)i ö = r02 æ 1 re-fi ö
è z0^z1 z1z1^ ø   è r0r1e(q1-q0)i r12 ø   è refi r2 ø
where z0 = r0eq0i , and z1 = r1eq1i. We have z1 = refi r0eq0i     whenever r0 > 0 .

    In a nonrelativistic Â3,1 model with     »   =   defined with regard to a given e4 we have
    y = ½(z0 + z1 s1)(1+ s3)   =   ½r0eq0i(1 + r(fi) s1)(1+ s3)   =   ½r0eq0e1234(1 + r(fe1234)e14)(1+e34)
    y» = ½(1+ s3)(z0^ + z1^ s1) = ½r0e-q0i(1+ s3)(1 + re-fi s1) = y .
      And so y»y = r02(1 + r2)½(1+ s3)  .
[ Proof :  Note first that (1+ s3)v(1+ s3)=(1+ s3)iv(1+ s3)=0 for any matrix v anticommuting with s3.
    y»y = r02½(1+ s3)(1 + re-fi s1) (1 + refi s1)½(1+ s3) = r02½(1+ s3)(1 + r2)½(1+ s3) = r02(1 + r2)½(1+ s3)  .  .]

    The normalisastion condition z02+z12=1 corresponds to r02(1+r2)=1 , which remains true for z0=0,r=¥ if we consider (1+¥2) = 0 .

    A unit inner product has representative a11=1,a12=a21=a22=0 corresponding to Â3,1+ multivector ½(1+e34) so the correct scaling for a normalised qubit ket of zero phase angle is given by
    y~   =   (1+r2)(1 + refie14)½(1+e34)   =   (1+r2)(1 + refie13)½(1+e34)     and the normalised pensity by
    y!   =   y~y~»    =   ½(1+Riem(refi)e4)   =   ½(1+wpe4)
where spacial unit 1-vector wp   =   Riem(refi) º efe12 e-2( tan-1(r))e31 e3 = (r2+1)-1( e1 2r cosf + e2 2r sinf + e3 (1-r2) ) is the   Riemann sphere representation of complex number z = refi = z1z0-1 and anticommutes with e4.
[ Proof : y! º y~y~» = (1+r2)-1 (1 + refie14)(½(1+e34))2(1 + re-fie14)   =  (1+r2)-1 (1 + refie14)½(1+e34)(1 + re-fie14)
    = ½(1+r2)-1 ((1 + refie14)(1 + re-fie14) +(1 + refie14)(1 - re-fie14)e34)   =   ½(1+r2)-1 ((1 + r2 +2r cosfe14) +(1 - r2 +2r sinfie14)e34)   =   ½(1+r2)-1 (1 + r2 + 2r cosfe14 + 2r sinfe24 +(1 - r2)e34)   =   ½( 1 + (1+r2)-1( 2r cosfe1 + 2r sinfe2 +(1 - r2)e3))e4  .]

    Since y~y~» = y~(½(1+ s3))2y~» = ½(1 + y~ s3y~») = ½(1 + y~e3y~»)e4 we have wp = y~e3y~» .

    Letting y~% = y~   =   (1+r-2)(1 + r-1((f+p)i)e13)½(1+e34) we have y~%» y   =   0     so a normlised ket "orthogonal" to y is given by inverting r and adding p to f .
[ Proof :  y~%» y   =   ½(1+e34)(1+r-2)(1 + r-1(-(f+p)i)e14») (1+r2)(1 + r(fi)e14)½(1+e34)   =   ½(1+e34)(1+r-2)(1+r2) (1 + (-pi)e142 + O(e14))½(1+e34) = 0  .]

    Restoring our p suffix, we therefore have y!p = rp2½(wpy-e4)e4     as the non-normalised pensity for the state y , eigenpensity of the multiplicative operator wpe4 with eigenvalue 1.
    We interpret rp2 as the classical probability of the "particle" being "at" p, and wp as its spin if it is indeed there. y!p does not encode a velocity or a momentum, however. Probability gradient Ñprp2 , for example, need not be timelike.
    The "opposite spin" state has pensity rp2½(-wp-e4)e4 and eigenvalue -1.
     wpy   =   yp~e3yp~»   =   Riem(rpyefpyi)     is the (e4 specific) spacial unit spin 1-vector 3-urbservable .
    We can thus tabulate the following correspondances between matrix and multivector representations.
Â3,1 » = model . Replace e4 with e45 for Â4,1 » = §# model
Ket SymbolKet MatrixKet Multivector r,f Eigenvalue    Pensity Matrix    Pensity Multivector
(the primary idempotent)
æ 10 ö    ½(1+e34) 0,any 1 for e34     æ 10 ö    ½(1+e34) 1
è 00 ø è 00 ø
ïñ æ 00 ö e14½(1+e34) ¥,any -1 for e34 æ 00 ö ½(1-e34)
è 10 ø =e13½(1+e34) è 01 ø
(the standard ket)
æ 20 ö 2(1+e14)½(1+e34) 1,0 1 for e14 æ ½½ ö ½(1+e14)
è 20 ø =2(1+e13)½(1+e34) è ½½ ø
ïñ=2(ïñ-ïñ) æ 20 ö 2(1-e14)½(1+e34) 1,p -1 for e14 æ ½ ö ½(1-e14)
è -20 ø =2(1-e13)½(1+e34) è ½ ø
ï´ñ=2(ïñ+iïñ) æ 20 ö 2(1+e24)½(1+e34) 1,½p 1 for e24 æ ½i ö ½(1+e24)
è 2i0 ø =2(1+e23)½(1+e34) è ½i½ ø
ï·ñ=2(ïñ-iïñ) æ 20 ö 2(1-e24)½(1+e34) 1,-½p -1 for e24 æ ½½i ö ½(1-e24)
è -2i0 ø =2(1-e23)½(1+e34) è i½ ø

