III. THEORETICAL STRUCTURES: Mp[QMS] [III]

Theoretical structures of QM are

[III-1]mp = < X, T, p, h ¢ .«Å▸, |.|, H, V >

where non-theoretical structures

[III-2]mpp = < X, T, p, h ¢ .«Å▸ > Œ Mpp[QMS]

are emended with a finite dimensional inner product space H [IX.5.4.8], which plays the role ofthe Hilbert space in familiar formulations of quantum mechanics [R-2]. |.| [III.1 ] is a lattice mor-phism [IX.4.1] which embeds LW

X [II.2.4] into LPH [IX.5.4.9.3], the projection lattice for H, and V ÃU[H] [III.2 ], is a finite set of unitary operators on H [IX.5.4.8]. This, together with |LW

PX|, providesus with the apparatus to define QM-states [III.3].

III.1 |.|-EMBEDDING: |.| [III.1]

First, we define the set of |,|-embeddings [III.1.1 ]. Then we consider some notation pertain-ing to them [III.1.2]. Finally, we consider as operator assignments [III.1.3].

III.1.1 DEFINITION [III.1.1]

The projection lattice LPH [IX.5.4.9.3] for an n-dimensional inner product space H [IX.5.4.8] isan orthomodular, orthocomplemented lattice (OMOLAT) [IX.4.7.4]. Every such lattice is the past-ing § [IX.4.7.4.3] of the set of its maximal sub-lattices, |O-BLOCK| [IX.4.7.4.3.1]. We require that

[III-3]dim LPH = dim LWX = n

and that LWX be isomorphic to the pasting of some subset of the set of maximal sub-lattices of LPH

[III-4]B` H

à |LPH-BLOCK|, #B` H

= #|OWX |

equi-numerous with the number of X-observables. More precisely, the set of |,|-embeddings is

[III-5] |LWX Å▸LPH| := { |»| Œ IOMOLAT(LW

X , §B` H

) | B` H

à |LPH-BLOCK|, #B` H

= #|OWX |}.

[E-III-5]

It can be shown that, for all |,| Œ |LWX Å▸LPH|,

[III-6]"o Π|LWX |, element-dim.OMOLAT(o) = sub-spaceH-dim(|o|).

[E-III-6]

This entails that the lattice upper bounds correspond under the |.|-mapping, i. e.

[III-7]|VWX | = |I|.[E-III-7]

Ultimately, the theoretical structure of QM, M[QMS] [IV], requires that |,|-embeddingstogether with the inner product on H determine BT result-observation sequence probabilities[II.4.4.6.1], [IV.1.1]. Only those members of LPH which are |,|-images of members of |LW

X | have"empirical significance" provided by the BT result-observation sequence probabilities they deter-mine. Generally, there will be many |,|-embeddings of LW

X into LPH and each one determinesthese probabilities in a different way.

Mp.nb - 1 - 10-15-14 -- 9:22

Ultimately, the theoretical structure of QM, M[QMS] [IV], requires that |,|-embeddingstogether with the inner product on H determine BT result-observation sequence probabilities[II.4.4.6.1], [IV.1.1]. Only those members of LPH which are |,|-images of members of |LW

X | have"empirical significance" provided by the BT result-observation sequence probabilities they deter-mine. Generally, there will be many |,|-embeddings of LW

X into LPH and each one determinesthese probabilities in a different way.

III.1.2 NOTATION [III.1.2]

For all H = < V, X.\ > |VECCX\| [IX.5.3],

[III-8]|»| Œ |LWX Å▸LPV| o Œ |LW

X |,

we let

[III-9]|o| := |»|(o) Œ |LPV|

.[IX.5.4.9].

For ui Π|O`

.WX | [II.2.1],

[III-10]|ui| := { |ioi| | ioi Πui }

is a set of mutually orthogonal elementary projection operators, atoms of LPV. These correspondone-one to one-dimensional subspaces of V, U io

`i

[IX.5.4.9]. We denote generic normalizedvectors (Π|N| [IX.5.3.4]) in these sub-spaces by

[III-11]|ioi\. ΠU io

`i.

[E-III-11]

so that

[III-12]|ioi| = |ioi\Xioi|

and, for all o Πu Π|O.WX |, ioi Πui Π|O

`.WX |, ui 2X u,

[III-13]|o| = |ioi\Xioi| ( i sum, j Œ I à I@1,nD; no j sum) o = ‹iŒI ioi.

[E-III-13]

For ui, uj ΠΠ|O`

.WX |, inner products appear as

[III-14]Xioi| joj\.

The normalization requires that

[III-15]Xioi|ioi\ = 1.

For ioi, koi Œ ui, i ¹≠ k, |ioi\ and |koi\ are orthogonal, i. e.

[III-16]Xioi|koi\ = 0.

Note that,

[III-17]|VWX | = |O| |LW

X | = |I|.

.[IX.5.4.9.3]. Recall, the |,|-mapping is always into |LPH| since |LW

X | is finite and |LPH| infinite. We willdenote the |,|-image of |LW

X | in |LPH| by '|LWPX|' and note that it is the domain of a finite sub ortho-

complemented lattice of LPH

QM-DRAFT

Mp.nb - 2 - 10-15-14 -- 9:22

Recall, the |,|-mapping is always into |LPH| since |LWX | is finite and |LPH| infinite. We will

denote the |,|-image of |LWX | in |LPH| by '|LW

PX|' and note that it is the domain of a finite sub ortho-complemented lattice of LPH

[III-18]LWPX mOMOLAT LPH.

[E-III-18]

[III-19]

Members of |LWPX| are QM-states of X. But, they do not exhaust the set of QM-states of X [IV.1.2].

