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} @ C_{4} 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+e_{345})
and ï↓ñ with ½e_{13}(1+e_{345}) 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 i^{2}=-1 ; i^{#}=i^{#}^{§}=-i ;
and i^{§}=i.
We thus associate "complex numbers" with central multivectors in Â_{4,1}.
Pensity y_{p}y_{p}^{§}^{#} will be self-scaling with complex frane-independant scaling facotr r_{p}
whose real positive modulus (r_{p}r_{p}^{#})^{½} provides
y_{p} = r_{p}y_{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_{k}a_{k} y_{p}_{k} 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_{p}y_{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_{e4}A = -½m^{-1} (Ñ_{p}_{[e4*]}A)^{2} + f(p)
or Dirac Equation Ñ_{p}y_{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 {e_{1},e_{2},e_{3},e_{4},e_{5}}
with e_{1}^{2}=e_{2}^{2}=e_{3}^{2}=e_{5}^{2}=1 ; e_{4}^{2}=-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+e_{5}d where c and d are in Â_{3,1} space e_{5}^{*}.
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_{ase}ÌÂ^{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:
Ideal Kets
Suppose h_{1},h_{2},...,h_{k} are commuting plussquare unit Dirac real multivectors so that h_{j}^{2}=1,
h_{j}^{»}=h_{j} and h_{i}h_{j}=h_{j}h_{i} . Suppose further that each h either commutes or anticommutes with every extended basis element.
Let u=½^{k}(1±h_{1})(1±h_{2})..(1±h_{k})
be one of the 2^{k} disinct annihilating idempotents in A_{lgebra}{h_{1},h_{2},..,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}()(h_{1},h_{2},..,h_{k})= C_{entral}(u) commutes with u
and a' is has ua'u=0 where
complex a = a(1^{0}) + a^{1}h_{1} + ... a^{k}h_{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-1}ua^{»}
and if we insist a^{»}a be central we have
(aua^{»})^{k} = (a^{»}a)^{k-1}aua^{»} .
Dirac conjugation provides a complex-valued inner product for composite states
c»y º
ò_{CM}d^{M}p c_{p}»y_{p}
where C_{M} is a particular M-curve of interest. Typically an e_{4} 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_{p}u'
and y_{p}=y_{p}u 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_{p}c_{p}^{»} need not vanish but is "null" in that (y_{p}c_{p}^{»})^{2} = 0.
Since (y_{p}c_{p}^{»})y_{p} = 0
while (y_{p}c_{p}^{»})c_{p} = y_{p} |c_{p}|^{2}
we have
(ay_{p}c_{p}^{»})^{↑} = 1 + a(y_{p}c_{p}^{»})
so
(ay_{p}c_{p}^{»})^{↑} c_{p} = c_{p} + a|c_{p}|^{2} y_{p}
and
(ay_{p}c_{p}^{»})^{↑} y_{p} = y_{p}
and we can regard (ay_{p}c_{p})^{↑} as introducing y_{p} linearly.
For kets based on the same idempotent we have
y_{p}^{»}c_{p} = uy_{p}^{»}c_{p}u
(y_{p}^{»}c_{p}) ^{†u} .
y_{p}c_{p}^{»} = y_{p}uc_{p}^{»} .
(y_{p}^{»}c_{p})^{2} =
uy_{p}^{»}c_{p}u y_{p}^{»}c_{p}u
= 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_{1}_{p}u_{1}
+ y_{2}_{p}u_{2} + ... +
+ y_{k}_{p}u_{k}
where u_{1},u_{2},...,u_{k} are k mutually annihilating Dirac real idempotents.
Transformation u_{1}_{»} = u_{1}_{=} has the effect of annihilating products like au_{i}
and u_{i}a for any i¹1. Its effect on au_{1} is to negate anything in a that
anticommutes with u_{1}.
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_{M}_{p}-normalisation
y^{~}_{p} º y_{p}
|ò_{CMp}d^{M}q
y_{q}^{»}y_{q})
|^{-½}
where C_{M}_{p} 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 a_{1}ï1ñ+a_{2}ï2ñ+...+a_{M}ïMñ
is also a ket for any complex a_{1},a_{2},... 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_{p}y_{p}^{»} =
y_{p}uy_{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_{p}c_{p}^{»})^{2} = (c_{p}»y_{p})(y_{p}c_{p}^{»})
and hence has complex selfscale
(c_{p}»y_{p}) which vanishes ((y_{p}c_{p}^{»})^{2} = 0)
if c_{p} and y_{p} are "orthogonal".
We will initially represent pensities with selfscaling (aka. "selfeigen") (y^{!}_{p}^{2}=l_{p}y^{!}_{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^{!}_{p}^{2}=l_{p}y^{!}_{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 = u_{0}^{-1} (yc^{»})_{<0,N>}
and in particular
y»y = u_{0}^{-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} = u_{0}^{-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|_{+}^{2}y)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
|y_{1}»y_{2}|_{+}^{2}
= ½(1+w_{1}¿w_{2}) .
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_{p}b)a_{<-§#>}b^{§}^{#})_{<0;N>}
with ^{»} = ^{§}^{#} provides
Ñ_{p}_{*} y_{»}(a) = 0 for all constant Dirac-real a
and
Ñ_{p}_{*} y_{»}(a) = 2((Ñ_{p}y)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_{p}y_{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_{i}y^{!}_{i}^{~} .
The invertible (ay^{!}^{~})^{↑} = 1 + ((±a)^{↑}-1)y^{!}^{~} according as
y^{!}^{~}^{2}=±y^{!}^{~} .
More generally
(ayc^{»})^{↑} = 1 + (ab_{c»y})^{↑}-1)
b_{c»y}^{-1}yc^{»}
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_{Ck}d^{k}p |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 ± l^{2})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 = a_{1}ï1ñ+a_{2}ï2ñ+...+a_{M}ïMñ)
for
M orthonormal kets ï1ñ , ï2ñ, ... ïMñ and central a_{1},a_{2},...
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^{»} =
(a_{1}ï1ñ+a_{2}ï2ñ+...+a_{M}ïMñ)
(a_{1}ï1ñ+a_{2}ï2ñ+...+a_{M}ïMñ)^{»}
= å_{i}|a_{i}|_{+}^{2}ïiñáiï
+ å_{i¹j}
(a_{i}a_{j}^{^} ïiñájï
+a_{j}a_{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_{p}y_{p}^{»} is real
when regarded as a linear geometric ket operator
y^{!}_{p}(c_{p}) º
y^{!}_{p}c_{p} = y_{p}(y_{p}^{»}c_{p}) =
(y_{p}»c_{p})y_{p} .
[ Proof :
(acb)»y = u_{0}^{-1}((acb)^{»}y)_{<0,N>}
= u_{0}^{-1}(b^{»}c^{»}a^{»}y)_{<0,N>}
= u_{0}^{-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_{p}y_{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} + ... +z_{1}¦ + z_{0}1 = 0
for some complex valued z_{0},z_{1},...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 l_{1},...l_{m} be normalised eigenkets associated with m discrete eigenvalues l_{1},
..l_{m}.
Linear operator å_{j=1}^{m} l_{j}l_{j}^{»} sends l_{k} to l_{k} " k
so if the m eigenkets are a complete set for K , å_{j=1}^{m} l_{j}l_{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 l_{}l_{}^{»} 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)^{»}
= u_{0}^{-1}(c^{»}y)(y^{»}c)
= u_{0}^{-1}(c^{»}yy^{»}c)_{<0>}
= u_{0}^{-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_{i}l_{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_{i}l_{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_{CM}d^{M}p y_{p}»(¦_{p}(y_{p}))
º y» ¦(y)
= (¦(y))»y
= u_{0}^{-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) = ¦_{y}u 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)
= u_{0}^{-1} (y^{»}fy)_{<0>}
= u_{0}^{-1} y^{!}_{*}f
= u_{0}^{-1} y'_{»}(u)_{*}f
= u_{0}^{-1}(u_{=}y'^{»}_{»}(f))_{<0>}
= ½ u_{0}^{-1}
(y^{!}_{<0>} + b_{*}y^{!})
when f=½(1+b).
Note that (y^{!}fy^{!})_{<0>}
= (y^{!}^{2}f)_{<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|_{+}^{2}u
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|^{2}y^{!} , 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) - ¦_{y}c , 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}+t^{2}g^{2}-it(¦g-g¦) .