    Using ½(1+e34)=e34½(1+e34) we can factor the ket representors in the form R½(1+e34) where R in Â3,1+ has RR§=1. We can replace every occurance of e12 in the factor with i=e1234  and any nonspacial bivector can be converted to a scalar and a spacial bivector. This provides the necessary reduction of  8-dimensional Â3,1 + multivectors into 4 dimensional qubits.

    It is easy to verify using either the matrix or multivector representations, for example, that
áï s3ïñ = ½(1+e34) , and that áï s1ïñ   =   áï s2ïñ = 0 ; corresponding to expected values of 1 for s3=e34 observations, and 0 for s1=e14 and s2=e24 observations of state ↑.
    Since áïe142ïñ = áïe242ïñ = áïe342ïñ = áïïñ = 1 the dispersion of e34 observations in state ↑ is 0 (always get +1) while the dispersion of e14 and e24 observations is 1 (always get either +1 or -1 with 50-50 probablities).

    The classical probablity of c®y follows immediately as
    Probabilty( (wpye4)?(c) = y )   =   ½(1+wpy¿wpc)   =   ( cosq))2     where q is the angle subtended by 1-vectors wpy and wpc .
[ Proof : The pensity scalar product is ¼ ((wpy-e4)e4)*((wpc-e4)e4) = ¼ ((wpy-e4)((-wpc-e4)e42)<0> = ¼(1+wpy¿wpc) while that for the -1 eignevalue eigenket is ¼(1-wpy¿wpc) . Dividing the first by the sum of both gives the result.  .]

     The Dirac inner product y~»c~ = (1+r12)(1+r22)(1 + r1r2e(q2-q1)i) ½(1+ s3)     so the complex conjugate ^ of inner products is again provided by for Â3,1 + .
[ Proof :   (½(1+r12)(1 + r1eq1i s1)½(1+ s3))§ (1+r22)(1 + r2eq2i s1)½(1+ s3)
    = ¼-1(1+r12)(1+r22) (1+ s3)(1 + r1e-q1i s1)(1 + r2eq2i s1)(1+ s3) = (1+r12)(1+r22) ½(1+ s3)(1 + r1r2e(q2-q1)ie1)½(1+ s3)     and s3=e34 commutes with both e1 and ie1  .]

    There is a tendancy in much of the literature to "strip idempotents" from ket representors. Moving from an ideal space Â3,1u into a subspace of Â3,1 in which elements in the left annihilation { a : au=0 } of u can be considered "irrelevant" .

   Moving into Â4.1
    The » = , ypÎÂ3,1+ model serves for modelling non-relativistic qubits but the above table and all subsequent discussion can be adapted for our   »   =   §#, Â4,1 trivector model by associating si with trivector ei45 and i with e12345,  and replacing ei4 by ei45 throughout the discussion.   
[ Proof : For i¹j si sj = eie45eje45 = ei4ej4 = = eij = eijkek    = eijk4ek4 = eijk45ek45 = eijke12345 sk     and so forth,  .]

    The general qubit ket is then ½r0(q0e12345)(1 + r(fe12345)e145)(1+e345)   =   ½r0(q0e12345)(1 + r(fe12345)e13)(1+e345) .
    Dropping the arbitary phase factor we have normalised ket
    w = (1+r2)(1 + r(fe12345)e145)½(1+e345)   =   S½(1+e345)     where S º (1+r2)(e3 + r cos(f)e1 + r sin(f)e2)e3 is a unit spacial 2-versor in e123 with S»   =   S#§   =   S§   =   S-1 ,     S2 = (1-r2)(1+r2)-1 .
    The normalised pensity w!   =   ½(1+Riem(r(fi))e45).
    With regard to a given e4, a qubit pensity is thus parameterised by a unit spacial 1-vector spin wpÎe123 and is the <0;3>-multivector ½(1+wpe45) as anticipated.            

    In particular we have
    ïñ   =   ½(1+e345) with pensity ½(1+e345) , ïñ   =   ½e13(1+e345) with pensity ½(1-e345) , the eigenstates of e345;
    ïñ and ïñ   =   2(1±e13)½(1+e345) with pensities ½(1±e145) , the eignestates of e145;
    ï´ñ and ï·ñ   =   2(1±e23)½(1+e345) with pensities ½(1±e245), the eignestates of e245.

    w% = S%½(1+e345) provides a ket orthogonal to w where S% º (1+r-2)(e3 - r-1 cos(f)e1 - r-1 sin(f)e2)e3 also satisfies S%§S%=1. When r=1 we have S%=S§ .