We will denote members of |LPV| that are not in |LWPX| by

[III-20]|LPH-X| Π|LPH| - |LWPX|.

Generic vectors in HX which need not be members of |LWEX| are denoted by |vX \.

The |,|-images of maximal observables ui are orthonormal bases

[III-21]Bi = {|{o i\}.{=1{=n Œ B[H],¶

[E-III-21]

where the ‘i’ index refers to the observable ui; the ‘{’ and ‘m’ refer to members of ui [III.1.2]. Theset of elementary projection operators corresponding to menbers of Bi is

[III-22]Vi = {|{o i\Xio {| |{o i\ Œ Bi }. à |LW

EX|[E-III-22]

.[IX.5.4.9.3]. The set of orthonormal bases that are images of maximal observables is

[III-23]E[H],¶ = { B = {|e {\}.{=1{=n Œ B[H],¶ | $ ui Œ |O

`.WX | ' |e {\ = |{o i\ }.

[E-III-23]

These are direct empirical bases.Non-elementary projection operators corresponding to non-maximal observables can be

expressed as sums of elementary projection operators as in

[III-24]|Loi| = ⁄iŒL |ioi\Xioi| L Õ I@1,nD.

Such sums areare projection operators because all summads are mutually orhtogonal. They arenot statistical operators because the sum of their "weights" doess not equal 1.

Sums of elementary projection operators corresponding to results of different maximal observ-ables, for example

[III-25] |ioi\iaiX

ioi| + | joj\jajX

jo j|,

are generaly not projection operators for H (Π|LPV|). Thyey may be statstical operators [III.3].Linear combinations of vectors corresponding to results of maximal observables such as

[III-26] a|ioi\ + b| joj\

correspond to sums of elementary projection operators via

[III-27] (aXioi| + b`Xjo j|) ( a|ioi\ + b| joj\)

QM-DRAFT

Mp.nb - 3 - 10-15-14 -- 9:22

[III-28]= |ioi\|a|.

2Xioi| + | joj\abXioi| + |ioi\b

`aXjo j| + | joi\|b|.

2Xjo j|.

QM generally avoids explicit use vectors like |ioi\ and their linear combinations, employinginstead the corresponding sum of elementary projection operators. Discussion of "interference"and "coherence" employ the corresponding sums of elementary projection operators.

III.1.3 |.|-OPERATORS [III.1.3]

In many discussions of quantum mechanics, what we present as the choice of a |.|-embed-ding is presented as the assignment of operators in H to "physical" observables. The physicalcontent of specific applications of quantum mechanics inheres in this operator assignment.

"It is not possible to solve a single physical problem with that formalism until one obtains corre-spondence rules that identify particular dynamical variables with particular operators.." ([R-2] p. 64).

For this reason it is useful to see how operator assignment appears in QM. Eigenvalues of opera-tors do not fall naturally out of the QM approach because QM-states are not associated withnumerical values. These can be added arbitrarily with a one-one function

[III-29]i Œ ISET(POT(X), I@1,ðPOT HXL).

Using this device, members of |LWX Å▸LPH| assign Hermitian operators to observables [III.1.3.1 ].

Functions of these operators are defined in the usual way [III.1.3.2 ]. We will subsequently [IV.4]relate these opertors to QM probabilities.

III.1.3.1 |.|-OPERATOR ASSIGNMENTS [III.1.3.1]

Observables u Œ |OWX | [II.2.1] are assigned Hermitian operators in H (members of |Hℌ[H]|

[IX.5.4.6]) in the following way. Consider

[III-30]|O| Œ SET( |LWX Å▸LPH| x |OW

X |, |Hℌ[H]|) := |Hℌ[H]|.WX

[E-III-30]

such that, for all |.| Œ |LWX Å▸LPH|, ui Œ |OW

X |,

[III-31]|Ou»»| := |O|(|.|, u)

is such that, for all uj Π|O`

.WX |, the spectral representation of |Ou

»»| in the orthonormal basis for H

[III-32]Bu` := { |koj

X\ }.i=1i=n ΠB[H].

[III-33]|Ou»»| = |koj

X\i(Ùji(koj

X))XjokX|

[E-III-33]

where Ùji is the super-set function [II.1.2.3] for uj and ui. Again we employ the usual convention

that repeated upper and lower indices 'i' indicate summation over i [IX.5.3.1].

Here |kojX\ is an eigenvector of linear operator |Ou

»»| with the eigenvalue i(Ùji(koj

X)). For uj Œ

|O`

.WX |, the eigenvalues are all distinct since

[III-34]Ùjj(koj

X) = kojX

and

QM-DRAFT

Mp.nb - 4 - 10-15-14 -- 9:22

[III-35]k ¹≠ { fi i(Ùji(koj

X)) ¹≠ i(Ùji({oj

X)).

Members of |O`

.WX | correspond to maximal compatible sets of observables. For u Π(|OW

X | - |O`

.WX |), it

may be the case that

[III-36]i(Ùji(koj

X)) = i(Ùji({oj

X)).

In this case the eigenvalues are "degenerate".

In the special case of u Π|O`

.WX |, [E-III-33] reduces to

[III-37]|O` u`»»| = |koj

X\i((kojX)Xjok

X|.[E-III-37]

III.1.3.2 |.|-OPERATOR FUNCTIONS [III.1.3.2]

Let

[III-38]F Œ SET( SET(I@0,I@0), |Hℌ[H]|.WX)

.[E-III-30] such that, for all

[III-39]F† = Œ SET(I@0,I@0), |Ou»»| Œ |Hℌ[H]|.W

X ,

[III-40] |F† (|Ou»»|)| := F(F† , |Ou

»»|) = |kojX\F† (i(Ùj

i(kojX)))Xjok

X|.