Whence
|(¦-itg)(y)|_{»}^{2}
= ((¦-itg)(y))^{»}(¦-itg)(y))
= y^{»}((¦-itg)^{»}(¦-itg)(y))
= y^{»}(¦^{2}+t^{2}g^{2}-2it(¦×g))(y)
= ¦^{2}_{y}+t^{2}g^{2}_{y}-2t(i¦×g)_{y}
Since |(¦-itg)(y)|_{»} ³ 0 the quadratic discriminant 4(i¦×g)_{y}^{2}
- 4¦^{2}_{y}g^{2}_{y} £ 0 so
¦^{2}_{y}g^{2}_{y} ³ (i¦×g)_{y}^{2}
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)_{y}^{2} 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) > e^{2}) . Then ò_{CM} |d^{M}p| y_{p}»((d^{-1}¿p)y_{p}) = ò_{CM} |d^{M}p| (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_{d}_{y} 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} = (e^{4}¿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^{M}p| y_{p}_{»}( p_{d}(y_{p}))
= 1, the expected measure will be
ò_{S} |d^{K}p| y_{p}_{»}( p_{d}(y_{p}))
= ò_{S} |d^{K}p| (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 e_{124} or e_{345} (ie. the "simplest" nonscalar multivector satisfying f^{2}=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 e_{145},e_{245}, and e_{345} 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} = z_{1}c_{p} + z_{2}x_{i}_{p} at a given p with complex z_{1},z_{2} independant of p
then
y_{p+d} = z_{1}c_{p+d} + z_{2}x_{i}_{p+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,d}y_{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 ¦_{p}y_{p} = c_{p} . 1-vector d induces a displaced observable
¦_{p}_{,d} defined by
¦_{p}_{,d}y_{p+d} º c_{p+d} = D_{p,d}c_{p} =
D_{p,d}¦_{p}y_{p}
.
From ¦_{p}_{,d}y_{p+d} = ¦_{p}_{,d}D_{p,d}y_{p} we can thus derive
¦_{p}_{,d} = D_{p,d}¦_{p}D_{p,d}^{-1} and so ¦_{p}_{,d}D_{p,d} = D_{p,d}¦_{p}
.
We have
¦_{p}_{,d} = D_{p,d}¦_{p}D_{p,d}^{-1} = ¦_{p} +
2e( Ð_{d}×¦_{p}) + _{O}(e^{2})
where Ð_{d}×¦_{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_{d}y_{p} º
Lim_{e ® 0} (y_{p+ed} - y_{p})e^{-1}
= Lim_{e ® 0} (D_{p,d}y_{p} - y_{p})e^{-1} .
Since we can multiply D_{p,d} by
e^{ap,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}_{,d}D_{p,d} = D_{p,d}¦_{p} we obtain ( Ð_{d}¦_{p}_{.d})D_{p,d} + ¦_{p}_{.d}( Ð_{d}D_{p,d}) = ( Ð_{d}D_{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}(e^{2})=0 .
Alternatively, differentiating normalisation condition y^{»}y = u gives
(Ð_{d}y^{»})y + y^{»}(Ð_{d}y) = 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_{d}y_{p} for a real scalar d-directed momentum m_{d}
such as y_{p} = (-m_{d}i(p¿d))^{↑} y_{0}.
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)^{-1}i ( 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)^{-1}h ¶/¶x_{j} is common in the literature.
]
We have undirected four-momentum operator m_{} º
å_{i=1}^{N} e^{i} m_{ei}
= å_{i=1}^{N} he^{i} 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_{b}y_{p} =
h Ð_{a}((b^{-1}¿p))y_{p})
= h(Ð_{a}(b^{-1}¿p))y_{p}
+ h(b^{-1}¿p)(Ð_{a}y_{p})
= h(b^{-1}¿a)y_{p} + p_{b} m_{a}y_{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(e_{4}¿Ñ) is traditionally
regarded as the "energy operator" or Hamiltonian at p. We say y_{p} is e_{4}-isolated
if H_{p} is independant of t=e^{4}¿p , ie. if the system evolution operator is time invariant.
An eigenket of the Hamiltonian is a solution to hÐ_{e4}y_{p} = E_{p}y_{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)=e_{4}t is considered to represent a particle "at rest with respect to e_{4}"
in that repeated observations of the instantaneous "velocity" of the particle average out to give e_{4}
and we regard the partcle as having momentum me_{4}.
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)e_{123}^{-1}
for spacial position
p and momentum m 1-vector operators within e_{123}.
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^{j}Ùe^{k}) p_{ej} m_{ek}
= å_{j<k=1}^{N} (e^{j}Ùe^{k})( p_{ej} m_{ek} - p_{ek} m_{ej})
= h^{-1}å_{j;k=1}^{N} (
e^{j}(e^{j}¿p)e^{k}Ð_{ek} -
e^{k}Ð_{k}e^{j}(e^{j}¿p) )
=
e^{j}e^{k}(e^{j}¿p)Ð_{ek} - e^{k}e^{j}((e^{j}¿e_{k})+(e^{j}¿p)Ð_{ek})
More generally, we have
L_{a,b} º
p_{a} m_{b} - p_{b} m_{a} =
h ((a^{-1}¿p)Ð_{b} -
(b^{-1}¿p)Ð_{a}) .
Even though L_{a+lb,b} = L_{a,b}
we have not here indexed L_{,} with a 2-blade
as L_{aÙb,} because the relative magnitudes of a and b
effect the position operators. Traditional QM
usually considers only orthonormal a and b within e_{123}
, regarding
L_{a,b} as the angular momentum about spacial 1-vector "axis" a×b = (aÙb)e_{123} .
We have important commutation relationships _{[ IQT 8.1.1 ]}:
For brevity and compatability with the literature we define L_{1} º L_{e2,e3} ; L_{2} º L_{e3,e1} ; L_{3} º L_{e1,e2}
Though the three L_{j} do not commute with eachother, they all commute with L^{2} º L_{1}^{2}+L_{2}^{2}+L_{3}^{2}
_{[ IQT 8.2.2 ]} . This means that we can find a simultaneous eigenket of L_{3} and L^{2} . It can be shown
that if y_{p} is mutual eigenket of eigenvalue mh for L_{3}
and lh^{2} for L^{2} then the (e_{3}-specific) angular momentum ladder operators
L_{±} º L_{1} ± iL_{2} act as L_{3} eigenvalue changers in that
L_{±}(y_{p}) remains a
lh^{2} eigenvalued eigenket of L^{2} but
is an (m±1)h eigenvalued eigenket for L_{3}.
The L_{±} commute with L^{2} but L_{3}×L_{±} = ±½hL_{±}.
We also have |L_{±}y|_{»}^{2} = (l-m(m±1)) h^{2} |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 e_{145},e_{245},e_{345} . We can extend it to include minussquare e_{125},e_{135}, and e_{235} by allowing one or both of the a and b in L_{a,b} to be timelike within e_{1234} but there is a complication. The commutation of L_{a,b} and L_{c,d} simplifies to the above result only if we have (c^{-1}¿a)=(c¿a^{-1}) and so forth, which essentially requires a^{2}=b^{2}=c^{2}=d^{2}
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)^{-1}h 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 e_{145}, e_{245}, e_{345} (dual to e_{23},e_{13}, and e_{12}) within the angular momentum
observable.
Electron Orbits
Y(x)
= x^{l} y(x^{~})
= r^{l} y(q,f)
with spherical harmonic
y(x^{~}) = y(q,f)
= ((4p)^{-1}(2l+1)((l-m)!(l+m)!^{-1})^{½}
P^{m}_{l}( cosq) (imf)^{↑}
solve Laplace equation Ñ_{x}^{2}Y(x)=0 and it can be shown
_{[ GAfp 8.155 ]} that in Â_{3}
y(x^{~}) = y(q,f)
= ( (l+m+1)P^{m}_{l}( cos(q))
+ P^{m+1}_{l}( cos(q))e_{f}e_{123}^{-1} )
((l-1)fe_{12})^{↑}
solves (xÙÑ_{x})y(x^{~}) = -ly(x^{~})
and hence Y(x) = |x|^{l}y(x^{~})
is monogenic (Ñ_{x}Y = 0).
[ where q is the polar angle within [0,p] ; f the longitudinal [0,2p] angle;
P^{m}_{l} a Legendre Polynomial.
]
l=m=0 is just the constant scalar idempotent y=1. l=0,m=-1 provides 2-vector solution y = e_{f}e_{123}^{-1}(-fe_{12})^{↑} = e_{f}(-fe_{12})^{↑} e_{123}^{-1} which we can regard as the free electron orbitting itself.