    Given our convention that la represents the same space as a , it is natural to regard idempotents ½(1±e345) as representating spacial 2-blade e12 together a boolean sign or orienation, ie. as a twist flag. Both have aa§=aa#=0 and a2=a»=a . If we consider e4=em and e5=e+ as generalished homogenous extenders of Â3 then e345=e¥0e3 represents a 1-plane (line) through 0 with direction e3.

    The component (1+e345)= = (1+e345)» of y»» annihilates all odd blades in e1234.

Sierpinski (<1K) Multiparticle Algebras

Multiparticle Systems
    A system of just three classical particles moving through vaccuum under mutual Newtonian inverse square gravitational attraction is mathematically intractible. Systems of QM particles are even more problematic.

    The state of a K-particle system is represented by a KN dimensional multivector field defined over a KN dimensional pointspace, ie. a function yp1,p2,...,pK mapping KN dimensional points onto a potentially 2KN dimensional product space UKN = Âp,q,rK comprising blades formed from blades from distinct participant's geometries.
    For our universe, this is a big geometric algebra! N=5 is manageable and perhaps reducable by "corelating" particular blades  of every participant,  but K » 2266 > 228 , the number of "particles" in the universe, is vast, and 25K > 22268 is incomprehensibly vast. [ 2[k] º 22[k-1] with 2[0]º1 so that 2[1]=2 ; 2[2]=22 2[3]=222 ]

    Fortunately, however, the dimension of y is frequently drastically reduced by considering the particles to be "inditinguishable", to the extent that it is not a product of K and consequently vastly more manageable. A collection of K indistinguishable spinless particles may be representd non-relativitistically, for example, by a complex-valued wavefunction   (ie. a scalar-pseudoscalar multivector) y(t,p1,p2,...pK)=r(t,p1,p2,...pK)(iq(t,p1,p2, following Shrodinger's multiparticle equation
    Ðe4ßy = (½m-1håj=1K Ñpj2 + h-1f(t,p1,p2,...,pK) )y ; where p1,p2,..,pK are the 3D spacioal positions of the K particles at universal time t. The nonnegative amplitude |y(t,p1,p2,...pK)|+2 is interpreted as the real probability density for particle one being at p1, particle two being at p2, and so on.

Local vs disperesed geometries

    To accomodate spin, and similar discrete properties we must choose between two distinct approaches. We can either interpret multivector amplitude r(t,p1,p2,...,pK)2 = |y(t,p1,p2,...pK)|»2 as the probability density function for the position of K particles and mE(t)[j](t,p1,p2,...,pK)   =   Ñpj qp(t,p1,...pj,..,pK) (evaluated at t,p1,p2,...,pK) as the momentum of the jth particle if the partciles have positions p1,p2,...pK at time t, or we can extend the parameter space of y to y(t,p1,S1,p2,S2,...,pK,SK) where S1,S2,.. are multivectors embodying spin, velocity and/or momentum, and any other "internal freedoms". This expanded parameter phase space typically has dimension 6K+1 or 7K+1 .  
    If the _VSi have discrete rather than continuous coordinates then when integrating over the y parameter space we have summations rather than integrations over those Si coordinates. If we are not worried about the momenta of the particles then each Si might reduce to two possible discrete spinstates Ù or Ú.

    Alternatively we assume y!p1,p2,...,pK2 = rp1,p2,...pKy!p1,p2,...,pK where y! is locally normalised y!p1,p2,...,pK» y!p1,p2,...,pK=1 " p1,p2,..,pk and real scalar rp1,p2,...pK³0 is globally normalised and interpretable as the classical probability density for particle 1 being at event p1 and particle 2 at event p2 and so on. For rp1,p2,...pK>0, the geometric content of y!p1,p2,...,pK~ = rp1,p2,...pKy!p1,p2,...,pK~ embodies the values of the properties (spin, velocity, momentum, mass, charge,...) of the particles, if they are all at the specified positions; or more generally is a superposition of possible states of these properties.
      For odd N we have i[j]  central in UN[j] but suffer the anticommution i[j]i[k] = -i[k]i[j] . The i[j] can be made to commute by imposing e5[i] = e5[1] = e5 . There is no "correlator" we can multiply by to ensure this, we must simply accept it as a given. The geometric spaces of each participant thus "share" a common spacial direction e5. This ensures that e12345[i] and e12345[j] commute, which would not otherwise be the case, and that the pseudoscalar i has odd dimension 1+(N-1)K for all K.
    If we also unify the e1234[j] by means of a correlator we reduce the algebra to 2N(2N-1-1)K-1 blades in an extended basis.

    We say the particles are independant if yp1,p2,...pK = yp1[1]yp2[2]...ypK[K] , more generally we have entangled systems which do not so factorise.