III.2 DYNAMIC OPERATORS: |D[H]| [III.2]

The set V porvides for the treatment of “time dependent” dynamic operators in QM. Norestriction are placed on the membership of V. Thus, the null operator |I| may, but need not, be amember of V.

An alternative, more austere, conception associates a single unitaty operator |U| with systemX. The unitary operator associated with system X is an intrinsic property of X determining it’sdynamic behavior as an isolated system. Changes in dynamic behavior of X must now beviewed as resulting from interaction with other systems. Thus, concatenation state sequencesare used to describe time dependent dynamic operators. This view reqiuires a more complicatedconception of single models for QM.

The set of mp-dynamic operators is

[III-41]|Dmp | := |LWPX| ‹ V.

Of interest is the action of these operators on QM-states [III.3 ], a subset of |W[H]|, statisticaloperators for H. This action is most conveniently described in terms of sequences of |W[H]|-functions [IX.5.4.13.6] acting on QM-states [III.3 ]. However, the set of |W[H]|-functions and theirsequences, resticted to those derived from |Dmp |, is needed to define QM-states.

We consider

[III-42]Pmp ΠSET(|LWPX| x SET(|W[H]|, |W[H]|) )

such that, for all |P| Π|LWPX|, |W| Π|W[H]|,

QM-DRAFT

Mp.nb - 5 - 10-15-14 -- 9:22

[III-43]Pmp(|P|, |W|) = |P|W|P`| / Tr |P|W|.

It can be shown that ([R-11], pp. 99-100) the factor (Tr |P|W|).

-1 "normalizes" |P|W|P`| so that

[III-44]Pmp(|P|, |W|) Π|W[H]|.

The notation here is redundant in that

[III-45]|P| = |P`|

so that

[III-46]Pmp(|P|)(|W|) = |P|W|P| / Tr |P|W|.

Intuitively, P(|P|) is a normalized version of |P|.Unitary operators are unproblematic. But, for conformity of notation, we define

[III-47]U ΠSET(|V]|, SET(|W[H]|, |W[H]|) )

such that, for all |U| Π|V|, |W| Π|W[H]|,

[III-48]U(|U|)(|W|) = |U|W|U`| Π|W[H]|.

For all m ΠI@1,, the set of m-|W[H]|-function sequences is

[III-49]LIST.

m[SET(|W[H]|, |W[H]|)].[E-III-49]

Of interest here are sequences of those functions that can be "derived" from mp-dynamicoperators

[III-50]|Dmp | := |LWPX| ‹ V.

via the P and U functions. To make this precise, we define mp-m-|W[H]|-dynamic functionsequences

[III-51]

D1... mmp (|W[H]) := { D1... m ΠLIST.

m[SET(|W[H]|, |W[H]|)] | t ΠI@1,mD, $ |P| Π|LW

PX| ' Dt = P(|P|) Ó $ |U| Œ |V[H]| ' Dt = U(|U|) }.

[E-III-51]Compositions of these functions are members of

[III-52]

|Dmmp[|W[H]|]| := { Dm

mp ΠSET(D1... mmp (|W[H]), |W[H]|) |

" D1... m ΠD1... mmp (|W[H]), |W| Π|W[H]|,

|D1... m(|W|)| := Dmmp( D1... m, |W|) ) =

D1(D2(...Dm-1(Dm(|W|)) }.[E-III-52]

The '| |' flanking 'D1... m(|W|)' serves to indicate that

Intuitively, |D1... m(|W|)| is just the result of composition of a sequence of m members ofD1... m

mp (|W[H]) acting on |W|.

QM-DRAFT

Mp.nb - 6 - 10-15-14 -- 9:22

Intuitively, |D1... m(|W|)| is just the result of composition of a sequence of m members ofD1... m

mp (|W[H]) acting on |W|. For «Å▸ Œ INT[I], #«Å▸ = m,

[III-53]|D«Å▸(|W|)| := |D1... m(|W|)| Œ |W[H]|.[E-III-53]

III.3 QM-STATES: |W[H]| [III.3]

OM-states are defined concepts for members of Mp[QMS]. They provide us with the fullapparatus required to formulate the fundamental law of QM [IV.1.1].

Roughly, |.|-images of atomic members of |LWX |, members |LW

EX| [IX.5.4.9.3], are analogous topure quantum mechanical states of physical systems. We will call them pure QM-states of X.Modern formulations of quantum mechanics ([R-2], [R-12]) take the set of quantum states to becoextensive with |W[H]| [IX.5.4.13]. Analogous to this approach, we consider QM-states, roughly,the set of all members of |W[H]| that that can be constructed from |LW

EX| and operations on themby members of D[W[H]] D@W@HDD.

It is convenient to break the definition of QM-states into two parts; first the subset of directQM-states [III.3.1 ], then the full set of QM-states [III.3.2 ]. Finally, we consider matrix representa-tions of QM-states [III.3.3].

III.3.1 DIRECT QM-STATES: |W[H]| [III.3.1]

The set of direct QM-states is

[III-54]|W[H]| := { |W

w| | |W

w| = | i oi\

iiwiiXi oi| (i, i sum),

w Œ SET(IB1,ð LWEX F,R@0, 1D), iiwii = 1, (i, i sum), | i oi\X

i oi| Π|LWEX| ) } .

[E-III-54]

whose members are “weighted sums” ("weights" iiwii) of elementary projection operators whichare |.|-images of atoms of LW

X . Intuitiely, direct generalized QM-states are statistical operators thatcan be constructed from elementary projection operators that are "empirically meaningful" in thesense that they are |.|-images of results of observables.

