A MATHEMATICAL STUlYi OF NON COMMUTATIVE
PROBABILITY THEORY
by
Thomas K. Louton
Institute of Statistics
Mimeograph Series No. 927
Raleigh - May 1974
iv
TABLE OF CONTENTS
Page
INTRODUCTION
. . . . . , . . . . .
0
•
1
A THEOREM ON SIMULTANEOUS OBSERVABILITY
13
TOPOLOGY ON OBSERVABLES AND CHANGE OF STATE
46
APPLICATION TO CLASSICAL PROBABILITY THEORY
79
LIST OF REFERENCES
100
INTRODUCTION
When we do an experiment we automati cally conside:r a set of
possible outcomes of' the experiment,
event.
Each such outcome is called an
Let us define a new event from two given events
a Ub
This new event,
is the event that either
a
or
a
and
b
is the
outcome (we indicate this new event by saying that either
occurs),
We also can generate another new event from
'viOhich is denoted by
b
a
nb
a
is an event we say that not
reasons, we wish to extend
collections of events.
see Halmos (1950).
or
and
and which is the event that both
8.re outcomes (:in this case we say that both
Also if
a
a
U
and
n
a
to
a
and
b
is an event.
encon~ass
b
b
b,
a
and
occur).
For various
countable
For a discussion of this extension process,
Thus, we assume that the events form a C)-algebra.
It follows from the Loomis representation theorem that this
C)-
algebra of events corresponds to a C)-algebra of subsets of some set,
Now we are often only able to learn of occurrences by stUdying
functions from
example,
events
function
to
a., die
¢'
f
~.,
that is a measurement of some object.
I
w:i.th six Bides,
(e }, [e '},
l
2
(e }, , .. , (e , e , ••. , e }
l
l
6
2
(e.)
l
=
i
throwing of the die.
tel' . -., e6}
=
0
0
.0', [e6}
generates
In general 'we consider a
and we study averages associated wi th the
This example is interesting in that the space
is a set we can not only conceptually consider
. but'-may phys.ically stUdy as well, that is, it is not an abstraction.
In c'ontrast, consider a marble.
Say we
cJ.o
all experiment wi th the
2
marble
by treating it as though it were
find ,vhich ",dde" comes upo
Now
0
[j
1.~.
:lie)
,we roll it and
'J.n Ci.bst:::'8.ction, a geometric
18
point is not something we can physically idt:nti fy, but the event
space generated by this is real and we can identify its elements, the
open fldis cs fl .
For example, the marble could be painted with different
colored patches and the color of each experiment. coule. be identified.
If we associate a number w'i th each colored
case of
th,~
die, Ire would be considering o. function
o -+
general Ive '\\Tant to consider all functions
f-l(E)
as we do wi th the
diSC,
is an event for every Borel set
reasons of completeness.
E
~
of
1': 0 -+ 6Y,'
~
NO"l'T
,in
such that
.We do this for
Such functions are called random variables.
Notice that given a random variable 1': 0 -+ IW. , and a probability
measure
cr
on
0
we have in a natural manner a measure
which is defined by
S(E) = cr(f-l(E))
distribution of the random variahle.
on
R,
and which is called the
Thus we study
random variables, whose distributions
S
S
0
by considering
are the actual objects of
observation.
We vTill use these brief remarks to inLrod;.:'c'c: a more general
theory of events associated Ivi th experiments.
JJut:Lcc that in the
classical 3i tuation which is described above,
:),
are always defined.
U and
n
n b,
and,
In our more general framc\·;-ork, t;'le opera ti OrtS
are not alvJays defined.
We will j'onnulace this nevI theory
via a sequence ofaxi oms due to Mackey and our cUscus:3ion is a
slight modification of his.
Let (9
and
a Ub
F be non-empty sets and
(9 X
6 X B(~,) -+ [O,lJ
p
8.
mapping from
3
B(~)
(here
elements of
states.
and
~3ubset.s
is the CT-algebra of all Borel
of
are called observables and those of
(9
~).
6
The
are called
The axioms which follow list various properties of
~,
6
which will be used throughout this thesis.
p
Axiom I:
If
x
~
€
B (~)
the algebra
some sense to
a
and
of
S(l)
==
6
€
the
R.
p(x,a,p) is a probability measure on
The measure
l
a(f- (. ))
p (x,a)
corresponds in
as disc:u:3sed above.
Axiom II:
If
= p(x,a~E)
p(x,a,E)
If
for all
E
and
a
x, then
a'.
p(x,a,E) = p(x',a,E) for all E and a then x = x'
This is clearly reasonable, since otherwise we have no other ways
to distinguish observables and states.
Axiom III:
Let
x
be an observable and
then there exists
y
a
BC~).
Y
and
€ (5
f
E
in
(9
€
f
a real valued Borel function,
p(y,a,E) = p(x,a,f-l(E))
such that
This unique
y
for all
will be denoted by
(x) •
Axiom IV:
Let
for all
(t.l }'-N
l·i
and
be a sequence of real numbers such that
l
there exists a state
E in
B (~)
and
1.
I: t.
x
a
in
a mixture of the states
If
(a i
such that
(9.
}
(a.}.
l
is a sequence of states, then
p(x,a,E)
We write
t.:<:
0
l
a =
=
I:
I: t.p(x,a.,E)
l
t.a.
l
l
l
and
a
for all
is called
4
Before
~iscussing
the remaining axioms
\erE:
first need to
distingui sh a subclass of the class of all obf;enrables and to derive
some of the properties of the elements of' thi
8
subclass,
The elements
of this subclass "are called "questions" and they correspond to
experiments which have only two values (yes-no or true-false),
precisely, an observable
x
is called a question iff
p(x,O',[O,l})
for each state
write
/.1
0'
(x)
When
0'.'
More
x
1
is a question and
Q(
is a state we
= p(x,O',[l1) •
The following proposition has proven to be quite useful.
Proposi tion:
If
x
is an observable then
x
2
x
iff
x
Proof:
x
First assume
P(x,O',E
if
0 <
n {tit <
a<
OJ)
2
=
x
Let
2
p(x ,O',E
E
n [t
be any Borel set.
Notice that
< OJ)
1, then
P(x,QI:VE
n
.L
p(x,QI, (an,l))
(0,1) )
5
Thus
1
p(x,~,E
n
(0,1)) ~
lim p(x,~, (on,l))
n-+oo
00
n
p(x,~,
n=l
o.
Finally if
°>
1 , then
1
p(x,a, (E
n
(6,+00)) = p(xn,~,E
n
(0,+00))
and
p(x,a,E
n (0,+00
)) ~
lim p(x,~, (on,+oo))
n-+oo
lim p(x,~,
n-+oo
n
n
(0 , + a:> ))
o.
Thus,
p(x,~,E)
for all Borel
E.
=
p(x,~,E
Therefore
x
n
(O,l})
is a question.
The converse is obvious.
Now assume that
C'
E: ~
.
is an observable and tllat
~
by
QX = f (x)
E
Observe that
QX
is actually a quc3tion
define a question
.->
x
E
where
f ( ,,)
E
E
B (rR,).
--- 1'13; (5)
Sl.rlce
We
for each
6
-l(~,_,\)
p(~,Q"P)
P ( x, Q' J X l-,
.L',
.t')
and
x
Q
E
'I'hus,
in
is the question, did th€ measurement of
x
lead to a value
E?
It follows that to each
r~x,
x, there corresponds a family of
and further, we claim tnat
crues t:i uriS,
l.\:tE J
by
By this ,ve mean that if
then
x
XlS
ueiquely determined
E ,
for every Borel set
(the proof is trivial, just write the definitions of
= X'
and
Notice tpat we may impose a partial order on the set of
Q s: Q
l
2
questions, that is, we say that
s: P(Q2'Q', {I})
for every
in
for each state
s: !1Q' (Q2 )
!1Q' (Ql)
Q'
6
iff
P(Ql,Q',{l})
(we often write
Q' )
0
Q ~ ~2
iff
We clearly lli:ive t.he following
properties for arbitrary questions:
Q s; Q
and
Q2
and
Define the observable
1
by
is a question and
1
~.~
Then
1
define
for all
1 - Q'
Q'
E
p(l,Q', (11)
for all
to be the question forwhicb
6 '
1
Q
l' or all
Q'
E
S .
For each question
p(l - Q,Q',E)
Q
c
= p(Q,Q',E )
7
c~aYB
On8
that two questions are disjoint iff'
for 8.11 ",tates.
be any observable, and let
E c: F ,
t_h en
Suppose
Q ~ Q'
1- 2
In this case we write
, ... QX
QXE',"
F
and
F
be Borel sets in
Moreover, if
and
-Q1,
E
For example, let
E
n :F'
Q "are questions,
2
~
=
,
then
R.· If
~ (; ~
If there is a question
Q such that
for every state
ex
then we denote
are Borel sets such that
Q by
E n F =¢
+ •Q"
•
c:.
Q
l
and tf
x
If
E
and
is an observable, we
note that
To see that this is sq merely observe that
y
''EUF
tbus,
for each state
ex,
It follows from these remarks that
question-valued measure,
B(R)
.:1:..
~.,
into the set of questions,
QX
is a
is a !'o~rlOmomorphism" from
QX
This stat'c:ment will bE" made more
precise in tbe discus;:don following Axiom IT arid ii: "L,scribed more
fully by Vardarajan (1962)0
]\Jotic(~
QxE + QJ;x,"
further that if
E
and
.
lS no 1,anger a ques t lon.
F
arC'
uyt,
F
di:3joint sets,
'I'!1ir; ,leadEl u,s to Axiom V.
x
8
Axiom V:
be a' sequence of' ques ti ons Guch thatQ. 0 Q. ,
. ""'i
J
Let
i ' 1= j
,
then
exists in the sense that tbere is a question
Q '" I:Q.
J.
which we denote by
such that
I:Q.
:1.
p(Q,QI,[l})
I: p(Q.,QI,{l})
l
QI c:
holds for all
Notice that
least
Q
.
l
(C; ,
Q
Q.1
~
all
It can
i
br~
;,',bmvrl that
Q
is the
SUCll quc8tion~
Also based on the above discussion, we introduce the following
axiom.
Axiom VI:
If
[QE}EE:B(~)
x ,
observable,
is a question valued measure then there exists an
such that
for every Borel set
E
This is in a c;ense a ,superfluous axiom, [',ince it is the case that
if
(QE}
.is a que,;Lion valued measure then or:
of observables to a new set
observable
x
for all
and
QE;
x
Q
E
Let
Q3' Rl
.L
and
in such a
',{:J.,V
U'Ht
(",I
contaim: an
such that
F
for ,::ach
Q_
(9'
can enlarge the set
an]
in
,;,e have
and
E: c B(6i) .
Q.)
be questions
such that
P'ett there exl,;t;:.; questions
~;uch
Q) 0 R.l .,
R
1
6 E) ,
We then say that such que,:ti ee!
an,,, .,irnultaneously
(9
9
ar.~~vlcrable.
x
Q-E '
If
x
is an observable, then it i;j trivial to show that
al'e simultaneousl.y answerable.
.
l ··p
..L
x
y
ar~d
are
OtmeryabJ.e;:; :.:uch that their question valued measures are
simultaneously awnverable,we say that
observalJle,
(we write
x
and
y are simultaneously
x ... y ) .
Theorem (Varadaraj an) :
<
If'
f'
x ... y
then there exists an observable
g, :mch that
D.D.d
x:= f(x)
z
and Borel functions
y= g(:,;)
and
In that which appears below we denote the mean of
state
by
ct
for all
If
ct
i1
fx ) •
and
where
in the
e0/x (E)
:=
p(x,ct,E)
E.
x:= f(z)
defined.
Thus,
01\
x
and
y:= g(z) , then it is clear that
One merely defines
x + y := (1' + g) (z,).
x + y
is
Observe that wi th
this definition we have that
i1
01
for each state
(x + y) :=
i1
01
(x) +
i1
ct
(y)
It is not, in general,
01.
:p()ss.iblf~
to define
x + y •
This leads to a modification of Axiom II,whichlrre c.all Axiom II'.
Ax:iom II':
If
i1
ct
(x) :=
i1
ct
(y)
for all
ct, then
x' y •
With the aid of thi:::: axiom, we can dt.'fic,p
observable such that
i1
01
(x + y) :=
(x) + 11 (y)
i1
01
ct
x. + y
I'or
to be an
each state
ct.
Of COUl:'Re, such an observable may not al-way:; e:x.i.Lt.
Lc:t
e
der:.ote the set of all qU<=:stions.
ordered set under
s:
defined above.
Then it is a partially
Morc;ovel', if fa::' each
Q,
lie
10
let
Q'
'-=
1 - Q "then
I
is a complemcLtation on
an orthoc:omplemented partially ordered
operator
'.)
e
In general if'
is
(~Y:Lth
s,~t.
e
and
f',
complementation
setwl.tb a partial order
8.
is an orthocomplementation on
de.fined on it we fiay that
becomes
e
:s:
iff
the fo.llmving conditions hold:
a)
If
aI' a 2 , ...
a.a a.