    For identical fermions we have antisymmetric y ( y(t,p1,S1,p2,S2,...pK,SK)= -y(t,p2,S2,p1,S1,...pK,SK) ) and y (changing sign if p1 and p2 are swapped and [1] and [2] are swapped) while for identical bosons we have symmetric y (y(t,p1,S1,p2,S2,...pK,SK)=y(t,p2,S2,p1,S1,...pK,SK)) and y. Indistinguishable particles for which wavefunction y and y are neither symmetric nor antisymmetric are known as anions and tend to arise in when particles are restricted to a spacial 2-plane. [  We have swapped the first two arguments P1 and P2 here but the basic idea is that y changes sign when any two distinct arguments are excahnged (_PP3 and PK say) for fermions, and is immune to argument exhanges for bosons ]

Example: Hydrogen Molecule Ground State
    Suppose we have two protons at locations P1 and P2 orbited in some unspecified way by two electrons at p1 and p2. Assuming P1 and P2 to be fixed, the ground state wavefunction for this model of a hydrogen molecule has the form y(p1,p2) = a(y1(p1-P1)y1(p2-P2) + y1(p2-P1)y1(p1-P2)) 2(↑[1][2]-↓[1][2]) where a is an arbitary nonzero constant and scalar y1(p)=y1(|p|)=y1(r)=(pa03)(-a0-1r) is the wavefunction for a spinless electron in the n=1 hydrogen atom ground state.
    The complex wavefuntion for this is y(p1,S1,p2,S1) = a(y1(p1-P1)y1(p2-P2) + y1(p2-P1)y1(p1-P2)) 2( dS1,+1dS2,-1 - dS1,-1dS2,+1) with scalar S1 and S2 restricted to ±1.
    The positional factor of this wavefunction is symmetric in p1 and p2, the electons' fermionic antisymmetry being provided by the spin correleation .

Einstein-Podolsky-Rosen Paradox
    Perhaps the most disturbing and counter intuitive notion in Quantum Mechanics, and the cornerstone of Quantum Cyptography, is manifest by the Einstein-Podolsky-Rosen experiment in which two (anti)correleated particles are allowed to travel a large distance apart before one of them is observed. This observation is predicted by QM to collapse the combined wave function for both particles so that if , for example, Alice observes an ↑ spin about the "vertical" axis of one of the particles, Bob is bound to subsequently observe a ↓ spin if he observes the spin of the corresponding particle about the same quaxis. This in itself seems unremarkable. One can CLASSICALLY postulate that Alice's particle was always ↑ and Bob's was always ↓ from the moment of their seperation. Bob would have measured ↓ for his paticle regardless of whether Alice measured hers or not and there is no need whatever to postulate any "magical instantaneous influence" on Bob's particle due to Alice's observation. This is sometimes known as the Bertleman's socks explanation of the single quaxis case. Both observations merely "reveal" a past connection and there is nothing mysterious about it. The opposite spins that the particles "always had" (as opposed to the spin observed at the moemnt of observation) are known as a hidden variable that is locally revealed by observation.
    When such "EPR experiments" are actually conducted, then Alice observes ↑ 50% of the time and ↓ the remaining 50%, with Bob always obtaining opposite readings. in accordance with both QM and the classical interpretation that each of Alice's particles was "created with" ↑ spin just 50% of the time with Bob's particle always "created with" opposite spin.

    The hidden variables local revelation hypothesis fails to concurr with experiment when we allow Alice and Bob to measure on non-parallel quaxies, however. QM predictions that explicitly contradict the localised hidden variables paradigm are vindicated.

    It is worth noting that in the above we postulate Alice measuring her perticle's spin before Bob's measurement, which we decribed as "subsequent". Since their readings are spacially seperated, however, then (relativistically) some observers may percieve Bob's reading as occuring after Alice's while others may percieve Bob as meaauring first. Our use of the word "subsequent" presumes a favoured reference frame, and while that of the particle generator is an obvious candidate, one cannot relativistically think of one reading "causing" a change in the other in the conventional sense of a cause preceding an effect.

Bell's Inequality

    Let us suppose we have a source of paired particles that generates particles with a potentially large number M of hidden variable scalar parameters m1,m2,..mM which we will denote as a single M-D 1-vector variable m , generating a particular parameter combination m with classical (real nonnegative scalar) probability desity function P(m) Î [0,1] integrating to 1 over the M-volume M of all possible values for m .
    Assume that Alice measures her particle along quaxis n1 whereas Bob uses n2. Let ¦?A(n1,n2,m) denote the scalar spin observed by Alice if she observes the spin of a particle parameterised by m about axis n1 given that Bob uses n2. We know from experiment that ¦?A(n1,n2,m)=±1 and our assumption of locality provides   that Bob's choice of n2 has no relevance to Alice's reading so we have ¦?A(n1,n2,m) = ¦?A(n1,m) . Similarly let ¦?B(n2,m) denote Bob's spin measurement along axis n2 . [  Note that it is not strictly rigourous to speak of n1 and n2 being the "same" because they are directions of measurements at different locations and comparing them requires notions of parallel transport, nontrivial in the presence of gravity. We here assume a flat spacetime so that n1 and n2 can be defined in a coordinate frame common to both Alice and Bob and products such as n1¿n2 are meaningful. ]   
    We know from experiment that Alice and Bob always observe opposite spins when they measure about the same quaxis so we must have
    ¦?B(n,m) = -¦?A(n,m)     for all m that have arisen in past experiments , so we assume it " mÎM .