Note that the i, i sum in [E-III-54] runs over all elementary projection operators in |LWEX|.

Intuitively, |Ww

| Π|W[H]| represents a probability distribution over |LWEX|. In most cases of interest,

only a small number of these will have non-zero weights iiwii. It is convenient to have somenotaition for these. Thus, for all

[III-55]V Ã |LWEX|

[III-56]|WVw

[H]| := { |WVw

| | $ |Ww

| Œ |W[H]| ' iiwii > 0 ï | i oi\Xi oi| Œ V, |W

Vw| = |W

w| }

[E-III-56]

'|Ww

|' denotes some direct QM state in which all members of |LWEX| may have non-zero weights;

'|WVw

|' denotes some direct generalized QM-state in which all and only members of V have non-

zero weights. For most purposes, the more general |Ww

| suffices. |WVw

| is only necessary when oneneeds to be specific about V.

Direct QM-states are a proper subset of of |W[H]|,

QM-DRAFT

Mp.nb - 7 - 10-15-14 -- 9:22

[III-57]|W[H]| Ã |W[H]|,

and thus have the properties described in [IX.5.4.13]. The w-function of [E-III-54] is generally notunique [IX.5.4.13.4]. It may be the case that there exist w, w' satisfying [E-III-54] such that w ¹≠ w',

[III-58]|Ww

| = |Ww'

|.

Thus, we denote the generic member of |W[H]| by '|W|' when it is not necessary to make the w-function explicit.

III.3.2 QM-STATES: |W[H]| [III.3.2]

We enlarge the set of direct QM-states [E-III-54] to include statistical operators resulting fromthe operation of sequences of dynamic functions [III.2], [E-III-53] on direct QM-states so that

[III-59]|W[H]| := { |W| | $ «Å▸ Œ INT[I], #«Å▸ = m-2, D«- -Å▸ Œ D1... m

mp (|W[H]) , |WÅ▸| Œ |W[H]| '

|W| = D«- -Å▸(|WÅ▸|) }.[E-III-59]

Here

[III-60]D«- -Å▸(|WÅ▸|)) = D«-(D«--(...D--Å▸(D-Å▸(|WÅ▸|)).

.[E-III-53].The interval notation '«Å▸' , '«- -Å▸', is chosen for compatability with subsequent uses[IV.1.1.1].

It can be shown that

[III-61]|W[H]| Ã |W[H]|,

.[IX.5.4.13.5] but generally

[III-62]|W[H]| ç W[H]|.

There may be QM-states that are not direct QM-states.Let

[III-63]|W\ X

[H]| := |W[H]| › |W\ X

[H]|.

It can be shown that, for all |W| Π|W[H]|,

[III-64]|W| Π|W\ X

[H]| ï |W|.

2 = |W| ï Tr(|W|).

2 = 1

.[IX.5.4.13.4]. Further,

[III-65]|LWEX| Ã |W

\ X

[H]|.

But, it may be that

[III-66]|W\ X

[H]| - |LWEX| ¹≠ L.

There may be pure statistical operators resulting from the operation of a sequence of dynamicaloperators that are not elementary projection operators in |LW

EX|, i. e. images of atoms of LWPX.

QM-DRAFT

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There may be pure statistical operators resulting from the operation of a sequence of dynamicaloperators that are not elementary projection operators in |LW

EX|, i. e. images of atoms of LWPX.

Simply by virtue of being statistical operators, members of |W[H] may be expressed(generally in several ways) as the weighted sum of elementary projection operators in the man-ner of [E-III-54] However there is no assurance that the elementary projection operators in theexpression are members of |LW

EX|. It does not appear to be the case that the members of E(|W|)

.[IX.5.4.13.2] of interest to QM are just those in which all the terms in |Wvw

| are pure QM-states.Indeed, the motivation for enlarging the set of direct QM-states is to provide for things like Bellstates [ ] which are entangled states produced by unitary transformations on product states.

Statistical operators appear in the context of compound systems [V.3.3.3] in a way that isquite different from that described by [E-III-54] and [E-III-59]. Whether the set of statistical opera-tors |W[H]| should be enlarged to account for this (and if so, how) is matter about which there issome difference of opinion [R-7].

In accord with the customary quantum mechanial termnology for statistical operators [ ],

[III-67]

|W\ X

[H]| := |LWEX| ~ pure QM-states

|W[H]| - |W\ X

[H]| ~ mixed QM-states

|WVw

[H]| - |W\ X

[H]| ~ incoherent QM-superpositions of V.

In all cases, 'QM' indicates that only elementary projection operators that are |.|-images of someatomic members of W-X-observable-state lattice LW

X (members of |LWEX| ) appear as terms.

III.3.3 MATRIX REPRESENTATIONS OF QM-STATES [III.3.3]

Matricies for statistical operators may be viewed abstractly [III.3.3.1 ], from the perspective ofthe definition [D-IX-108] and explicitly from the perspective of incoherent superpositions of purestatistical operators [E-IX-400], [III.3.3.2].

III.3.3.1 ABSTRACT [III.3.3.1]

For maximal observable ui with corresponding direct empirical basis [E-III-23]

[III-68]Bi = {|{o i\}.{=1{=n Œ E[H],¶,

the matrix representation of |W| Π|W[H]| in the basis Bi, is .(Bi, |W|)., whose elements are

[III-69]{.(Bi, |W|).m = Xio {|W|mo i\ (no i, {, m sum)

where the ‘i’ index refers to the observable ui; the ‘{’ and ‘m’ refer to members of ui [III.1.2]. Thus,

[III-70]|W| = |mo i\{

.(Bi, |W|).mXio {| ({, m sum, no i sum)

.[E-IX-312] where the summation over repeated indicies is operator sum +† . In this case, the diago-nal elements {

.(Bi, |W|).{ have no special significance.