Consider taking a directed derivative (d¿Ñ_{x})(afe_{12})^{↑} for unit d.
It will depend exclusively on the change d¿e_{f} in f on moving from x to x+d and
have value
((a(d¿e_{f})e_{12})^{↑}-1)(afe_{12})^{↑} which for small d approaches
(a(d¿e_{f})e_{12})(afe_{12})^{↑}
= a(d¿e_{f})(½p+afe_{12})^{↑}
providing
Ñ_{x} (afe_{12})^{↑}
= Ñ_{x}¿(afe_{12})^{↑} =
a e_{f} (½p+afe_{12})^{↑}
= (½p-afe_{12})^{↑} e_{f}.
Now Ñe_{f}=(r sin(q))^{-1}e_{f}=R^{-1}e_{f} .
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)h^{2} for L^{2} ; an orbital magnetic moment quantum number m with |m|£l
associated with eigenvalue mh for L_{3}; 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 L_{3} = L_{e1,e2} observable m^{2}h^{2} £ l^{2}h^{2}
< l(l+1)h^{2} so the L_{3} observable is always less that the "L magnitude" observable regardless of how
"e_{12}-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 2n^{2}
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 Â_{p,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 C^{K}
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^{-½} 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 C^{K} 1-vectors
and is representable with regard to the eigenbasis by a C_{K×K} matrix in the conventional way,
with l_{i}^{»}(g(l_{j})) providing the i^{th} element of the j^{th} 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 :
l_{i}^{»}¦_{1}l_{j}
= l_{i}^{»} l_{1 j}l_{j}
= l_{1 j}d_{i j}
.]
Note that any unitary (U^{†} = U^{-1}) matrix can be expressed as (iH)^{↑} º
e^{iH} 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 C_{K×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 ïl_{1} l_{2} ..ñ
is | ál_{1} l_{2} ..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 y_{p}
with a 1-D complex-cordinate 1-vector representation
defined over pÎB_{ase} .
As a 1-D complex 1-vector, y_{p} can be regarded as a complex scalar field y_{p} over B_{ase}, our primary idempotent
u is simply the real scalar 1.
Since, at a given p, y_{p} is "impervious" to multiplication by "complex numbers", all nonzero y_{p}
represent the same state at p. Thus there are only two distinguishable states at a given p, characterised by
y_{p}=0 and y_{p}¹0. 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 y_{p} (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 B_{ase}.
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 y_{p} 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" p_{0} we collapse the y_{p} waveform
to a ket satisfying y_{p0} = 1.
Geometrically, we can consider the "complex" ket wavefunction to be y_{p} = r_{p}(-hq_{p})^{↑} = r_{p}(iq_{p})^{↑} in natural units , for scalars r_{p} and q_{p}, with i=e_{12345} taking the i role. y_{p}y_{p}^{»} = r_{p}^{2}. The generalised spacial momentum is provided by m_{p} = Ñ_{[e123]}q_{p} and q_{p} is known as the phase. Continuity assumptions in y_{p} mean that the phase q_{p} must vary continuously except over nodal K-curves over which r_{p}=0
Such complex wavefunctions frequently satsify (to good approximation) Schrodinger's equation
Ð_{e4ß}y_{p}
= (½m^{-1}hÑ_{p}_{[e123]}^{2} + h^{-1}f(p))y_{p}
[ ie.
Ð_{e4ß}y_{p} = (-½m^{-1}Ñ_{p}_{[e123]}^{2} + f(p))^{*}y_{p} in natural units ]
for real scalar potential f(p) .
Bohm Quantum Potential
Bohm inserts complex y_{p} = r_{p}(-q_{p}h^{-1})^{↑}
into the nonrelativistic Schrodinger equation to obtain
a "conservation equation"
¶r_{p}^{2}/¶t + Ñ_{[e123]}¿(r_{p}^{2}(Ñ_{[e123]}q)m^{-1})=0 ;
and a modified Hamilton-Jacobi equation
¶q_{p}/¶t + ½m^{-1}(Ñ_{[e123]}q_{p})^{2} + V_{p}
+ Q_{p} = 0
[ Holland 3.2.17 ]
where real scalar quantum 0-potential
Q_{p} = ½m^{-1}h^{2} (Ñ_{[e123]}^{2}r_{p})r_{p}^{-1}
is dependant only on r_{p} in a way independant of the magnitude of r_{p}
(ie. |Q_{p}| can be large for small |r_{p}|)
and so is highly nonlocalised.
The phase q_{p} also provides the action, with orbit momentum 1-vector m_{p} = Ñ_{p} q_{p}
independant of probability wieghting r_{p}.
More generally we have a quantum 1-potential a_{p} = a_{p}(q_{p}, Ñ_{p}q_{p}, ...) 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+a_{p}) where a_{p} is a 3-vector with
a_{p}^{2}=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 | 1 | 0 | ö ; | ↓ | = | æ | 0 | 0 | ö ; | y | = | æ | z_{0} | 0 | ö ; | y^{»} | = | æ | z_{0}^{^} | z_{1}^{^} | ö for complex z_{0}=r_{0}e^{q0i} , z_{1}=r_{1}e^{q1i} | |
è | 0 | 0 | ø | è | 1 | 0 | ø | è | z_{1} | 0 | ø | è | 0 | 0 | ø |
↑ and ↓ can be regarded as eigenkets of eigenvalue 1 and -1 respectively for the linear operator | s_{3} | º | æ | 1 | 0 | ö . |
è | 0 | -1 | ø |
y^{»}y | = | æ | z_{0}^{^}z_{0}+z_{1}^{^}z_{1} | 0 | ö | = | æ | r_{0}^{2}+r_{1}^{2} | 0 | ö | = | æ | 1 | 0 | ö |
è | 0 | 0 | ø | è | 0 | 0 | ø | è | 0 | 0 | ø |
ab | a |
which we can summarise as s_{j} s_{k}= e_{ijk}i s_{i}
for distinct ijk ; s_{i}^{2}=1 where "imaginary scalar" i
commutes with the s_{i} and has i^{2} =-1. Thus the s_{i} anticommute with the product of any two being
the dual of the third , signed cyclicly.
Note that s_{1} s_{2} s_{3}=i s_{3}^{2} = i 1. Physicists often write a.s = a_{1} s_{1}+ a_{2} s_{2}+ a_{3} s_{3} 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} = a^{2}1 and hence (i(a.s)^{↑} = cos(|a|) + i(a.s) Sin()(|a|) | ||||
1 | s_{1} | s_{2} | s_{3} | |||
b | 1 | 1 | s_{1} | s_{2} | s_{3} | |
s_{1} | s_{1} | 1 | -i s_{3} | +i s_{2} | ||
s_{2} | s_{2} | +i s_{3} | 1 | -i s_{1} | ||
s_{3} | s_{3} | -i s_{2} | +i s_{1} | 1 |
Such a basis is provided by Pauli matrices
1 | = | æ | 1 | 0 | ö | ; s_{1} | = | æ | 0 | 1 | ö | ; s_{2} | = | æ | 0 | -i | ö | ; s_{3} | = | æ | 1 | 0 | ö |
è | 0 | 1 | ø | è | 1 | 0 | ø | è | i | 0 | ø | è | 0 | -1 | ø |
æ | a_{11} | a_{12} | ö | = ½(a_{11}+a_{22})1 + ½(a_{12}+a_{21}) s_{1} + ½i(a_{12}-a_{21}) s_{2} + ½(a_{11}-a_{22}) s_{3} |
è | a_{21} | a_{22} | ø |
Thus | æ | a_{11} | a_{12} | ö | = ½(a_{11}+a_{22}) + ½(a_{12}+a_{21})e_{14} - ½(a_{12}-a_{21})e_{13} + ½(a_{11}-a_{22})e_{24} |
è | a_{21} | a_{22} | ø |
The (non-normalised) pensity matrix yy^{»} is given by singular Hermitian matrix
yy^{»} | = | æ | z_{0}z_{0}^{^} | z_{0}z_{1}^{^} | ö | = | æ | r_{0}^{2} | r_{0}r_{1}e^{(q0-q1)i} | ö | = | r_{0}^{2} | æ | 1 | re^{-fi} | ö |
è | z_{0}^{^}z_{1} | z_{1}z_{1}^{^} | ø | è | r_{0}r_{1}e^{(q1-q0)i} | r_{1}^{2} | ø | è | re^{fi} | r^{2} | ø |
In a nonrelativistic Â_{3,1} model with ^{»} = ^{†} defined with regard to a given e_{4} we
have
y
= ½(z_{0} + z_{1} s_{1})(1+ s_{3})
= ½r_{0}e^{q0i}(1 + r(fi)^{↑} s_{1})(1+ s_{3})
= ½r_{0}e^{q0e1234}(1 + r(fe_{1234})^{↑}e_{14})(1+e_{34})
y^{»} = ½(1+ s_{3})(z_{0}^{^} + z_{1}^{^} s_{1})
= ½r_{0}e^{-q0i}(1+ s_{3})(1 + re^{-fi} s_{1})
= y^{†} .
