A MATHEMATICAL STUD'! 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 automatically consider 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 U b
This new event,
is the event that either
a
or
outcome (we indicate this new event by saying that either
occurs),
We also can generate another new event from
which .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).
and
b
is the
a
U and
n
a
a
and
b
is an event.
b
or
and
and which is the event th8.t both
are outcomes (:in this case we say that both
Also if
a
a
b
b,
a
and
occur).
For various
to encompass countable
For a discussion of this extension process,
Thus, we assume that the events form aCT-algebra.
It follows from the Loomis representation theorem that this CTalgebra of events corresponds to a CT-algebra of subsets of some set,
Now we are often only able to learn of occurrences by studying
functions frQm
0
to
~.,
that is a measurement of some object.
I
generates
example, a die "'iv:ith six sides,
#
events
function
¢'
(e }, , .. , (e , e , ••. , e }
l
6
l
2
f(e.) = i
.1
throwing of the dieo
[e , ... , e,.-} =
.1
0
(1
0
In general ·we consider a
and we stUdy averages assodated 1,'ith the
'fhis example is inteTes ting in that the space
is a set we can not only cOilceptua.lly consider
. but"may phyc;ically study as well, that is, it is not an abstraction.
In contrast, consider a marble.
Say we do all experiment wi th the
2
marble
by t.reating it as though it were
find whj.ch "side" comes upo
Now
'3.
~:!::..
cUe,
,we roll it and
is iJ.n o.cst:::>8.ction, a geometric
(2
point is not something we can physically id"ntify, but the event
space generated by this is real and we can identify its elements, the
open Ildiscs ll •
For example, the marble could be painted with different
colored patches and the color of each experiment could be identified.
If we associate a number with each colored disc, as vre do iii th the
case of the die, wc,; would be considering a. lunctioXl
general wevrant to consider all functions
f-l(E)
reasons of completeness.
on
~
(2
~.
of
such that
We do this for
Such functions are called random variables.
Notice that given a random variable f:
measure
E
is an event for every Borel set
(2 -+~.
f:
R , and a probability
(2 ....
we have in a natural manner a measure
which is defined by
Now in
f: (2 .... bt,.
SeE)
= ~(f-l(E))
distribution of the random variahle.
R,
on
and which is called the
Thus we study
random variables, whose distributions
S
S
n
by considering
are the actual objects of
observation.
We vrill use these brief remarks to introdclce a more general
theory of events associated wi tll experiments.
l'Jotice that in the
classical situation which is described above,
.i
In our more general framc':lOrk,
are always defined.
U and
n
n
are not always defined.
and,
OTlera
ti ons
.t
We will fO:l'Innlate thi.s new theory
via a ;;equence ofaxi oms due to Mackey and our cli[;cu:3sion is a
slight modification of his.
Let
(9
and
F be non-empty sets and
(9
X
(5
X
B(~)
a Ub
-. [O,lJ
p
a mapping from
3
B(~)
(here
elements of
states.
and
R).
is the o--algebra of all Borel SUbficts of
are called observables and
(9
tbo~:;e
of
(5
The
are called
The axioms which follow list various properties of
p
e
(9,
which will be used throughout this thesis.
Axiom I:
If
x
B(~)
the algebra
some sense to
a
and
(9
€
the
6
~.
of
S(l)
E
a(f
==
p(x,a,p)
The measure
-1
(.»
is a probability measure on
p(x,a)
corresponds in
as discussed above.
Axiom II:
p(x,a,E)
If
If
p(x,a,E)
=
= p(x,a~E)
p(x/,a,E)
for all
for all
E
E
and
and
a
a
x, then
then
x
a' .
= 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
ex
E
B(~).
y
= f (x)
6
and
E
in
(9
€
f
a real valued Borel function,
such that
p(y,a,E)"" p(x,a,f-l(E»
This unique
y
for all
will be denoted by
0
Axiom IV:
Let
for all
{ti}i=N
i
and
be a sequence of real nmnbers such that
L: t.
there exists a state
E
in
B(~)
and
1
1
x
a
in
a mixture of the states
0
If
(ex.}
1
such that
(9 •
(ex.}.
1
1
~
0
is a sequence of states, then
p(x,a,E)
We wri te
t.
a =
=
2~
L: t.p(x,ex.,E)
1.
t.a.
1 1
1
and
a
for all
is called
4
Before
~iGcussing
the remaining axioms We first need to
distingui sh a subclass of the class of all obE;ervables and to derive
some of' the properties of the elements of this 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,I})
~_.
for each state
write
(x)
f.1
~
=
When
More
x
1
ex
is a question and
is a state we
p(x,~,[l1) •
The following proposition has proven to be quite useful.
Proposi tion:
If
x
is an observable then
x
2
x
x
iff
is a
guesti~:m·.
Proof:
First assume
P(x,~,E
n (tit
x
2
= x
< o}) =
Let
p(x2,~,E
p(x,~,¢)
if
0 <
P(x,~,E
° < 1,
n
E be any Borel set.
n (t
< O})
0
then
(0,1)1
nn
L/
\x,c¥,'V.r.c
u'
.r
1
.
I n , ~)
p ( x,~, ,0 ,1.;
··_»
(b,1.
Notice that
5 .
Thus
1
p(x,~,E
n
(0,1)) ~
lim p(x,~, (on, 1))
co
n
p(x,Q',
n.... oo
n=l
°.
Finally if
0 > 1 , then
1
p(x,~, (E
n
(6,+00))
= p(xn,~,E n (6,+00))
and
p(x,~,E
n (6,+00
)) ~
n
lim p(x,Q', (6 ,+oo))
n.... co
lim p(x,Q',
n
n
(0 ,,+(0))
n.... oo
°.
Thus,
p(x,~,E)
for all Borel
E.
p(x,~,E
Therefore
x
n
CO,l})
is a question.
The converse is obvious.
Now asswne that
define a question
s
E ~
.
~
Observe that
x
is an observable and that
E
E B(~).
by
QX = f(x)
E
x
QE
is actually a question sl.nce
\IJtwre
f (s)
Yn; ( 5)
We
for each
6
and
x
E
'I'hus,
in
Q
i,s the question, did th€ measurement of
lead to a value
x
E?
It follows that to each
qv_\::'stions,
{Q,~1
x, there corresponds a family of
and further, we claim tf.lat
is uniquely determined
x
~J
by
X
By this we mean that if
(QE}
x = x'
then
x'
QE
.-
x
QE for every Borel set
(the proof is tTi vial, just write the defini ti ons of
E
x
QE
and
Notice trat we may impose a partial order on the set of
questions, that is, we say that
Q ~ Q
l
2
~ P(Q2,QI, (l})
in
E5
for each state
01
for every
~ QI (Ql) ~ ~QI (Q2 )
QI
iff
P(Ql,QI,(l})
(we oftenl'iTi te
) ,
Q,
s:
~2
iff
We clearly uave the following
properties for arbitrary questions:
QS:Q
and
Define the observable
'I'hen
1
define
for all
is a question and
1
by
1
~.~
p(l,QI,tl})
for all
1 - Q' to be the question for which
QI E
E5 •
Q
1
for all
QI
E
is .
For each question
,
Q
7
One says that two questions are disjoint iff
for all .s tates.
In this case we write
be any observable, and let
E C}!' ,
and
F
be Borel sets in
Moreover, 1. f
then
Suppose
E
.Q
and
1·
For example, let
n F'
E
Q .' are ques ti ons.
2
==
m
'r ,
1tLlen
R.· If
QX 8 Q)(.
"F
E
•
If there is a question
Q such that
for every state
01
Q by
then we denote
are Borel sets such that
nF
E
==¢
and .tf
Q + .Q
l
x
2
If
E
and
is an observable, we
note that
~UF
==
~
~
+
•
To see that this is sq merely observe that
x
'"EUF
thus,
for each state
01.
It follows from these remarks that
question-valued measure, !.~.,
B(~)
QX
into the set of questions.
QX
is a
is a ''c''.;:1Omomorphism lt from
This statem,:'Dt vrill be made more
precise in the discussion following Axiom 11 arid ie cL,scribed more
fully by Vardaraj an (1962).
Notice further that if
Q~ + ~
E
and
is DO longer a question.
F
F
arc [lot di:3joint setf3,
T~is leads us to Axiom V.
x
8
Axiom V:
be a' sequence of ques ti ons Guch thatQ. {)
Let
i
":j:
j
,
.
Q
then
l
Q
,Of,(l})
L: p(Q.
.
l
J.
Of
J
such that
L:Q.].
P(Q,Of,[l})
holds for all
Q. ,
ex:ists in the senSE, that there is a question
== L:Q.
whi ch we denote by
. """.i
l: (Cj ,
Q
Notice that
~
e-
Cl.
:L
all
i
It can be ,:hawn that
Q
is the
least such question.
Also based. on the above discussion, we introduce the following
axiom.
Axiom VI:
If
fQ';
~)
E EcB(;rr
is a question valued measure then there exists an
1.
x ,
obsen/able,
QE
such that
x
==
QE
for every Borel set
This '-3 in a sense a ,superfluous axiom,
if
i Dec it is the case that
{QE}i2' a question valued measure then 01>, can enlarge the set
of observable;:; to a new set
observable
x
for all
QX
E
Qy R
R
2
r
for ('ach
and
and
l
n
and
Let.
E~
+ Qt,
/
(9'
in such a !,{a.y t.h:,t.
(~I
contains all
(9'
'Ne have
such that
"
'
-l(F))
p (X,Of,X
E
QE
E
=
P '(Q,XE'Of,]''")
.i n
Th,.,':' in
and
E c B (6t.) .
c,"-{,2
be questions
such that
Q 6 R.l
3
,;UCll
,
We tben say that. such
R
l
UJtt ther'c exi,:u; qucsL:Lons
I)
R
que~. ti Oc!
Q.,1
a.r::-:
=-
R + Q
I
3
"
and
imultaneously
(9
9
an~~"jerablb
x
x
(J--
is an observable, then 1. tl:; trivial to show that
.
are sl.multaneously answerable.
(\-
~'
x
If'
~,
If'
cirmervat::,le:3 ;:;:uch that their question valued
x
simulta.r..eom,ly answerable, we say that
(we write
olJser nib le,
x
yare
a:ld
mE.ca~mres
and
are
y are simultaneously
x ... y ) .
Theorem· (Varadara,j an):
If
x ... y
such that
g ,
a!".!.d
then there exists an observable
=
x
f(x)
z
and Borel functions
and
In chat which appears below vIe denote the mean of
state
by
ex
f or all
If
ex
iJ
Ot
(x) •
and
where
in the
ex0/ (E) = p(x,Ct',E)
E.
x = fez)
defined.