Is this true?

If {.(Bi, |W|).m is diagonal [IX.5.2.8] then

QM-DRAFT

Mp.nb - 9 - 10-15-14 -- 9:22

[III-71]|W| = |{o i\{

.(Bi, |W|).{Xio {| ({ sum, no i sum)

and |W| is an incoherent QM-superposition of Vi [E-III-22]. The {w{ may be viewed as a probabil-ity distribution over Vi.

Thus, the probability of {o i resulting from an observation of ui on a system in QM-state |W|is {

.(Bi, |W|).{ since this probability is 1 for an observation on the pure QM-state |{o i\Xio {| and this

state has probability {.(Bi, |W|).{. If one has a relative frequency interpretation of probability

and envisions |W| as representing an "ensemble" (set) of systems in diffferent pure QM-states, then {

.(Bi, |W|).{ is the fraction of the ensemble in QM-state |{o i\Xio {|. For this rea-

son, eigenvalues of |W| are sometimes called populations ([R-13] p.163). But, recall thereneed be no direct empirical basis in which |W| is diagonal. So this interpretation is limited in QM.

Note that there need be no direct empirical basis in which |W| is diagonal.## ## ##

Put in IV and clean up.

## ## ##

III.3.3.2 EXPLICIT [III.3.3.2]

For

[III-85]|Ww

| = | { oi\i{w{iX

i o{| (i, { sum) Π|W[H]|

.[E-III-54],

[III-86]Bi = {|{o i\}.{=1{=n, Bj = {|mo j\}.m=1

m=n Œ E[H],¶,

expressing everything in Bj,

[III-87]|{o i\ = Xjo m|{o i\|mo j\ X i o{| = Xjo m|{o i\Xj om| (m sum)

[III-88] |Ww

| = | { oi\i{w{iX

i o{| (i, { sum)

[III-89] = Xjo m|{o i\|mo j\ i{w{i Xjo m|{o i\Xj om| (i, j, { sum)

[III-90] = |mo j\ Xjo m|{o i\i{w{iX

jo m|{o i\ X j om| (i, j, { sum)

[III-91] = |mo j\ i{w{i|X i o{|mo j\|.

2 X j om| (i, j, { sum).

For fixed i and |mo j\, and all vectors normalized

[III-92]⁄{=1n |X i o{|mo j\|.

2 = 1

[III-93]i{w{i|X i o{|mo j\|.

2 = 1 (i, j, { sum).

Thus |Ww

| is an incoherent superposition of Vj.

Is this true? How does one show it?

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The matrix

[III-94](Bj, |Ww

|)

is such that

[III-95]n.(Bj, |W

w|).m = X j on|W

w|mo j\

[III-96] = X j on| { oi\

i{w{iXi o{|mo j\ (i, { sum)

and the diagonal elements are

[III-97]n.(Bj, |W

w|).n = X j on| { oi\

i{w{iXi o{|no j\

[III-98] = iiwii |X i o{|no j\|.

2.

For

[III-99]|WVw

| = | i oi\i{w{iX

i oi| | i oi\Xi oi| ΠV

.[E-III-56], i{w{i sums are restricted to V.For singleton

[III-100] V = {| { oi\Xi o{|}. Ã |LW

EX| ,

[III-101]n.(Bj, |W

Vw|).n . = |X i o{|no j\|.

2.

If |WVw

| is diagonal in Bi then

[III-102]|WVw

| = | n oi\n

.(Bi, |WVw

|).nXi on| (n sum)

[III-103] (i, {, j sum) i{w{i = n.(Bi, |W

Vw|).n = 1. (n sum).

but, there need be no other relation among the "weights", i{w{i (no sum) and the eigenvaluesn

.(Bi, |WVw

|).n (no sum). But, if

[III-104]V = Vi .,

it may (but need not) be that

[III-105] i{w{i = {.(Bi, |W

Vw|){ (no sum).

Note that there need be no direct empirical basis in which |Ww

| is diagonal.As in [III.3.3.1], the i{w{i may be viewed as a probability distribution over Vi.

## ## ##

III.3.3.2.1 OFF DIAGONAL ELEMENTS OF W IN BASIS B [III.3.3.2.1]

For X-statistical operator |W| Π|W[H]|, and orthonormal basis

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[III-106]B† = {|e† i\}.i=1

i=n ΠB[H],

[III-107] |W| = |e† j\Xie† |W|e† j\Xie

† |

[III-108]= |e† j\iW†jXie† |.

where

[III-109]iW†j = Xie† |W|e† j\.

are the B† matrix elements of W

† and the summation over repeated indicies is operator sum +† [ ].

For |e† k\ Œ B†,

[III-110]|koé i| = |e† k\Xke† | (no sum)

[III-111]

p(koé i «» × × × Å▸) = Tr |e† k\Xke† |W| = Tr |e† k\Xke

† |e† j\iW†jXie† |

= Tr |e† k\kW†kXke† |

= kW†k.

[E-III-111]

Thus, the diagonal elements kW†k (k Œ I@1,nD) are a probability distribution over results of a obser-

vation of the observable ui associated with B°. The are sometimes called populations ([R-13]

p.163), a term usually associated with a relative frequency interpretation of probability.