And so y^{»}y = r_{0}^{2}(1 + r^{2})½(1+ s_{3}) .
[ Proof : Note first that (1+ s_{3})v(1+ s_{3})=(1+ s_{3})iv(1+ s_{3})=0 for any matrix v
anticommuting with s_{3}.
y^{»}y =
r_{0}^{2}½(1+ s_{3})(1 + re^{-fi} s_{1})
(1 + re^{fi} s_{1})½(1+ s_{3})
= r_{0}^{2}½(1+ s_{3})(1 + r^{2})½(1+ s_{3})
= r_{0}^{2}(1 + r^{2})½(1+ s_{3}) .
.]
The normalisastion condition z_{0}^{2}+z_{1}^{2}=1 corresponds to r_{0}^{2}(1+r^{2})=1 ,
which remains true for z_{0}=0,r=¥ if we consider (1+¥^{2})^{-½} = 0 .
A unit inner product has representative a_{11}=1,a_{12}=a_{21}=a_{22}=0 corresponding to Â_{3,1+}
multivector ½(1+e_{34}) so the correct scaling
for a normalised qubit ket of zero phase angle is given by
y^{~} = (1+r^{2})^{-½}(1 + re^{fi}e_{14})½(1+e_{34})
= (1+r^{2})^{-½}(1 + re^{fi}e_{13})½(1+e_{34})
and the normalised pensity by
y^{!} = y^{~}y^{~}^{»} =
½(1+Riem(re^{fi})e_{4})
= ½(1+w_{p}e_{4})
where spacial unit 1-vector
w_{p} = Riem(re^{fi})
º e^{fe12} e^{-2( tan-1(r))e31} e_{3}
= (r^{2}+1)^{-1}( e_{1} 2r cosf + e_{2} 2r sinf + e_{3} (1-r^{2}) )
is the
Riemann sphere representation
of complex number z = re^{fi} = z_{1}z_{0}^{-1}
and anticommutes with e_{4}.
[ Proof :
y^{!} º y^{~}y^{~}^{»} = (1+r^{2})^{-1}
(1 + re^{fi}e_{14})(½(1+e_{34}))^{2}(1 + re^{-fi}e_{14})
= (1+r^{2})^{-1} (1 + re^{fi}e_{14})½(1+e_{34})(1 + re^{-fi}e_{14})
= ½(1+r^{2})^{-1}
((1 + re^{fi}e_{14})(1 + re^{-fi}e_{14})
+(1 + re^{fi}e_{14})(1 - re^{-fi}e_{14})e_{34})
= ½(1+r^{2})^{-1}
((1 + r^{2} +2r cosfe_{14})
+(1 - r^{2} +2r sinfie_{14})e_{34})
= ½(1+r^{2})^{-1}
(1 + r^{2}
+ 2r cosfe_{14}
+ 2r sinfe_{24}
+(1 - r^{2})e_{34})
= ½(
1 + (1+r^{2})^{-1}(
2r cosfe_{1}
+ 2r sinfe_{2}
+(1 - r^{2})e_{3}))e_{4}
.]
Since
y^{~}y^{~}^{»} =
y^{~}(½(1+ s_{3}))^{2}y^{~}^{»} =
½(1 + y^{~} s_{3}y^{~}^{»})
= ½(1 + y^{~}e_{3}y^{~}^{»})e_{4}
we have w_{p} = y^{~}e_{3}y^{~}^{»} .
Letting y^{~}^{%} =
y^{~} = (1+r^{-2})^{-½}(1 + r^{-1}((f+p)i)^{↑}e_{13})½(1+e_{34})
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+e_{34})(1+r^{-2})^{-½}(1 + r^{-1}(-(f+p)i)^{↑}e_{14}^{»})
(1+r^{2})^{-½}(1 + r(fi)^{↑}e_{14})½(1+e_{34})
= ½(1+e_{34})(1+r^{-2})^{-½}(1+r^{2})^{-½}
(1 + (-pi)^{↑}e_{14}^{2} + _{O}(e_{14}))½(1+e_{34})
= 0
.]
Restoring our _{p} suffix, we therefore have
y^{!}_{p} = r_{p}^{2}½(w_{p}_{y}-e_{4})e_{4}
as the non-normalised pensity for the state y , eigenpensity of the multiplicative operator
w_{p}e_{4} with eigenvalue 1.
We interpret r_{p}^{2} as the classical probability of the "particle" being "at" p, and w_{p}
as its spin if it is indeed there. y^{!}_{p} does not encode a velocity or a momentum, however.
Probability gradient Ñ_{p}r_{p}^{2} , for example, need not be timelike.
The "opposite spin" state has pensity r_{p}^{2}½(-w_{p}-e_{4})e_{4} and eigenvalue -1.
w_{p}_{y} = y_{p}^{~}e_{3}y_{p}^{~}^{»}
= Riem(r_{py}e^{fpyi})
is the (e_{4} specific)
spacial unit spin 1-vector
3-urbservable .
We can thus tabulate the following correspondances between matrix and multivector representations.
Â_{3,1} ^{»} = ^{†} model . Replace e_{4} with e_{45} for Â_{4,1} ^{»} = ^{§}^{#} model | |||||||||||||
Ket Symbol | Ket Matrix | Ket Multivector | r,f | Eigenvalue | Pensity Matrix | Pensity Multivector | |||||||
ï↑ñ
(the primary idempotent) | æ | 1 | 0 | ö | ½(1+e_{34}) | 0,any | 1 for e_{34} | æ | 1 | 0 | ö | ½(1+e_{34}) | 1 |
è | 0 | 0 | ø | è | 0 | 0 | ø | ||||||
ï↓ñ | æ | 0 | 0 | ö | e_{14}½(1+e_{34}) | ¥,any | -1 for e_{34} | æ | 0 | 0 | ö | ½(1-e_{34}) | |
è | 1 | 0 | ø | =e_{13}½(1+e_{34}) | è | 0 | 1 | ø | |||||
ï→ñ=2^{-½}(ï↑ñ+ï↓ñ)
(the standard ket) | æ | 2^{-½} | 0 | ö | 2^{-½}(1+e_{14})½(1+e_{34}) | 1,0 | 1 for e_{14} | æ | ½ | ½ | ö | ½(1+e_{14}) | |
è | 2^{-½} | 0 | ø | =2^{-½}(1+e_{13})½(1+e_{34}) | è | ½ | ½ | ø | |||||
ï←ñ=2^{-½}(ï↑ñ-ï↓ñ) | æ | 2^{-½} | 0 | ö | 2^{-½}(1-e_{14})½(1+e_{34}) | 1,p | -1 for e_{14} | æ | ½ | -½ | ö | ½(1-e_{14}) | |
è | -2^{-½} | 0 | ø | =2^{-½}(1-e_{13})½(1+e_{34}) | è | -½ | ½ | ø | |||||
ï´ñ=2^{-½}(ï↑ñ+iï↓ñ) | æ | 2^{-½} | 0 | ö | 2^{-½}(1+e_{24})½(1+e_{34}) | 1,½p | 1 for e_{24} | æ | ½ | -½i | ö | ½(1+e_{24}) | |
è | 2^{-½}i | 0 | ø | =2^{-½}(1+e_{23})½(1+e_{34}) | è | ½i | ½ | ø | |||||
ï·ñ=2^{-½}(ï↑ñ-iï↓ñ) | æ | 2^{-½} | 0 | ö | 2^{-½}(1-e_{24})½(1+e_{34}) | 1,-½p | -1 for e_{24} | æ | ½ | ½i | ö | ½(1-e_{24}) | |
è | -2^{-½}i | 0 | ø | =2^{-½}(1-e_{23})½(1+e_{34}) | è | -½i | ½ | ø |
Using ½(1+e_{34})=e_{34}½(1+e_{34}) we can factor the ket representors in the form R½(1+e_{34}) where R in Â_{3,1+} has RR^{§}=1. We can replace every occurance of e_{12} in the factor with i=e_{1234} and any nonspacial bivector can be converted to a scalar and a spacial bivector. This provides the necessary reduction of 8-dimensional Â_{3,1 +} multivectors into 4 dimensional qubits.
