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 Â_4,1 @ 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 ï↑ñ with ½(1+e₃₄₅) and ï↓ñ 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 y_p in Â_4,1 defined over particular 1-vector pointspaces and satisfying frame-invariant condition y_p^§y_p = 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 Â_4,1. Pensity y_py_p^§# will be self-scaling with complex frane-independant scaling facotr r_p whose real positive modulus (r_pr_p^#)^½ provides y_p = r_py_p^~ for a "normalised" y_p^~ having idempotent pensity y_p^~y_p^~#§ .
    A composite state is then represented as a complex weighted superposition y_p = å_ka_k y_pk over some possibly infinite functional basis where the a_k are independant of p
    A composite state can then be represented as a event-dependant complex-valued function of a ket-type multivector, r_p(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 y_p at a particular event p. We will also be unconcerned with any "propagation equation" like Ñ_py_p = F_p(y_p) to which permitted states are "solutions". How the system ecolves while unobserved is implicit in our defining y_p over a spacetime p. Such Hamilton-Jacobi equations Ð_e4A = -½m^-1 (Ñ_p[e4^*]A)² + f(p) or Dirac Equation Ñ_py_p = (m-qa_p^*)y_p or Hamiltonian form Ð_e4 y_p = g(Ñ_p[e123], y_p) as may be solved by y_p 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 Â_{4,1 +}  multivectors, and also as c+e₅d where c and d are in Â_3,1 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 y_p 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 B_aseÌÂ^3,1 as a ket y representing the state of the "entire" or "composite" system "across" B_ase . We denote the local state "at" event p by y_p and the "composite" state across B_ase 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 Â_4,1 spacetime multivector algebra.
    A key Dirac hypothesis is that ketvector  r e^fi y_p represents the same local state as does ketvector y_p for any (potentially p-dependant) r>0, f Î Â . Consequently ket r e^fi 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 y^{T^} and the Dirac product is value of the sole non-zero element of the complex matrix psi^{T^}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 2^N 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:

• y^»» = y
• y^»c = (c^»y)^»
• y^»_*y = y^»_*y ³ 0
• ((fi)^↑y)^» = (-fi)_expnj (y^»)

Ideal Kets

    Suppose h₁,h₂,...,h_k are commuting plussquare unit Dirac real multivectors so that h_j²=1, h_j^»=h_j and h_ih_j=h_jh_i . Suppose further that each h either commutes or anticommutes with every extended basis element. Let u=½^k(1±h₁)(1±h₂)..(1±h_k) be one of the 2^k disinct annihilating idempotents in A_lgebra{h₁,h₂,..,h_k} .
    Anything that anticommutes with an h_j is anihilated by u₌ so we can decompose a = a + a' where a Î C_entral()(h₁,h₂,..,h_k)= C_entral(u) commutes with u and a' is has ua'u=0  where complex a = a(1⁰) + a¹h₁ + ... a^kh_k 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 C_entral(u) and a' satisfies ua'u=0.
    Ket products simplify as aubucu...gu = 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 y_p = y_p_mvu has y_p^» = _mvuy_p^».
    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  ò_CMd^Mp c_p»y_p     where C_M 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 y_p 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 c_p=c_pu' and y_p=y_pu are kets based on the distrint idempotents then y_p^»c_p = c_p^»y_p = 0 and the two kets trivially satisfy the ket rules.
    y_pc_p^» need not vanish but is "null" in that (y_pc_p^»)² = 0. Since (y_pc_p^»)y_p = 0 while (y_pc_p^»)c_p = y_p |c_p|² we have (ay_pc_p^»)^↑ = 1 + a(y_pc_p^») so (ay_pc_p^»)^↑ c_p = c_p + a|c_p|² y_p and (ay_pc_p^»)^↑ y_p = y_p and we can regard (ay_pc_p)^↑ as introducing y_p linearly.
    For kets based on the same idempotent we have y_p^»c_p = uy_p^»c_pu (y_p^»c_p) ^†_u .
    y_pc_p^»  = y_puc_p^» .
    (y_p^»c_p)² = uy_p^»c_pu y_p^»c_pu = u (y_p^»c_p)(y_p^»c_p)) ^†_u = u (¯_u(y_p^»c_p) + ^_u(y_p^»c_p)) (¯_u(y_p^»c_p) - ^_u(y_p^»c_p)) u = u (¯_u(y_p^»c_p)² + ^_u(y_p^»c_p)) + 2(^_u(y_p^»c_p)×^_u(y_p^»c_p) ) u
   
    Thus a more general ket can be regarded as y_1pu₁ + y_2pu₂ + ... + + y_kpu_k where u₁,u₂,...,u_k are k mutually annihilating Dirac real idempotents.
    Transformation u_1» = u₁₌ has the effect of annihilating products like au_i and u_ia 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 B_ase), 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 e^fi 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 e^fi as a phase rotation.
    Note therefore the important distinction between local normalisation y_p^~ º y_p|y_p»y_p|^-½ so that y_p^~»y_p^~ = ±u     " p ; and C_Mp-normalisation
    y^~_p º y_p |ò_CMpd^Mq y_q^»y_q) |^-½     where C_Mp is an understood possibly p-dependant M-curve in B_ase over which we wish y^~_q^»y^~_q to integrate to ±1 .
    Even when B_ase 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 y_p and y_p+d .    