The off-diagonal elements iW†j play no role in determining p(koé i «» × × × Å▸). In all bases in which

sum of the diagonal elements are identical with kW†k (" k Œ I@1,nD) the probabilities [E-III-111] will

be identical. The observable ui is |.|-associated with the basis B

†. The preparation ··· Å▸ is associated with

|W|. Other preparations are associated with other statistical operators |Wé |. But, if

[III-112]kW†k = kWé

†k " k Œ I@1,nD,

the same probability distribution over results of a observation of the observable ui is determinedby [E-III-111]. The matricies W

† and Wé

† having identical diagonal elements will differ in their off-

diagonal elements. What do the off-diagonal elements tell us? Consider the example for n =2

[III-113]†v\ = 2-1ê2(|e† 1\ + exp(if)|e† 2\)

[III-114]|W| = †v\Xv| = 2-1(|e† 1\ + exp(if)|e† 2\)(X1e† | + exp(-if)X2e† | )

[III-115] = 2-1(|e† 1\X1e† | + exp(-if)|e† 1\X2e

† | + exp(if)|e† 2\X1e† | +

|e† 2\X2e† |)

[III-116]1W†1 = X1e† |v\Xv|e† 1\ = 1

2(X1e† |e† 1\X1e

† |e† 1\) = 12(1)

[III-117]1W†2 = X1e† |v\Xv|e† 2\ = 1

2(X1e† |e† 1\exp(-if)X2e† |e† 2\) = 1

2(exp(-if))

[III-118]2W†1 = X2e† |v\Xv|e† 1\ = 1

2(X2e† |e† 2\exp(if)X1e† |e† 1\) = 1

2(exp(if))

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Mp.nb - 12 - 10-15-14 -- 9:22

[III-119]2W†2 = X2e† |v\Xv|e† 2\ = 1

2(X2e† |e† 2\X2e

† |e† 2\) = 12(1)

[III-120]|W| = †v\Xv| = 12

1 exp H-ifLexp HifL 1

[III-121]|W| = †v\Xv| = 12

1 e-if

eif 1Generally

[III-122]|v\ = Xie† |v\|e° i\ Xv| = Xje† |Xv|e° j\

[III-123]|v\Xv| = Xie† |v\|e° i\Xje† |Xv|e° j\

[III-124]|W| = |v\Xv| = Xie† |v\|e° i\Xie† |v\Xv|e° j\Xje

† |Xv|e° j\

[III-125]= |e° i\Xie† |v\Xv|e° j\Xje

† |

[III-126] = |e° i\iaajXje† | where ia = Xie† |v\ aj = Xv|e° j\

[III-127]iW†j = Xie† |v\Xv|e° j\ = Xie† |e° i\Xie

† |v\Xv|e° j\Xje† |e° j\ = Xie† |v\Xv|e° j\ = iaaj

[III-128]iW†i = Xie† |v\Xv|e° i\ = |Xv|e° i\|.

2 (no sum)

For |e† k\ Œ B†,

[III-129]|W| = †v\Xv|

[III-130]= |e° i\Xie† |v\Xv|e° j\Xje

† |

[III-131]|koé k| = |e† k\Xke† | (no sum)

[III-132]

p(koé k «» × × × Å▸) = Tr |e† k\Xke

† |W| (no k sum)

= Tr |e† k\Xke† |e° i\Xie

† |v\Xv|e° j\Xje† | (no k sum)

= Tr Xke† |e° i\Xie† |v\Xv|e° j\Xje

† |e† k\ (no k sum) = Xke† |v\Xv|e° k\ (no k sum)

= |Xv|e° k\|.

2 = kW

†k. [E-III-111]

What about [III-133]|W| = †v\Xv| + †u\Xu| ?

[III-134]|W| = |e° i\Xie† | + |e° j\Xje

† | ?

[III-135]|W| = |ioi\Xioi` | + | joj\X

jo j` | ?

## ## ##

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III.3.3.2.2 INTERPRETATION [III.3.3.2.2]

SOME OF THIS MOVED TO III?

In QM, the QM-state of a system is represented by some |W| Π|W[H]|. Orthonormal bases

[III-141]Bi = {|ioi\}.i=1i=n Œ B[H],¶.

are assigned to maximal observables ui. The statistical (density) matrix relative to Bi, is

[III-142](Bi, |W|)

and i .(B, |W|).i is taken to be the probability of obtaining the result ioi associated with |ioi\Xioi| from

an observation of u on the QM-state associated with |W|. Thus, the diagonal elements iWi are aprobability distribution over possible results of a observation of the observable associated with Bi.

More preciely,

[III-143]p(ioé i « v`éi «' × × × Å▸) = Tr|ioi\X

ioi|W|.

This interpretation motivates defining the expectation function [III-144]P T Œ SET(|Hℌ[H]| x |W[H]|, |R|)

such that, for all |A| Œ |Hℌ[H]|, |W| Œ |W[H]|,

[III-145]PATW = Tr(|A|W|).

[III-146]PATW := X|A|, |W|\ := Tr(|A|W|) = `|A|W|t.

.[E-III-XXX]. Here we use the more cumbersome notation 'PATW' in preference the more common'XA\W' to avoid ambiguity in the usage of angle brackets 'X' and '\'.

In an orthonormal basis

[III-147]B = {|ei\}.i=1i=n ΠB[H]

in which |W| is diagonal

[III-148] PATW = i .(B, |W|).iXie|A|ei\ (i sum).

[III-149]= i .(B, |W|).iPAT|ei\Xie|.

If |A| is diagonal in B,

[III-150]PAT|ei\Xie| = iAi.