    Consider the correlation C(n1,n2)  º  òMdm P(m) ¦?A(n1,m) ¦?B(n2,m)     =   -òMdm P(m) ¦?A(n1,m) ¦?A(n2,m)  
which is the expected value of the random variable given by 1 if Alice and Bob measure the same scalar spin value (along their different axies) or -1 if they measure opposite spin values. We clearly have C(n,n)  =  -1 stating that Alice and Bob always get opposite readings if they use the same axis and also have a scalar Bell Inequality |C(n1,n3) - C(n1,n2)| + C(n2,n3) £ 1 which we expect to hold for any possible schemata and distribution of hidden variables m as an inevitable consequence of the locality assumption.
[ Proof : C(n1,n3) - C(n1,n2)   =   òMdm P(m)¦?A(n1,m) (¦?A(n2,m) - ¦?A(n3,m))
       = òMdm P(m) ¦?A(n1,m)¦?A(n2,m) (1-¦?A(n2,m)¦?A(n3,m))     since ¦?A(n2,m)2 = 1
    Þ |C(n1,n3) - C(n1,n2)|   =   |òMdm P(m) ¦?A(n1,m)¦?A(n2,m) (1-¦?A(n2,m)¦?A(n3,m))|
    £ òMdm P(m) | ¦?A(n1,m)¦?A(n2,m) | (1-¦?A(n2,m)¦?A(n3,m))   =   òMdm P(m) (1-¦?A(n2,m)¦?A(n3,m))   =   1 - C(n2,n3)  .]

    Under the QM paradigm, however, we have no hidden variables other than the implied existance or absence of a particle pair, representable by M={ 0,1 }, but no locality assumption. Instead of an integral over M we assume m=1 and set   C(n1,n2) º ¦?A(n1,n2,1) ¦?B(n1,n2,1)   = - cosq where q is the angle subtended by n1 and n2.
[ Proof : Suppose Alice measures +1 for spin axis n1 with probability ½, collapsing Bobs particle to spin -n1. Bob will then measure -1 spin for axis n2 with probability ( cosq))2 and +1 with probability 1-( cosq))2=( sinq))2 so the expected value of C(n1,n2) given that ¦?A(n1)=+1 is ( sinq))2-( cosq))2 = - cos(q)  . It has the same value when ¦?A(n1)=-1 and the result follows.  .]
    Suppose n1,n2, and n3 are coplanar with n1 and n2 subtending 2q and n3 bisecting their exterior angle and so subtending p-q with both n1 and n2. Then C(n1,n3)=C(n2,n3)= cosq while C(n1,n2)=- cos2q . For q < ½p we have |C(n1,n3) - C(n1,n2)| + C(n2,n3) = 2 cosq- cos(2q) which comfortably exceeds 1, violating the Bell Inequality, for many q.

    However, as Christian observes, Bell's inequality is predicated on a scalar spin measurement, typically parameterised by a 3D spacial 1-vector direction n1 ; and we typically think of the spin being forced to ±n1 by the n1-directed obsservation. It is more natural to regard Alice as measuring unit 2-blade spin ±n1i3 where i3=e123 is the unit spacial pseudoscalar and hence 2-blade ¦?A(n1,m)=±n1* where * denotes duality in i3, rather than a scalar observable.
    Since ¦?A(n1,m) and ¦?B(n2,m) no longer commute in general, it is natural to consider a symmetrised geometric correlation
    C(n1,n2)  º  òMdm P(m) (¦?A(n1,m) ~ ¦?B(n2,m)) . [  Where a~b º ½(ab+ba) as usual ]
    Taking m to be an arbitarily scaled 3-blade m=mi3 we have (n1¿m)(n2¿m) = m2n1i3n2i3 = -m2n1n2 so that taking ¦?A(n1,m)=n1¿m gives
    C(n1,n2) º òM dm P(m) (¦?A(n1,m) ~ ¦?B(n2,m))     =   -i3òM|dm| P(m) m2(n1¿n2) .
    Taking m=±1 with equal probabilities gives C(n1,n2)   =   -i3(n1¿n2) = - cos(q)i3 which is dual to the QM paradigm scalar value.

    Thus what is arguably the simplest possible geometric "hidden variable" (the orientation (sign) of a pseudoscalar), provides the correct QM "violation" of the Bell inequality provided we take 2-blade spin observations and the dualed symmteric geometric correlation i3-1òMdmP(m)(¦?A(n1,m) ~ ¦?B(n2,m))   rather than commuting scalar spin measures.

Impossibility of FTL quantum signalling
    The reason that we cannot use EPR phenomena to create an FTL signalling device is that entangled particles are disentangled by observation, 2(ï[1][2]ñ -ï[1][2]ñ ) collapsing to ï[1][2]ñ=ï[1]ñï[2]ñ for example. While it is true that Alice's observation of her particle "effects" Bob's particle "via " the  "co-collapse" of their waveforms on the first observation, subsequent observations by Alice of her particle have no effects on Bob's counterpart particle.