    If ï1ñ , ï2ñ, ... ïMñ are M kets then a₁ï1ñ+a₂ï2ñ+...+a_MïMñ 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.

Bras
    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 } .

Pensities

    We here regard multivector local pensity y^!_p  º  y_p(y_p^»)   =   y_py_p^»   =   y_puy_p^§   =   y_p§(u)     as being more fundamental than y_p.
    Because y_p expresses itself as y_p§ , ay_p acts with ay_p^§ "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 (y_pc_p^»)² =  (c_p»y_p)(y_pc_p^») and hence has complex selfscale (c_p»y_p) which vanishes ((y_pc_p^»)² = 0) if c_p and y_p are "orthogonal".
     We will initially represent pensities with selfscaling (aka. "selfeigen") (y^!_p²=l_py^!_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 l_p = |y^!_p|_s as the selfscale of the pensity at p, negative when y_p is an antiket. y^!_p²=l_py^!_p implies y^!_p is either singular (ie. noninvertible) or the "scalar system" y^!_p=l_p.

    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   e_ijk*y^! = y_»(e_ijk)

    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 Ñ_p*(bab^§#)   =   2((Ñ_pb)a_<-^§#>b^§#)_<0;N>     with ^» = ^§# provides
    Ñ_p* y_»(a) = 0 for all constant Dirac-real a   and Ñ_p* y_»(a)   =   2((Ñ_py)ay^»)_<0;N> for all constant imaginary a .
    In particular Ñ_p*y^!_p = 0       for any pensity (since 1^»=1) and we have the ket kinematic rule
    Ñ_p*(y_p^»ay_p)   =   2((Ñ_py_p^»)ay_p)_<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^~₌(y^S)   =   l_iy^!_i^~ .

    The invertible (ay^!~)^↑ = 1 + ((±a)^↑-1)y^!~ according as y^!~2=±y^!~ .
    More generally (ayc^»)^↑ = 1 + (ab_c»y)^↑-1) b_c»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|₊ = (y_p^»y_p)^½ divided by   ò_Ckd^kp |y^!_p|₊ over some k-curve C_k 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+...+a_MïMñ) for M orthonormal kets ï1ñ , ï2ñ, ... ïMñ and central a₁,a₂,... then
    y^»y   =   å_i |a_i|₊² áiïïiñ   =   å_i |a_i|₊² u   so positive real scalar y»y   =   å_i |a_i|₊² ; and
    y^! º yy^»   =   (a₁ï1ñ+a₂ï2ñ+...+a_MïMñ) (a₁ï1ñ+a₂ï2ñ+...+a_MïMñ)^»   =   å_i|a_i|₊²ïiñáiï + å_i¹j (a_ia_j^{^} ïiñájï +a_ja_i^{^} ïjñáiï)
    But y^!2 = y(y^»y)y^» = y»yy^! so the normalisation condition for both y and y^! is å_i |a_i|₊² = 1 .

Observables

    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   =   y_py_p^» is real when regarded as a linear geometric ket operator y^!_p(c_p) º y^!_pc_p = y_p(y_p^»c_p) = (y_p»c_p)y_p .
[ 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 y_p is a eigenket of linear ket operator ¦_p if ¦_p(y_p)=a_py_p     " p for some "complex" eigenvalue scalar field a_p. If a_p=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 l_i 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 + z_m-1¦^m-1 + ... +z₁¦ + z₀1 = 0
for some complex valued z₀,z₁,...z_m-1 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,...l_m be normalised eigenkets associated with m discrete eigenvalues l₁, ..l_m. Linear operator  å_j=1^m l_jl_j^» sends l_k to l_k  " k so if the m eigenkets are a complete set for K , å_j=1^m l_jl_j^» 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.

Probabilities
    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.
    P_robabilty(¦^?(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
     P_robabilty(¦^?(y)=l_j)   =   (y^»l_j)(l_j^»y) ( å_i(y^»l_i)(l_i^»y) )^-1
      =   l_j^!_*y^! (å_il_i^!_*y^!)^-1     where the summations are over all eigenkets of ¦ (and may include integrations for continuous eigenspaces). The l_i 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 ¦.     