It is easy to verify using either the matrix or multivector representations, for example, that
á↑ï s_{3}ï↑ñ = ½(1+e_{34}) , and that
á↑ï s_{1}ï↑ñ
= á↑ï s_{2}ï↑ñ = 0 ; corresponding to expected values
of 1 for s_{3}=e_{34} observations, and 0 for s_{1}=e_{14} and s_{2}=e_{24} observations of state ↑.
Since
á↑ïe_{14}^{2}ï↑ñ
= á↑ïe_{24}^{2}ï↑ñ
= á↑ïe_{34}^{2}ï↑ñ
= á↑ïï↑ñ
= 1
the dispersion
of e_{34} observations in state ↑ is 0 (always get +1) while the dispersion of e_{14} and e_{24} observations is 1
(always get either +1 or -1 with 50-50 probablities).
The classical probablity of c®y follows immediately as
P_{robabilty}( (w_{p}_{y}e_{4})^{?}(c) = y ) =
½(1+w_{p}_{y}¿w_{p}_{c}) =
( cos(½q))^{2}
where q is the angle subtended by 1-vectors w_{p}_{y}
and w_{p}_{c} .
[ Proof :
The pensity scalar product is
¼ ((w_{p}_{y}-e_{4})e_{4})_{*}((w_{p}_{c}-e_{4})e_{4})
= ¼ ((w_{p}_{y}-e_{4})((-w_{p}_{c}-e_{4})e_{4}^{2})_{<0>}
= ¼(1+w_{p}_{y}¿w_{p}_{c})
while that for the -1 eignevalue eigenket is
¼(1-w_{p}_{y}¿w_{p}_{c}) .
Dividing the first by the sum of both gives the result. .]
The Dirac inner product
y^{~}^{»}c^{~} =
(1+r_{1}^{2})^{-½}(1+r_{2}^{2})^{-½}(1 + r_{1}r_{2}e^{(q2-q1)i})
½(1+ s_{3})
so the complex conjugate ^{^} of inner products is again
provided by ^{†} for Â_{3,1 +} .
[ Proof : (½(1+r_{1}^{2})^{-½}(1 + r_{1}e^{q1i} s_{1})½(1+ s_{3}))^{§}
(1+r_{2}^{2})^{-½}(1 + r_{2}e^{q2i} s_{1})½(1+ s_{3})
= ¼^{-1}(1+r_{1}^{2})^{-½}(1+r_{2}^{2})^{-½}
(1+ s_{3})(1 + r_{1}e^{-q1i} s_{1})(1 + r_{2}e^{q2i} s_{1})(1+ s_{3})
= (1+r_{1}^{2})^{-½}(1+r_{2}^{2})^{-½}
½(1+ s_{3})(1 + r_{1}r_{2}e^{(q2-q1)i}e_{1})½(1+ s_{3})
and s_{3}=e_{34} commutes with both e_{1} and ie_{1}
.]
There is a tendancy in much of the literature to "strip idempotents" from ket representors.
Moving from an ideal space Â_{3,1}u
into a subspace of Â_{3,1} in which elements in the left annihilation
{ a : au=0 } of u can be considered "irrelevant" .
Moving into Â_{4.1}
The ^{»} = ^{†} , y_{p}ÎÂ_{3,1+} model serves for modelling
non-relativistic qubits
but the above table and all subsequent discussion can be adapted for our
^{»} = ^{§}^{#},
Â_{4,1} trivector model by
associating s_{i} with trivector e_{i45} and i with e_{12345}, and replacing
e_{i4} by e_{i45} throughout the discussion.
[ Proof :
For i¹j s_{i} s_{j} = e_{i}e_{45}e_{j}e_{45}
= e_{i4}e_{j4} =
= e_{ij} = e_{ijk}e_{k}
= e_{ijk4}e_{k4}
= e_{ijk45}e_{k45}
= e_{ijk}e_{12345} s_{k}
and so forth,
.]
The general qubit ket is then
½r_{0}(q_{0}e_{12345})^{↑}(1 + r(fe_{12345})^{↑}e_{145})(1+e_{345})
= ½r_{0}(q_{0}e_{12345})^{↑}(1 + r(fe_{12345})^{↑}e_{13})(1+e_{345})
.
Dropping the arbitary phase factor we have normalised ket
w = (1+r^{2})^{-½}(1 + r(fe_{12345})^{↑}e_{145})½(1+e_{345})
= S½(1+e_{345})
where
S º (1+r^{2})^{-½}(e_{3} + r cos(f)e_{1} + r sin(f)e_{2})e_{3}
is a unit spacial 2-versor in e_{123} with S^{»} = S^{#}^{§} = S^{§} = S^{-1} ,
S^{2} = (1-r^{2})(1+r^{2})^{-1} .
The normalised pensity w^{!} = ½(1+Riem(r(fi)^{↑})e_{45}).
With regard to a given e_{4}, a qubit pensity is thus parameterised by a unit spacial 1-vector spin w_{p}Îe_{123}
and is the <0;3>-multivector ½(1+w_{p}e_{45}) as anticipated.
In particular we have
ï↑ñ = ½(1+e_{345}) with pensity ½(1+e_{345}) ,
ï↓ñ = ½e_{13}(1+e_{345}) with pensity ½(1-e_{345}) , the eigenstates of e_{345};
ï→ñ and ï←ñ = 2^{-½}(1±e_{13})½(1+e_{345}) with pensities
½(1±e_{145}) , the eignestates of e_{145};
ï´ñ and ï·ñ = 2^{-½}(1±e_{23})½(1+e_{345}) with pensities
½(1±e_{245}), the eignestates of e_{245}.
w^{%} = S^{%}½(1+e_{345}) provides a ket orthogonal to w where S^{%} º (1+r^{-2})^{-½}(e_{3} - r^{-1} cos(f)e_{1} - r^{-1} sin(f)e_{2})e_{3} 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±e_{345}) as representating spacial 2-blade e_{12} together a boolean sign or orienation, ie. as a twist flag. Both have aa^{§}=aa^{#}=0 and a^{2}=a^{»}=a . If we consider e_{4}=e_{m} and e_{5}=e_{+} as generalished homogenous extenders of Â_{3} then e_{345}=e_{¥0}e_{3} represents a 1-plane (line) through 0 with direction e_{3}.
The component (1+e_{345})_{=} = (1+e_{345})_{»} of y^{»}_{»} annihilates all
odd blades in e_{1234}.
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 y_{p1,p2,...,pK} mapping KN dimensional points onto a potentially
2^{KN} dimensional product space
U_{KN} = Â_{p,q,r}^{K}
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
» 2^{266} > 2^{28}
, the number of
"particles" in the universe, is vast, and 2^{5K} > 2^{2268}
is
incomprehensibly vast.
[
2^{[k]} º 2^{2[k-1]} with 2^{[0]}º1
so that 2^{[1]}=2 ; 2^{[2]}=2^{2}
2^{[3]}=2^{22}
]
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,p_{1},p_{2},...p_{K})=r(t,p_{1},p_{2},...p_{K})(iq(t,p_{1},p_{2},...p_{k}))^{↑}
following Shrodinger's multiparticle equation
Ð_{e4ß}y = (½m^{-1}hå_{j=1}^{K} Ñ_{pj}^{2} + h^{-1}f(t,p_{1},p_{2},...,p_{K}) )y ;
where p_{1},p_{2},..,p_{K} are the 3D spacioal positions of the K particles at universal time t.
The nonnegative amplitude
|y(t,p_{1},p_{2},...p_{K})|_{+}^{2} is interpreted as the real probability density for particle one being at p_{1}, particle two being at p_{2}, 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,p_{1},p_{2},...,p_{K})^{2} = |y(t,p_{1},p_{2},...p_{K})|_{»}^{2} as the probability density function for the position of K particles
and m_{E}(t)^{[j]}(t,p_{1},p_{2},...,p_{K}) = Ñ_{pj} q_{p}(t,p_{1},...p_{j},..,p_{K}) (evaluated at t,p_{1},p_{2},...,p_{K})
as the momentum of the j^{th} particle if the partciles have positions p_{1},p_{2},...p_{K}
at time t, or we can extend the parameter space of y to
y(t,p_{1},S_{1},p_{2},S_{2},...,p_{K},S_{K}) where S_{1},S_{2},.. 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 S_{i} coordinates. If we are not worried about the momenta of the particles then
each S_{i} might reduce to two possible discrete spinstates Ù or Ú.