],
for
J
i
are members of
*j
(a 0 b
i::'!'
Z such that
a:S:
tr:ere exis tE~ a uniqu.e least element
that
for all
denoted by
a U a'
b)
a - a
= bUb'
U a
l
for all
c)
If
U a
for all
denote this element by
then
a such
a
The element
i
2
1)')
3
a
1
an<1
Thus
and we
b
a U a' .. 1 ,
a.
a:s: b , then
a U (b' U a)' .
b
That these axioms hold for the set of questions is
Mackey.
is
~hown
by
Moreover when we consider strongly convex sets of probabili ty
measures on
e
we can show that
e-valued mea.sures correspond to
observables and that the two systems arc t:':qui'Je,h:nt.
Thus we may
consider partially ordered sets with orthocomp}_ementation and
strongly convex set of probabili ty
going Axioms
The set
meaEmre~~iw-;tead
a
of' tbe fore-
This is the method used by Val'aders,jan (1962).
0
e
is called the logic of the
Olll' next axiom is a strong restricticG
allowed in our theory.
~:;m t:'cm.
Oil
ttle type of logics
11
Axiom VII
e
~
E,t of closed sub-
is :i somorrhic to the partially orcleT,-,j
spaces of
where
H
Ivhere
H
is some Hilbert s:pGtce of' dimension
iJ
X C1
[.1(,.)
corresponds to the usual orthogonal complementati Ol! ope ['cl,or
defined on subspaces of
H.
Axiom VIII:
Q L:; a non-zero question, then there exists a state
If
that
fl
Ct
ex
such
(,~) == 1
Axiom VIII is a powerful axiom which allows us to obtain a reformulaUon of the axioms in terms of Hilbert space theory.
For
Q is a question then 1 t may be identified
ex anrple, observe that if
with a closed subspace of the Hilbert space
was asserted by Axiom VII.
H
whose existence
Moreover, the closed subspaces of
H
"
are in one-one correspondence with the set of continuous projections
of
H
into itself.
projections of
H
Thus, questions can be ident.if:ied with the set of
into
H.
Moreover we have already indicated that
there is a one-one correspondence between obse.rvo.bles
question-valued measures
(~}EEB(R)'
x
and their
The upbhot is that there is a
one-one correspondence between the class
(9
the set of all projection-valued measure::.; on
of all observables and
H
On the other hand,
every projecti on valued measure corresponds, 'vi a the spectral theorem,
to a seJ.f-adjoint operator on
H
and
conver~.(:ly.
'l'J:r.J.s, if
observable, then there is a unique self-g.dj oint o:perator
which corresponds to
x
and conversely.
A
x
x:
j,s an
on
H
12
We use Gleason's theorem (see Mackey (1963)) to show that the
states cOYTespord one-one to positive definite trace one operators.
Moreover) if
A is the positive definite trace one operator
corresponding to the state
f'ormulatJ..ol~
for the mean
Ire particular) if
dimensional and where
IJ.
01
Ct, then vie
IJ.
(x) :
correSpOnd[3 to
01
AT]
01
=:
Ulfc following
havE~
AT]
for some
A
where
T] E~ Hand
A .is one-
IAI
=:
1) then
(x)
We refer the reader to Mackey
(1963)
for the details.
Wewil.l novr proceed to use the structure developed here to
consider a question involving statistical em;embles.
Namely, to
explore the common technique of resolving a statistical ensemble to
component parts, each of which possesses certain properties with
probability one.
It is shown in Chapter 1 that'N"hen such techniques
can be carried out simultaneously with hra observables, the
observables have a joint distribution.
We will then proceed to
consider the strong operator topology as a cc:m.c:eqlLcnce of assuming
that only a rini te number of experiments vIi tb a.cbiGrary accuracy.
Several
a:::~I)ects
of this topology are considc;f.'E'd.
t.he thj.ri chapt:;cr to consider what this
probahili ty.
1I!C'8.t1;,
i
11
He then proceed in
t",rms ai' classical
It is shovm that COll'vergencfC i.1l t.h.e str,:)ng operator
topology corl'ei3ponds to convergence in dj.stribution (of a random
variable).
This structure is used to pose que,stioLS in terms of
stochastic processes.
13
A THEOREM ON SIMULTANEOUS OBSER'/ALEl'TY
(,
A common technique often employed\,hf::'v],
phenomena
,:tl1JiE~s
which he would like to observe accurately, i:: to cotj·':ider a statisUcal
ensemble to which a sorting mechanism i
c
appliE,d iel order to isolate
~.~.,
subens::mbles which possess the property of interl:;:]t,
the ensemble
is reso,Lved into component parts.
is of interest.
E,
~~
Let
a
a Borel subset of
r(A)
property
~,
a
:- X,
'f: ,
then the measurement of
,
Val' f(A) == 0
Q'
A
where
D
i
t-111.:'
we view a state of an ensemble
u
8°
Lit?
which i:3
yiE',lcis
Q'
t
,3
'
.
prob~lb~Lt
is equal to the
If we extract a sUbensem'ch'
over, ..,e find that
ensemble
Thu,::. if
Q. •
probability of having property
question
mew:~~reml~n:J(
be the property that a
OLL'
.ore-
v:d,lt'
,
"
a mixt
L
"
,j
Lt--lC;
"
\J_:_
S Ld)-
tDre::
i'C;
01
the various component systems.
NOI, suppose it is wished to find accuratt.:' value,: for the
observable
B
at the same time that we knOl' exact
The E;mne procedure (as applied to
the subensemble
j
A
n terms of the observab Ie
A
and
B
A.
O.l.ve
B
A
anrj
I;
We wi 11 show that in the event such a program
it can be concluded that
ror
above) may I:e w;ed to
general such a resolution is not feasible for
taneously.
value;~
,;ilJluJ~vork;,
then
have joint probability
distributions in each state.
Assume that
A
and
B
take an arbjtrary ensemble,
SN
is a measurable value of
and
A
tu~
are related .in
,
I'ollm{j
and arbj
c > 0
related in such a way that there is a
(:
A.~;~)ume
6 > 0
nr:,~
> 0
that
A
manlJ.e:,
\'le
Suppose
A
ar.d
B
and a resolution of
are
SN
14
into sunensembles
i3N
such that whenever
Prob(A E (A - 0, A + o)} ~ Prob{B E (~
0'.
Var A < 0 ,
O'i
- E,
~
l
the state of
A~ B
then
spectral theorem).
+
E)}
where
When
A
is
0'.
l
l
is its average value.
and
are so related we write
that if
Q'.
and
B
The main result of this chapter is
B is functionally dependent on
In particular this implies that
joint probability distribution.
A (via the
A and B have a
We proceed to develop the apparatus
needed to prove the theorem.
We will continue to refer to our axioms by the labeling system of
Mackey (1963).
For the following discussion we adopt all of them
except Axiom VII, which states that the logic in question corresponds
1:1
to the set of closed subspaces of a Hilbert space.
following discussion, we let
observables, and
e
E5 be the set of states,
In the
(9
the set of
the logic.
Recall that when
probability measure on
0' E
E5
and
x
E (9 ,
and the mean of
~
O'(x)
=p(x,O',.)
x
in the state
is a
0'
is
defined to be
when
In a natural manner we define the variance of
as
Var (x)
0'
2
~ (x ) 0'
(~ (x»
0'
2
x
in the state
0'
15
Proposi tion 1:
et
If
is a state in which
t E.~
and, conversely, if
varet(x)
such that
=
0
et(x) (,uet(x))
t:C>.;n
1
Q'(x) (t) '" 1 , then
Var (x) := 0 .
et'
Proof:
o
2
Var (x) = J(A - ,u (x)) det(X)(A)
:=
et
iff
et
Q'(x)(,u (x)) := 1
et
Recall that the joint distribution of
I
exists for
every state
et, iff the family
observable.
In this case the joint distributi on is defined by
z:
A
B(R)
~
e,
z(n
-1
A
(Varadarajan (1962).
When
{x. Ii
l
(E))
A
(E)
A
We let
1,2}
=
x
=
tXA A lOt..}
If
Xl
*
x
~
2
and
and
2
define the spectrum of
open in
{l, 2}
A E
A
and
EE
B(~)
for our further discussion.
is a simultaneously observable family we
and we say that
observable.
=
for each
i:3 simultaneously
Xl
z
and
are simultaneously
is their joint distribution, we
cr(z), to be the set
Z
zeN) = OJ)
C
cr(z):= (L(NIN
is
•
Proposi ti on 2:
If
x
1.
*
x
2
and
z
is their joint distribution, then
Moreover, in general,
u(z):j: cr(x ) X cr(x ) •
l
2
Proof:
The inclusion follows directly from the
cr(z) :j: cr(x ) X cr(x )
1
2
d(o~fini tion.
we consider the followirg:
To see that
16
Example 1:
e
IJet
.)
denote the logic of clm-',ed s~Jbspau's
structure,: e.,,) an orthocomp1emented latt.ice,
L.~-,
1.
is the usual complementary subspace operat.or
and, for 8ubspaces
b.
iden+~ify
We
a
e
and
b
~
'-
t~,e
s
\.'i th its usual
operation
is set inclusion,
a Ubi:; the 1i. Qf':ar span of
with the set of all pY·oj'.octions of Dt 2
and let
P
Let
Extend
x
and
and
Z("1'''3)
("2'''4)}
x
y
Pl ,
y
rE~al
numbers,
".
l
Pick a
i
,
=
1, 2,
be the observables corresponding to
in the usual manner to homomorphisms.
so that we have
= P2
Z("2'''4)
* [("1'''2)}
6
be th.e COITcsponding projecti on
l
We choose di s ti. net
3, 4.
and
and
wi th the set of posi ti ve self adjoint OI!erators of trace 1.
subspace of dimension one; let
a
(J
X ("3'''4}
(x) X
(J
Now
CJ(z) = (("1'''3)'
(y) .
Proposition 3:
SUPIJOSe
e
is a logic which satLsfies Ax.i oms I-VI a.rlCi that it,
in addition, satisfies
Axi am VIII:
If
x
and
a
exists
E I1==i
Q'
'f
e,
Q'
x
Eo
and
6
y
then for each
such that
a i= 0 , there
QI(a)'~
such that
yare observables such that
distribution of
there exi st.s
For each
x..
E
1 ,
y
and
Z
is the joint
> 0
and
("1'''2) E a{z)
17
Var (x) < E
Moreover,
Var (y) <
and
~
~
E .
.
Proof:
(A 1 ,A )
If
E
2
\FE
(A 2 - ,~,
o-(z) , then
\fE
A + L~))
2
A +
1
:j:
'fr)
4
0
Z((A
1
-
~,
A
l
and thus there exists
X (A
2
- '[i
4'
A2
+
+~)
~
If)]) = 1
X
such that
(Axiom VIII) .
and that
Since
(a +b )
222
~ 2
(a +b ) ,
a,b E R , we have
The corresponding inequa.li ties
£'01'
y
follow· in a similar fashion.
18
Corollary 1:
Let
~
E
6
x
be an observable, then for every
such"that
~
IMa(X)-AI<
~(X(A
and
~,
-
vara(x) <
..
Proof:
A+
E
~) = 1
> 0,
E
A
E ~(x)
, and
, it follows that
0
~.
Same as Proposition 3.
Of course Proposition 3 will not hold for arhi trary elements
E ~(x)
(A l ,A2)
X ~(y).
To see this consider the following example,
based on .Example 1.
Example 2:
Choose an aribtrary orthonormal basis for
elements
x
and
y.
Let
x
~(x)
X
Y. in such a way that
~(z) =
distribution, we see that
consider
(0,0) E
~(x)
X
~(y).
and if
{O,l), (1,0)}
tl
(t
a
tc)
)
t
<-
be such a state.
Var (x)
a
t
Var (x) <
a
E
2
and
then
l
+ t
,
1
2
('j
t
z
y
to subspace
{O,l}.
Then
is their joint
by Example 1.
t
Let
1
0 .'
~
1
1
Var (y) > l-E
('j
and
t cr •
~
0
t
M (y)
M (x) = t
2
a
"L
a
Now it is c: lear that if
Let us compute
Var (y)
and, call its
States vnll correspond +0 an
arbi trary, posi ti ve matrices of trace 1.
a
x
~(x)= ~(y) =
= {(0,1)(1,0), (0,0), (l,l)}
~(y)
2
denote the observable corresponding to
the projection on the subspace generated by
generated by
R
0
We
19
If we are doing some experiment, we will observe the mean values
of our observables.
Importantly, we will ofteG O(l,SerVe yes or no
observables, questions in Mackey's vernacular.
lI~implest!l
Que,stions are the
observables from which all others can be constructed as
functions of many simultaneous questions,
~.~.,
3.3,
Theorem
Varadarajan (1962).