1-Observables

    We will initially regard a 1-observable as a p-dependant Dirac-real linear ket operator ¦_p : K ® K taking y_p to ¦_p(y_p). 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 l_i of ¦ satisfying l_-j^»y ¹ 0 , returning as measure ¦^?(y) the associated real scalar eigenvalue l_j .
    If y = å_i=1^m z_il_i for orthonormal real eigenvalued eigenkets l_i of ¦, then ¦^?(y)=l_j and ¦^?(y)=l_j with probability
    |l_j»y|₊²(å_i=1^m |l_i»y|₊²)^-1 = |y»l_j|₊²(å_i=1^m |y»l_i|₊²)^-1
    We can theoretically retrieve the eigenvalue measure ¦^?(y) from the eigenket l_i¦^?(y) as (e_ij..l* ¦(l_i)) (e_ij..l* l_i)^-1 where e_ij..l is any blade present in l_i, and we will typically use the scalar part via ¦^?(y) = (¦(l_i))_<0> l_i<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 ò_CMd^Mp y_p»(¦_p(y_p)) 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 E_y(¦) 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
      f_y 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)²_y º ((¦-¦_y)²)_y = (¦²)_y - (¦_y)² the variance of ¦ in y and its positive square root provides a real postive scalar
    D_y(¦) º (¦-¦_y)²_y^½     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 D_y(¦) = 0 and uncertain if D_y(¦) > 0.

    For general possibly noncommuting 1-observables ¦ and g we have the uncertainty principle      D_y(¦)D_y(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)   =   ¦²_y+t²g²_y-2t(i¦×g)_y
    Since |(¦-itg)(y)|_» ³ 0 the quadratic discriminant 4(i¦×g)_y² - 4¦²_yg²_y £ 0 so ¦²_yg²_y ³ (i¦×g)_y² with equality only when repeated root t occurs with (¦-itg)(y)=0, It is easily verified that (¦-¦_y)×(g-g_y) = ¦×g so substituing the the 0-centred observables for ¦ and g we obtain
(¦-¦_y)²_y (g-g_y)²_y 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 D_y(¦)D_y(g) ³ 0 but unless ¦ commutes with all the 1-observables of a system we must have D_y(¦) > 0.  

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

k-Observables
    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 y_i such that each y_i is an eignket of all k operators, having associated eigenvalue l_{i 1} for ¦₁ , l_{i 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 l_j denote a particular eigenvalue combination. and l_j 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.

k-Urbservables
    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 p_d,p(y_p) º p_d(y_p) º (d^-1¿p)y_p. 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 i^th coordinate denoted p_i 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 l_d with eigenvalue l_d for p_d would perforce satisfy ((d^-1¿p)-l_d)l_d(p) = 0     " pÎB_ase     forcing l_d(p)=0 outside the hyperplane      (d^-1¿p)=l_d) which would typically restrict a spacial 3-plane volume integral expression for l_d^»l_d to a 2-plane and so cause it to vanish.  .]
    However p_d has a single local eigenvalue (d^-1¿p) at p so p_d^?(y_p) = (d^-1¿p) is certain for all y.

    If p_d returns measure l when acting on y the collapsed state theoretically satisfies p_d(c_p) = lc_p " p which can only hold if c_p=0 whenever d^-1¿p ¹ l . Within the hyperplane all states share the p_d eigenvalue, so the Neumann-Luders postulate suggests that p_d^?(y_p) collpases y_p   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 å_j=1^N e^j p_ej so that p_d = (d¿p) so that py_p = py_p . Rather than eigenvalues, geometric observable p has 1-vector eigenvalues corresponding to the measured position.

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

    Clearly p_ei and p_ej commute so event p is an N-observable. T º T_e4 º p_e4 = (e⁴¿p) is known as the time observable.

    So what happens when we "measure" p_d?  We will have some kind of apparatus capable of returning an event coordinate. A K-curve screen S Ì C_M Ì B_ase perhaps. Assuming y»y ò_CM |d^Mp| y_p»( p_d(y_p)) = 1, the expected measure will be ò_S |d^Kp| y_p»( p_d(y_p))   =   ò_S |d^Kp| (d¿p) |y_p|_»²

Spin Observable

    Another simpler example of a geometric observation is provided by taking f_p = 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 C_M 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 C_4×4 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 Â_4,1,.