Alternatively we assume
y^{!}_{p1,p2,...,pK}^{2} = r_{p1,p2,...pK}y^{!}_{p1,p2,...,pK}
where y^{!} is locally normalised
y^{!}_{p1,p2,...,pK}»
y^{!}_{p1,p2,...,pK}=1 " p_{1},p_{2},..,p_{k} and real scalar
r_{p1,p2,...pK}³0 is globally normalised and interpretable as the classical probability density for
particle 1 being at event p_{1} and particle 2 at event p_{2} and so on.
For r_{p1,p2,...pK}>0,
the geometric content of
y^{!}_{p1,p2,...,pK}^{~} =
r_{p1,p2,...pK}^{-½}y^{!}_{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 U_{N}^{[j]} but suffer
the anticommution i^{[j]}i^{[k]} = -i^{[k]}i^{[j]} . The i^{[j]}
can be made to commute by imposing e_{5}^{[i]} = e_{5}^{[1]} = e_{5} . 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 e_{5}.
This ensures that e_{12345}^{[i]} and e_{12345}^{[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 e_{1234}^{[j]} by means of a correlator we reduce the algebra
to 2^{N}(2^{N-1}-1)^{K-1} blades in an extended basis.
We say the particles are independant if y_{p1,p2,...pK} = y_{p1}^{[1]}y_{p2}^{[2]}...y_{pK}^{[K]} , more generally we have entangled systems which do not so factorise.
For identical fermions we have antisymmetric y (
y(t,p_{1},S_{1},p_{2},S_{2},...p_{K},S_{K})=
-y(t,p_{2},S_{2},p_{1},S_{1},...p_{K},S_{K}) )
and y
(changing sign if p_{1} and p_{2} are swapped and ^{[1]} and ^{[2]} are swapped)
while for identical bosons we have symmetric y
(y(t,p_{1},S_{1},p_{2},S_{2},...p_{K},S_{K})=y(t,p_{2},S_{2},p_{1},S_{1},...p_{K},S_{K})) 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 P_{1} and P_{2} here but the basic idea is that y changes sign
when any two distinct arguments are excahnged (_PP3 and P_{K} say) for fermions, and is immune to argument exhanges for bosons ]
Example: Hydrogen Molecule Ground State
Suppose we have two protons at locations P_{1} and P_{2} orbited in some unspecified way by two electrons at p_{1} and p_{2}.
Assuming P_{1} and P_{2} to be fixed, the ground state wavefunction for this model of a hydrogen molecule has the form
y(p_{1},p_{2}) = a(y_{1}(p_{1}-P_{1})y_{1}(p_{2}-P_{2}) + y_{1}(p_{2}-P_{1})y_{1}(p_{1}-P_{2}))
2^{-½}(↑^{[1]}↓^{[2]}-↓^{[1]}↑^{[2]})
where a is an arbitary nonzero constant and scalar y_{1}(p)=y_{1}(|p|)=y_{1}(r)=(pa_{0}^{3})^{-½}(-a_{0}^{-1}r)^{↑}
is the wavefunction for a spinless electron in the n=1 hydrogen atom ground state.
The complex wavefuntion for this is
y(p_{1},S_{1},p_{2},S_{1}) =
a(y_{1}(p_{1}-P_{1})y_{1}(p_{2}-P_{2}) + y_{1}(p_{2}-P_{1})y_{1}(p_{1}-P_{2}))
2^{-½}(
d_{S1,+1}d_{S2,-1}
- d_{S1,-1}d_{S2,+1})
with scalar S_{1} and S_{2} restricted to ±1.
The positional factor of this wavefunction is symmetric in p_{1} and p_{2}, 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 m_{1},m_{2},..m_{M} 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 n_{1} whereas Bob uses n_{2}.
Let ¦^{?}_{A}(n_{1},n_{2},m) denote the scalar spin observed by Alice if she observes the spin of a particle parameterised by m
about axis n_{1} given that Bob uses n_{2}. We know from experiment that ¦^{?}_{A}(n_{1},n_{2},m)=±1
and our assumption of locality provides that Bob's choice of n_{2} has no relevance to Alice's reading so we have
¦^{?}_{A}(n_{1},n_{2},m) = ¦^{?}_{A}(n_{1},m) . Similarly let
¦^{?}_{B}(n_{2},m) denote Bob's spin measurement along axis n_{2} .
[ Note that it is not strictly rigourous to speak of n_{1} and n_{2} 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 n_{1} and n_{2} can be defined in a coordinate frame common to both Alice and Bob and products such as n_{1}¿n_{2} 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(n_{1},n_{2})
º ò_{M}dm P(m) ¦^{?}_{A}(n_{1},m) ¦^{?}_{B}(n_{2},m)
= -ò_{M}dm P(m) ¦^{?}_{A}(n_{1},m) ¦^{?}_{A}(n_{2},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(n_{1},n_{3}) - C(n_{1},n_{2})| + C(n_{2},n_{3}) £ 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(n_{1},n_{3}) - C(n_{1},n_{2})
= ò_{M}dm P(m)¦^{?}_{A}(n_{1},m)
(¦^{?}_{A}(n_{2},m) - ¦^{?}_{A}(n_{3},m))
= ò_{M}dm P(m)
¦^{?}_{A}(n_{1},m)¦^{?}_{A}(n_{2},m)
(1-¦^{?}_{A}(n_{2},m)¦^{?}_{A}(n_{3},m)) since ¦^{?}_{A}(n_{2},m)^{2} = 1
Þ
|C(n_{1},n_{3}) - C(n_{1},n_{2})| =
|ò_{M}dm P(m)
¦^{?}_{A}(n_{1},m)¦^{?}_{A}(n_{2},m)
(1-¦^{?}_{A}(n_{2},m)¦^{?}_{A}(n_{3},m))|
£ ò_{M}dm
P(m) | ¦^{?}_{A}(n_{1},m)¦^{?}_{A}(n_{2},m) |
(1-¦^{?}_{A}(n_{2},m)¦^{?}_{A}(n_{3},m))
= ò_{M}dm
P(m) (1-¦^{?}_{A}(n_{2},m)¦^{?}_{A}(n_{3},m))
= 1 - C(n_{2},n_{3})
.]
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(n_{1},n_{2}) º ¦^{?}_{A}(n_{1},n_{2},1) ¦^{?}_{B}(n_{1},n_{2},1)
= - cosq where q is the angle subtended by n_{1} and n_{2}.
[ Proof :
Suppose Alice measures +1 for spin axis n_{1} with probability ½, collapsing Bobs particle to spin -n_{1}. Bob will then measure -1 spin for axis n_{2} with probability
( cos(½q))^{2} and +1 with probability 1-( cos(½q))^{2}=( sin(½q))^{2}
so the expected value of
C(n_{1},n_{2}) given that
¦^{?}_{A}(n_{1})=+1 is
( sin(½q))^{2}-( cos(½q))^{2} = - cos(q) .
It has the same value when ¦^{?}_{A}(n_{1})=-1 and the result follows.
.]
Suppose n_{1},n_{2}, and n_{3} are coplanar with n_{1} and n_{2} subtending 2q and n_{3} bisecting their exterior angle
and so subtending p-q with both n_{1} and n_{2}. Then
C(n_{1},n_{3})=C(n_{2},n_{3})= cosq while
C(n_{1},n_{2})=- cos2q .
For q < ½p we have
|C(n_{1},n_{3}) - C(n_{1},n_{2})| + C(n_{2},n_{3}) = 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 n_{1}
; and we typically think of the spin being forced to ±n_{1} by the n_{1}-directed obsservation.
It is more natural to regard Alice as measuring unit 2-blade spin ±n_{1}i_{3} where i_{3}=e_{123} is the unit spacial pseudoscalar
and hence 2-blade ¦^{?}_{A}(n_{1},m)=±n_{1}^{*} where ^{*} denotes duality in i_{3}, rather than a scalar observable.
Since ¦^{?}_{A}(n_{1},m)
and ¦^{?}_{B}(n_{2},m) no longer commute in general, it is natural to
consider a symmetrised geometric correlation
C(n_{1},n_{2})
º ò_{M}dm P(m) (¦^{?}_{A}(n_{1},m) ~ ¦^{?}_{B}(n_{2},m)) .