Thus,
x
and
y
= g(z)
One merely defines
, then it is clear that
= (f'
x + y
+ g) (2,).
x + y
is
Observe that with
this definition we have that
iJ
for each state
0/
Ct'.
(x + y) =
iJ
0/
(x) +
iJ
0/
(y )
It is not, in general, possible to define
x + y •
This leads to a modification of Axiom II, which we call Axiom II'.
Axiom II':
If
iJ
Ct'
(x) = iJ (y)
Ct'
for all
Ct', th<'.;n
x
vJich the aid of this axiom,\'le can dt:fiile
observable such that
iJ
0/
(x + y)
=
iJ
Ct'
(x) + iJ (y)
y.
x + y
to be an
for f.;ach state
Ct'.
Ct
Of c.our,,,;<'::, such an observable may not al\l8.Ys ex.i t.
L::;t
ordered
e
;:;(~t
del:ote the set of all questions.
under
~
defined above.
'TClen i t is a partially
Moc:.'bJVer, if for each
Q, we
10
let
QI
~
1 - Q
then
J
an orthocorrrplemented partially ordered set.
I.)
opera.tor
defined on
j
t
e
In general if
VIe
e
is a complementation on
say that
e
and
becomes
(With complementation
~
is a set with a partial order
is an orthocomplementation on
e
iff
the following conditions hold:
a)
If
are merribers of
aI' a 2 , ...
a.6 a.
i '*' j
for
J
l
(ao b
.i:'1'
I:
such that
a~b/)
there exists a unique least element
that
a.
l
~
for all
a
denoted by
a
a U a' = b U b
'
b)
a
1
c)
If
a such
The element
for all
a
anei
a. U a I
Thus
1
we
ar.~d
b
..
1 ,
a.
a ~ b , then
b
a U
Co'
U a) I
•
That these axioms hold for the set of questions is
Mackey.
is
a
U a2 U a
3
denote this element by
for all
.
i
then
~hown
by
Moreover when we consider strongly convex sets of probabili ty
measures on
e
we can show that
e-valued me&.:c:ures correspond to
observables and that the two systems are equi valf~nt.
Thus we may
consider partially ordered sets with orthocompicmentation and
a
strongly convex set of probability mea:cmres lr:.stead of the foreThis is the method used by Varadarajan (1962).
going Axioms.
The set
e
is called the logic of the
Our next axiom is a strong restricti on
aLlowed in our theory.
~yutem.
0(1
trw ty:pe of logics
11
Axiom VII:
e
is isomorphic to the partially
spaces of
where
I
II
where
H
order,~j :,',~~t
of closed sub-
is some Hilbert space of dimension
Llu
corresponds to the usual orthogonal complementatlOt!
defined on subspaces of
0Vl':J (·01'
H.
Axiom VIII:
If
ex
Q is a non-zero question, then there ezists a state
such
f.l.et(Q) '" 1 .
that
Axiom VIII is a powerful axiom which allows us to obtain a reformulation of the axioms in terms of Hilbert space theory.
For
Q is a question then it may be identified
example, observe that if
with a closed subspace of the Hilbert space
was asserted by Axiom VII.
Moreover, the
H
whose existence
clo~;ed
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 identified with the set of
into
H.
Moreover we have already indicated that
there is a one-one correspondence between observables
question-valued measures
(~}EEB(R).
one-one correspondence between the class
x
and their
The upshot is that there is a
<9
the set of all projection-valued measures on
of all observables and
H,
On the other hand,
every projection valued measure corresponds, via the spectral theorem,
to a self-adjoint operator on
Hand cOrlVe.r2:ety.
1'b'L:.s, if
observable, then there is a unique self-<:j.dj oint oper ator
which corresponds to
x
and conversely.
A
x
x
is an
on
H
12
We use Gleason I s theorem (see Mackey (1963») to show that the
states cOITeSpoGd one-one to posi tive definite trace one operators.
Moreover, if
A
il3
the posi ti ve definite trace one operator
corresponding to the state
formulat:i"on for the mean
Qt,
J.1
ex
then we
hav(~
t.he following
(x) :
fAdp(x,ex,A) •
In. particular, if
dimensional and where
J.1
ex
ex
corresponds to
A~ = A~
A
~
for some
€
where
A
Hand
is one-
IAI
=
1 , then
(x)
VIe refer the reader to Mackey
(1963) for the details.
We 1'1111 nmrproceed to use the structure developed here to
consider a question involving statistical
en~:;embles.
Namely, to
explore the common technique of resolving a statisbcal ensemble to
component parts, each of which possesses certain properties with
probability one.
It is shown in Chapter 1 that when such techniques
can be carried out simultaneously vvi th two
observable~:;
have a joint distribution.
cib:c~el'v8~bles,
the
We will then proceed to
consider the strong operator topology as a
con~:equence
of ass·wning
that only a finite number of experiments wi tri arbi Crary accuracy.
Several aSliects of this topology are
considE::r(~ci.
We then proceed in
the third chapter to consider what this mean;: in terms of classical
probabi.li ty.
It is shown that convergence, in the' strong operator
topology corresponds to convergence in di.stri bution (of' a ra...'1dom
variable).
TLh~ structure is used to pose quec.;tior:.s in terms of
stochastic processes.
13
A THEOREM ON SIMULTANEOUS OBSER'!ABILEY
~,tl.ldi~~s
A common technique often employec1 ,vhen or;
i
,,,hich he would like to observe accurately, b
ensemble to which a sorting mechanism
l~,
to cor,sicler a statistical
applied in order to isolate
~.,::.,
subenmribJEs which possess the property of interest,
is resolved into component parts.
is of interest.
jJl
E,
E
Let
a
property
~,
ensemble
A
Thu,s if
a
If we extract a
prob8b.ll~ t
is eq1.J.al to the
subenseJilLh~
then the measurement of
A
Val' f(A) = 0 , where
over, we find that
ensemblt~
Let us say that the ob u'vable
~,.
probabili ty of having property
f (A)
the
be the property that a mem,urement. oj'
a Borel subset of
question
phenomena
Q'
which
plcls D
i:3
Ilctve
OUrl
llu:~
_o.r\~:')
is the
a
we view a state of an ensemble as a mixtn'c
~r:HC
Ul
01
the various component sys tems.
Now suppose it is wished to find accurate value
observable
B
at the same time that we know exact value.3 lor
The same procedure (as applied to
A
A,
above) may be used to n'1::01ve
the subensemble in terms of the observable
general such a resolution is not feasible
taneously.
ror the
B
A
faY'
and
SilliU
1-
We will show that in the event clUch a program works, Ulen
it can be concluded that
A
and
have jcint probability
B
distributions in each state.
Assume that
A
and
B
are related in
t(y
f'ollO',.ring marmer.
take an arbj_ trary ensemble,
SN' and arbi tra"y
(: > 0
is a measuraole value of
and
> 0 .
that
A
E
related in such a way that there is a
As~::ume
6 > 0
A
and
Hf~
Bare
and a resolution of
SN
-
14
into subensemb.les
i SN
such that whenever
A< 0 ,
Var
Q'i
Prob(A E (A - 0, A + c)} ~ Prob(B E (~
the state of
and
A ~ B then
spectral theorem).
- E,
~
Q'.
l
+
is its average value.
are so related we write
that if
Ci.
l
E)}
where
When
A
is
r:x.
l
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
6
be the set of states,
In the
0
the set of
the logic.
Recall that when
probability measure on
r:x E 6
~
and
x
E
0 , Q'(x)
and the mean of
x
=p(x,r:x,.)
in the state
is a
Ci
is
defined to be
~
(x)
when
Ci
In a natural manner we define the variance of
as
Var (x)
r:x
~
2
r:x
(x ) -
(~
r:x
(x»
2
x
in the state
Ci
15
Proposition 1."
~
If
is a state in which
and, con.vE:~r;:"ely, if
Vax (x)
:=
~
t E.~
var~(x) = 0
a(x) (t)
such that
~(x)(~~(x»
tten
:=
1. , then
Q'
(x) (~ (x»
1
0 .
Proof:
iff
:= 1
~
Recall that the joint distribution of
,
[x II.. E 1\} E'.xists for
A
1\} :1:.:; ;::;i.multaneously
every state
Ol
observable.
In this case the joint distribution is defined by
z:
A
B(~) ~
e,
iff the family
z(n
-1
A
(Va.radarajan (1962).
When
wri te
xl" x
observable.
We let
If
xl'" x
for each
1\:= (l,2}
and we say that
2
E
AE
1\
and
E E B(~)
for our further discussion.
is a simultaneously observable family we
l
2
define the spectrum of
open in
:= xI.. (E)
(E»
[x. Ii := 1,2}
[xAIA
and
z,
Xl
z
and
x
2
are simultaneously
is their j oint distribution, we
o-(z), to be the set
o-(z):= (L{NIN
is
~2 and z(N):= O})C •
Proposition 2:
If
xl'" x
2
and
cr(z) C o-(x ) X o-(x ) .
1
2
z
is their joint distribution, then
Moreover, in general,
u(z):j:: o-(x ) X o-(x ) .
1
2
Proof:
The inclusion follows directly from the d.'c:!fi ni tion.
o-(z) :j:: o-(x ) X o-(x )
l
2
we consider the following:
To see that
16
F.:xample 1:
:Let
strnetuno
.)
e
denote the logic of
a,C:;
an orthocomplemented lattice,
clm;E~d s~b~,pa(:f'~,';
l: ~~.,
.1..
is the usual compleme.ntary subspace operator
and, for i3ubspaces
b .
a
e
vIe ide:'ltify
and
R'-
t~~e
s
',vi th its usual
operation
is set inclusion,
a U b i s the l i near span of
b
proj'~ct.ions
with the set of all
~2
of
and
wi th t.he set of posi ti ve self adj oint. oper-ator3 of trace 1.
P,
st,bspace of dimension one; let
be the
cor.Te~;ponding
a
and
(i3
Pick a
projecti on
.L
i3.nd let
~.
1'2:;:C
3, 4.
Let
x
We choose distinct real
and
y
X(A )
l
x
and
Z(A l ,A3)
("2'''4)}
:j:
y
i
=
1, 2,
be the observables corresponding to
P
:=
x("2)
Ex.tend
number~:,
l
P2
in the usual manner t.o homomorphisms.
Now
so that we have
Pl , z("2'''4) = P2
[(A l ,A 2 )} X (A3'''4}
o-(x)
X
o-(y) .