At this point we will consider some exampleEi which point out
difficulties arising from the fact that
thc~
axioms of Mackey are not
enough to insure that there exists sufficient states to lIseparatell the
logic.
aur first example deals wi th a particular logic, our old friend
~2
, wi th the usual structure on the logic.
in this example
6
IG contrast to Example 1,
is not the trace one operators, but a vastly
reduced set, which is in no way connected to trace 1 operators.
~2 but simply take
not take all subspaces of
Example 1.
Now choose some subspace
i ::: 1, 2.
(s
S
l
such that
is of course one-dimensional).
(1,a,p ,p ,s,sl} and
1 2
lattice of six elements
P
,
P
S
as in
2
n P.1
e
No\'i let
lt~t ~,
We do
p
(
2
)::: (a} ,
denote the
and
y
be
~,
t3
and
the states defined by
~(Pl)
~(~)
Further, let
y.
Q'
-
(S )
~ (sl)
is
1
y(P )
1 ,
'2 '
l
y (S)
1 ,
1
3"
/(~)
1
.2
)' (81)
2
3
.
be the set of all convex combinations of
We define observables
x
and
y
by
20
x(i)
i
1
i
2
We extend these to make them
Now
x ~
(£.!.,
y
VarS(x)
0
=
iff
(1969,
1
j
2
p.
199))
but we have
o
Vex (y)
Var (x)
r:x
thus
j
~-homomorphismso
Varadarajan
varf3 (x)
s
y(j)
r:x
= Varf3 (y) = 0
varS(Y)
0,
=
S
E
6 .
Consequently we have no way to distinguish the fact that
yare not simultaneously observable.
which
x
can be measured accurately,
automatically
precision as
y
x
x
and
Thus, if we can find a state in
i.~.,
zero variance, then
is known and further it is known with the same
and
conversely.
At this point we wish to use the above discussion to enlighten
various aspects of Mackey's Axiom VIII.
Note that at no point are we
assuming Mackey's Axiom VIII (the Hilbert space axiom).
Henceforth,
we refer to Mackey's Axiom VIII as Axiom VllIm and we introduce a new
axiom due to Varadarajan which we refer to as Axiom Vlllv.
This
axiom is:
if
a,
such that
bEe,
r:x(a)
a
*b
* r:x(b)
In accordance with Gudder
then there exists
r:x E 6
•
(1966)
case Axioms VIllm and VlIIv obtain.
we will call a logic full in
21
S
a
e
For each
a
[ala
0'(0) = l}
E ~J
E
let us define a set,
.
S
a
,
as follows.
a ,.,. (9
In a full logic,
implies
S
a
We say, following Gudder again, that a logic (state set) is
*¢ .
qui te full in case,
Sa c Sb
implies
~
a
b .
In view of our discussion we will now consider the six element
lattice
e
e
following Example 2
e
is full logic, but that
above.
is not quite full.
a U b
(By the way, if we let
be the span of
above, then we have an anti-lattice.
(1966,
~.!., Gudder
We note that in that example,
a
and
b
It is not a qUite full logic,
88 )
p.
We will now prove a formula dealing wi ttl the variance of
a, where
mixed state
a
n
= I:
1
t.a. .
l
,
t.a.
l
l
n
I: t. == 1
l
1
Var (x)
a'
,
t.
~
l
t. t . (/-1
l
J
ai
Then
0
(x)
Proof:
We will prove this strange formula by a llstraight forward"
computation.
Var (x)
a
We have
2
= /-1 (x ) - (/-1 (x))
a
2
l
2
n
?
(I: t./-1
(x))
1 l ai
n
n
2
(x) - t./-1.) + I: t /-1i - ~
I: t.t./-1./-1.
ail l
1 i
i == 1 j =1 l J l J
I: (t./-1
1
n
I: t./-1
(x) 1 l ai
a
n
2
n .
2
x
in a
This will, in turn, be used to
l
consider some consequences of assuming a quite full logic.
Variance Formula:
n
Let a == I:
i=l
as
22
n
~.
where we let
~
l
L: t.
Since
(x)
ct.
1
l
1 , the latter expression
l
is equal to
n
L: t. Var
1
ct ,
l
(x) +
l
2
L: t . t . (~. l
J
l
l,J
..
~. ~
l
L: t
.)
J
l
.
(x)
Var
ct.
1.
n
t. t . (~. -
L:
+
l
i<j
J
l
~.)~.
J
l
t. t . (~. - ~.)~. .
L:
+
i>j
1
J
J
l
l
After relabeling, we get
n
(x) +
I: t. Var
1
1
cti
n
L: t. Var
1
(x) +
i<j
cti
l
L:
t. t . (~. - ~.)
l
J
l
2
J
.
We wi 1.1 now make a definition of a relationship between two
observables in the general case.
The following discussion will lead
to a demonstration that two observables with discrete spectra in a
quite full logic which are related in this way are indeed
simultaneously observable.
Assume
x
and
yare observables.
provided the follrrwing condition holds:
such that for each
E
such that whenever
ct
> 0 and each
"E
We will write
There is an x-null set
o-(xJ\N , there exists
is a state for which
and
Var (x) < 6
ct
"
x -+ y
N
6" > 0
23
Proposi tion 4:
Suppose
x
~
y
(0' n } is a sequence of states such that
and that
lim IJ. 0' (x)
AE
for some
~(x).
If
lim Var
n~ (X)
then
(y)}
(1J.Q1
"
n
n~(X)
(x)
0,
0' n
is a Cauchy sequence.
n
Proof:
E
Say we are given
E
> 0 , then choose
l
be chosen as the definition of
let
N
be chosen such that
Var
(x) < -1. .
2
n
t ,t > O.
l 2
Let
QI
=
x
tlQl
n
+ t
Q1
2 m
~
E
y
above.
and let
Moreover
S"
(x) - /...1< ~
and
Qln
for some m, n > N and for
n > N implies
5
0'
o<
2
< (.-1.)2
IIJ.
By our variance formula, we have
VarQI (x) =
Since, also,
we conclude that Var (y) <
0'
that
and
E.
tlt21IJ.QI (y) - IJ. (y)/2 < E
n
Q1m
t 2 we have that \IJ. (y) Qln
This implies by our variance formula
and, by an appropriate choice of
(y)l<
m
1J.Q1
VE
2 <
E
l .
t
l
24
Proposi tion 5:
Let
x
and
y
be observables with
x-null set as in the definition of
fa}
assume that
~
x
x
y.
~
y
Let
and let
A
E
N be an
0"(x)\N
and
is a sequence of states such that
n
lim
11
Q'n
(x)
A
and
o.
lim Var (x)
an
If
fA}
I-'n
is a sequence of states such that
lim 11~ (x)
n
=A
and
lim var~ (x)
n
0
then
lim 11~ (y)
lim
n
11
an
(y)
Proof:
Choose
N
l
A
(x) < 2
so that
implies
and
6
Choose
N
2
var
Q'n
{~ }.
n
Let
N be
above.
Let
Q' = tlan + t_B
•
Zn
max[N ,N }.
l 2
vara(x) < 6
< tl
for
in similar fashion for the sequence
• Therefore,
A
n,m > N. Thus
lim 11~ (y)
n
We uce an argument
similar to that
111a (x) - AI < A and
Then
vara(y) < t
lim 11a (y)
.
n
and we have
25
Proposition 6:
When
lim /-l
x,
~n
y
(y)
(~n}
and
then
1.
are as in Proposition 5 and
<J(Y) •
€
Proof:
Choose
t
definition of
> 0
x
~
and for each
y.
'A
We choose
N
l
measure in
be chosen as in the
n > N
l
such that
l/-l~n (x)- 'A I < .6'A and var~n (x) < 0'A .
~probability
0A > 0
let
Recall that
implies
~n (y)
is a
Then by Chebysheves' inequality, we
B(6\',).
have
~ (y)((zIIZ-/-l
. ~n
n
(Y)\ ~.t}) s: 1 2 Var"" (y) •
t
.... n
Therefore,
~ [Y((/-l
n
Find
N
2
~n
-2\
(y) -
/-l
~n
n > N
2
such that
(1 - t,1 + t)
~
n
(y('A-t,A+t))
-
Var
~n
(y) < t
* O.
1 - 4
t2
2
14
implies
~ (/-l~
and we want to show that
~) lJ ~
(y) +
(y) -
n
~n
~'/-l~
Var .. (y)
~n·
(y) -
11
(y) +
~)
n
~n[(/-l~ (y) - /-l~ (y) +
n
·n
Now note that if
t
"2)J
t
< -.
2
Then
*
implies
0
n > max (N ,N ) , then
l 2
and
t
~ n (Y[(/-l ~....
(y) - -2'/-l""
n
(y) +
n
~2)J) ~
1 -
~
t2
Var
(y)
~
1 -
~
~n ·
.t2
t > 0 .
Proposi ti on 7:
f
is cont.inuous on
t apology on
~,.
(J(x}\N
in the topology induced by the usual
26
Proof:
Let
Al
Cf(x)\N
E
and
t
> o.
1 > 0 > 0
Choose
corresponding
2
to
% as in the definition of
%.
IA - All <
x -+ y.
Let
A
Cf(X)
E
[~n}
Choose two sequences of states
such that
[~n}
and
such
that
lim iJ,
Q'
lim iJ,~ (x)
(x)
n
n
lim VarQ' (x).
lim Var (x)
~n
n
Let
=
0
n > N implies
N be a positive integer large enough so that
o
o
<3
Var
Q'n
<-
3
8
var~ (x) <
(x) < 2
o
8
'2 •
n
0
< -+-<~
2
3
u,
and
where we have used the fact that
1iJ,
(x) Q'n
follows from the definition of
x -+ y
iJ,,)
~ n
(x) I <
o.
and the choice of
Now it
0
that
27
2
Var
In
(y) <
Using the variance formulae for
t
3b
t1
Y with
t2
:=
1
:=
'2 ' we see that
or
n
l~ow l'OX'
slJ.f'f'iciently large it is clear tbat
If(A) - f(Al)1 ~ If(A) -
11
13 n
(y)1 + 111
13 n
(y) -
11
01
(y)1 + 111
n
01
(y) - f(Al)1
n
t
t
t
<-+-+-==t
333
The proposition follows.
This implies that
If(A) - f(Al) < If(A) Choose
N
l
n > N
such that
N )
3
such that
11
01
n
(y)1 + If(A l
n > N
l
If (A)
n
Val'
01
(y) n
and
above inequality are less than
When
-
11
13 n
01
x -+ y,
(y) I <
~
01
-
I
3"t
11
and
is a state, and
and
(y)1 •
(x) < 8
and choose N
2
t
A
< =.3. If N = max (N, N ,
)' n
\f(A ) - :r(A)1 < t •
1
Proposi tioD 8:
If (A)
11
(y)
13 n
n > N , then all three members of the right side of the
2
implies
implies
(y)1 + 111
) - 11f)
Var01 (y) <
A
E
! ' then
o-(x )'\N
such that
2
28
Proof:
f (f(A
) -
1.1,
Q'
(y) + ~1 (y) Ct
We will now consider two cases of where
r)
x -+ y
implies
x .. y .
This result is aimed at bringing some link between the abstract
theory and the actual observations of an experiment.
can measure
x
sense measured
A
E
Thus, when we
accurately we then know that by doing so we have in a
y.
This is done by taking the measured value of
(-lIxll, IIxll) then the corresponding
y
value is
f (A)
x,
with
arbitrarily small variance.
A further application will be discussed
at the end of this chapter.
The first case deals with an observable
which has a discrete spectrum.
First we prove a preliminary proposition.
Proposi tion 9:
If
x
is an observable and
then each point of
N is an x-null subset of
N is a limit point of'
cr(x),
cr(x).
Proof':
A EN.
Let
would exist
x((:\})
=
0
E
> 0
If
A were not a limit point of
such that
(A - E, A + E)
it would follow that
X((A -
E,
n cr(x)
A+
cr(x)
=
fA}
E» = 0,
there
and since
But by the
29
very definition of .(J(x), X((A -
(A -
E,
A+
Q,,\CJ(X)
E) E
For the following
tAlA
N
E,
=
0
implies that
A
contrary to the fact that
dis~ussion
is a limit point of
Observe that since
11.'+ f))
CJ(x)
let
CJ(X)
CJ(x)
=
and
E
N ~ CJ(x)
tA.}.
N
l lE
and let
x((A.})
O} .
is ~losed, we have that
=
N ~ CJ(x)
Proposi tioE 10:
,N
is
x-nulL
Proof:
x
is a'CJ-homomorphism and
N is countable, therefore
N is
x-nulL
Assume that
x
is the set of all
CJ(X)
and
yare arbitrary observables and that
A € CJ(x)
for which
is countable and,thus that
suppose that
x -+ y
on
X(A)
=
N is x-null.
N
We assume that
0
Furthermore, we
CJ(x)'\N.