[ Where a~b º ½(ab+ba) as usual ]
Taking m to be an arbitarily scaled 3-blade m=mi_{3} we have
(n_{1}¿m)(n_{2}¿m) = m^{2}n_{1}i_{3}n_{2}i_{3} = -m^{2}n_{1}n_{2}
so that taking
¦^{?}_{A}(n_{1},m)=n_{1}¿m gives
C(n_{1},n_{2})
º ò_{M}
dm P(m) (¦^{?}_{A}(n_{1},m) ~ ¦^{?}_{B}(n_{2},m))
= -i_{3}ò_{M}|dm| P(m) m^{2}(n_{1}¿n_{2}) .
Taking m=±1 with equal probabilities gives
C(n_{1},n_{2})
= -i_{3}(n_{1}¿n_{2}) = - cos(q)i_{3} 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
i_{3}^{-1}ò_{M}dmP(m)(¦^{?}_{A}(n_{1},m) ~ ¦^{?}_{B}(n_{2},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^{-½}(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 = y_{1}^{[1]}y_{2}^{[2]}...y_{K}^{[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 s_{i}^{[1]} , s_{j}^{[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]}=e_{1234}^{[1]} and i^{[2]}=e_{1234}^{[2]} by means
of a further idempotent " correlator" geometric multiplier
½(1-i^{[1]}i^{[2]})=½(1-e_{1234}^{[1]}e_{1234}^{[2]}) so that our
Â_{3,1+}^{k} , ^{»} = ^{†} model
normalised 2-quregister is represented by
w = (1+r^{[1]}^{2}))^{-½}
(1+r^{[2]}^{2}))^{-½}
(1+r^{[1]}e^{f[1]i[1]} s_{1}^{[1]} )
(1+r^{[2]}e^{f[2]i[2]} s_{1}^{[2]} )
½(1+ s_{3}^{[1]})
½(1+ s_{3}^{[2]})
½(1-i^{[1]}i^{[2]})
= (1+r^{[1]}^{2}))^{-½}
(1+r^{[2]}^{2}))^{-½}
(1+r^{[1]}e^{f[1]i[1]}e_{14}^{[1]} )
(1+r^{[2]}e^{f[2]i[2]}e_{14}^{[2]} )
a_{12}
where
b º
½(1+e_{34}^{[1]})½(1+e_{34}^{[2]})½(1-e_{1234}^{[1]}e_{1234}^{[2]})
=
½(1+e_{34}^{[1]})½(1+e_{34}^{[2]})½(1-e_{12}^{[1]}e_{12}^{[2]})
is the product of three commuting idempotents
acting as a source or sink (and so also a converter between) of e_{34}^{[1]} and e_{34}^{[2]}, as a converter of e_{1234}^{[2]}
and e_{12}^{[2]} into e_{1234}^{[1]} s (or e_{12}^{[1]} s)
and satisfying b^{2}=b^{»}=b ; and bb^{§}=0 .
Multiple 5D Qubits
We can extend to an even multiparticle algebra Â_{3K+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 e_{5} but distinct e_{1},e_{2},e_{3} and e_{4}.
All spaces share the same scalar 1^{[i]}=1^{[1]}=1 and we correlate the i^{[i]}=e_{12345}^{[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 e_{1234}^{[i]} given that e_{5}^{[i]}=e_{5}^{[1]}.
c=(1-e_{1234}^{[1]}e_{1234}^{[2]}) (1+e_{1234}^{[1]}e_{1234}^{[3]})...(1+e_{1234}^{[1]}e_{1234}^{[K]}) = (1-i^{[1]}i^{[2]})(1-i^{[1]}i^{[3]})...(1-i^{[1]}i^{[K]}) has e_{5}^{[i]}c = e_{5}^{[1]}c but cannot be used to "correlate" the e_{5}^{[i]}. We must impose e_{5}^{[i]}=e_{5}^{[1]}=e_{5} to ensure commutativity of s_{i}^{[k]}=e_{i45}^{[k]} and s_{j}^{[m]}=e_{j45}^{[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}))^{-½}
(1+r^{[2]}^{2}))^{-½}
(1+r^{[1]}(f^{[1]}i^{[1]})^{↑}e_{145}^{[1]} )
(1+r^{[2]}(f^{[2]}i^{[2]})^{↑}e_{145}^{[2]} )b
where
b = ½(1+e_{345}^{[1]})½(1+e_{345}^{[2]})½(1-e_{12}^{[1]}e_{12}^{[2]})
= ½(1+e_{345}^{[1]})½(1+e_{345}^{[2]})½(1-e_{12345}^{[1]}e_{12345}^{[2]})
= ½(1+e_{345}^{[1]})½(1+e_{345}^{[2]})½(1-e_{1234}^{[1]}e_{1234}^{[2]})
( given e_{5}^{[i]} = e_{5}^{[1]} = e_{5} )
hs b^{2}=b^{»}=b^{#}=b and bb^{§}=0 and commutes with e_{5} .
be_{14}^{[i]}b
= be_{13}^{[i]}b
= be_{24}^{[i]}b
= be_{23}^{[i]}b
= 0
Fermionic Correlations
Anticorrelated Fermionic singlet ket
h º 2^{-½} (ï↑^{[1]}↓^{[2]}ñ -ï↓^{[1]}↑^{[2]}ñ )
= 2^{-½}(e_{13}^{[2]}-e_{13}^{[1]})b
has h^{»}h = h^{§}h = b
while h^{2} = 0 .
ï→^{[1]}←^{[2]}ñ
-ï←^{[1]}→^{[2]}ñ
= -h and
ï´^{[1]}·^{[2]}ñ
-ï·^{[1]}´^{[2]}ñ
= -e_{12345}^{[1]}h
= 2(e_{23}^{[1]}-e_{23}^{[2]})b
are easily verified.
[ Proof : ((1+e_{23}^{[1]})(1-e_{23}^{[2]})-
(1-e_{23}^{[1]})(1+e_{23}^{[2]}))b
= 2(e_{23}^{[1]}-e_{23}^{[2]})b
= 2(e_{245}^{[1]}-e_{245}^{[2]})b
= 2(e_{13}^{[1]}-e_{13}^{[2]})e_{12345}^{[1]}b
.]
h is a -1 eigenvalue eigenstate of s_{3}^{[1]} s_{3}^{[2]} = e_{345}^{[1]}e_{345}^{[2]} = e_{34}^{[1]}e_{34}^{[2]} [ Since e_{345}^{[1]} commutes with the e_{13}^{[2]} while negating the e_{13}^{[1]} before being absorbed by b while e_{345}^{[2]} negates the the e_{13}^{[2]} ] and also a -1 eigenvalue eignstate of s_{1} s_{1}.
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 e_{1234}^{[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 (e_{14}^{[2]}-e_{14}^{[1]})e_{5} factor means that e_{14}^{[1]}h = - e_{14}^{[2]}h .
We can send an e_{34}^{[1]} through
(e_{145}^{[2]}-e_{145}^{[1]}) negating one term, convert it to an e_{34}^{[2]} and send it back, negating the other,
whence e_{34}^{[1]}h = -e_{34}^{[2]}h . Similarly e_{12}^{[1]}h = -e_{12}^{[2]}h ,
while e_{24}^{[1]}h can be converted as e_{21}^{[1]}e_{14}^{[1]}h
= -e_{21}^{[1]}e_{14}^{[2]}y
= e_{14}^{[2]}e_{21}^{[2]}y
= -e_{24}^{[2]}y. Spacial bivector e_{ij}^{[1]}y can be converted as e_{i4}^{[1]}e_{j4}^{[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 Â_{1,3+}, ^{»} = ^{†} Multiparticle SpaceTime Algebra of
Doran et al
in which the relativistic fermionic singlet ket is
h = (e_{13}^{[1]}-e_{13}^{[2]})½(1-e_{12}^{[1]}e_{12}^{[2]})½(1-e_{0123}^{[1]}e_{0123}^{[2]})
_{[ GAfp 9.93 ]}
, which commutes with e_{4}^{[1]}e_{4}^{[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 + (fe_{12345})^{↑} sin(a)h
= S^{[1]}S^{[2]}( cos(a) + (fe_{12345})^{↑} sin(a)(e_{13}^{[2]}-e_{13}^{[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-e_{1234}^{[1]}e_{1234}^{[2]})(1+e_{345}^{[2]})S^{(2)}^{§}S^{(1)}^{[2]} (1+e_{345}^{[1]})h
Pensity h^{!} = (e_{13}^{[2]}-e_{13}^{[1]})_{§}(
(1+e_{34}^{[1]}e_{34}^{[2]})½(1-e_{1234}^{[1]}e_{1234}^{[2]})
)
with h^{!}^{2}=h^{!}^{§}=h^{!} and
e_{i}^{[1]}h^{!}
= e_{i}^{[2]}h^{!} e_{4}^{[1]}e_{4}^{[2]}
= e_{i}^{[2]} e_{4}^{[1]}e_{4}^{[2]} h^{!}
for i=1,2,3 .