Thus, if both |A| and |W| are diagonal in B,

[III-151]PATW = iAii .(B, |W|).i (i sum).[E-III-151]

which, under the above interpretation of the i .(B, |W|).i as a probability distribution over B, is theusual "expectation value" of the eigenvalues of |A|. Note that |.|-operators in QM [III.1.3] need notbe diagonal in the same basis as an arbitrary |W| Π|W[H]|.

The variance function is[III-152]PP TT Œ SET(|Hℌ[H]| x |W[H]|, |R|)

QM-DRAFT

Mp.nb - 14 - 10-15-14 -- 9:22

[III-153]PPATTW := PA2TW - (PATW).

2

[III-154] = Tr(|A2|W|) - (Tr(|A|W|)).

2.

[III-155] = `|A2|W|t - `|A|W|t.

2

.[E-III-XXX].For

[III-156]|W| = |v\Xv| Xv|v\ = 1,

this becomes, by [E-III-XXX],

[III-157]XA\v = Tr(|A|v\Xv|) = Xv|A|v\,

[III-158]XXA\\v = Xv|A2|v\ - Xv|A|v\.

2.[E-III-158]

It can be shown that, for all |A|, |B| Œ Hℌ, |v\ Œ |V\|, Xv|v\ = 1,

[III-159]2XXA\\vXXB\\v ³≥ Xv|[A ˝ B]|v\.[E-III-159]

III.3.4 EXAMPLE [III.3.4]

The point of the example is that incoherent superpositions of Vi have diagonal matricies inbasis Bi. Coherent superpositions have matricies with non-zero off-diagonalel elements. Is thisgeneralizable? 'oherent superposition' needs to be defined. Is this generalizable?

A coherent superposition of basis Bi is any vector v ù Bi expressed in Bi.

III.3.4.1 dim H = 2 [III.3.4.1]

For dim H = 2, we consider direct empirical basis

[III-160]Bi = { |{oi\ }.{=1{=2 = { |1oi\, |2oi\ },

and

[III-161]|W| = 12( |1o i\X

io 1| + |2o i\Xio 2| ).

The matrix elements in Bi are

[III-162]{.(Bi, |W|).m = Xio {|W|mo i\.

Explicitly,

[III-163]1.(Bi, |W|).1 = Xio 1|W|1o i\

[III-164] = 1

2( Xio 1|1o i\X

io 1|1o i\ + Xio 1|2o i\Xio 2|1o i\

)

[III-165] = 12

[III-166]1.(Bi, |W|).2 = Xio 1|W|2o i\

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[III-167] = 1

2( Xio 1|1o i\X

io 1|2o i\ + Xio 1|2o i\Xio 2|2o i\

)

[III-168] = 0.Likewise

[III-169]2.(Bi, |W|).2 = 1

2 2 .(Bi, |W|).1 = 0.

Thus,

[III-170] .(Bi, |W|). = 12

1 00 1

.

Next consider[III-171]

[III-172]|W† | = 1

2.

## ## ##

III.3.5 PHASE [III.3.5]

WHERE DOES THIS GO?

Relevant to [ ] is the somewhat ambiguous concept of "phase" ([R-11], p 93). We distinguishglobal phase and relative phase .

III.3.5.1 GLOBAL PHASE [III.3.5.1]

The global phase equivalence relation is [III-196]Q := { < |u\, |v\ > Π|H|.

2 | $ q Π|R| ' |u\ = exp(iq)}|v\ }.

It can be shown that Q Π|EQ|. Q-equivalent vectors correspond to identical elementary projectionoperators in the sense that

[III-197]|u\ Q |v\

fi|u\Xu| = exp(iq)|v\Xv|exp(-iq) = |v\Xv|.

III.3.5.2 RELATIVE PHASE [III.3.5.2]

The relative phase equivalence relation is

[III-198]Q` := { < |u\, |v\, {|ei\}.i=1

i=n > Π|H|.

2 x B[H\] | $ q Π|R|.

n ' " i Π[1,n], Xie|u\ = qiXie|v\ }.

It may be the case that

[III-199]|u\ Q` |v\, |u\Xu| ¹≠ |v\Xv|.For example, n = 2, consider

[III-200]ui` = {1oi, 2oi}

[III-201] |1oi\ = |e1\ |2oi\ = |e2\

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Mp.nb - 16 - 10-15-14 -- 9:22

[III-202] |1oj\ = |e° 1\ = 2-1ê2(|1oi\ + |2oi\) = 2-1ê2(|e1\ + |e2\)

[III-203]|2oj\ = |e° 2\ = 2-1ê2(|1oi\ + exp(ip)|2oi\) = 2-1ê2(|e1\ exp(ip)- |e2\)

[III-204]|2oj\ = |e° 2\ = 2-1ê2(|1oi\ - |2oi\) = 2-1ê2(|e1\ - |e2\)By, [E-III-195]

[III-205]p(1oj v`j1oj v

`j) = |Xjo1|1oj\|.

2 = 1

[III-206]p(1oj v`j2oj v

`j) = |Xjo1|1oj\|.

2 = 0.

Thus, observations of ui` on and |1oj\ and |2oj\ "give opposite results" ([R-XXX], p. 51). Note that

this example works for real vector space as well. "Phase" does not appear to be essential to theexample.

III.4 Mp[QMS] [III.2]

First, we define Mp[QMS] [III.4.1]. Then we consider some notation pertaining to it [III.4.2].

III.4.1 DEFINITION [III.4.1]

Theoretical structures (potential models) for QM are defined below.

D-III-1] [D-III-1]

For system structure

S = < S, ª, i, Î, E, W >.