Proposition 3:
Suppose
e
~ld
is a logic which satisfies Axioms I-VI
that it,
in addition, satisfies
Axi om VIII:
For each
exists
If
x
and
'f
E
«5
e,
ex
x
E
and
6
y
then for each
such that
a
'*
0 , there
ex(a) '" 1 .
such that
yare observab1es such that
distribution of
there exists
ex
a
x ...
E
y
and.
z
is the joint
> 0
and
("1'''2) e o-(z)
17
Var (x) < E
Moreover)
Cit
and
<
Var (y)
Cit
.
E •
Proof:
If
(A2 -
(Al)A 2 )
~)
A2 +
E
<Y(z) ) then
~)) * 0
z((A
l
-
.~)
Al
and thus there exists
+~) X
Cit
such that
Thus,
and that
The corresponding inequali ties
1'01'
y
follow in a similar fashion.
18
Corollary 1:
Let
~
E
6
x
be an observable, then for every
such"that
I~~(X)-AI< ~
~(X(A - ~, A + ~» = 1
Var~(x) < E
and
> 0,
E
A E o-(x) , and
, it follows that
•
Proof:
Same as Proposition 3.
Of course Proposition 3 will not hold for arbitrary elements
(A l , A2)
E
o-(x) X o-(y).
To see this consider the following example,
based on Example 1.
Example 2;
2
R
Choose an aribtrary orthonormal basis for
elements
x
Y...
and
Let
x
denote the observable corresponding to
the projection on the subspace generated by
Y
generated by
in such a way that
o-(x)
o-(x) X o-(y) = [(0,1)(1,0), (0,0), (l,l)}
consider
(0,0)
E
t
l
(b
o-(x) X o-(y).
)
be such a state.
Var (x)
t
Var (x) <
~
E
ot
2
t
2
and
then
l
+ t
t
Let us compute
Var (y)
ot
t
ot
~
z
Then
is their joint
by Example 1.
Let
ot
(x)
1
.-
t
,
0
~
1
,
t
!-1
ot
2
~
0
(y) _. t
2
Now it is clear that if
l
VaX' (y) > l-E
to subspace
States will correspond ro an
- 1
2
y
= u(y) = [O,l}.
a
t
and
and if
arbitrary, positive matrices of trace 1.
a
x
o-(z) = [0,1), (1,0)}
distribution, we see that
and, call its
0
,
We
19
If we are doing some experiment, we will observe the mean values
of our observab les.
Importantly, we wi 11 often obc,:erve yes or no
observables, questions in Mackey's vernacular.
Qu.e.stions are the
"simplest!! observables from which all others can be constructed as
functions of many simultaneous questions,
~.g.,
3.3,
Theorem
Varadarajan (J_962).
At this point we will consider some
example~iwhich
difficulties arising from the fact that the axi
om~;
point out
of Mackey are not
enough to insure that there exists sufficient states to "separate" the
logic.
Our first example deals wi th a particular logic, our old friend
~2
, wi th the usual structure on the logic.
in this examplf3
~
Ie. 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.
i
=
1, 2.
Now choose some subspace
(8
8
l
such that
is of course one-dimensional).
[1,O,P ,p ,s,s11 and
l 2
lattice of six elements
P
,
P
8
as in
2
n Pi (
e
Now let
lt~t
ex,
We do
2) = (O} ,
denote the
P
and
)'
be
be the set of all convex combinati ons of
ex,
13
and
the states defined by
QI
~ (ri)
13 (sl)
Ftlrther, let
)'.
(8 )
ex(P )
l
~
1 ,
l(8)
1 ,
We define observables
1
)'(P )
l
x
and
"2 '
1
2
,)
1
)'(gl)
3
y
1(4)
by
(-
3
.
20
x(i)
P
co
2
i
1
i
2
We extend these to make them
Now
x
*
y
Var (x)
01
thus
VarS(x)
j
1
j
2
~-homomorphisms.
(~.!., Varadarajan
Var~ (x)
s
y(j)
(1969, p. 199)) but we have
Vax (y)
o
Var~ (y)
0
01
=
= 0 iff 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
~.~.,
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 I s Axiom VIII as Axiom -{111m and we introduce a new
axiom due to Varadarajan which we refer to as Axiom Vlllv.
This
axiom is:
if
a,
such that
bee,
OI(a)
a
*b
* OI(b)
then there exists
01 E
6
.
In accordance with Gudder (1966 )we will call a logic full in
case Axioms VIllm and VIllv obtain.
21
For each
{O'IO'
Sa
E
a
e
E
let us define a set,
~, 0'(0) = l}.
S
, aB follows.
a
In a full logic,
a
EO (S
implies
¢.
Sa*
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 Exmrrple 2
e
is full logic, but that
(By the way, if we let
above.
is not quite full.
U b be the span of a and b as
a
above, then we have an anti-lattice.
~.!.,
We note that in that example,
It is not a qUite full logic,
Gudder (l966, p. 88 )
We will now prove a formula dealing with the variance of
x
in a
n
mixed state
0', where
0'
=
~
1
t.O'..
l
This will, in turn, be used to
l
consider some consequences of assuming a quite full logic.
Variance Formula:
n
Let 0' = L: t.O'.
l l
i=l
n
L: t.
l
1
1
,
t.
l
~
Then
0
Var (x) 0'
Proof:
We will prove this strange formula by a " ssraight forward"
computation.
We have
n
~ t./-l.
Var (x)
0'
1
l O'i
0
(x~)
2
n
?
t./-l..) + ~ ti/-l. i II
1
n
(~t./-l.
-
1
n
~
l O'i
(x))
2
n
~ t.t./-l../-l..
i=lj=llJlJ
22
11.
where we let
11
1
Since
(x)
Ct.
n
L
1
1
t.
1 , the latter expression
1
is equal to
n
L: t. Var
1
(x) +
L: t. Var
Ct i
1
Ct i .
1
(x)
n
L:
t. t . (11. - 11.. )11. +
L:
+
J
1
i<j
J
1
1
t. t . (11. - 11. )11. •
1 J
1
J 1
i>j
After relabeling, we get
n
L: t. Var
1
Ct.
1.
(x) +
t. t . (11. - 11.)11. + (I-l. - 11. )fJ . )
J
1
1
J
1
J
1
J
1
n
L:
1
t. Var
Ct '
1
(x) +
L:
.
.
l<J
l
t. t . (11. - I-l.)
1
J
1
2
J
•
We will now make a definition of a relationship between two
observable.::: 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 following condition holds:
such that for each
E
such that whenever
Ct
> 0
and each
'"
Var (x) < 0
Ct
Var y <
Ct
€
•
A
x -+ y
There is an x-null set
E
er(x)\N , there exists
is a state for which
and
then
We wi llwri te
N
6", > 0
23
Proposi tion 4:
Suppose
x
~
y
lim 11
n~CD
A E ~(x).
for some
(~}
and that
~
(x)
. lim Var
(11
~n
(y)}
==
A
n
If
n~CD
then
is a sequence of states such that
n
~n
(x)
0,
is a Cauchy sequence.
Proof:
E
Say we are given
E
> 0 , then choose
l
o<
be chosen as the definition of
let
Var
~
n
x
E
-1)2
< (2
y
~
above.
and let
Moreover
o
N be chosen such that n > N implies 111~ (x) - Al< 21..
and
n
01..
(x) < - • Let ~ = tl~n + t2~m for some m,n > N and for
2
t ,t > O.
l 2
By our variance formUla, we have
Var (x)
Var~
~
tl
n
(x) + t 2
Var~
+
m
+ t 1 t 2[ (11
20
2
+ 20
l 2( A 4
~n
tlt2(11~
(x)
n
(x) - A) -
~ 11~
)2
m
(11
(x) - A)]
()I
2
>t
In
2
A)
+ t t
Since, also,
we conclude that Var (y) < E.
This implies by our variance formula
~
that
tlt2111~ (y) -
and
t2
(y)/2 <
11
n
we have that
Q'm
III
~n
E
and, by an appropriate choice of
(y) - 11~ (y)l<
m
VE
2 < El •
t
l
24
Proposition 5:
Let
x
and
y
be observables with
x-null set as in the definition of
(a}
assume that
x
~
x ~ y.
y
Let
and let
A
E
N be an
u(x)\N
and
is a sequence of states such that
n
lim
11
an
(x)
"-
and
o.
lim Var (x)
an
If
(f3}
n
is a sequence of states such that
lim
11
13 n
(x)
A
and
lim varf3 (x)
n
0
then
lim
11
13 n
(y)
==
Proof:
Choose
a
N
l
var (x) < 2A
an
Choose
(13 n }.
Let
N be
ab ove.
Let
a
Vara(x) < a •
A
< t1
for
==
N
2
max(N ,N } .
1
2
t 1an + t , n·
Therefore,
n,m > N.
Thus
lim 1lf3 (y)
n
III
and
(x) - AI
an
in similar fashion for the sequence
n > N
l
so that
implies
We use an argument
Then
IIIa (x)
vara(y) < t
similar to that
- A I < A and
and we have
25
Proposition 6:
When
x,
y
lim 11 (y)
Q'n
and
then
are as in Proposition 5 and
(Q'n}
1.
E
<r(Y) •
Proof:
Choose
t
definition of
> 0 and for each
x
IIlQ'n(X)- AI < .O.A
y.
~
and
We choose
N
l
B(~).
be chosen as in the
n > N
l
such that
varQ'n (x) < 0A'
probability measure in
0A > 0
A let
Recall that
Q'n(Y)
implies
is a
Then by Chebysheves I inequality, we
have
Q' (y)((zllz-1l (y)1
n
. cv n
~ t}) s: 1 2 Var (y).
.
t
cv
n
Therefore,
Find
N
2
such that
n > N
2
and we want to show that
Q'n (y(A-t'A+t))
Var
CV
(y) < t
* O.
implies
luQ' (Y) n
cv n [(11cv (y)
n
Now note that if
t
11
< -.
t
*
2
Then
0 implies
cv n (y) + ~)J
n > max(Nl,N) , then
- Il
and
n
Proposi tion 7:
f
is continuous on
topology on
~"
<r(x.J\N
in the topology induced by the usual
26
Proof:
Let
Ai
o-(x)\N
E
and
t
> o.
2
to
5b
as in the definition of
IA - All <
%.
x
Choose
~
1
corresponding
0
A € o-(x)
Let
y
> 6 >
t~n}
Choose two sequences of states
and
such that
(~n}
such
that
lim
J.l
~n
lim Var
(x)
~n
Let
lim
(x)
J.l[3
n
(x)
u
lim Var[3 (x)
==
0
n
n > N implies
N be a posi ti ve integer large enough so that
6
<-
6
<-
3
Var
6
~
n
(x) <2
3
Var[3 (x) <
n
6
2 .
and
where we have used the fact that
follows from the definition of
1J.l~ (x) - J.l p (x)l < 6. Now it
n
n
x .... y and the choice of 6 that
27
2
Val'
In
(y) <
Using the variance formulae for
t
3b
.