    Having observed along ↑, Alice is free to rerandomise the up/down spin by a ←/→ observation of only those particles which she observed as being ↓. If this caused Bob's corresponding particles to be rerandomised, Bob would observe (on average) 75% of his particles to be ↓ and onlt 25% to be ↑. This Bob would notice given enough particles - deducing that something was effecting a supposedly 50-50 particle stream - and so a communications protocol could be estabalished and an instantaneous signal could be sent by Alice to Bob.

    But since Alices's subsequent observations have no effect on Bob's corresponding particle then, whatever Alice does, all Bob will ever see is a set of particles half of which are ↑ with an apparently random 50-50 distribution. Hence no signal can be sent.

    The question remains as to whether Alice could express herself non-discretely by "partially observing" and so only "partially collapsing" her particle, not fully into an eignket, but merely "closer to" one. This fails because there are no "partial observations". One cannot get any measure from an experiment having distinct eigenvalues but one of those eigenvalues. In order to differentiate her particle stream according to a spin measure, Alice has to observe that measure at least "partially". She has to resolve some of its ambiguity. But when considering her particle there is are no observations possible but the full one returning the full measure, or a zero observation corresponding to there not being a particle at all.

Multiple 4D Qubits

    The multiparticle spacetime algebras in the literature tend to consider only on internal ("spin") freedoms without reference to positions and velocites.

    We say qubits are independant if the ket representing the full quregsiter geometrically factorises as y = y1[1]y2[2]...yK[K] , more generally we have entangled systems which do not so factorise.

    For K=2, the product of two 4-D multivector spaces exists in the 16-D geometric algebra generated by the six bivectors si[1] , sj[2] . This is twice the dimension of the conventional complex 4D 1-vector Dirac space. We can halve the dimension by forcing an equivalence between i[1]=e1234[1] and i[2]=e1234[2] by means of a further idempotent " correlator" geometric multiplier   ½(1-i[1]i[2])=½(1-e1234[1]e1234[2])     so that our Â3,1+k ,  »  =   model normalised 2-quregister is represented by
    w = (1+r[1]2)) (1+r[2]2)) (1+r[1]ef[1]i[1] s1[1] ) (1+r[2]ef[2]i[2] s1[2] ) ½(1+ s3[1]) ½(1+ s3[2]) ½(1-i[1]i[2])
     = (1+r[1]2)) (1+r[2]2)) (1+r[1]ef[1]i[1]e14[1] ) (1+r[2]ef[2]i[2]e14[2] ) a12     where
    b º ½(1+e34[1])½(1+e34[2])½(1-e1234[1]e1234[2])   =   ½(1+e34[1])½(1+e34[2])½(1-e12[1]e12[2]) is the product of three commuting idempotents acting as a source or sink (and so also a converter between) of e34[1] and e34[2], as a converter of e1234[2] and e12[2] into e1234[1] s (or e12[1] s) and satisfying b2=b»=b ; and bb§=0 .

Multiple 5D Qubits
    We can extend to an even multiparticle algebra Â3K+1,K + for a K-particle system and retain commutability of distint partcicle kets by allowing distinct qubits to come from spaces "sharing" a single common e5 but distinct e1,e2,e3 and e4. All spaces share the same scalar 1[i]=1[1]=1 and we correlate the i[i]=e12345[1] by multiplication by (1-i[1]i[2])(1-i[1]i[3])...(1-i[1]i[K]) which also has the effect of correlating the e1234[i] given that e5[i]=e5[1].

    c=(1-e1234[1]e1234[2]) (1+e1234[1]e1234[3])...(1+e1234[1]e1234[K]) = (1-i[1]i[2])(1-i[1]i[3])...(1-i[1]i[K]) has e5[i]c = e5[1]c but cannot be used to "correlate" the e5[i]. We must impose e5[i]=e5[1]=e5 to ensure commutativity of si[k]=ei45[k] and sj[m]=ej45[m] for k¹m.

    Thus, neglecting a phase factor, the general normalised ket for two independant spin-only particles is       
    w   =   w[1]w[2]   =   (1+r[1]2)) (1+r[2]2)) (1+r[1](f[1]i[1])e145[1] ) (1+r[2](f[2]i[2])e145[2] )b     where
    b   =   ½(1+e345[1])½(1+e345[2])½(1-e12[1]e12[2])   =   ½(1+e345[1])½(1+e345[2])½(1-e12345[1]e12345[2])
      =   ½(1+e345[1])½(1+e345[2])½(1-e1234[1]e1234[2])     ( given e5[i] = e5[1] = e5 ) hs b2=b»=b#=b and bb§=0 and commutes with e5 .

    be14[i]b = be13[i]b = be24[i]b = be23[i]b = 0

Fermionic Correlations
    Anticorrelated Fermionic singlet ket h º 2 (ï[1][2]ñ -ï[1][2]ñ )   =    2(e13[2]-e13[1])b     has h»h = h§h = b while h2 = 0 .
    ï[1][2]ñ -ï[1][2]ñ   =   -h     and ï´[1]·[2]ñ -ï·[1]´[2]ñ   =    -e12345[1]h   =    2(e23[1]-e23[2])b     are easily verified.
[ Proof :  ((1+e23[1])(1-e23[2])- (1-e23[1])(1+e23[2]))b   =    2(e23[1]-e23[2])b   =    2(e245[1]-e245[2])b   =    2(e13[1]-e13[2])e12345[1]b  .]