Lemma 1:
In a logic satisfying Axioms I-VI and VllIm, vnIv and whic.h is
quite full,
A
E
CJ(x)\N
implies
xC,,)
~
y(f(A)) •
Proof:
By the quite full assumption and the fact that
implies
x((,,1)
*0
E
e,
A
E CJ(x~
30
"
"vI"
J:--,ave that there exists
a
such that
a(x([A.}))
Therefore, by Proposition 1,
we have
if
01
Var (y) = 0
a'
1 .
=
Var
(x) =
0
a·
and. ~ (x) = A •
a
and, thus, by definition,
is any state such that
~
a
By x -+ Y ,
(y) = f(A.).
Vara(x) = 0 , then. vara(y) = 0
Now,
and so
impl.ies
Therefore,
which implies
Proposition 11;
Assume
x
and
yare as above.
vnrm and v and is quite fUll, then
If
y
=
e
satisfies Axioms I-VI,
f(x)
Proof:
By Lemma 1, and the fact that
for all
~
E
x
is a o-nomomorphism, we have
cr(y) , (.£.!., Varadarajan, 1962).
31
Now both
1
=
x
and
y
U(X(f"-l(\))I\
are cr-homomorphisms, so we have that
E
cr(y)} s: U(y([I})I~
E
cr(y)}
and thus,
Further,
and
Thus,
and we have
Therefore
y
=
f(x) .
We would clearly like to obtain a theorem of the above nature in
the case of an observable having a continuous spectrum.
We will
present such a theorem for the case of a quite full logic.
The theorem
we have, however, requires a stronger relation between the observables
x
and
y
than the relation
x
~
y
we have been considering.
Althrnlgh we do not have an explicit counterexample, it is quite likely
32
~hat
~
x
y
does not generally imply that
y
~
f(x).
We will show
that our new relation is fully equivalent to the equation
and thus that if
x
~
is to imply
y
f(x)
y
then
~
x
y
y
f(x)
=
must be
equi valent to the new relation which we are about to introduce.
Wi th-
out further ado, our new relation is defined as follows.
Assume that
x
(read
x
~
every
E
> 0 and
y
and
yare observables.
strongly) iff there is an x-null set
such that
s
Intuitively x
~
there is a tolerance
y
J.1.
01
(y) +
€ )
5" .
Val' (x) <
such that
01
x~y
N such that for
there exists
E,
for every state
s
We say that
means that for an arbitrary error
8 > 0
> 0
€
such that the probability that
y
is in
the interval
(y) + €)
J.1.
01
dominates the probabili ty that
for any state
01
particular, if
01
in which
x
is in the interval
(" - 8, " + 8)
can be measured with accuracy
is a state in which
also be certain in that state.
certain as
x
x
is certain then
Even more crudely,
y
5.
y
will
is at least as
x.
Proposition 12:
If
x
and
are observables then
S
x-+y
In
implies
x
~
y .
33
Proof:
Let
E
> 0 and
definition of
exists
x
s
~
for every state
~
y.
where
We show that
x
N is the x-null set in the
~
y
on
~(x)\N.
var~(x) < 0A .
for which
~
For any state
.
~(x)\N
€
> 0 such that
0A
var~(y)
A
such that
=
2
jTs - /.l~(Y)J da(y)(s)
S;
€
QI
(y)( (/.l (y)-I/€,
QI
/.l
QI
var~(x) < 0A '
we have
(y )+\[E))
On the other hand, Chebyshev's inequality implies that
and thus that
Var",{y)
""
S; E
+
411yll 2
1
-""""'""="2 Var (x)
(0)
~
A
Now there
34
~
. [~
1
1. U~ 2 €
} , th en
A
A = mln u A' 4 ----2
Ilyll
which varCl(x) < SA we see that
'Now if we let
state
Q'
and for any
u
Var (y) s:
CI
E + 411Yl12
8
1
(& )2 A
<
2E .
A
It
follows that
x ... y .
Proposi ti on 13:
Let
x
be an observable and
which is cont.Lnuous on the
f:
comp~ement
o-(x) ...
~,
a bounded function
of an x-null set.
Then
x ~:+ f(x) .
Proof:
Let
E
>0
and
N denote the set of discontinuities of
A
E
o-(xJ\N
there exists
&A > 0
f.
such that
jmplies
If
Q'
is any state, we have by Chebyshev's inequality,
If..l (f(x)) - f(A)1
QI
E
< -3 + 2M 1
&2
A
Var (x)
Q'
For each
35
where
If
a
If I
M is a bound for
is a state such that
2M
L
52
on
~(x)
Vara(x) <
Now let
5C,
then
Var (x) < ~
a
),
3
,I
and
I~a (f (x))
Let
C =
~(x)\N
and
y
-' f (), ) I <
32€ .
= f(x) , then
~ f
-1
) (~(y
a
€, ~
a
(y )+ )
€
and
= f (x ) (~ (y) a
Thus,
and
s
x -+ y .
€,
~
a
(y) + E) •
36
Proposi ti on 14:
Let
x s-+ y
x
and
y
be observables on a quite full logic 'such that
and let
a-(x) ... a-(y)
f:
be the map defined immediately following the proof of Proposition 6.
If
> 0
E
and
A E a-(x) is a point at which f
is continuous, then
such that
there exists
Proof:
Observe first that the map
a-(x) ... a-(y)
f:
exists since
s
x -+ Y by Proposi tion 12.
continuous (cr. Proposition
E
> 0
0' > 0
A
and
A E a-(xJ\N.
such that if
a
7)
Also we know that
except on an x-null set
By the definition of
is a state such that
and
then
E
<2
f
N.
f
Let
there exists
is
'-37,
Moreover, since
x
~(X(A - 6~"
for every state
s
~
y , there exists
A + 6~')) ~ ~(y(~~(y)
for which
~
~ - 1
uA. - 4"
6~' ~
- ~, ~~(y) + ~))
Var (x) < 6'S
A
~
0 such that
Let
. l ~, ~ , , I}
nun
u ' uA,'
.
A
We intend to show that
by showing that for every state
~
for which
It also follows that
a(y(f(A.) -
€,
f(A) + E))
l.
The desired conclusion then follows from the fact that our logic is
qui te full.
This is easy, if
then we see that
a
is a state such that
38
Thus
VarQ' (x) .< 0"A
and
But
thus
E
<2
and
It follows that
and thus that
Q'(y(f(A) - E, f(A)
The proposition follows.
+
E))
=
1 .
39
Proposition 15:
Let
x
and
y
be observables such that
s
x ..... y •
Then there
is a function
f:
o-(x) ... o-(y)
which is continuous on the comp:;L.ement of some x-null set and which
has the property that
y = f(x),
y(E)
for every Borel set
!.~.,
f has the property that
f(x)(E) =.x(f-1(E))
E.
Proof:
Let
o-(x) ... o-(y) ~ R
f:
be the function defined after the proof of Proposition 6,
f ( lim 11
n... oo
(x) )
lim
!.~.,
(f(x))
11
n... 00 01 t1
OI n
We first show that
for
If
E
any open set.
For
A
€
E
C denotes the set of points of
continuous then, for each
choose·
o-(x)
€"
> 0
at which
such that
f
is
40
there exists
at > 0
such that
By use of Proposition 14, we see that we can (by choosing
smaller if necessary) assure that
where
But
and thus
s: y(E) •
Now
and thus there exists a sequence
(t.}
l
in
6t
41
such that
r-I(A - E~, ~ + €~)
nC~
U (t i - St.' t i + St.)
i
l
l
(recall that
is separable since
~
is).
Thus,
~ ~ x(t i - St.' t i + St.)
l
l
l
s; y(E) .
Now
E is separable and
thus there exists a sequence
(~iJ
E such that
in
Then
f-l(E)
=
U f-l(~i
i
and
-
€
A.
l
,
A. +
l
E
~.
l
)
42
X(f·l(E»
x(~ f-l(A i - ~h.' hi
1
1
+ EA.»
~ ~ X(f·1(hi - EA.' Ai
+ EA.»
=
1
~ y(E) •
1 1 1
It follows that
Now let
a
€
~ y(E)
X(f-l(E»
for every open set
E.
We will show that
E.
x(f
-1
~
(a»
y({a}) .
If
then by Proposition 14 there exists
Since
f-l(a)
f-l(a)
nC
nC
6
A
> 0
such that
is separable, there is a sequence
(Ai}
such that
f-
1 (a)
n C ~ U (A.
»
i
1
- 6, , A. + 6,. )
~i
1
~1
Thus,
~
yea -
Now
. '-I
x(f ·(a»:s: yea - E, a + E)
€,
a + €) .
in
43
for arbitrary
x(f
-1
E
(a))
Finally, let
thus
~
co
1
1
n y(a - -, a + -) = y({aJ) .
n=l
n
n
E = (a,b)
be any open interval.
Thus
e
E = (- co, a) U [ a} U (b} U (b, + co )
and if
u = (- 00, a) U (b, + co )
then
U is open and
It follows that
Thus
-1)
(U)
U x(f -1 (a)) U x(f -1 (b))
=
x(f
~
y(U) U y((a}) U y(t b })
44
and
It follows that
for every open interval
E.
Clearly
for any open interval (finite or not) by a minor modification of the
above argument.
Since every open set is a disjoint union of open
intervals
for every open set
E.
To complete the proof of the proposition, observe that the
collection
€
.
B(~)/Y(B)
= x(f
is easily shown to be a
~-algebra
which contains all open sets.
A = [B
Since
B(~)
is the smallest such
for every Borel set
B.
-1
(B»}
~-algebra
and thus that
y(B) = x(f
-1
(B»
we have that
A = B(~)
45
Aththis point we summarize the findings of this chapter in one
theorem.
Theorem:
Let
x
and
y
be observables with values in a quite full logic
and let
cr(x) ... cr(y)
f:
be the naturally induced mapping.
(1)
f
Then
is continuous on the complement of an
x-null set,
(2)
the relation
x ~ y
is equivalent to the
y = f(x) ,
functional calculus equation
(3)
the relation
(4)
if
x
x ~ y
implies
x ... y , and
has a countable spectrum and
x ... y
on the complement of the x-null set
N
=
(Alx([A})
=
O} , then
y
=
f(x) •
Finally we note that we do not know, in general, whether or not
x ... y
.i.~.,
implies
y = f(x)
but we rather suspect that this is not so,
we believe that the relation
relation
x ... y .
x -...
y
is distinct from the
46
TOPOLOGY 011
()E3fH\/J~BtE0
MiD:~i{ANGE
OF STATE
When one studies a system he sets up several'experiments and
proceeds to measure aspects of the
finite number of experiments.
sy~tem.
c&n do at most a
He
Eacll expcc;.ri;ncrlc wj_ll consist of
studying a probability distribution on the.real line - or moce
generally on
say
e.
1
~,
N
.
Ttle observer
C[Hl
do this wi th limi ted accurac:,'.
ttl
is the error of the i _. experiment.
This leads us to stU'lY
a topology on ei ther the observables or the ;;tates of our model.
Since both the observables and the states are constructs, it is not
clear which should have the topology.
obtain
11.]
We do
N :experiments and we
. th- experlmen
.
t , and
the mean of the 1
Var. < e
i
1
(we mean
here that we have done many experiments, see for example von
Neumann, 1955).
'1'0 impliment these thoughts on our model, we need to
consider two possibilities.
Are we doing
n
x , and considering
experiments with the sysLe", in oue state, say
n
observables in that state?
would consider as neighborhoods of the state
the
e.
l
Ai
(i
th
x
observable, corresponding to the i
In this case, we
those determined by
th
experiment)
an~
the
By this we mean that
I ,Al.x,
x) -
P.
L
'_
e.
1
determines the semi-norm on the states.
Our other alternative is to consider that each experiment
corresponds to a state,
hoods of observables.
Xi' and that we ar.e interested in neighborWe are thus studying one observable
A and we
are interested in neighborhoods of it.
is contained in a neighborhood of
((B
-*
In this case we say that
A determined by
B
Lx..1. }
A)2x ., x.). < e.
l
1
l
The former construction yields a model in which the observables
correspond to experiments and the state is, in essence, the
the study.
This is called the weak
for example Haag, 1972).
*
obj t:ct
ot'
topology on the states (see
We will not pursue this any fur,tber," In the
latter structure the emphasis is shifted to the observables and the
states correspond to the experiments.
It is this;second model which
is of interest to us.
The next chapter will consider in more depth the compa.risori of
putting the strong operator topology on the observables vIi ttl the TNeak
topology on distributions.
Proposi tion 1:
Suppose
A
n
~
A in the strong operator topology.
'I'hen
A
n
in the weak operator topology.
Proof:
1
((A
n
- A)x, y)
~
((A
1
222
n
- A) x, x) (y, y)
.
This means that if two observables are near eqch other"
TJien
their' average values are also near each other.
