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 ¦.