    h is a -1 eigenvalue eigenstate of s3[1] s3[2] = e345[1]e345[2] = e34[1]e34[2] [  Since e345[1] commutes with the e13[2] while negating the e13[1] before being absorbed by b while e345[2] negates the the e13[2] ] and also a -1 eigenvalue eignstate of s1 s1.

    h satisfies the frame-independant property that a[1]h = a[2]§h     ( and also h»a[1] = h»a[2]§ )     " even a[1] Î e1234[1] . Since bh = 0 we have (S[1]S[2]b)»h = 0 for any r=1 rotor S with S§S=1 corresponding to orthogonality of h to "same spin" kets such as ï[1][2]ñ and ï[1][2]ñ  
[ Proof : The (e14[2]-e14[1])e5 factor means that e14[1]h = - e14[2]h . We can send an e34[1] through (e145[2]-e145[1]) negating one term, convert it to an e34[2] and send it back, negating the other, whence e34[1]h = -e34[2]h . Similarly e12[1]h = -e12[2]h , while e24[1]h can be converted as e21[1]e14[1]h = -e21[1]e14[2]y = e14[2]e21[2]y = -e24[2]y. Spacial bivector eij[1]y can be converted as ei4[1]ej4[1]y, again with a sign change. Hence all bivectors are "converted" with negation and the first result follows. (S[1]S[2]b)»h = bS[2]§S[1]§h = bS[2]§S[2]h = bh = 0 gives the second.  .]
    Our h differs from the idempotent-stripped Â1,3+, » = Multiparticle SpaceTime Algebra of Doran et al in which the relativistic fermionic singlet ket is h = (e13[1]-e13[2])½(1-e12[1]e12[2])½(1-e0123[1]e0123[2]) [ GAfp 9.93 ] , which commutes with e4[1]e4[2]

    We can superpose a "same spin" 2-quregister S[1]S[2]b (where S[1] and S[2] are the "same" rotor in different spaces) with orthogonal h as
    y   =   cos(a)S[1]S[2]b + (fe12345) sin(a)h   =   S[1]S[2]( cos(a) + (fe12345) sin(a)(e13[2]-e13[1]))b     with entanglement angle a and arbitary "singlet phase" f.

    (S(1)S(2)b)»h   =   bS(2)§S(1)§h   =   bS(2)§S(1)[2]h   =   (1-e1234[1]e1234[2])(1+e345[2])S(2)§S(1)[2] (1+e345[1])h

    Pensity h!   =   (e13[2]-e13[1])§( (1+e34[1]e34[2])½(1-e1234[1]e1234[2]) )     with h!2=h!§=h! and ei[1]h!   =   ei[2]h! e4[1]e4[2]   =   ei[2] e4[1]e4[2] h!     for i=1,2,3 .
[ Proof : 4(e13[2]-e13[1])b(e13[2]-e13[1])»   =   4(e13[2]-e13[1])§(b)   =   4(e14[2]-e14[1])§(b) . Any blade in b that commutes with e13[1] while anticommuting with e13[2] (or vice versa) will be annihilated by (e13[2]-e13[1])§ , so within the pensity y! we can replace the (1+e345[1])(1+e345[2]) factor in b with (1+e345[1]e345[2]) =(1+e34[1]e34[2]) . Now (1+e34[1]e34[2])½(1-e1234[1]e1234[2]) commutes with any "balanced" blade of the form eij..l[1] eij..l[2] so
     ei[1]h!   =   ei[1] e4[2]e4[1] e4[1]e4[2]h!   =   - e4[2] ei4[1] h!e4[1]e4[2]   =   - e4[2] ei4§[2] h!e4[1]e4[2]   =   ei[2]h!e4[1]e4[2]  .]

Bosonic Correlations

    Correlated Bosonic ket w º 2 (ï[1][2]ñ -ï[1][2]ñ )   =    2(1-e14[1]e14[2])b   =    2(1-e13[1]e13[2])b     has w»w = h§h = w2 = b . Like h, w "transplants" bivectors between [1] to [2] spaces but reverses (negates) only e14 and e23.

    w is a +1 eigenvalue of s3[1] s3[2] [  Since e345[1]e345[2] commutes with e14[1]e14[2] ] and a -1 eigenvalue of s1[1] s1[2]

    Anticorrelated Bosonic singlet ket z º 2(ï[1][2]ñ + ï[1][2]ñ) = 2(e13[1]+e13[2]))b also has z»z = z§z = b and bz = 0 . It transplants bivectors, negating only e12 and e34.
    z is a +1 eigenvalue eigenstate of s1[1] s1[2] = e145[1]e145[2] = e14[1]e14[2] and a -1 eignvalue eignstate of s3[1] s3[2]

    Correlated Bosonic ket m º 2(ï[1][2]ñ + ï[1][2]ñ)   =   2(1+e13[1]e13[2])b   =   2(1+e14[1]e14[2])b has m»m = m§m = m2 = b . It transplants bivectors, negating e13 and e23.
    m is a +1 eigenvalue eigenstate of bothe of s1[1] s1[2] and s3[1] s3[2].