Proposi ti on 2:
As before, let
. t eger
In
m,
A
m
Am-"A
n ~
n
~
A
strongly, then, for each positive
s t rong ly .
~
A
48
Proof:
First we note that there exists a real
IIAnl! s: M
M such that
(see, for example, Riesz and sz.-Nagy, Functional Analysis, p. 200).
Then
~ MI. I (An - A)xll + II (A n - A)xll •
Now use induction and the result follows.
Proposition 3:
Suppose, as above, that . ~n
that each
~
A strongly and assume, as usual,
is an observable (~.~., is se~f-adjoint).
An
continuous function,
f:
R
~ ~,
, then
f(A ) ~ f(A)
n
If
f
strongly.
Proof:
Let
domain
M be as above, then we may clearly assume that
[-M, M]
Thus,
f
is the uniform limit of
(stone-Weierstrass Theorem).
f
polynomi~ls
Now, since we have
and since addition is continuous in this topology,
strongly, where pm
is any polynomial of degree
This completes the proof.
Dunford and
m
So now
This proposition may be found in
Schwartz, 1963, (Vol. II), pp. 922-23.
has
is a
49
Thus, we have that we study functions (90ntinuous functions, that
is) of the observable and the proper relations are pres-erved,
We
should also notice that it is important to be able to do experiments
with small variances.
If we ignore variances and consider only means,
we would then _wish to show
weakly,
An ... A weakly
. implies
f(A)
n ... f(A)
We think that this is false,
Parenthetically, we should note that we may always consider
bounded observables,
interested.
If
Say that
A is an observable in which we are
A is unbounded, we are really only interested in
the spectral measure which corresponds to
define functions of
A.
We consider the
for that matter, any homeomorphism
f
interval
f (A)
(a, b)
Further,when
We notice that
t
E
have
Var
x
A < E.
be continuous,
of
from
there exists
< 8
urctan
and
~
function, or
onto a bounded
is defined and is bounded.
& > 0
such that if
x
is
varx((f(A)) < 8 , then we
This comes from the fact that
f
is assumed to
Thus we can be assured of making accurate measurements
A by studying
from a study of
cr(f(A))
l~x(f(A)) - tl
a state such that
A) through which we may
f(A) ; that is, we can, in a sense, recover
f(A).
Actually we can recover
any invertible Borel function of
A
A when we study
A.
We recall that when one does an experiment and gets a certain
result the state of the system is changed,
That is, it is now in a
state such that the new state will yield the same average with the
same observable and wi th small
(z~ro) vad ance.
Of course, one does
not really observe an observahle with a continuous spectrum, he
observes discrete values
(!.~.,
something is in an interval or it is
50
not in the interval - a question).
fuller discussion of' this.
See von Neumann,
19~i5,
for a
We will discuss this at greater length in
conjunction \vi th the discussion of classical models'.
We now prepare to consider some interesting points that deal with
(observations) observables causing a change of state.
Before doing this, we will first consider an example of a
rJf't
which converges in the strong operator topology to the "position"
observable.
This particular example is quite important due to the
fact that all observables are (in a sense to be made precise below)
position observables.
The following notation will be used in sequel.
Let
A be a
self-adjoint operator with a continuous spectrum on the Hilbert space
H.
Recall that we may resolve
~(E)
wi th
f
H with respect to
= (PAE
being a cyclic vector for
is not uni que si nee
unitary operator
U:
f
f, f)
A.
is not uni que.
H -+
A, so that
In general, the
meaStLrt~
0-
Moreover, there c:,l,,_
.£2(-IIAI\, lI Anll) such
that
UAU-l(f)(t) = tf (t) •
This is, of course, a result of the spectrum theorem.
As was
mentioned above, we may view the spectral theorem as -celling us that
in our model, every observable is a position observable - that is,
every observable is uni tarily equivalent to a posi tion observable (\ve
cannot distinguish between two unitarily equivalent observables).
When we ask where something is, we really ask whether or not it is in
51
some Borel set
observable,
This leads us to consider the two valued
E
o on
PE ' which is
and
on
1
E,
~ (t)'f(t) .
This is the spectral
above.
meq~ure
corresponding to the operator
A
This is important since as we well know, every measurement
corresponds to, the position of some object.
(See Mackey's
"Induced Representations of Groups and Quantum Mechanics", 1968,
pp. 62-63, for a discussion of the nature of the position
observable.
Also, see Nelson I s llDynamical Theories of Brownian
1967, where the correspondence between observables and
Motion",
posi tion is mentioned.)
We now proceed to the example.
Proposi tion 4:
Let
I
~ ~
be a closed interval and
probability measure on
given by
partition
(Af)(t)
P
of
I
Let
tf (t )..
I
J'I[tf(t)
where
X.
l
a continuous Borel
A be the operator on
Then for each
E
> 0
such that if
E
is any refinement of
~
P
=
(E.}.
Z
l lE
then
- I: . a.c.X.
(t)J2d~(t) < (2
l
.J. l
2
S,
(I,
~)
there exists a
52
C
[j'E. f(t)dCJ(t)]/er(E i )
i
l
[j'E.tder(t)]/o-(Ei )
ai
l
Proof:
First we show that for each
that if
E = (E.}
l
refines
E >,0
there is a partition
P
P then
(1)
with
for
c.l
andi
X
i
JCf(t)
E
as above.
To see this, let
Z and note that for each f
E
~2(I, er)
- E c.x.]2der
(t) =(f - E
c.X.,
f - E
c.X.)
II
.
II
.
l1.
l
1.
2
(f, f) - 2 E c.(X., f) + E c.er(E.)
l
.
l
1.
1.
1.
2
2
- (f, f) - 2 E
c.er(E.)
+ E c.er(E.) = ( f, f ) .
l
l
.
l
l
l
=
l
~ fE. f(t)2 der (t)
l
= E
.
1.
-
r
E.
~
i
1.
cjj'E. f(t)der(t)
J
[f(t)2 - c.f(t)]der(t)
l
l
= ~ JE.f(t)[f(t) - ci]der(t) .
.
But if
l
A
E
1.
E.l , then
2 (
~ c.rr
l
.
l
.
E. )
.l
so
53
(2)
If(A) - cil.= ,\[SE.f(A)dcr(t) - JE.f(t)dcr(t)]/o-(Ei )\
1
1
lIE. [f(A) - f(t)]dcr(t)I/cr(E i
)
l
[IE. € dcr(t)]/cr(Ei )
=
€
l
Thus,
As a consequence, we have for
I [t
f(t)
=
t ,
2
2
- L: a.X.] dcr(t) < €cr(1) •
l
1
Observe that our proof above works for any refinement of the
specific
for
{E }
i
t , t
l
2
f € :2(1, cr)
(E.} .. Z
l
l€
chosen,since the crucial property is that
in the same member of the partition.
and
€ > 0
is any refinement of
E. ~ f
l
for each
and choose a partition
i € Z.
-1
P
P
[(i - l)E, iE]
Now recall that if
2
so that if
then (1) and (3) hold and
a, b
2
(a + b)- ~ 2(a2 + b )
and thus
Now fix any
E ~
then
,
Jl t f (t)
(t) . ~ I", l. tf - I:
D.,
1
fX.
" L a. fX.
1
J.
-
1
J
I
"...
::;
-z
~'.J·L·L
,C
J.l \t
,. J' (
2 L:
_ \,
,2
I
L
--'
r)
(t
a, )X. )"dCJ(L)
-
1
lE. f(t).2(t
i
c .x. -l~'·I.·i
. 1
- a )2dCf (t)
L
L
+ 2I:a;SE,[f(t) - c i
J2dCf(t)
1
By (2') for
:f(t)
:=
Similarly, for each
t
we have for each
tEE.
1
,
we have
tEE.
1
T'hus,.
and
(t) < c
2
2 I'
2
L: a.er\E.) ~ c L
:I
1
i
0
(E.
)
. 1.
L. 'I. C. X.. r-cio
j
1
i
'.')
,,-)
- I, a.c.X.
(t)j"'j\T(i)
<
] 1 l·
.
("-lll.1.'I"I':'
.
(J (
r )] .
T)w pI'oposiUon foLlo\13.
He=
'W\:I
introduce a bit of notation.
of the interval
For each partition
define
I
t,y
where
[J'E. tdu(t)]/u(E i
ai
)
l
and
C
[JE. f(t)do-(t).J/u(Ei )
i
1
are defined as before.
If
F_~
.I:!,;.
is defined by
1
FE.f
[[J~.f(t)du(t)J/cr(Ei)}XE.
=
1 1 1
then we see that
is a projecti on on the (;:e djmensi onEil
FE
J.
spanned by
X .
E
and
.l
A
'E;
'3ub:~:c.·.·(
56
(compare with the remarks introducing Proposition 4).
each
E,
~
!..~.,
is an observable,
~
Note that for
is a self-adjoint linear
transformati on.
Proposition 6 may now be rephraqed as follows:
Proposi ti on 5:
The net
[~}
converges to
in the strongest operator
A
topology.
Proof:
We assume that
(f.}.
h.
l lE
converges.
is a sequence in
We show that
A - A
E
i
semi-norm determined by
E
> 0
If.}. h'
l
there is a partition
2
(r, 0-) such that
goes to zero in the
Thus·we must show that for each
lE
P
l
such that
(1)
for each partition
E which refines
P (cf. Sakai, 1971, p. 34, for
the definition of the strongest operator topology).
it is useful to show that
(2)
for each
n.
To see this, observe that
':There
ai
- CJE. tdo-(t
l
)J/ O-(Ei
.
)
Before doing this
f:7
),
and
c?
= [J'E . f n (t)d~(t)J/~(E.)
l
l
l
Thus,
~ IIf 11
n
Thus, (2) ho14s.
2
L: ~(Ek) = II f
k
n
2
II .
We now prove (1).
Let
E
> 0
and choose
N
E
h
such that
co
L:
n=N+1
2
IIfnl12 < E/2[1 +~!IIAII + IIAII J
Then
N
:s;;
I;
n=l
co
0
II(AE -
A)(f n)lI"-
+
L:
n=N+1
[Il~fnll
r
I-
IIAf'!)1!]'"
58
N
L:
lI(A
n:::.:l
E
- A)(f
n
)112
+
CD
2
IIAI1
+
t
L:
11"j\J+ 1
N
< L:
I!(AE - A)(f'n)11
n=l
Now for each
1
n
~
f'or each refinement
Proposition
4).
~
2
E
+2
N choose a partition
E
of'
P
n
n
such that
(that this is possible ip due to
The proposition clearly follows since we now see that
Since we have observables such as
.
observable
P
~
near the position
,
A, it is interesting to ask the question as to whether
the new states arising from measurements of the
some appropriate
topology~
~
.'
will converge .in,
If so, then vie could define the limit of
this new sequence to the new state resulting from a measurement of
A
•
It appears that this is a futile program but \ve are unable to produce
the necessary counterexample.
.~
We wili discuss this in more detail in
the final paragraphs of this Chapter.
conc~rned
The next five proposi tions are
with elaborating these ideas.
Note that
~
is not strictly speaking an observable\vi th a
simple discrete spectrum, but we can modify its lightly so that its
modification has' a simple discrete spectrum and in such a way that any,
'.
59
limi t of the net is not changed. . For eachparti tion
E
{E.}
of
l
I
let
A(E)
max
m(E.)
l
ki~n
where
is th~ Lebesque measure of
m(E.)
. l
E..
If
l·
X.l =
~.,
is the
K
l
E , then
characteristic function of
2
of . £ (r, 0-)
[XJ
i
and, for e,ach
is an orthogonal. ~~ubset
E, we choose an orthonormal ~)asis
,
for the orthogonal complement of the subspace of
I::J
£~(l,
0)
Finally let
P
where
.
£2(r, 0-)
is the projection of
q;>j (E)
dimensional subspace spanned by
q;>.(E).
onto the one-
Observe that
J
It is cl5!ar that
lim A(E)
=
0
E
and, thus, the nets
{~}
(BE}
and
have the same limits (in norm),
We now oonsider the new observables obtained after a
.Let
state
U
E
be the new state arising from
U.
BE
given that it .was::
Then
n
L:
i=l
Let
tr (UPX. )pX. + L: tr (UPrI'lJ' (E) )1'rl'l:). (E)
l
l
j
...
'I"
measur~menL
'1" •
60
n
DE
=:
.L: tr(UP X. )P ..
x1
i=l
~
We first show that the sets
CUE}
and
(DE}
have tl;1e same limi ts in
the .weak topology..
ProEosi ti on 61
LetQ
parti tiOD
be any projection and
such that if
EJ:
E
E
> O.
Then there exists a
is any finer parti tion then
Proof:
~
First write
D·=;.L: aiP .
f
r-2
projection of
may assume that
where
a
i
1
~
onto the subspace. spanned by
'.
co
IIfill =.:1
i;lnd that
P .
and
0
is the
f
1
.L: a
n=l n
f..