Y with
t
l
= t
2
=
21 '
we see that
or
O\{
i'(.lr
n
suf'f'i.ciently large it is clear that
If("J - f(Al)\ ~ If(A) -
(y)1 +
{i(3
(y) -
1{i(3
n
(y)1 +
{iQ(
n
(y) - f(Al)1
!{iQ(
n
n
The proposition follows.
This implies that
\f(A) - 1'(A l ) < If(A) Choose
N
such that
N )
3
and
l
such that
IJ-Q(
n
(y)1 + 11'("1) - IJ-(3 (y)1 + 1IJ-Q( (y) - IJ-(3 (y)\ •
n
n
n
n > N
l
implies
(x) < 0
Val'
In t
I
A
and choose
(y) <'5". If N = max(N, N ,
2
I-'n
n > N , then all three members of the right side of the
n > N
2
implies
11'(,,) -
above inequality are less than
3"t
{iR,
and
If (A ) ~ l' (A)
l
I<t
.
Proposi ti on 8:
When
N
2
x ... Y
Q(
is a state, and
AE
v(x)~N
such that
28
Proof:
J(f ("
) -
j.J,
01
(y) +
j.J,
01
(y) -
We will now consider two cases of where
!)
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
,,€
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
(-lIx\l,lIxll) then the correspondingy
value is
f(,,)
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.
Proposition 9:
If
x
is an observable and
then each point of
N is an x-null subset of
N is a limit point of
~(x),
~(x).
Proof:
Let
A
would exist
X(tA}) = 0
N.
€
€
> 0
If
A were not a limi t poi nt of
such that
(A - E, A
it would follow that
X((A -
+
E,
E)
n dx)
A + E)
~
=
(x)
tA}
= 0"
there
and since
But by the
29
very definition of .CT(X), x((" (" -
E,
"
+
E) E
~,'CT(X)
A '+ f))
E,
CT(X)
[AlA is a limit point of CT(X)
CT(X)
Observe that since
implies that
0
contrary to the fact Umt
For the following discussion let
N
_CC
is
~losed,
and
A
[A.}.
N
l lE
N ~ CT(X)
E
and let
X({A}) = O} .
we have that
N ~ CT(X)
Proposi tiOIl 10:
:N
is
x-null.
Proof:
x
is a "CT-homomorphism and
N
is countable, therefare
N
is
x-null.
Assume that
x
is the set of all
CT(X)
and
A E
yare arbitrary observables and that
CT(X)
for which
x -+ y
on
We assume that
= 0
N is x-null.
is countable and,thus that
suppose that
x('A)
N
Furthermore, we
CT(xl\N.
Lemma 1:
In a logic satisfying Axioms I-VI and VlIIm, VIIlv and whic.h is
quite full,
'A
€
CT(x)\N
implies
Proof:
By the quite full assumption and the fact that
implies
x(('An
*0
€
e,
'A
€
CT(X~
30
"I"
}-,ave that there exis'ts
0/
Therefore, by Proposi tion 1,
we have
if
Var (y)
01'
=
0
Var (x)
0/
= .0
and.
and, thus, by definition,
is any state such that
01
such th8.t
iJ,
0/
iJ,
varO/(x) '" 0 , then,
(x)
0/
(y)
=
A'
=
By
f(A).
varO/(y) = 0
x ... y ,
Now,
and so
implies
Therefore,
which implies
Proposi tiOD 11;
Assume
x
and
yare as above.
If
e
satisfies Axioms I-VI,
y '" f (x)
VIIrm and v and is qui te full, then
Proof:
By Lemma 1, and the :fact that
for all
A
E
cr(y) ,
x
(~.'!:, Varadaraj an,
is a c-nomomorphism, we have
1962).
31
Now both
x
and
y
are O"-homomorphisms, so we have that
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.
Although we do not have an explicit counterexample, it is quite likely
32
that
~
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
Y is to imply
~
y
f(x)
then
~
x
y
y
f(x)
=
must be
equivalent to the new relation which we are about to introduce.
With-
out further ado, our new relation is defined as follows.
Assume that
(read
x .... y
every
E
and
x
are observables.
y
strongly) iff there is an x-null set
> 0
and
A
E
CT(X~
Var(x) <
such that
ex
s
Intuitively x --+ y
there is a tolerance
N
x-+y
such that for
such that
there exists
E,
for every state
S
We say that
IJ.
ex
(y) +
E)
5" .
means that for an arbitrary error
5 > 0
E
such that the probability that
y
> 0
is in
the interval
IJ.
dominates the probability that
for any state
particular, if
ex
ex
in which
x
x
(y) + E)
is in the interval
(" - 8, A + 8)
can be measured with accuracy
is a state in which
also be certain in that state.
certain as
ex
x
is certain then
Even more crudely,
y
y
5.
will
is at least as
x.
Proposi ti on 12:
If
x
and
are observables then
S
x-+y
In
implies
x .... y.
33
Proof:
Let
E
> 0 and
defini tion of
exists
0A > 0
for every state
E
~(x)\N
€
s
x ...... y.
where
We show that
N is the x-null set in the
x ... y
on
~(x )\N.
such that
a
for which
For any state
~
A
a
Vara(x) < 0A •
such that
Varry(x) < \\ ' we have
Qi(Y)((1l (y)-I{E, Il (y)+\[E))
01
Qi
On the other hand, Chebyshev's inequality implies that
and thus that
Var (y) ~
a
E
+ 411yll
2
1
2 Var (x) .
(6)
a
A
Now there
34
~
Now if we let
state
Q'
. r~
n 1. u '" '
u '" = ffil
which
<
VarQi(x)
1
1
'4 II yll 2
8",
1 2
(8",)
Q'
x
~
h
ten
,
8A. >
°
and for any
we see that
Val' (y) s: E + 411yll 2
It follows that
~2}
u '" €
a",
<
2E •
y .
Proposi tion 13:
Let
x
be an observable and
comp~ement
vlhich is continuous on the
s
.~ f
x
f:
cr(x) ~~,
a bounded function
of an x-null set.
Then
(x) .
Proof:
Let
>0
E
j
and
N
denote the set of discontinuities of
'" E cr(x)\N
there exists
8", >
°
f.
such that
mpli es
If
cr
is any state, we have by Chebyshev's inequality,
II-L (f(x)) - 1'(")1
()I
E
< -3 + 2M 1 2 Var (x)
8
Qi
'"
For each
35
where
M is a bound for
~I
_
If I
on
~(x)
l~
1
~2_}
•
Now let
vA - nun vA' 6M vA E
If
ex is a state such that Varex(x) < 8(", then
2M
L2
6
A
Var (x) < §.
ex
3
.I
and
Let
C=
~(x)\N
and
y
= f(x) , then
and
= f(x)(J.l (y) ex
'I'hus,
and
s
x -. y .
E,
J.l
ex
(y) + E) •
36
Propos:l. tion 14:
Let
s
x -+ y
x
and
y
be observables on a quite full logic 'such that
and let
J(x) -+ J(Y)
f:
be the map defined immediately following the proof of Proposition 6.
If
> 0
E
and
there exists
A
€
8A > 0
J(x)
is a point at which
f
is continuous, then
such that
Proof:
Observe first that the map
J(x) -+ J(Y)
f:
exists since
s
x -+ Y by Proposi tion 12.
continuous (cf. Proposition
E
> 0
0' > 0
A
and
AE
J(x~,N.
such that if
~
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
Moreover, since
x
for every state
~
s
~
y , there exists
for which
6~' ~
var~(x) < 6~;
0 such that
Let
We intend to show that
by showing that for every state
~
for which
It also follows that
~(y(f(A)
-
€,
f(A) + E)) = 1
The desired conclusion then follows from the fact that our logic is
qui te full.
This is easy, if
then we see that
~
is a state such that
38
Thus
VarQI (x) .< 0"A
and
But
thus
E
<2
and
It follows that
and thus that
QI(y(f(A) - E, f(A)
The proposition follows.
+
E))
=
1 .
39
Proposi tion 15:
Let
x
and
y
be observables such that
s
x--.y.
Then there
is a function
1':
CJ(x) -+ CJ(y)
which is continuous on the co.mp:;L.ement of some x-null set and which
y = fex), l.~., l' has the property that
has the property that
y(E)
for every Borel set
f(x)(E) ='X(f-1(E)
E.
Proof:
Let
be the function defined after the proof of Proposition 6,
1'( limj.1 (x»)
n-+ CO ri n
limj.1
l.~.,
(f(x))
n-+co ri n
We first show that
for
If
E
any open
set.
For
A € E choose'
C denotes the set of points of
continuous then, for each
CJ(x)
EA,
> 0
at which
such that
l'
is
40
there exists
0t > 0
such that
By use of Proposition 14, we see that we can (by choosing
smaller if necessary) assure that
where
But
and thus
~
Now
and thus there exists a sequence
(t.}
l
in
y(E) •
0t
41
such that
f-
1
(A -
L"
~
A+
€,) n c ~ U (t. - at.' t.
~
i l l
1
+
at )
i
(recall that
is separable since
is).
R
Thus,
u x (t.
~
~
Now
E
at.'
-
t
t. )
y(E)
is separable and
(Ai}
thus there exists a sequence
E =
U (A.. i
1
€ \.'
11.
1
E such that
in
+€
1.. 1.
Ai
).
Then
r-,l(E)
=
~ r-1(A i
1
and
+ 6
..
1
i
1 1 1
- EA.'
1
Ai
+ CA.)
.1
42
It follows that
Now let
a
E
E.
x(f-l(E)) ~ y(E)
for every open set
E.
We will show that
x(f-1 (a))
~
y( ( a}) .
Ii'
0A > 0
then by Proposition 14 there exists
Since
f-l(a)
f-l(a)
nC
nC
such that
is separable, there is a sequence
(Ai}
such that
f -l(a)
n C ...~ U (A.
i
~
- 0A ' Ao + 0)..
~
i
)
•
1
Thus,
~
yea - E, a + €) •
Now
. ·1
x(f- ·(a)) ~ yea - E, a + E)
in
43
for arbitrary
-1
x(f
E
(a)) ~
Finally, let
C
E
thus
CXJ
n
1
1
yea - -, a + -) = y((a}) .
n=l
n
n
E = (a,b)
= (- CXJ,
be any open interval.