Bell States
    The four correlations m , w , z , and h form the Bell Basis or Bell States conventionally denoted ïF+ñ ,    ïF-ñ ,    ïY+ñ ,    and ïY-ñ respectively or ïb00ñ , ïb10ñ , ïb01ñ , and ïb11ñ respectively in the literature.
KetMultivector Ket s1[1] s1[2]
s3[1] s3[2]
m    ïF+ñ    ïb00ñ     2(ï[1][2]ñ + ï[1][2]ñ)     2(1+e14[1]e14[2])b    +1+1 Bosonic
wïF-ñïb10ñ 2 (ï[1][2]ñ - ï[1][2]ñ) 2(1-e13[1]e13[2])b-1+1 Bosoninc
zïY+ñïb01ñ 2(ï[1][2]ñ + ï[1][2]ñ) 2(e13[1]+e13[2]))b+1-1 Bosoninc
hïY-ñïb11ñ 2 (ï[1][2]ñ - ï[1][2]ñ ) 2(e13[2]-e13[1])b -1-1 Fermionic

    Note that s1[1] s1[2] ïbijñ = (-1)iïbijñ while s3[1] s3[2] ïbijñ = (-1)jïbijñ known as reading the phase and parity bits respectively.

Quantum Teleportation
    Suppose Alice and Bob each posses one of a correlated qubit pair, say the bosonic correlation m º m[1.2] º 2(ï[1][2]ñ + ï[1][2]ñ) where [1] denotes Alice's qubit and [2] denotes Bob's.
    Suppose further that Alice has a third qubit y º y[3] = aïñ[3] + bïñ[3] = (a+be13[3])(1+e345[3]) for unknown complex a, b (combinations of 1 and i[3]).

    The ket for this three-qubit state is   the geometric product ym(1-i[1]i[3]) = my(1-i[1]i[3]) where the (1-i[1]i[3]) factor has been added to extend the correlator in m[1.2] over the three-particle algebra so that all the i[j] can be freely replaced by i. Via somewhat tedious algebra, ommitted here, this can be shown to be expressible as
    y[3]m[1.2]   =   ½( m[1.3]y[2] + w[1.3] s3[2]y[2] +   + z[1.3] s1[2]y[2] +   - h[1.3]i s2[2]y[2] )(1-i[1]i[2])     where y[2] º aïñ[2] + bïñ[2] ; m[1.3] º 2(ï[1][3]ñ + ï[1][3]ñ) = 2(1+e14[1]e14[3])b[1.3] lacks the (1-i[1]i[2]) component of the correlator; and so forth.
    If Alice measures the phase and parity bits of her two qubits using s1[1] s1[3] and then s3[1] s3[3] she will collapse the three-qubit state into (a complex multiple of) one of m[1.3]y[2] ; w[1.3] s3[2]y[2]; z[1.3] s1[2]y[2] ; or    -h[1.3]i s2[2]y[2] (ie. a complex multiple of h[1.3] s2[2]y[2]) with equal ¼ probablities and her two bit measurement will tell Alice which state her (and Bob's) qubits are now in.
    Let us suppose she measures eignevalues of +1 and -1 for s1[1] s1[3] and s3[1] s3[2] respectively. She then knows that the three-qubit system is now in state z[1.3] s1[2]y[2] whcih means that Bob's qubit is no longer entangled with hers and has aquired state s1[2]y[2] = bïñ[2] + aïñ[2].

    Alice then contacts Bob via an insecure classical channel and tells him to "use s1" whereupon Bob aplies unitary transformation s1 to his qubit driving it to s1[2]2y[2] = y[2].
    Note that Bob is not making a s1[2] observation and collapsing his qubit bïñ[2] + aïñ[2] = 2(ïñ[2](b-a) + ïñ[2](b+a)) into either ïñ[2] or ïñ[2] with probablilites in proportion to the complex amplitudes a ± b. Rather, he must use s1 (or whichever Pauli operator Alice tells him to use) as a unitary operator to "rotate" or "evolve" his qubit into state y[2].

    Though this phenomena is referred to in the literature as quantum teleportation this is arguably something of a misnomer. Alice still has two qubits and Bob still has only one. What has happened is that the state of qubit [3] has been transfered to qubit [2]. Although there was some information transmitted (at potentially sublight speed over a classical channel, such as via carrier pidgeon) by Alice to Bob, this was only of two bits, far from adequate to convey the ratio of the two complex weightings a and b that specifes state y.  The entanglement of qubits [1] and [2] has effectively been exploited as an information channel, conveying state y[3] from [3] to [1].
    Note also that qubit [3] no longer contains any vestige of its original y[3] state. The information has been "moved" rather than "copied". Furthermore, if [3] was itself entangled with some other (distant) qubit [4], that entanglement will also have been transfered to [2]. Such is the wierdness of quantum mechanics.

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Copyright (c) Ian C G Bell 2003
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