1
CJ,;;.rly we
converges.
We have tbat
.L: tr(DPcp. (E) )tr(Pcp. (E)Q)
j
=1
J
J
- ~ tr (DPx.) .
~
To show that the latter is less than
choQse any
case when
f
D
n
~
€
for appropriate
,re-label it more simply by
= Pf
.
Then
E', we
f , and consider the
61
=
((pfP
Xi
)(X.),
J.
. 2
x.J. )!lIx.1I
.
J.
:: ((X., f)f, X. )!r:r(E. )
1 1 1
and
(f, f - I: [(r, \)!r:r(Ei)]X )
i
i
s: Ilfll'llf - I: [(f,
i
X.J. )!r:r(E.J.
)]x.11
J.
=llf - D
C.x·1I
.11
1
where
c.J. = (f, X.J. )!cr(E.)
::: [j'E . f(t)dcr(t)J!cr(E.)
.
J.
.
1
J.
By (1) of the proof of Proposi tion If we see that there is a
parti tion E *
of
I
such that whenever
Thul? for any refinement
E
of
I: tr(P f P (E»:::
j
n CPj
N
such that
refines
E*,
E*
\I'f
- E c.x.1I
nil 1
This concludes the proof for the case that
case,' choose
E
<
E
U::: P
f
.
n
For the general
62
Apply the proof given above to obtain a parti tion
every refinement
E
of
E*
and for
1
s: n
:s;;
such that for
E"
N ,
€
N
2( I:
n=l
a +1)
n
Thus,
I: I: a
n
j n
N
;::
I: [ I:
j
i
n=l
an
co
tr (p f, P . (E»
n cpJ
a tr(P P .(E»]
+ 'I:
f n cpJ
n
n=N+l
co
N
:s;;
I:
a
n=l
t
a < E
tr (pfP . (E) ) + I:
n
n
n tl'J
j
n=N+l
.
From this we see that
tr(CU E - UE)Q):S;; ~ tr(upcp.(E»
J
<
E
•
J
The proposition follows.
Proposi tion 7:
I
Let
I
I
A be an observable such that
exists for ef).'ch projection
Then
Q in the spectral resolution of
A.
63
exists and
(~}FEB(~)
where
is the projection valued measure of
A.
Proof:
Since
B
~
Q
E
Thus if
A
..
is,. self-a,djoint there is .a projection-valued measure
such that
E
>
0
then there is a parti tion
that for any refinement
E
of
E*
of
[ -"All,
IIAIIJ
such
E*
We 4ave assumed that
.
..-
eXists'fQr each
exists.
Ei
'
&nd we wish to show that
We Will show that
(tr(U:iff)}
showing tha.t it has a limit.
partitions of
[-IlAII, IIAII]
is a Cauchy n<2t tlJerer,y
To do this assmne,that
and consider the inequality:
(1)
III
Iii
+ltr(UF(~ A·QE )) - tr UK(I. A·QE
1
F
l
i l l
i
)1
and
K
are
64
ill
0"
+·ltr(UK(~ Ai~i)~o-' tr(U0)1,·
We wish to show that each o.f the three swnmands on the right-hand
side of the inequality goes to O.
n°
Itr(UK(i Ai~i)) - tr(U0)lo= 'i:i Ai j:l t:r(UPKj)tr(PKjQEi)
ill
(2)
First observe that if
ill
n
-
0'
.L:
J=l
I
tr(UPK)tr(PK.A) I
J
oJ
n
m
tr(UP K )[ L: A.
j=l
j
i=l l
L:
The latter inequality is independent of
Finally,
°
K', thus
(4)
and since the map
J ...
tr(UJ~.)
is a Cauchy net (for each
l.
have that, for each
Ri
i , there exists a partion
:i
),
we
such that
Itr(UF~ ) - tr(UK~.)1 < ;m
.
i
l.
for arbitrary refinements
common refinement of
than
R
l
F
,
R
and
2
,
i
R .
K of
m
If
••• , R
R be a
Now let
F
and
K are finer
R then
Adding the three inequali ties (2), (3), and (4) and substituting. into
(1), we get
TtlUS. J ... tr(UJA)
is a Cauchy net and hence has a limit.
inequality (2) shows that for each partition
Moreover,
K,
and, 'thus,
The proposition follows.
In the next proposition we assume
O'
is a continuous measure.
66
Proposi ti OQ 8:
If
'-J
P:
j'-
~J...
Ci, 0') ..t S-
(T, 0)
pCl)
is a projection whose image is a finite dimensional subspace
of
~
2
~)
(I,
then
lim tr(UEP)
E
~
0
Proof:
---For any partition
E
J
(E.
. l
=
of
let
I
x.l =
denote the
X--E.
l
characteristic
function of
_
E.l , let
be the projection of
P
xi
outo the one dimensional subspace spanned by
x. , and let
l
P
l
~2
be
defined by
Ii'
f'¥.
- l
l
J
i3 an orthonormal basis of the orthogonal complement of the
spanned
by
.
'UU2pace of
tr(UP. )
l
X.l , the observe the
= L: ((UP.)'f.,
'¥.) + ((UP. )(X. ), X.)
l
J
J
l
l
l
j
and
(UX., X.) •
-l
Thus,
tr(UP.)
l
~
(UX., X.)
l
l
tr
(UP x. )
l
l-
Now suppose
and let cp
P
is a projection onto a one dimensional subspace
be a unit vector in
PU?).
We have for any parti tion
I,
of
(~ tr(uP . )P . (cp), cp)
X
X
l
l
l
NoW; we claim that if we define
for each Borel set, B ,then the map
is a probability measure.
continuous with·resp~ct to
that if
o,
~(B)
P == 0
B
Now if
Bet and
E
~(B)
>
~(E. )
l
< 6
R
To see that this is so, simply n0te
0 ,
(~
B
there is
6
for every
>
Since
IT
i.
tr(UP. )
l
such that if
0
tr(UP ) <
B
such that if
I
almos t everywher~
tr(UP ) = 0 •
and
< 6 , then
of
this measure is absolutely
then
Measure Theory, 1950).
parti tion
~.
= ~(t)f(t) = 0
(PBf)(t)
and thus
Moreove~
E
B
is a Borel
(see page 125 of Balmos'
is continuous there exists a
E
is any refinement of
'l'hus, for any
E
which refines
I
.~~.
thi:n
E
68
and
~
l
tr(UPi )j'E.CP(A)2 dCJ (f.J
l
~
L:
i
E
tr(uEP) < E
It follows that
J:t<~
cp(A)2 dO-(A)
i
E •
. .
and
lim tr(UEP)
=
0 •
E
Finally if
P
is a projection onto a finite dimensional subspace
and
j.s
spanned by an orthnormal set
••• ,
rn..
1 then
'Y[l-
and
n
L:
lim tr (UEP )
E
CPi
i=l
o.
Theorem 1:
i
Let
~ =S,
2
(I, CJ)
any observable on
~
CJ
where
is a continuous measure.
If
A is
whose spectral resolution is a series
00
A
L:
i=l
in which each of the projections
A.P.
l
l
P.l
has a fini te dimensional range,
then
lim tr (UEA) = 0 .
E
In particular this is true for all compact operators and all traCE: .
class operators.
Proof:
It follows from Proposition 7 tha.t
lim tr(UEA)
E
where
>A
{PBht: B(fil.)
UE,P~JJ
j\d[lim[tr
E
=
~
is the spectl'al measure of
A.
But
lim
E
and
by Proposi tion 8.
Thus
00
lim tr(UEA) = L: Ai lim.tr(UEP.)
E
i=l
E
l
O.
The theorem follows.
Pro;eosi tion 9:.
As before, for each partition
Xi
=
'»:.
and
PX.
].
].
spanned by
X.
].
uE
and
(E }
i
=
of an interval
the projection onto the subspace of
Define a map
(P.f)(t)
].
Let
E
=
P.:
l
'2
So
... So
X. (t)f(t) .
].
U be the operators given by
E
"'.
~.
DE
~ tr (UJ\. )px.
l
~
= L:
i
].
tr(UP. )P
l
l
xi
2
by
I
S?(I, (J)
let
If
Q:
s.2 .... ;.2
is a projection, then ei ther both the limits
eWc;l
lim tr(UEQ)
rj"l
exist or both fail to exist.
Moreover, if both limits exist;. then
they are equal.
Proof:
Let
[~j}
denote the orthonormal basis of the orthogonal
cOmplement of the subspace of
=
2
~ (I, ~)
L: [( Q~ ) (X. ), X.)/
1·
1
i
(X.} .
spanned by
I' X.111'2-J
1
+ L:
j
C)
x. )/lI x·II'-
= L: L: tr(uP ) ((QPX ) (X.),
i
k
k
k
1
1
,:;,
= L:
.
1
=
and thus
tr(UP.) (QX., X. )/llx.ll~
111
1
L: tr(UP. )[ (QX., X. )/~(E. )J
1111i
1
Then
71
= E tr (up.).
i
l
(QX., x.)
(QX., 'X.)
l
~
~ (E ) ~ ~ tr(UP . )
(J(E.
)
x
(J i
l
l
~
f [tr (upi)
$
(QX., X. )
(J(~i ) ~
- tr (UP.\)J
E [tr(up ) - tr(UP .)
i
Xl
i
= 1 - E tr(UP
i
Xi
).
But ip the proof of Proposition 6, it was shown that there is a
partition
E*
sQ
E of
that for every refinement
~ tr (up'1') <
J
€
E*
•
J
(Note that in Proposition 6 the family
['1'j}
is denoted by
(CP,i (E)} ).
It is clear that
1 - E tr(UP . )
X
i
~
~ tr (UP'1' . )
J
J
thus we see that
I:.r0;E0-?i ti ~
If
measure
A is the multiplication operator on
(J
and if
P
is a projection of
then there is a Borel subset
f E l
2
(I, cr) .
F
of
I
2
~,
flu,
0)
such that
such that
Pf
==
for som(~
FA
X.F·f
== A}'
for all
72
Proof:
We will first show that
P(f)
= f· P(l)
for all polynomials
and use the fact that the set of all polynomials is dense in
to conclude that
that for each
P(f)
= f'P(l)
for all
-
2
f E S. (0-)
f
s.2(0-)
But observe
n
Thus
for all
n.
From this i t is easily seen that
all polynomials
f
Thus
P(f) = f· P(l)
p(f) == f· P(l)
for every
f
'1'01'
i~(o-).
E
In
particular'
P(l)
and thus
Thus
P(l)
P(l) = ~
(P'P)(l)
P(l)P(l)
P(P(l))
is a function having only
for some Borel set
F
0
and
and
P(f)
1
as its values.
~'f
.
The author is indebten to Stephen Campbell for shovring him the
,)roof of this "well-known" proposition (Proposition 10).
Proposi tion 11:
Let
B
be a Borel subset of
,)
01:" (I, 0-)
,
into
2
£ (I, 0-)
~
defined by
and
P
B
the map from
73
If
P
r2
is a projection of
Borel set
B
which commutes with
P
B
for every
then
exists and
for some non-negative function
subset
L
of
~
such tqat
f
E
rl(K,cr)
and for some Borel
Pg
Proof:
By the previous proposition we see that
some Borel subset of
P
=
P
where
L
L is
By Proposition 9 we see that
.~
lim tr(UEP) = lim tr (ifp )
E
E
(P(XE.), ~.)
~ tr (UP . )
E
i
1.
is a partition of
I
1.
l
and if we take
a refinement of this partition we see that for each
L
n E.
cp
l
or
L
n E.l
=
E.l
E
i
=
{E.}
l
either
to be
Thus
lim
2:
i,E.~
E
1
1
is absolutely continuous with
Since the measure
respectto(J
tr(UPE . )
we know that there exists a function
f E~l(I, rr)
such that
for
~very
Borel set
B.
It follows ·that
Thus,
lim tr(UEP)
=
J~fd(J
E
apd the
~roposition
follows.
Theorem
2;
,
i
If
~
is the mUltiplication operator on
lim tr(UEA)
E
e;Kists.
Moreover, lim tr (DEA) '"' tr (UA)
~
<to ~
E
Proof:
If
A is the multiplication operator And
A
= J).dP~
~
2
(I, rr) , then
75
if the spectral resolution of
A
(PB}BE E(~)
measure
A then the projection values
is given by
(P~f)(t) = ~(t)f(t)
If
B is
W1Y
fixed
.
Borel set it is clear that
all the other projections
~ for F
€
B(~).
~ commutes with
Thus, for fixed
B,
it follows from Proposition 11 that
exists.
This holds for every
exists.
Moreover,'
B, thus Proposi tion 7 implies that
where
for every B E
B(~,).
The Theorem follows.
At this point we summarize and interpret our findings.
Throughout this thesis we have viewed the projections as being
"events" in our generalized probabili ty' theory.