Thus
a) U [ a} U (b} U (b, + co )
and if
u = (- 00, a) U (b, + CXJ
then
)
U is open and
It follows that
= x(f -1(U)) U x(f -1 (a)) U x(f -1 (b))
~
Thus
y(U) U y((a}) U y(tb})
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 a.rgument.
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
E
B(~)ly(B)
=
is easily shown to be a
~-algebra
which contains all open sets.
A = [B
Since
B(~)
is the smallest such
X(f-l(B»}
~-algebra
and thus that
y(B) = x(f
for every Borel set
B.
-1
(B»
we have that
A
= B(~)
45
At·· this 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
f:
o-(x) ... o-(y)
be the naturally induced mapping.
(1)
f
Then
is continuous on the complement of an
x-null set,
(2)
the relation
S
x-+y
is equivalent to the
functional calculus equation
(3 )
the relation
(4 )
if
x
S
x-+y
implies
y = f(x) ,
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
!.~.,
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
un
TOPOLOGJ:'
()BSEJiVABl,~S
OF STATE
AriD CHANGE
When one studies a system he sets up several experiments and
proceeds to measure aspects of the systern.
finite number of experiments.
Each
can do at most a
He
e;~perimCi1l> \~ill
consist of
studying a probability distribution on the. real line - or more
generally on
say
~,
N
.
The observer eGn do this wi th limi ted accuracy'.
th
is the error of the i-' experiment.
e.
1
'Ehis leads us to stUll)'
a topology on ei ther Ule observables or the ptates of our model.
Since both the observables and the states are constructs, it is not
clear which should have the topology.
obtain
/.L
i
We do
N :experiments and we
.th
. n t , and
the mean of the 1
- exper1me
Var. < e.
1
1
(we mean
here that we have done many experiments, see for example von
Neumann, 1955).
To impliment these thoughts on our model, we need to
consider two possibilities.
Are we doing
n
x , and considering
experiments with the sysLeiL in oue state, say
n
observables in that state?
would consider as neighborhoods of the state
the
e.
1
Ai
(i
th
correspo~ding to
observable,
x
the i
In this case, we
those determined by
th
experiment)
an~
the
By this we mean that
I tA.x,
. 1
x) - ))..1 I~. 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
A
In this case we say tbat
determined by
B
Lx.1 }
The former construction yields a model in which the observab.l.c;;::,
correspond to experiments and the state is, in essence, the ob,jc,ct
the study.
This is called the weak
for example Haag, 1972).
*
or
topology on the states (see
We will not pursue this any fur,ther.·' 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 comparisori of
putting the strong operator topology
On
the observables\,ri ttl the
~tTeak
topology on distributions.
Proposi tion 1:
A
Suppose
n
~
A
in the strong operator topology.
Then
A
n
in the weak operator topology.
Proof:
1
((A
n
- A)x, y) ~ ((A
n
1
222
- A) x, x) (y, y) .
This means that if two observables are near eqch other"
t.tli~ll
their'average values are also near each other.
Proposi tion 2:
As before, let
integer
m,
An
-" Am
Am ....
n
~
A
strbngly, then, for each posi ti-,'e
s t rong ly •
~
A
48
Proof:
First we note that there exists a real
M such that
\lAnll:o;; M
(see, for example, Riesz and sz.-Nagy, Functional Analysis, p. 200).
Then
11
2
2
(An - A )xll ~
II [An (An
~
Mil (An
- A) + (An - A)A]xll
- A)xll +
II (An
- A)x\l •
Now use induction and the result follows.
Proposi tion 3:
Suppose, as above, that . ~n
that each
~
A strongly and assume, as usual,
is an obseEvable (!.~., is se~f-adjoint).
An
continuous function,
f:
~ ~ ~
, then
f(A) ~ f(A)
If
f
is a
stronglY.
n
Proof:
Let
domain
M be as above, then we may clearly assume that
[-M, MJ
Thus,
f
is the uniform limit of
(stone-Weierstrass Theorem).
and since addition is continuous in this topology,
strongly, where pm
is any polynomial of degree
This completes the proof.
Dunford and
Schwartz,
m
~
Am
.
.
strongly
c
pm(An ). ~ pm(A)
So now
This proposition may be found in
1963, (Vol. II), pp. 922-23.
has
polynomi~ls
Am
n
Now, since we have
f
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 weak+y 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.
f
interval
f (A)
Further, when
We notice that
t E ~(f(A))
have
Var x A <
be continuous.
of
E
x
•
tl
< 5
and
~
function, or
onto a bounded
is defined and is bounded.
5 > 0
such that if
x
is
Varx ((f(A)) < 5 , 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
from
there exists
I~ (f(A)) -
a state such that
arctun
We consider the
for that matter, any homeomorphism
(a, b)
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 with small
(zE~ro)
variance.
Of course, one does
not really observe an observaple 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, 1955, for a
We will discuss this at greater length in
conjunction wi 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 n;.,t
which converges in the strong operator topology to the "pasi tion"
observable.
This particUlar example is quite important due to the
fact that all observables are (in a sense to be made precise below)
posi tion 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)
with
f
H with respect to
= (PA
f, f)
E
being a cyclic vector for
is not unique since
uni tary operator
U:
f
A.
is not unique.
H
-+
A, so that
In general, the meilsure
Moreover, there
c:,~,
~
'J
.£2( -IIAI\' IIAnll) such that
tf (t) .
This is, of course, a result of the spectrum theorem.
As was
mentioned above, we may view the spectral theorem as telling us that
in our model, every observable is a posi tion observable - tbat j.s,
every observable is uni tari ly equivalent to a posi tion observable (\,'e
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
PE ' which is
o on
and
on
1
E ,
~ (t)'f(t) .
This is the spectral measure
corresponding to the operator
..
above.
A
Tl1i.s is important since as we well know, every measurement
t~
corresponds
the position of some object.
(See Mackey's
"Induced Representations of Groups and Quantum Mechanics I' .• 1968,
pp.
62-63, for a discussion of the nature of the position
observable.
Motion",
Also, see Nelson I s "Dynamical Theories of Brownian
1967,
where
the correspondence between observables and
posi tion is mentioned.)
We now proceed to the example.
Proposition 4:
Let
I
~
R be a closed interval and
probability measure on
given by
partition
of
Let
tf (t )...
(Af)(t)
P
r
I
J'r[tf(t)
A be the operator on
Then for each
> 0
E
such that if
E
is any refinement of
a continuous Borel
~
P
= (E.}.
Z
1 lE
then
- L: . a.1c.X.
(t)J2d~(t) <:
11
E
2
2
S- (r,
~)
there exists a
52
[J~.f(t)dcr(t)J/a(Ei)
C
i
1
ai
[JE. tda( t)J/ a-(Ei )
1
Proof:
First we show that for each
that if
E = (E.}
1
refines
E >,0
there is a partition
P
P then
(1)
with
c.1
and i
X as above.
To see this, let
E.1 = [(i - l)e, iel
for
i
E
Z and note that for each
f
E
£2(1, a)
2
[[f(t) - ~ c.X.J
da(t) =(f - ~. 1
c.X.,1f .- 1
~ c.x.)
11
1
1
(f, f) - 2
~ c.
(X., f)
1
2
+ ~ c.a(E.)
.1111
1
~ (f, f) - 2 ~. c~a(E.)
+ ~ c~a(E. )
1
1
.
1
1
1
J.
r
= ~
[f(t)2 - c.f(t)]da(t)
. E.
1
1
1
= ~ JE.f(t)[f(t) - ciJda(t) .
J.
-
But if
1
A E E.1 , then
( f, f ) -
~
2 ( E. -)
c.a
.11
J.
so
53
(2)
If(t..) - c i
I. =
-leSE. f(t..)dcr(t) - IE. f(t)dcr(t)]/o-(E I
j)
1
1
lIE. [f(t..)
f(t)Jdcr(t) l/cr(E-j)
1
= [fE.
€
1
dcr(t)]/cr(Ei ) =
€
•
Thus,
S[r - ~
C
1
dcr
i \ ]2 (t) <
As a consequence, we have for
~
JE. If (t )\If (t)
1
1
f(t)
=
- c i Idcr(t)
t ,
(3 )
Observe that our proof above works for any refinement of the
specific
for
(E }
i
t , t
l
2
chosen, since the crucial property is that
in the same member of the partition.
f € l2(I, cr)
and
> 0 ffild choose a partition P so that if
€
(E.}. Z is any refinement of
1 l€
E. h f
1
for each
i
€
Z.
-1
P
then (1) and (3) hold and
[(i - l)E, i€]
Now recall that if
a, b
22
(a + b)-2
::;; 2(a
+ b )
and thus
Now fix any
€ ~
then
,;
'[
1
t [' \, t)
c.
I'
(t. )l
!'L U'
:t )
- L: a. fX.
L a.
f-
1
1
rx.'
,:1, C. \.
j
Ii
;)
]c du
I
~)
tlL
fX.
oJ
1
;:J
~ 2Jr(t)-C~
1,
+ 2JT~
+
?
a (f(t) -
C
i
2L:a:SE.[f(t)
i
)Xj (t)rdu(t)
ciJ2d~(t)
-
l
By
(2) for
we have for each
f(t) = t
Similarly, for each
tEE.
we have
~)
~t
l
tEE. ,
l
Thus,
L fE.
1
?
f(t)~(t - a i-duet) <
i
_?
~
(-Jr:. f(t)'-dolt.)
J
and
':)
c'-er(D)
1 ,jo
jltf (t)
- ~
C.
.I
x,
':-l
(t
n-(1'T
(t)
(2[.
<'
"
.1
Ii!--:-:('1,2
.1
0(1)] .
The propos i ti on follows.
We now introduce a bit of notation.
E
of the interval
for each partition
lE.}.
Z
1 J.E
define
I
by
I:
. Z
lE
a.c.X
].
1
E.
·
:1
where
[J~.td~(t)J/~(Ei)
ai
1
and
[JE. f(t)d~(t)J/~(Ei)
ci
1
are defined as before.
If
FE.
is defined by
1
[[JE.f(t)d~(t)J/o(Ei)}XE.
PE. f'
1
then we see that
J
p
~E.
~.
spanned by
X .
E
~
and
1
is a projection on the one dimensional
56
(compare with the remarks introducing Proposition
each
E,
~
!..~.,
is an observable,
~
4).
Note that for
is a self-adjoint linear
transforlIlati on.
Proposition 6 may now be rephrased as follows:
Proposi tion 5:
The net
[~}
A in the strongest operator
converges to
topology.
Proof':
We assume that
(f.}.
h.