[ Proof :
4(e_{13}^{[2]}-e_{13}^{[1]})b(e_{13}^{[2]}-e_{13}^{[1]})^{»}
= 4(e_{13}^{[2]}-e_{13}^{[1]})_{§}(b)
= 4(e_{14}^{[2]}-e_{14}^{[1]})_{§}(b) .
Any blade in b that commutes with e_{13}^{[1]} while anticommuting with e_{13}^{[2]}
(or vice versa) will be annihilated by (e_{13}^{[2]}-e_{13}^{[1]})_{§} , so within the pensity
y^{!} we can replace the
(1+e_{345}^{[1]})(1+e_{345}^{[2]}) factor in b with
(1+e_{345}^{[1]}e_{345}^{[2]})
=(1+e_{34}^{[1]}e_{34}^{[2]}) .
Now (1+e_{34}^{[1]}e_{34}^{[2]})½(1-e_{1234}^{[1]}e_{1234}^{[2]}) commutes with any
"balanced" blade of the form
e_{ij..l}^{[1]}
e_{ij..l}^{[2]}
so
e_{i}^{[1]}h^{!}
= e_{i}^{[1]}
e_{4}^{[2]}e_{4}^{[1]}
e_{4}^{[1]}e_{4}^{[2]}h^{!}
=
- e_{4}^{[2]}
e_{i4}^{[1]}
h^{!}e_{4}^{[1]}e_{4}^{[2]}
=
- e_{4}^{[2]}
e_{i4}^{§}^{[2]}
h^{!}e_{4}^{[1]}e_{4}^{[2]}
= e_{i}^{[2]}h^{!}e_{4}^{[1]}e_{4}^{[2]}
.]
Bosonic Correlations
Correlated Bosonic ket w º 2^{-½} (ï↑^{[1]}↑^{[2]}ñ -ï↓^{[1]}↓^{[2]}ñ ) = 2^{-½}(1-e_{14}^{[1]}e_{14}^{[2]})b = 2^{-½}(1-e_{13}^{[1]}e_{13}^{[2]})b has w^{»}w = h^{§}h = w^{2} = b . Like h, w "transplants" bivectors between ^{[1]} to ^{[2]} spaces but reverses (negates) only e_{14} and e_{23}.
w is a +1 eigenvalue of s_{3}^{[1]} s_{3}^{[2]} [ Since e_{345}^{[1]}e_{345}^{[2]} commutes with e_{14}^{[1]}e_{14}^{[2]} ] and a -1 eigenvalue of s_{1}^{[1]} s_{1}^{[2]}
Anticorrelated Bosonic singlet ket
z º 2^{-½}(ï↑^{[1]}↓^{[2]}ñ
+ ï↓^{[1]}↑^{[2]}ñ) =
2^{-½}(e_{13}^{[1]}+e_{13}^{[2]}))b
also has z^{»}z = z^{§}z = b and
bz = 0 .
It transplants bivectors, negating only e_{12} and e_{34}.
z is a +1 eigenvalue eigenstate of s_{1}^{[1]} s_{1}^{[2]} = e_{145}^{[1]}e_{145}^{[2]}
= e_{14}^{[1]}e_{14}^{[2]} and a -1 eignvalue eignstate of s_{3}^{[1]} s_{3}^{[2]}
Correlated Bosonic ket
m º 2^{-½}(ï↑^{[1]}↑^{[2]}ñ
+ ï↓^{[1]}↓^{[2]}ñ)
= 2^{-½}(1+e_{13}^{[1]}e_{13}^{[2]})b
= 2^{-½}(1+e_{14}^{[1]}e_{14}^{[2]})b
has m^{»}m = m^{§}m = m^{2} = b .
It transplants bivectors, negating e_{13} and e_{23}.
m is a +1 eigenvalue eigenstate of bothe of s_{1}^{[1]} s_{1}^{[2]}
and s_{3}^{[1]} s_{3}^{[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
ïb_{00}ñ ,
ïb_{10}ñ ,
ïb_{01}ñ , and
ïb_{11}ñ respectively in the literature.
Symbols used | Ket | Multivector Ket | s_{1}^{[1]} s_{1}^{[2]} eignvalue | s_{3}^{[1]} s_{3}^{[2]} eigenvalue | Nature | ||
m | ïF^{+}ñ | ïb_{00}ñ | 2^{-½}(ï↑^{[1]}↑^{[2]}ñ + ï↓^{[1]}↓^{[2]}ñ) | 2^{-½}(1+e_{14}^{[1]}e_{14}^{[2]})b | +1 | +1 | Bosonic |
w | ïF^{-}ñ | ïb_{10}ñ | 2^{-½} (ï↑^{[1]}↑^{[2]}ñ - ï↓^{[1]}↓^{[2]}ñ) | 2^{-½}(1-e_{13}^{[1]}e_{13}^{[2]})b | -1 | +1 | Bosoninc |
z | ïY^{+}ñ | ïb_{01}ñ | 2^{-½}(ï↑^{[1]}↓^{[2]}ñ + ï↓^{[1]}↑^{[2]}ñ) | 2^{-½}(e_{13}^{[1]}+e_{13}^{[2]}))b | +1 | -1 | Bosoninc |
h | ïY^{-}ñ | ïb_{11}ñ | 2^{-½} (ï↑^{[1]}↓^{[2]}ñ - ï↓^{[1]}↑^{[2]}ñ ) | 2^{-½}(e_{13}^{[2]}-e_{13}^{[1]})b | -1 | -1 | Fermionic |
Note that s_{1}^{[1]} s_{1}^{[2]} ïb_{ij}ñ = (-1)^{i}ïb_{ij}ñ
while
s_{3}^{[1]} s_{3}^{[2]} ïb_{ij}ñ = (-1)^{j}ïb_{ij}ñ 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^{-½}(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]} =
aï↑ñ^{[3]} + bï↓ñ^{[3]}
= (a+be_{13}^{[3]})(1+e_{345}^{[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]} s_{3}^{[2]}y^{[2]} +
+ z^{[1.3]} s_{1}^{[2]}y^{[2]} +
- h^{[1.3]}i s_{2}^{[2]}y^{[2]} )(1-i^{[1]}i^{[2]})
where
y^{[2]} º aï↑ñ^{[2]} + bï↓ñ^{[2]} ;
m^{[1.3]} º
2^{-½}(ï↑^{[1]}↑^{[3]}ñ + ï↓^{[1]}↓^{[3]}ñ)
= 2^{-½}(1+e_{14}^{[1]}e_{14}^{[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 s_{1}^{[1]} s_{1}^{[3]}
and then
s_{3}^{[1]} s_{3}^{[3]} she will collapse the three-qubit state into (a complex multiple of) one
of
m^{[1.3]}y^{[2]} ;
w^{[1.3]} s_{3}^{[2]}y^{[2]};
z^{[1.3]} s_{1}^{[2]}y^{[2]} ; or
-h^{[1.3]}i s_{2}^{[2]}y^{[2]}
(ie. a complex multiple of
h^{[1.3]} s_{2}^{[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 s_{1}^{[1]} s_{1}^{[3]} and
s_{3}^{[1]} s_{3}^{[2]} respectively. She then knows that the three-qubit system is now in state
z^{[1.3]} s_{1}^{[2]}y^{[2]} whcih means that Bob's qubit is no longer
entangled with hers and has aquired state
s_{1}^{[2]}y^{[2]} = bï↑ñ^{[2]} + aï↓ñ^{[2]}.
Alice then contacts Bob via an insecure classical channel and tells him to "use s_{1}" whereupon Bob aplies unitary transformation s_{1}
to his qubit driving it to s_{1}^{[2]}^{2}y^{[2]} = y^{[2]}.
Note that Bob is not making a s_{1}^{[2]} observation and collapsing his qubit
bï↑ñ^{[2]} + aï↓ñ^{[2]}
= 2^{-½}(ï←ñ^{[2]}(b-a) + ï→ñ^{[2]}(b+a))
into either ï←ñ^{[2]} or ï→ñ^{[2]} with probablilites
in proportion to the complex amplitudes a ± b.
Rather, he must use s_{1} (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.
Next : The Dirac Particle