As in ordinary
probabili ty theory we are interested in the set of probabili ty
measures on the event space (see the introduction).
theorem,
th~
Now by Gleason's
set of probability measures on closed subspaces of
Hilbert space correspond to the set of P9sitive definite trace class
one operators .. We have interpreted
th~se
operators as being the
states'of our thlilory.
In this chapter, we have supposed that we are given a physical
system whose state is represented by the trace class one
op~rator
U
It is assumed that a measurement is made on the system and that, as
a result, the system is forced into a new physical state.
It.has
been our object: to determine the trace class one operator corres",:
ponding to this new state.
Our point of departure has' been to notice'
that von Neumann (1953). has already defined a "new state" operator
provided one JIlakes only observations whose corresponding observaples
have a simple discrete i;ipactrum.
We are also motivated by the
Compton-Simons experiment in which the position observable was
measured.
Prom the conclusions of this experiment von NeUJt1ann
formulated a new axiom.
This axiom asserts that
~hen
a measurement is
carried out, the sys,tem is in a new state which has the property that
its mean is the value obtained by the measurement and its' variance'
is zero.
The basic idea of the latter part of this chapter is to deterJlline
whether or not there exists
operator
VA
a positive definite trace class one
corresponding to the change of physical
when one measures the position operator
whether it was possible to define
A.
sta~e
We wanted to
obtained
de~erndne
UA . in such a way that the process
changing state is continuous in a certain weak topology. . To
aCQomplish this we found a_net
spectrum converging to
(BE}
of operators with discrete
A in the strong topology.
We define the
77
change of state operator due to the measurement
U
E
BE
to be an operator
as did von Neumann (recall that the spectrum of
We then attempted to find an operator
U in the same weak topology.
A
BE
U such. that
A
is discrete).
lU }
A
has limit
(The topology which we have been
referring to is the weakest one defined on the state space which makes
the functions defined by
v ...
tr(VB)
continuous for every observable
exis ts.
Theorems
1:.
and
U exist$ and
A
To make it absolutely clear that
g imply tha t there is
~
Theorems
1:.
We have shown by Theorems .:!:
B.)
and
g
££ such
UA
~
will assume that
to ge! a contradiction.
By
Theorem 1 we see that
B such that· B =
for any observable
then
for every
~.P . .
l
B = LA.P . •
l
l
l
But
and thus is itself a trace class one operator.
tr(UAUA) = 0 ,
and· U = 0 •
A
Thus if
U exists
A
should be a state
Thus, for
B= U '
A
On the other hand, Theorem 2 implies
that
for appropriate
U
A
=0
and
f
.
tr(UAA)
Thus
f
tr(UAA) = O.
0 , a clear contradiction.
He can only conclude that :i,f
Thus we have that
78
exists for every observable
B
measure on the event space.
Observe that the mapping
B ....
then it will not arise from a
lim tr (DEB)
E
is a linear functional (provided that the limit exists for all
and thus it can be viewed as a IIgeneralizedll state.
For further
discussion of these states and other matters, see Sa.kai (1971).
B )
79
APPLICATION TO CLASSICAL PROBABILITY THEORY
The causal view of the world is that there is some Borel space,
x,
of impossibly complete descriptions of states, such that when a
point,
x,
is chosen in the space
X, it corresponds to an
impossibly complete description of some state and the results of any
observation of the system are .predetermined.
Now, as we all know, we
cannot actually attain such a description, we can only in some sense
approx~mate
it.
So we assume we can factor
X into
way that we can determine values of states in
AX B in such a
B and that we have no
exact information regarding the description of states in
A
Thus,
in some sense, we assume that we can "split" the space into a part
where values can be determined and another part
not determined.
Since in
A whose values are
A, the values are not determined, it is
natural to associate to it a probability structure.
Now an
observation or measurement is usually a number of an a-tuple of
numbers.
n
So we ideritify the results with elements of
Thus, to synopsize we assume that
X
=
~n
for some
A X B and that there is
a function
h:
X ...
Q,
n
•
We will develop another viewpoint of this collection of ideas,
starting With the same idea of the space of impossibly complete.
descriptions.
In the end, we will have a measure space in a sense,
and there will be a natural association between the classes of
experiments and the measures on one hand and between the observables
and the real Borel functions on
X on the other.
B
80
So let us assume that there exists such a set
X of
impossibly complete descriptions and that a point of
determines the results.
of nature are causal.)
X again
(This is, of course, assuming that the laws
Now let us make the following observation.
When we set up a class of experiments, we are in effect partitioning
X into a known (almost known) and an unknown part, and further we
then have a probability measure corresponding to our lack of
i nformati on ab ou t the remai nder of
Since
X
=
X.
A X B , we fix some
some probability measure,
I-L
b '
on
b
€
A.
B which in turn determines
Thus, since we cannot really
find the point
b
"exactly", we -should also have a probabi l i ty
measure
B
This yields a candidate for a probabili ty
~
measure on
on
X, namely
We could consider that we can enlarge
B at
Als expense, and hence
we really should consider measures on
X itSi31f.
We are led tn a natural fashion to make the followi'ng con··
vention.· A clapS of experiments will correspond to a probabi l i ty
measure on
.
X.
The variables whose values are of interest to us and
which we measure are real valued Borel functions on
X·.
We do not,
of course, know if all measures correspond to classes of experiments.
We now proceed to develop a Hilbert space, caLl,.ed the space of
cr-functions on
X and which we denote by
recover all probability measures on
H(X)
From
H(X)
we can
X and it yields a natural way to
interpret the cor-respondence between classe$ of experiments and the
81
probability measures and between the observab1es (random variables)
That which follows is from E. Nelsan (1969) and
and Borel functions.
is really added fQr completeness.
(f,~)
We consider equivalence classes of ordered pairs
is a real valued Borel measure on
~
val-ued function on
such that
X
f
and where
X
E
where
is a complex
l'
The relation ....
£2 (X, ,u.).
is
defined between pairs of such elements as 1'o1101'l's:
(f J jJ.) ....., (gJ v)
iff there exjsts
A
» \)
J
real valued Borel measure
8
~
such that
A»
~
J
and
~
,(A a).mos t everywhere) .
Claim:
....
is an equivalence relation.
Proof:
vIe need to SIlO\: t:llil:~j
is a Borel measure
"j
AI
j-
»jJ.
(",):
s Lndeoenclent of
and Ai »\!.
( X~'.El'
Let
A"» AI
and
A"» A ,
~'~'J
}"
rhen He show that
)
A'
(A ......a. -e. )
+
Assume
A ~ A".
Then
A
82
The rest of the proof is clear.
An equivalence class of such pairs is called a cr-function and we
wri te
fYdJ.L
for the class
(r, IJ.).
We denote t,he set of cr-
functions with finite Borel measures by
We can make
H(X)
H(X) •
into an mner-product space a,nd we will show
that it is complete with respect to the inner-product.
finiteness of the measures to do this, which for us i
We use the
q
no limitation,
since we are interested in probabilities which a,re finite measures.
Addition is defined as follows:
where
A»
IJ.,
A» v.
This is clearly independent of the choice
of representation and also of
Thus
H(X)
X.
If
~
,
then we define
is indeed a vector space.
Finally we define the inner-product,
(,recall that the measures are all real).
~t
a E
«f, !J.), (g,v», by
This is well defined, ....,
i: -e. ,
is clearly independent of choice of representatives and of
A.
83
It remains only to show that
H(X)
is complete.
Assume that
( f ~ } 00
n
n n= 1
is a Cauchy sequence.
n ,
We first find
)., »IJ,
n
for all
~.g:,
(this is ,\There
He
used the fini teness of the measures).
is a Cauchy sequence in
!2(X, A)
Thus,
and thus it has a limit, say
!2(X, A)
~ in
f
and we notice that
lim f
n
in
A such that
vdil n
n
H(X)
Let us consider some examples.
Ilni te set.
Then
is clear that as
tractable.
(0, 1).
sort of
H(X)
=
en
Let
X = N , then
If
X becomes bigger,
We believe that
H(X)
X = (1, 2, 3, ... , n}
H(X)
H(X) =
rapidly
.e 2 (1N).
become~
It
un-
is not separable in the cas'e
We will see later that in some sense,
a
A is
H(X), will be some
2
L , but that the measure will be a direct sum of measure~
and the space a product space.
Let us proceed to consider random variables on
X,
a random variable or observable is a Borel function from
Ordinarily
X into
~~.
In this case we need to extend the concept to take into account the
fact that we do not have a fixed measure on
X but an entire family
84
8*(X)
of them.
Here
8*(X)
valued Borel measures on
denotes the set of all finite real
X.
We define an observable or random
variable in this context to be a family
11 €
A»
a*(x)
11
is a Borel function from
h
)
11
X into
bt
and where
implies that
h
almos t everywhere) .
(11
11
We denote the set of all random variables on
to show that
is an algebra over
<9
defined operations.
the set
IR,
X by
(9
It is trivial
with the usual obviously
For the purpose of simplicity we extend
<9
of all "random variables" which are complex-valued)
<9
(§
is in
(hll )Il€8*(X)
Borel map on
above.
where) for each
(hll ) 1l€8* (X)
Now
iff) for each
(h
X and the family
<9
)
ll
is an algebra over
multiplication of elements of
11
)
h
11
1:.'~')
is a complex-valued
is subject to the condition (* )
C since we may admit scalar
by complex numbers.
<9
to
Moreover
(§
admi ts an involution namely complex conjugation of the elements of a
family belonging to
Thus
(§.
More than this is true.
a norm
for all
lion
11
Actually
<9
<9.
x
E
<9
We define
is a
(§
C*-al~ebra) !'~') there is
and
Ilxx*1I
for all
is a complex algebra wi th J.nvolution.
such that
(§
in
x) y
(§
'1'0
IIxll
see this) let
II hII H(X)
by
2
h
(hll )Il E 8*(X)
be any element of
85
(Actua1~
& doesn't
consist of all
(h)
-
must be cut down to include only those h
finite.)
as defined earlier but
/.1
One must check to see that
(§
for whiohllhllli(x)
is indeed complete with
respect to this norm and that the conditions required of a
algebra are met.
In particular we will show that
Ilhh*II
=
sup
/.1
= sup
/.1
')
==
L
Ilhil
•
sup
II f"/dZLil =1
s~
II Ndilll =1
J(h h )(ff)di1
/.l /.1
-1
=
sup
/.1
~ sup
/.1
sup
II:Nd';.LII =1
sup
-1
J(h h )(ff)2(ff)2di1
/.1 /.1
[J(h
"f\{d;:ill =1
h )1
.1
1:
ff )di1)2 Cj(ff)di1]2
/.1 /.1
1
=
sup
/.1
sup
11:f\fd.i111=1
((h
Thus,
and since
Ilhll
==
Ilh*11
we see that
h )f\Idij, (h h )f\f'dil)2
/.1 /.1
is
/.1 /.1
C*-
86
Thus
(§
is a C*-algebra apd is merely the complexification of
the real algebra of observables,
is actually a W*-algebra,
~.~.,
(9.
We wi 11 show shortly that
(§
that is is a C*-algebra which is the
dual of some Banach space (see Sakai (19'72) for an exposition of W*algebras from this point of View).
Befo~e
doing this we wish to draw
a parallel between the causal description of reality we have been in
the process of describing and in the description of reality as
presented in previous chapters.
As we have already indicated the observables in this context of
this chapter are fami lies
of the operator on
The operator
only for
h
A
h
E (9 ,
H(X)
state
01
is
h
IJ.
For each SUQh
h,
let
~
be
defined by
may be defined for any
indeed, if
Moreover, the elements of
and
h ::: (h ) •
h
H(X)
€ (9
h
€
then for
~
but is self-adjoint
A»
IJ.,
A.»
\) ,
are the states 9f this theory and if
= (h) is an observable, then the mean of h in the
IJ.
In our operator theory we had
which shows the consistency of the parallel between the two theories.
We now show that
~
an algebra over
is a W*-algebra.
~
and thus has a trival involution
Moreove~ it is a C*-algebra since
algebra of
~
J
First observe that
L.,:::.
Thus, to complete the proof that
is and since
~
bas the required properties.
~
~
h)
~
the norm of
J
(h*
is a *-sub-
is a W*-algebra, we have only to
show how to obtain a space which we will show is the predual of
We call this space
1
1
The definition of
.
H(X).
parallel manner to that of
X
Borel measures on
g E
via
r 1 (X,
(f,~)
that
A»
Thus
1,1
(f
V)
"'" (g, V)
,
~
g
1
1
~
Suppose
and further that
and
f E rl(X,~)
~
dA
dv
= g-
dA
("
is the set of equivalence classes of
1
1
1
f E S, (X, ~) .
X and
finite Borel measure on
shm'ls that
V are finite
and
and
We define
"'"
A such
iff there exists a finite Borel measure
f -
~
procedes in a
are real-valued functions).
A »V , and
almost everywhere).
(f, ~)
for
~
a
The procedure which
is a Banach space is carried out just as wi th
0"-
functions.