I IE
converges.
is a sequence in
We show that
i
semi-norm determined by
E
> 0
(f.}. h'
I
there is a partition
co
(1)
L:
i=l
for each partition
A - A
E
(r, 0-) such that
goes to zero in the
Thus·we must show that for each
IE
P
2
.£
such that
II(~ - ~)fi,,2
E which refines
<
E
P (cf. Sakai, 1971, p. 34, for
the definition of the strongest operator topology).
it is useful to show that
for 'each
n.
To see this, observe that
IILf
-~ n
II
=
IIkE:Z
L:
akcknx. II
I
'orhere
a.
1
[j'E. tdo-(t)J!o-(E i )
I
'.
Before doing this
')'(
and
c?l = [J'E . f n (t)drr(t)J/rr(E.)
l
l
Thus,
Thus, (2) ho1qs.
such that
Then
We now prove (1).
Let
E
> 0
and choose
N
E
h
58
N
s:
L:
neeJ.
C:D
r)
!1All c.
+
L:
n"N+J.
N
< 2:
0==1
Now for each
!I(AE
ls::ns:N
- A)(f
)11
n
2
E
+-.
2
choose a partition
P
n
such that
E
<-2N
for each refinement
Proposition
4).
E
of
P
n
(that this is possible if due to
The proposition clearly follows since we now see that
Since we have observables such as
observable
~
near the position
A, it is interesting to ask the question as to whether
the new states arising from measurements of the
srune appropriate topologYf
~
will converge
~n
If so, then we 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 we are unable to produce
the necessary counterexample.
.•
;
We will discuss this in more detai 1 in
the final paragraphs of this chapter.
conc~rned
The next five propositions are
with elaborating these ideas.
Note that
~
is not strictly speaking an observable wi th a
simple discrete spectrum, but we can modify it slightly so that its
modification has' a simple discrete spectrum and in such a way that any.
'.
59
limit of the net is not changed . . For each partition
E
(E.}
of
1
1
let
A(E)
where
m(E.)
1
=
1
is th~ Lebesque measure of
characteristic function of
of . £2 (I, cr)
max m(E.)
l~'
..L':::>ls:n
and, for e;ach
E.:I.'
then
E.•
1
Lx.1 }
If
,1
is an orthogonal Imbset
E, we choose an orthonormal oasis
\
Lcpj(E)}
for the orthogonal complement of the subspace of
spapned by
(Xi}
is the
X. = X,
1
'E,.
')
.}~L(l, 0)
Finally let
2
vihere Pcp. (E) is the projection of £ (I, cr) onto the one.
J
dimensional subspace spanned by cp.(E).
Observe that
J
It is clear
that
..
lim A.(E) ,~ 0
E
and, thus, the nets
and
(B } have the
E
saml~
11mi ts (i:1 (LOrm,:.
We now consider the new observables obtained after a meaLur';cmert.
U
E
state
be the nevi state arising from
U.
L
Then
n
L:
1=1
tr(UP . )P . + L: tr(UP 'E')P. (w)
X
X
j
.
CPj'
CPj ~J
1
1
(.
Let
13
60
n
UE ~
L tr(UP . )p .
X~ x~
i:::;l
(U }
We first show that the sets
E
and
hav~ tpe same limits in
(U }
E
the .weak topology:.
Pro~osi tion
Let
parti Lion
6:
Q be any projection and
such that if
£i:
E
> O.
Then there exists a
:is any finer parti tion then
E
Proof:
---.,
First write
2
projection of
U·, L aiP f .
where
a
~
i
~
IIfill
=.:
1
and that
a
L
TO show that the latter is less than
case when
U
n
.
Then
C.Lc::rly we
converge,,:
We have that
n
J.
E
for appropriate
, re-label it more simply by
= Pf
f..
- ~ tr (UPX. )
~
f
is the
l
n=l
=1
Pf .
l
00
choqse any
and
onto the subspace. spannEld by
S,
may assume that
0
E') we
f , and consider the
61
.
. 2
X. ) (X.), x. )/IIX.II .
;:: ((P1'P
~
~
~
~
::: ((X., 1')1', X. )/cr(E.)
111
and
1- ~ tr(UP .)::: (1', f) - 2::, [(1',
X
1
(f, \)Xi)/cr(Ei)J
1
1
(1', l' - 2:: [(f, \)/cr(E )]X )
i
i
i
~ 111'11'111' - 2:: [(1',
i
x.)1 cr(E.1 )]X.l1
1
~
1I1'-2::c.x·1I
.11
1
.
where
c.~ ;:: (1', X.1 )/cr(E.)
;:: [J'E . 1'(t)dcr(t)J/cr(E.)
•
~
.
1
1
By (1) of tn,e proof of Proposi tion If we see that there is a
partition E*
Thu~
of
I
such that whenever
for any refinement
E
I: tr(P f P . (E)
j
n cpJ
of
'" lI'f
E
refines
E*
- 2:: c.x.11 <
n i l 1·
This concludes the proof for the case that
E
U;:: P ,
f
n
case,' choose
N
such that
E*,
For the general
62
co
L:
n=N+l
an <
E
"2
Apply the proof given above to obtain a partition
E of
every refinement
L: tr(p f
E*
~
n
N
~
such that for
J
:P . (E» < -=N....;E~-
n cpJ
j
1
and for
S\
2( L: a +1)
n
n=l
Thus,
j
= L: L:
j n
(E)
L: tr (up
Cl'j.
a
n
N
= L:
[L: ti
n
j i n=l
N
s:
L:
n=l
co
a
~ tr(PfP .(E»
n J
. n tl'J
+
L:
n=N+l
an <
E
•
From this we see that
tr((UE - UE)Q):S; ~ tr(UPcp.(E»
J
<
E
•
J
The prmposition follows.
Proposi ti on 7:
i
Let
A be an observable such that
exists for each projection
Then
Q in the spectral resolution of
A.
63
exists and
(~}FEB(~)
where
is the projection valued measure of
A.
Proof:
i
Since
A is self-a.djoint there is 9- projection-valued measure
'"
B -.
~
such tha,t
Thus if
~
>0
then there is a partition
that for any refinement
IIA -
E
of
E*
of
[-\lAII,IIA1IJ
such
E*
m
l: Ai ~.II <
i=l
1
E
3"
We qave assumed that
.
exists for each
exists.
E.
1
and we wish to show that
(tr (U~)}
We Wi 11 show that
showing that it has a limit.
parti tions of
(1)
[-IIAII, IIA1I]
ttr(U~) - tr(U0)1
is a Cauchy net tl1ere'l:'y
To do this assume that
F
and consider the ineqlla.lity:
s;
1lI
m
Itr(UFA) - tr(UF(i AiQEiJ)1
III
+ .ltr(UF(l: A.Q~ )) - tr UK(r, A·Qv
1 1 ~Ei
1 1 .c'i.
)1
and
K
are
64
m
+ Itr(UK(~ Ai ~i)? -' tr(U0)
I·
We wish to show that each oJ' the three swnmands on the right-hand
side of the inequality goes to O.
(2)
m
Itr(UK(~ A.Q1
l
"'E
i
» - tr(U_A)1
K-
First observe that if
m
n·
I ~. A. '~l tr(up K. )tr(PK.QE. )
i=l l J=
J
J l
-
n .'
- .~ tr(UPK)tr(PK.A)I
J=l
I
J
.J.
n
m
~
tr(UP
j =1
(A(X
K
Kj )[ i ~=1 A.l
), X )
K
a1K. )
j
JI
J
~ Ai~. (X K .) - A(XK .),
n
=s;
•
~l tr (UP K. )[ I
J=
. L:
J= 1
tr (up K. )
J
The latter inequality is independent of
Finally,
J
J
n~ Ai~.
n
~
(J"~K.)
l
J
An II x K.11 2
-
.~ (K. )
J
K', thus
J
(4 )
and since the map
J ~ tr(UJ~.)
is a Cauchy net (for each
~
have that, for each
for arbitrary refinements
cQWmon refinement of
than
i
R
i , there exists a partion
R1 ,
i
K of
R •
R2 , ••• , Rm
If
F
and
such that
R be a
Now let
F
i ), we
and
K are finer
R then
Itr(UF~.) - tr(uK~.)1 < ;m •
1
1
Adding the three inequalities (2), (3), and (4) and substituting. into
(1), we get
ThUS, 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
0-
i" a continuous measure.
66
Propasi tion 8:
If
is a projection whose image is a f'ini te dimensional subspace
2
of
S-
(1, <J)
p(g,2)
then
o.
lim tr (UEP)
E
For any parti ti on
of
E = (E.
'}
. J.
X.J. = )L
. 'K
let
1
denote the
J.
Ei , let
characteristic function of
be the projection of
PX.
J.
x.l ,
onto the one dimensional subspace spanned by
and let
P.
1
S-
2
be
defined by
I;'
!\Y
i
J
i3 an orthonormal basis of the orthogonal complement of the
spanned by
'ulJi3pace of
tr(UP. )
J.
X. , the observe the
l
l: ((UP.)'f.,
j
l
J
'if.) + ((UP. )(X.), X.)
J
J.
l
l
and
(UX., X.) •
tr(UP . )
X
l
1.
Thus,
tr(UP.) ~ (UX., X.)
l
J.
J.
tr(UP x .)
l
l.
Now suppose
a,nd let
P
is a projection onto a one dimensional subspace
be a unit vector in
q>
p(.£2).
We have for any partition
I,
of
(~ tr(uP . )Px . (q», q»
x
1
No~
1
1
we claim that if we define
for each Borel set, B, then the map
is a probability measure.
Moreove~
continuous with·resp~ct to
that if
P
B
Now if
set and
=0
€
~(B)
0
(~
> 0 , there is
< 0 , then
parti tion
R
~(E. )
for every
< 0
To see that this is so, simply n0te
of
0 > 0
tr(UP ) <
B
Since
~
such that if
I
almos t cverywherEJ
tr(UP ) = 0 •
B
and
Measure Theory, 1950).
~
~.
= 0 , then
~(B)
(PBf)(t) = ~(t)f(t) =
and thus
this measure is absolutely
i .
tr(UP. )
1
such that if
B is a Borel
(see page 125 of Halmos'
E
is continuous there exists a
E
is any refinement of
Thus, for an.y
tr(UPE )
1
<
E
E
vrhich refines
th(~rJ
I
n
1. \
~
E
68
and
~
1
tr(UPi )j'E.CP(A)2 d O-(A)
~
~
E
1
tr(uEP) < E
It follows that
JE.CP(A)2 dO-(;\)
E .
11·
and
Finally if
P
is a projection onto a finite dimensional subspace
and
is spanned by an orthnormal set
+ •.. + P
CPn
then
and
n
L:
i=l
lim tr (UEP )
E
CPi
o.