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 y_p with the state of a system at a given spacetime event p. y_p is then a ket-valued function of Â^3,1 .
    We make the assumption of universal superpositions, postulating that if y_p   = z₁c_p + z₂x_ip at a given p with complex z₁,z₂ independant of p then y_p+d   = z₁c_p+d + z₂x_ip+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 D_p,dy_p º y_p+d is then linear .
    Assuming that y_p+dd ^» y_p+dd = y_p^»y_p     provides the ket displacement operator normalisation condition D_p,d^»D_p,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 ¦_py_p = c_p . 1-vector d induces a displaced observable ¦_p,d defined by
    ¦_p,dy_p+d º c_p+d = D_p,dc_p = D_p,d¦_py_p .
    From ¦_p,dy_p+d = ¦_p,dD_p,dy_p we can thus derive     ¦_p,d = D_p,d¦_pD_p,d^-1     and so     ¦_p,dD_p,d = D_p,d¦_p .

    We have ¦_p,d = D_p,d¦_pD_p,d^-1 = ¦_p + 2e( Ð_dצ_p) + _O(e²)     where Ð_dצ_p º ½( Ð_d¦_p-¦_p Ð_d) .
    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 Ð_d to be the ket operator Ð_dy_p º Lim_{e ® 0} (y_p+ed - y_p)e^-1   =   Lim_{e ® 0} (D_p,dy_p - y_p)e^-1 . Since we can multiply D_p,d by   e^a_p,di for arbitary real a_p,d we obtain an arbitary imaginary additive term ai  where a = ( Lim_{e ® 0}a_p,ed)i in the derivative (somewhat akin to the arbitary "additive constants" pertaining in indefinite integrals).

    Differentiating ¦_p,dD_p,d = D_p,d¦_p we obtain ( Ð_d¦_p.d)D_p,d + ¦_p.d( Ð_dD_p,d) = ( Ð_dD_p,d)¦_p

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

    When acting as an observation, we thus expect i Ð_d^? to collapse a state y_p to an eigenket of iÐ_d , ie. to a solution of iÐ_d y_p = m_dy_p for a real scalar d-directed momentum m_d such as y_p = (-m_di(p¿d))^↑ y₀.
    It is customary to define a real directed four-momentum operator by m_d º h Ð^p_d   =   h(d¿Ñ_p) 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 Ð_d =  h^-1 m_d .
    [  Note that m_ej = h ¶/¶x^j = -h ¶x/¶x_j for j=1,2,3 in Â_1,3 timespace hence i^-1(2p)^-1h ¶/¶x_j is common in the literature. ]

    We have undirected four-momentum operator m º å_i=1^N eⁱ m_ei   =   å_i=1^N heⁱ Ð_ei   =   hÑ_p . Physicists usually think of the momentum operator as a 1-vector, using "four" only in the sense of having four dimensions. In our Â_4,1 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 Ð_ei 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.

    m_a× p_b = ½h(b^-1¿a) .
[ Proof :    m_a p_by_p   =   h Ð_a((b^-1¿p))y_p)   =   h(Ð_a(b^-1¿p))y_p + h(b^-1¿p)(Ð_ay_p)   =   h(b^-1¿a)y_p + p_b m_ay_p  .]
    Applying the uncertainty principle to m_a and p_b thus gives Heisenberg's uncertainty principle D_y( m_a)D_y( p_b) ³ ½h(2p)^-1|b^-1¿a|   with coordinate form D_y( m_ei)D_y( p_ej) ³ ½d_ij 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.

Hamiltionian
    H_p º m_e4 º h(e₄¿Ñ) is traditionally regarded as the "energy operator" or Hamiltonian at p. We say y_p is e₄-isolated if  H_p is independant of t=e⁴¿p , ie. if the system evolution operator is time invariant. An eigenket of the Hamiltonian is a solution to hÐ_e4y_p = E_py_p and if H_p is independant of t so to will be the scalar energy eigenvalue E_p.

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

    Applying the uncertainty principle to H= m_e4 and T=p_e4 gives the time-energy uncertainity D_y(H)D_y(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 M_p=mV_p 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 hÑ_p so we have 3-blade
    L_, = p.(hÑ_p) = (pÙÑ_p)h = h(pÙÑ_p) = 2h(p×Ñ_p) = 2p×(hÑ_p) = 2p×m ; and also the more exotic operator
    pÙm   =   (å_j=1^N e^j p_ej)Ù (å_k=1^N e^k m_ek)   =   å_j;k=1^N (e^jÙe^k) p_ej m_ek
      =   å_j<k=1^N (e^jÙe^k)( p_ej m_ek - p_ek m_ej)   =   h^-1å_j;k=1^N ( e^j(e^j¿p)e^kÐ_ek - e^kÐ_ke^j(e^j¿p) )   =   e^je^k(e^j¿p)Ð_ek - e^ke^j((e^j¿e_k)+(e^j¿p)Ð_ek)