~ let
t
It is easily shown that
t
Now if
h
E
linear functional of
(g, v)
are in
1
L
1
L , then
h:
h
L
l
is
~
~ ~ be defined by
is well defined, and is a continuous
It is linear, since if
(f,~)
and
88
cW.
( f dA +
(f, /-1) + (g, \)
(for
d\)
g dA' A
).» /-1 , A » \) ) and
=
,d/-1
dv
-- + g --JdA
dA
d).
Sh A[1
Similarly,
Conversely, let
£ be any linear functional on L1 .
'I'hen, for each
/-1 , the map
is a linear functional on
~l(x, /-1)
and thus by the Riesz
Representation 'I'heorem there is a Borel map
that
h
/-1
in
F(X, /-1)
such
Now we show that
to be an observable.
(hJ.L )J.LEB* (X)
satisfies the
require~ent
needed
A» J.L , then
Let
1
1
L (J.L) £; L (v)
and on
L 1(J.L )
we see that
1, (
(g, J.L ) )
Moreover,
and thus
dJ.L
)
= 1, ( (g dA'
A) =
Since the latter equation holds for all
(J.L almost
everywhere).
.
Thus,
g, we see that
h = (hJ.L)
is in
that
1, = 1,
is
isometric isomorphism from the Banach space
~l
h
IS
h
A
= hJ.L
and it is clear
One now shows that the correspondence
Banach space of cont:ltlUOUS linear functionals of
IS
l
L
onto the
To do this,
recall that the R.iesz theorem gives an isometric isomorphism between
the dual of
and
ex:)
S- (X, J.L) •
is the essential supremum norm.
Moreover, the norm on
'rhus wi th minor modifications of
this proof.. We can show that the correspondence
(*)
above is an
90
isometric isomorphism provided that we shovf
1\
'IH(x)
\I
0=
1\ 00 .
We proceed to prove thi s equali ty.
No,'v-i t is clear that we couJ.:d impose another norm on
positioD as the dual of
and for
measure
.
This new norm is denoted by
via its
II II 00
h E (9 ,
II Il v 00
where
1
L
C9
denotes the essential supremum norm relative to the
V.
We proceed to show that these norms are identical.
Clearly,
Ilh·1I
2
V L (11)
rrhus,
Ilhll 00
IlhIIH(x) s:
To show that they are equal, we let
3:
V
su.ch that
IlhV"V co> C Thus, there exists
(1)
and
Ilhllco'
C
N c; X
such that
. V (N)
:1=
0
E
'
then if
E
> 0,
en
(2)
N
a
If we define
c Ix \ Ihv (x) I
H(X)
E
c -
>
c) .
as
x
N
\) (N) 172
'
thenwe·have
j 'h\) S\)
x
I)
N
dv
(.N,)112
.
~ (C - e)
2
hence
Hence we have two.vie.is of
The elements of
may be identified on the on~ hand as self-
(9.
adjoin~ operators on
H(X)
and on the other hand as continuous
Ll .· We will use the view of
linear functions on
on' .H(X)
which are clearly equiva,lent.
.(9.
(9.
as opei'atoL
for a proposition dealing with mixtures of states.
context of causality, a mixture of
tate
It, t
behaves as a pure strJU
again.
NoW suppose
H(X).
A
is a posi [i 'lie
Then in the usual manner
subspace of
H(X)
Ii
A
[jni
te [race one operat01'
is a measure on the closed
and hence on the event space.
Pr,2P0si tion l:
Let
A
be a posi ti ve definite trace me opera tor on
Then there exis ts
CI E
H(X)
tr(xA)
such tJlat if
(XCI,
Qi),
x
t:
(9. ,
then
Ii (Y )
0:;
92
Proof:
First we consider the observable
x
P
where
E
P
E
is defined
by
Then the map
E ... tr(X:EA)
is a measure on
X.
Call this measure
S
lVdm
€
m.
Let
H(X) •
Then
II sll HeX)
= Jldm = m(X)
tr (A)
1.
Moreover,
Thus,
x
x =
P
E
satisfies 'the conclusion of' the proposition.
be any observable,
= J\d[tr (~A)J
tr[(JAdP~)Al
= tr (xA) .
The proposition follows.
Now let
93
We will now relate the theory we have developed with the
classical theory of convergence of distributions.
As has been.
mentioned, the real interestfun measures on the real line, since our
space
X is but a construct - an abstraction.
More precisely, VVe
consider the measure defined by
where
x
is an observable and
~
is a state.
We thus proceed to
the following definition.
We say that a sequence of distributions converges,
F -+ F , iff
n
Fn(r} -+ F[r}
for every bounded interval of continuity of
(a bounded interval
F
is an interval of continuity iff the endpoints of
This definition is from Feller
r
are non-atomic).
(1966, Vol. 2).
We have the following theorem which will be useful to us in the
sequel.
Theorem 1:
.. Let
tF}
n
be a sequence of proper probability distributions (a
probability distribution
(a)
is proper iff
F
n
F
n
(~) =
1) , then
In order that there exists a proper or defective
distribution
F
such that
1<'
n
-+:F'
'
itis
necessary and sufficient that the sequence of
. expectati ons
E (u)
n
JUdFn
I
converges for
UEC(-oo,+oo).
o
In this case
E (u) ... E (u) •
n
(b)
If the convergence is proper
(*)
holds for
all
U E C(-oo, +00) •
Here proper convergenc_e means that
is a proper probabiLi ty
The above theorem can be found in Feller (1966, VoL 2,
distribution.
p.
F
243).
Nov.; let
x
E (e)
and
x
the associated mUltiplication op',cr'J.t,'/'
x(f\fd(J)
Recall that if
/-La(E)
=
(¥,
a)
a
= x <Y:Ndci •
/-La
is a state and
where
is the measure defined by
a = f\Id(i , then
Jx
<Y
(t)du (t)
a
= td/-l.a (x <Y-1 (t )) •
S
Thus if
x
/-La
is the spectral measure of
x
in the state
~
95
~~(E)
==
(x -1
ex, ex) •
x (E)
0"
(Observe that
~
x
ex
has compact support since
is bounded. )
x
Using
this notation we have:
Theorem 2:
If
(x } is a sequence in
n
~
then
x
n
-
-+ x
in the strong
operator topology iff there is one interval which contains the supports
x
x
of the distributions (~n} and (~n} converges in distribution to
ex
x
~ex
ex
for every ex·
Proof:
First assume that
x
n
-+ x
strongly.
Then, by Proposition 3,
Chapter 2,
f(x) -+ f(x)
strongly for each continuous bounded function
f.
f(x) -+ f(x)
n
in the weak operator topology and
Thus,
n
(f(x )ex, ex) -+ (f(x)ex, ex)
n
for each state
ex.
Note, however, that if
then
(Here we have used the fact that the projection valued measure of a
multiplication operator is given by mUltiplication by characteristic
functions.)
Thus
x
Jf(t)d~~n(t) ~ Jf(t)~;(t)
x
and
~~
~
n
x
4 ~
as distributions.
~
x
Conversely, assume that
~~
n
~~
x
as distributions for each
~
~.
By assumption, there is a bounded interval which contains the supports
x
[~~n}
of all the measures
interval.
Let
and also that of
I
be such an
We have that
x
fItd~an(t)
4
fIt~:(t)
(Here we use (a) of the Theorem cited from Feller and the fact that
x
(~~n}
the supports of the
lie in
I.)
It follows from the spectral
theorem that
Since
[Xn}
converge to
converges weakly to
-2
x
x
(x nx}
and
(cf. Naimark (1970), Chapter 7).
(xx nJ
Thus,
and
t::.
-
limlil/'- n - x)~11
2
o.
n
It follows that
x
n
4
x-
strongly.
The theorem follows.
both
97
We have seen that considering convergence of distributions on
R
is the same as imposing a topology on the observables which is a
strong opel'ator topology.
Thus, convergence of distributions is
equi valent to the viel'l that only a finite nwnber of experiments wi th
an arbitrary but not absolute, accuracy can be done.
(By accuracy,
we mean that the variance of the difference of two random variables·
is smalL)
Now a fBlnily of random variables which is parameterized by time
is called a random process.
given a finite set
[h }
t
Say
is such a process.
[ti}i~l we will call the associated
Then,
[ht.}~ a
l
finite dimensional distribution of the process.
When an experiment
is performed and followed in time, a finite dimensional distribution
is usually the object of study.
somehow recover
is clearly, in
tht}
Thus, the interest of a study is to
from a finite dimensional distribtition.
genera~
not possible.
{h }
t
asking what we may say about
dimensional distribution.
This
We must content ourselves with
when we know only some finite
This is really a rephrasing of the third
problem of van del' Vaart (1973).
That is, given a difference
equation find a differential equation whose solutions behave in a
simi lar manner.
He know of one result along these lines, which is found in
Gikhamn and Skorokhod (1969), which will be stated without proof.·
This theorem requires some bit of introduction.
basic probability space and
if
{~t}
01
0
by our
the measure under consideration.
is a process, we call the function,
sew):
Let
[a,b] ... ~ ,
Then
a sample function
(w
E
0
is fixed as t cbanger.j,
theorem we will consider a special class
(Sn (t)}
processes, where the sample functions are in
probability 1.
x
= St(w),
(ulu(t)
w E
(St}
of
with
where
o}
and further we have a natural measure for
!-l(A) = a((wlst(w) = u(t), u
and
C[a,b]
X ~ C[a,b]
We have now a space
For this
X.
E
An
If
A eX,
0
This is called the measure corresponding to the process.
By our discussion on page 97 , we are interested in trw
convergence of finite dimensional distributions of
process of
(St}
(Sn(t)}
to the
(in the strong operator topology or equivalently as
convergence in distribution).
Theorem
3:
Suppose that the finite-dimensional distributions of' the
processes
(sn (t)}
converge (as above) to the fini te dimensi_ onal
Then for the sequ,,:nce of
distributions of a :process
(St}
distributions of
to converge to the distriuuhon of'
f(Sn(t))
for all functionals
f
that are continuotw on
C[a,b]
necessary and sufficient that
lim supa(
h~O
n
for every
E
sup
It/-t l l I~h
> 0 .
Isn(V) - sn(tll)1 > t}
0
J
it is
f(St)
99
If
f
is a continuous functional on ; C[ a, bJ , it corresponds via
the Riesz Representation Theorem to a function
g, of bounded
variation (or as often stated, to the measure generated by
The functionals,
averages.
f , of
£t
g).
correspond in some sense to time
One could presumably check these.
Thus,
100
LIST OF REFERENCES
Dunford, N. and J. Schwartz. 1963. Linear Operators, Part II.
Interscience PubL, New York, New York.•
Feller, W. 1966. An Introduction to Probability Theory and Its
Applications, VoL II. John Wiley and Sons, Inc., New York,
New York.
Gikhman, Skorokhod. 1969. Introduction 'ro"The 'l']:leory of Random
Processes. W. B. Saundexs Co.', Philadelphia, Pa.
Gudder, S. 196}' Spectral methods for a generalized probabifity
theory. Trans. American Math. Soc., 119:423-442.
- GUdder, S. 1966.
ol;Jservab1es.
Uniqueness and existence properties of bounded
Pacific J •. of Math., Vol.. 19, No.1, pp. ·81-93.
Haag". R. 1972. Quantum field theory.
pp. 1-14. (R. Streater, Editor).
New York.
Math. of Cont. Physics,
Academic Press, New York,
Mackey, George W. 1963. -Mathematical Foundations of Quantum
Mechanics. Benjamin Publ., New York, New York.
Mack~y,
George W. 1968. Induced Representations of Groups and Quantum
Mechanics. Benjamin Pub 1. , New York, New York.
Naimark, M. A. 1970. Normed Rings.
Groningen, Holland.
Walters-Noodhoff Publ.,
Nelson, Edward. 19b7. Dynamical Theories of BrownWn Motion.
frinceton Univ. Press, Princeton, New Jersey.
Nelson, Edward. 1969. Topics In Dynamics, I:
Univ, Press, Princeton, New Jersey.
Flows.
Riesz, F. and B. Sz.-Nagy. 1955. Functional Analysis.
Ungar Publ., New York, New York.
Saka.i, S. 1971. C*-Algebras and W*-Algebras.
Publ., New York, New York.
Princeton
Frederick
Springer-Verlag
van der Vaart, H. R. 1973. A comparative investigation of certain
difference equations and related differential: Implications for
model-building. BulL of Math. Biology, VoL 35, pp. 195-21L
101
Varadarajan, N. S. 1962. Probability in physics and a theorem or.
simUltaneous observability. Corom. of P. Appl. Math., Vol. XV,
pp. 189-217·
von Neumann,
Pr~nceton
1953. Mathematical Foundations of Quantum Mechanics.
University Press, Princeto~ New Jersey.
Yosida, A. M. 1962.
West Germany.
Functional Analysis.
Springer-Verlag, Berlin,