Theorem 1:
Let
If
= r 2 (I,
any observable on
0-)
If
where
0-
is a continuous measure.
If
A is
whose spectral resolution is a series
CD
A
L:
i=l
in which each of the projections
A.P.
1 1
P.1
has a fini te dimensional range,
then
lim tr (U A) = 0 .
E
E
In particular this is true for all compact operators and all trace .
class operators.
Proof:
It follows from Proposition 7 tha.t
UEP~·JJ
Hm tr (UEA) == j\d[lim[tr
E
>A.
(PB}Bf: B(~,)
,;",here
E
is the spectral measure of
A.
But
CD
L:
i==l
Ai lim tr (UEPi )
E
and
by Proposition
8.
Thus
CG
lim tr (UEA) ==
L:
i==l
E
A. lim. trCU",P. )
1
E
O.
.c, 1
The theorem follows.
Proposi tion 9:.
As before, for each partition
Xi
==~.
1
X.
1
1
(P.f)(t) == X. (t)f(t) .
1
Let
vB
and
of an interval
the projection onto the subspace of
1
'')
2
by
Define a map P.: s.,'- -+ s.,
Px .
and
spanned by
E == (E }
i
U
E
1
be the operators given by
"',
L: tr(TJ1)x.)p .
x
i l l
L: tr(UP. )P
i
1
i
x
..
;
I
s.,2 (1, 0-)
let
70
If
Q:
:2 ... s.2
is a projection, then ei ther both the limi tt
lim tr (uEQ)
E
Cl.nd
lim tr(UI8)
fli
exist or both fail to exist.
""--l
Moreover, if both limits .exist;. then
they are equal.
-Proof:
Let
[1 j J denote the orthonormal basis of the orthogonal
cOmplement of the subspace of
2
S- (I, ~)
spanned by
:;:; l: l: tr(UPk)((Q)?x )(X.), X.
i k
k
1
1
,)
x. )/llx.ll~
111
1
== l: tr(UP.) (QX.,
.
1
== l: tr(UP.)[(QX.,
.
1
and thus
x.)/~(E.)J
111
1
)/llx.!j2
1
{x.}
.
1
Then
71
==
f tr(UP i
)
(QX.,
X.)
J.
1
cr(E )
.
(QX.,
X.)
1
1
cr(E.1 )
i
(QX., x. )
~ [tr(up i ) - tr(UP~.)J -cr-(=~-.~)l-i
J.
J.
~ ~ [tr(UP ) - tr(UP .)
i
X
i
1
== 1 -
~
.
1
tr (upX.
).
1
But in the proof of Proposition 6, it was shown that there is a
pal'ti tion
E*
E of
so that for every refinement
E*
~tr(up'i'.)<E'
J
J
(Note that in Proposition 6 the family
('i'j}
is denoted by
[cp.(E)}).
.J
It is olear that
1 - ~ tr(UPX.)
J.
1
thus we see that
__
Proposi
.. ti on 10:
. _ ~ _
.
_
~
')
If
measure
A is the mUltiplication operator on
£C(I, (J)
cr
such that
FA
Pf =
X:F: f
Gtnd if
P
is a projection of
then there is a Borel subset
f
€
s-2 (I,
CT) •
F
of
I
£2
such that
fOI·
c;onh:
==
AP ,
for all
72
Proof:
We will first show that
P(f) == f· pel)
for all polynomials
and use the fact that the set of all polynomials is dense in
to conclude that
that for each
f
s..2(eJ)
But observe
P(f) == f'P(l)
n
Thus
for all
n.
From this it is easily seen that
f.
all polynomials
P(f)
'rhus
==
f. pel)
P(f) == f· pel)
for every
f
E
'for
s..2.((J).
In
particular'
P (1 )
and thus
Thus
pel)
pel)
(P'P)(l)
P(l)P(l)
P(P(l)
is a function having only
== ~
for some Borel set
F
0
and
and
1
as its values.
P(f)
'1'11e author is indebted to Stephen Campbell for shovring him the
. ',JToof of this
"well-knoWtl" proposition (Proposition 10).
Proposi tioD 11:
Let
B
be a Borel subset of
into
~
2
(I,
eJ)
~
defined by
and
P
B
the map from
73
If
P
is a projection of
Borel set
s..2
which commutes wi th
P
B
for every
B then
exists and
for some non-negative function
subset
L of
.~
such tQat
1
f E s.. (K,
Pg
X·g
L
cr)
and for some Bore}
for every
g
s..2( I, ,
E
'
0')
Proof:
By the previous proposition we see that
some Borel subset of'
.~
lim tr (UEP)'
E
P
=
P
where
L
L is
By Proposi tion 9 we see that
= lim
tr (UEp )
E
(P(XE .), ~.)
~ tr(UP .)
E:I.
1
is a partition of
I
1
1
cr(E.1 )
and if we take
a refinement of' this partition we see that for each
L
n E.1
cp
or
L
n E.1
E.
1
E
i
=
[E.}
1
either
to be
Thus
lim
I:
i,E.g,
1
E
Since the measure
respect to
.(J
tr(UPE. )
1
is absolutely continuous with
we know that there exists a function
f E!hI, (J)
such that
for every Borel set
B.
It follows·that
Thus,
aqd the proposition follows.
Theorem 2:
=+,
I
If A is the mUltiplication operator on
lim tr(u~)
E.
exists.
Moreover, lim tr(u~) ~ tr(UA)
~
f.:. ~
E
.
Proof:
If
A is the multiplication 'operator And
A
:=
J"dP~
2
l (I,
(J)
,
then
75
if the spectral resolution of
measure
[ pA}
.
B BE: B(~)
A then the projection values
is gi ven by
(P~f')(t) = ~(t)f(t)
If
B is any fixed
Borel set it is clear that
all the other projections
~ for F
E
B(~).
p
A
B
cormnutes 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
lim (UEA)
E
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
probability theory we are interested in the set of probability
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 positive definite trace class
one operators •. We have interpreted thl'!se operators as being the
states 'of our theory.
In this chapter, we have supposed that we are given a physical
system whose state is represented by the trace class one operator
U
:It is assumed that a measurement is made on the system and that, as
a result, the system is
forc~d
into a new physical state.
It.has
been our object. to determine the trace class one operator corres,":
pjnding to this new state.
that von
N~umann
OUr point of departure has' been to notic,e'
(1953),has'already d,efined a "new state" operator
provided one makes only
observation~
have a simple di screte spectrum.
whose corresponding observables
We are also motivated by the
Campton-Simons experiment in which the position obserVable was
measured.
From the conclusions of this experiment von Neumann
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 determine
whether or not there exists a positive definite trace class one
operator
o~tained
U corresponding to the change of physical state
A
when one measures the position operator
whether it was possible to define
UA . in
A.
We wanted to
,s~ch
de~er~ne
a way that the process
changing state is continuous in a certain weak topology. ,To
acqomplish this we found a.net
spectrum converging 110
(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
as did von Neumann (recall that the spectrum of
We then attempted to find an operator
U in the same weak topology.
A
to be an operator
BE
U such. that
A
is discrete).
tUA}
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
We have shown by Theorems
B.)
.1
exists.
Theorems
1.
and
U exists and
A
g
imply tha t there is .Q£ such
~
Theorems
1.
and
~
UA
~
will assume that
to ge! a contradiction.
By
Theorem 1 we see that
B such that
for any observable
for every
then
B :=
B := L:A.P. •
II
~A.P
l
..
l
But
and thus is itself a trace class one operator.
tr (UAUA)':= 0,
and· UA
=
O.
Thus if
U exists
A
should be a stafe
Thus, for
B
=
U '
A
On the other hand, Theorem 2 impli es
that
for appropriate
U
A
=0
and
f
Thus
.
tr(UAA)
f
tr(UAA)
=
O.
0 , a clear contradiction.
He can only conclude that :i,f
Thus we have that
exists for every observable
B
measure on the
Observe that tbt: mapping
(~vent
space.
then it wi 11 [iot arise from a
B ... lim tr (DEB)
E
is a linear functional (provided that the limi t exists for all
and thus it can be viewed as a "generalized" stateo
For further
discussion of these states and other matters, see Saka.i (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 deterniine 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 n-tuple of
numbers.
n
So we ideritify the results with elements of
Thus, to synopsize we assume that
X
=
Dt
n
for some
A X B and that there is
a function
h:
X -+ fir,
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
and the real Borel functions on
bet~veen
X on the other.
the observables
B
80
So let us assume that there exists such a set
X of
impossibly complete descriptions and that a point qf
determines the results.
of nature are causal.)
X again
(This is, of course, asswning that the laws
Now let us make the
fol~owing
observation.
mlen 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
tpen have a probability measure corresponding to our lack of
information about the remainder of
Since
X.
X = A X B , we fix some
b
€
B which in turn determines
A.
some probability measure,
Thus, since we cannot really
find the point
b
"exactly", we should also have a probability
measure
B
This yields a candidate for a probability
~
measure on
on
X, namely
We could consider that
~e
can
~nlarge
we really should consider measures on
We are led
~n
B at Als expense, aqd hence
X itself.
a natural fashion to make the followi'ng con··
vention.· A class of experiments will correspond to a
measure on
.
X.
The variables whose values are of
which we measure are real valued. Borel functions on
probabi~ity
~nterest
X·.
to us and
We do not,
of course, know if all measures correspond to classes of experiments.
We now proceed to develop a Hilbert space, called the space of
(J.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 correspondence between classes of experiments l';j.lld the
81
probability measures and between the observables (random variebles)
That whi~h follow~ is from E. Nelson (1969) and
and Borel functions.
is really added for completeness.
We consider equivalence classes of ordered p~lrs
!J.
X and where
is a real val.ued Borel measure on
X such that
valued function on
f
E
£2(X. !J.).
(f,!J.)
f
where
is a complex
The relation
~
is
defined between pairs of such elements as follows;
(r, 11) "'" (g, v)
A such that
iff there exjsts a real valued Borel measure
A»
A»!J.,
\) , and
~
. (A. almos t everywhere) .
Claim:
_
is an equivalence relation.
Proof;
; ,s i nclepellclent of
"'Ie need to s heM !:llU; :i:
is a Borel measure
:j
rrr;;dd~,
f :v~
Let
A"» A'
and
A.! »!J.
==
and AI »\).
ldu'd~ I
g.j~
A"» A ,
( XI ~.~.
~.~.,
/',
.
l.hen ,ve show that
)
A'
+ A
==
A".
-
()., a. e. )
,....
.\
(A" ~.~.)
Then
82
fdv
(A" ~. S· )
gV~"
(A I !:!:.~. )
The rest of the proof is clear.