Mp[QMS] = { < X, T, p, h ¢ .«Å▸, |.|, H, V > |1. < X, T, p, h ¢ .«Å▸ > Œ Mpp[QMS];

2. H = Π|VECCX\|.

n, n = dimLWX

3. |.| Œ |LWX Å▸LPH|;

4. V Õ |U[H]| }.Note that, for a given

[III-207]< X, T, p, h ¢ .«Å▸ > Œ Mpp[QMS];

there will be many isomorphic |.|, H, V satisfying the above conditions since dimension deter-mines an inner product space only up to an isomorphism. For H fixed, LPH and |LW

X Å▸LPH| aredetermined [E-III-5]. In specializations of QM [VI] H will be further restricted.

Since the theoretical structures of QM require only finite dimensional Hilbert spaces over Cand all these of the same dimension are isomorphic, one might suspect that we could make dowith a single specific Hilbert space for each dimension n, i. e. one member of |VECCX\|.

n. How-ever, some applications of QM [V.4.3] require specific realizations of the Hilbert spaces in theform of tensor products which are isomorphic, but not identical with those in which the vectorsare n-tuples of complex numbers.

QM-DRAFT

Mp.nb - 17 - 10-15-14 -- 9:22

Since the theoretical structures of QM require only finite dimensional Hilbert spaces over Cand all these of the same dimension are isomorphic, one might suspect that we could make dowith a single specific Hilbert space for each dimension n, i. e. one member of |VECCX\|.

n. How-ever, some applications of QM [V.4.3] require specific realizations of the Hilbert spaces in theform of tensor products which are isomorphic, but not identical with those in which the vectorsare n-tuples of complex numbers.

III.4.2 NOTATION [III.4.2]

We consider notation for projections on Mp[QMS] [III.4.2.1 ] and for substructures ofMp[QMS] [III.4.2.2 ]. We will employ parallel notation for M[QMS] [IV] as well. Similar notationfor Mpp[QMS] has been provided in [II.5.2].

III.4.2.1 PROJECTIONS [III.4.2.1]

As in [II.5.2], we denote the generic member of Mp[QMS] and its sub-sets by

[III-208]mp Œ Mp[QMS] Mp Õ Mp[QMS]

respectively. We employ notation to that of [II.5.2] for projections, e. g.

[III-209]mp57 = p57(mp).

III.4.2.2 SUB-STRUCTURES [III.4.2.2]

For system structure S, for all Mp ΠPOT (Mp[QMS]), we let

[III-210]Mp1 := { X | $ < X, T, p, h«Å▸, |.|, H, V > Œ Mp }

[III-211]Mp5, 6, 7 := { < |.|, H, V > | $ < X, T, p, h«Å▸, |.|, H, V > Œ Mp }

[III-212]Mp1,5, 6, 7 := { < X, |.|, H, V > | $ < X, T, p, h«Å▸, |.|, H, V > Œ Mp }

[III-213]Mp@1Dª,5, 6, 7 := { < [X] ª › Mp

1, |.|, H, V > | < X, |.|, H, V > ΠMp1,5, 6, 7 }

where

[III-214][X] ª Œ S ê ª .

is a º-equivalence [II.1.3] class of S.

III.4.3 ƒAND Mp[QMS] [III.4.3]

The relation

[III-215]{ < mp1 , mp

6> | mp Œ Mp[QMS] } à S X |VECFX\|.

is generally not a function and there is no requirement that system concatenations be paired withH's that are tensor products of those assigned to system concatenates. This requirementappears as constraint C3 [V.3].

QM-DRAFT

Mp.nb - 18 - 10-15-14 -- 9:22

III.5 REFERENCES [III.5]

Agarwal, Ravi, Martin Bohner, Donal O'Regan, Allan Peterson " Dynamic Equations on Time Scales: a Survey" Journal of Computational and Applied Mathematics. 41 pp. 1-26, 2002.[R-1]Ballentine, Leslie E., Quantum Mechanics: A Modern Development. World Scientific Publishing

Co. Singapore, 1998.[R-2]Birkhoff, Garrett, Lattice Theory, American Mathematical Society, Providence RI, USA,

1967.[R-3]Caves, Carlton M. Multiple Systems, the Tensor-Product Space and the Partial Trace.

http://info.phys.unm.edu/~caves/courses/qinfo-f05/lectures/tensorproduct.pdf.[R-4]Dirac, Paul A. M., The Principles of Quantum Mechanics. Clarendon Press, Oxford, UK,

1958.[R-5]Feynman, Richard P., Robert B. Leighton and Matthew Sands, The Feynman Lectures on

Physics vol. III: Quantum Mechanics. Addison-Wesley, Reading MA, 1965.[R-6]Fortin, Sebastian, and Olympia Lombardi, "Partial Traces in Decoherence and in Interpreta-

tion: What do Reduced Sates refer to?", Foundations of Physics, (DOI 10.1007/s10701-014-9791-3), March, 2004 [R-7]

Halmos, Paul R., Finite-Dimensional Vector Spaces. Van Nostrand, Princeton NJ, 1958 .[R-8]

Joos, Georg, Theoretical Physics. 3rd ed. Hafner Publishing Company, New York, 1958.[R-9]Messiah, Albert, Quantum Mechanics, Vols. I ; II. North-Holland, Amsterdam, NE, 1961.[R-10]

Nielson, Michael A., Isaac L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK, 2000.[R-11]Preskill, John, Lecture Notes. California Institute of Technology, Pasadena CA,http://www.theory.caltech.edu/people/preskill/ph229/#lecture 2004.[R-12]

Schumacher, Benjamin, Michael D. Westmorland, Quantum Processes, Systems and Information, Cambridge University Press, Cambridge, UK, 2010 .[R-13]

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