1m
equivalence class of such pairs is called a O"-function and we
f~
write
(f,~).
for the class
functions with finite Borel
We can make
H(X)
We denote the set of
~easures
by
0"-
H(X).
into a1f fnner-product space and we will show
tqat it is complete with respect to themner-product.
finiteness of the measures to do this, which for
~s
We use the
is no limitation,
since we are interested in probabilities which are finite measures.
Addition is defined as follows:
A»
where
A» v.
~,
This is clearly independent qf the choice
of representation and also of
a(f, ~)
=
(af, ~)
Thus
H(X)
~.
If
gldv) =
€ ~ ,
then we define
is indeed a vector space.
Finally we define thefuner-product,
(fVd~,
a
«f, ~), (g,v»
, by
Jfg JfJ. 1ft dA
(recall that the measures are all real).
This is well defined,
it is clearly independent of choice of representatives and of
.:!:.-'~.,
A.
83
It remains only to show that
( 1' ~ }
n
is a Cauchy sequence.
n ,
H(X)
Assume that
CX)
n n==l
A such that
We first find
A
»
~
n
for all
~.g,.,
(this isvihere
lie;
usC-od the fini teness of the measures).
is a Cauchy sequence in
s.,2(X, X)
2
s., (X, X)
'l'hus,
and tbus it has a lirni t, say
1', in
and we notice that
lim l'
n
in
is complete.
'fdii n
n
H(X)
Let us consider some examples.
Lini te set.
Then
is clear that as
tractable.
(0, 1).
sort of
H(X) ==
en
2
L
If
X
X becomes bigger,
We believe that
\'le
Let
H(X)
X == (1, 2,
3, ... ,
n1
a
')
== ~ ,
then
H(X)
rapidly becomes un-
fj:(X);= .t"-(IN).
is not separable in the case
wi 11 see later that in some sense,
It
X is
H(X) , wi 11 be some
,but that the measure wi 11 be a direct sum of measures
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
In this case we need to extend the concept to
fact that we do not have a fixed measure on
tak~
Ordinarily
X into
~.
into account the
X but an entire family
84
3*(X)
of them.
Here
3*(X)
valued Borel measures on
denotes the set of all fini~e real
X.
We define an observable or random
variable in this context to be a family
iJ.
€
a*(x))
(h)
where, for each
iJ. /..eB* (X)
is a Borel function from X into ~ and where
h
iJ.
A »iJ. implies that
h
h
/.L
(iJ. almos t everywhere) .
A
We denote the set of all random variables on
to shOW that
~
an algebra over
j_8
defined operations.
~
the set
~
Now
iff, for each
and the family
X
multiplication of elements of
iJ.
(hiJ. )
is an algebra over
~
It is trivial
For the purpose of simplicity we extend
is in
Borel map on
~
VIi th the usual obviously
of all "random variables" which are
(hiJ. )iJ.E3* (X)
above.
~
X by
~
C
C
~
O111P lex-valued,
to
!..~.,
,
is a complex-valued
h
iJ.
is subject to the condi tion (* )
since we may admit scalar
by complex numbeDs.
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
II
lion
x, y
Actually
(9
-
(9.
x
E (9
We define
~
is a
C*-algebra,
~.~.,
there is
and
Ilxx*1I
for all
is a complex algebra with inVOlution.
such that
(9
in
~
II x ll
To see this, let
II hll H(x)
by
2
h
(hiJ. )iJ.e3*(X)
be any element of
85
(Actually
~
doesn't consist of all
(h)
-
must be cut down to inc lude only those 1'1
finite.)
as defined earlier but
IJ.
~
One must check to see that
for which
IIhllH(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
IIbhl\II
:0,
2
111'111
.
-1
sup
IJ.
~
sup
IJ.
sup
II:NCiUII =1
sup
II f\[dj11l =1
-1
J(h h )(ff)2(ff)2dIJ.
IJ. IJ.
-1
-1
[J(h h )~ff)dIJ.]2[J(ff)dlJ.J2
IJ. IJ.
1
= sup sup
IJ.
((1'1
II f\fdj111=1
Thus,
and since
111'111
111'1*11
we see that
h )~,
IJ. IJ.
is
(1'1
h )fYdIJ.)2
IJ. IJ.
G*-
86
~
Thus
is a C*"-algebra and is merely the complexification of
the real algebra of observables,
is actually a W*-algebra,
~.~.,
(9.
We will show shQrtly that
~
that is is a C*-algebra which is the
dual of some Banach space (see Sakai (1972) 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 prev.iouB chapters.
As we have already indicated the observables in this context of
this chapter are fami lies
H(X)
of the operator on
The operator
only for
h
A
h
€ (9 ,
h
state
01
h
(h ) •
/.l
For each such
may be defined for any
indeed, if
=
h,
h
H(X)
€ (9
h
€
then for
~
/.l
is
/.l
01
be
but is self-adjoint
A»
/.l ~
A»
(h)
v ,
are the states of this theory and if
is an observable, then the mean of
(h )
Pn
let
defined by
Moreover, the elements of
and
=
h
in the
In our operator theory we had
IJ-
Of
(A1J
0=
u
(A1-(-Hdii), NdiJ-)
Sh
=
IJ-
11
{fdg
which shows the consistency of the parallel between the two theories.
We now show that
an algebra over
is a W*-algebra.
(9
First observe
and thus has a trival involution
1«,
th~t
(h*
~
h)
Moreover, it is a C*-algebra since
(§
is and since
1.. .£., the norm of
(9
hag the required properti es.
algebra of
(9,
Thus, to complete the proof that
is a *-sub-
is a W*-algebra, we have only to
(9
show how to obtain a space which we will show is the predual of
l
L
We call this space
Borel measures on
g
1
E ~
via
(X, v)
(f
and
A » IJ- ,
Thus
Ll
A
g
IJ-
Suppose
X and further that
(f, IJ-) '" (g, v)
that
H(X).
f
E
finite Borel measure on
l
L
V
~l(X, IJ-)
are finite
and
are real-valued functions).
and
diJ-
f -
dr..
= gdv
- (l
dt,.
is the set of equivalence claSSeS of
shows that
and
X and
f
E
1
S- (X, IJ-) •
~
We define
iff there exists a finite Borel. measure
»\) ,
(9
L1 procedes in a
The definition of
parallel manner to that of
"
sucb
almost everywhere).
(f, IJ-)
for
IJ-
a
The procedure which
is a Banach space is carried out just as with ~-
functions.
It is easily shown that
linear functi onal of
(g, \)
are in
L
.t
1
1
L , then
•
h
is
(9
is well defined, and is a continuous
It is linear, since if
(f, IJ-)
and
88
(f, /.1) + (g, V)
(for
A.»
{L
,
A. »v)
and
=
,d/.1
dv
-- + g --JdA.
JhA[f
dA
dA
= £h e (f, /.1) + (g, V ))
•
Similarly,
Conversely, let
J.1 ,
£ be any linear functional on
Then, for each
tbe map
is a linear functional on
~leX, /.1) and thus by the Riesz
Representation Theorem there is a Borel map
that
1,1.
h
/.1
in
Fex, /.1)
such
Nmv we show that
to be an observabl.e.
(h'
11 )Il€a* (X)
satisfies the requirement needed
A» 11 , then
Let
1
1
L (11) £; L (\I)
and thus
dll
t( (g dA' A)
)
g, we see that
Since the latter equation holds for all
(11 almost
that
t =-;.
everj'\"i'here).
h
.
OllC
Thus,
h
=:;
(h/.l)
is in
A
=- h
ll
and it is clear
(9
now shows that the correspondence
is an isometric isomorphism from the Banach space
Banach space of continuous linear functionals of
recall that the
h
Rj.em~
onto the
(9
L
l
.
To do this,
theorem gives an isometric isomorphism beb-reen
and
is the essential supremum norm.
Moreover, the norm on
S,
co
(X,Il)
Thus with minor modifications of
this proof~ We can show ,that the correspondence
(*)
above is an
90
isometric isomorphism provided that we show
We proced to prove this equali ty.
II II\) co
where
measure
denotes the essential supremum norm relative to the
V .
We proceed to show that these norms are identical.
Clearly,
Thus,
To show that they are equal, we let
3: V
such that
Thus, there exists
(1)
N c::; X
such that
. \) (N)
'*' 0
C - llhll
OJ
,then if
E
>
°,
91
(2)
Nc
If we define
Q
H(X)
E
Ixllh\.I (x)1
>C - c)-
as
then we . have
hence
11hll 00 = IlhIIH(x) Hence we have two. views of -(9
The elemE':nts of
may be identified on the on~ hand as self-
(9
adjoint operators on
linear functions on
on ',H(X)
which are clearly equiva.lent.
H(X)
and on the other hand as continuous
Ll .· We will use the view of
(9
as operator,
for a proposition dealing with mixtures of states.
In t!
context of causality, a mixture of stutes behaves as a pure staLe
again.
Now s1.lppose
II(X)
A
is a posi tj ve
Then in the usual manner
subspace of
H(X)
;c,li ni te crace one operator on
A
is a measure on the closed
and hence on the ev·ent space.
!:,roposi tion 1:
Let
A
be a posi ti ve definite trace
Then there· exists
Ct E
H(X)
tr (xA)
such that if
O'}t,
operator on
x"
(9 ,
then
H
92
Proof:
First we consider the observable
x
P
where
E
P
E
is defined
by
Then the map
E .... trey)
is a measure on
X.
Call this measure
S
l'fdiii:"
E
m.
Let
H(X) •
Then
IIsII H (x)
= Jldm = m(X)
tr(A)
1.
Moreover,
Thul3,
x
x
=
P
E
satisfies the conclusion of the proposition.
be any observable,
= J\d[ tr (~A)J
tr[ (JAdP~ )A]
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"
mentioned, the real
space
interest~n measures
As has been
on the real line, since our
X is but a construct - an abstraction.
More preciseiy, we
consider the measure defined by
where
x
is an observable and
a
is a state.
We thus proceed to
the following definition.
We say that a sequence of distributions converges,
F
n
~
F , iff
F tI} ~ F[I}
n
for every bounded interval of continuity of
F
(a bounded interval
is an interval of continuity iff the endpoints of
I
are non-atomic).
This definition is from Feller (1966, Vol. 2).
We have the following theorem which will be useful to us in the
sequel.
Theorem 1:
~
Let
[F }
n
be a sequence of proper probability distributions (a
probability distribution
(a)
F
n
F (~)
is proper iff
=
n
1) , then
In order that there exists a proper or defective
distribution
F
such that
F'
n
~
F'
'
it is
necessary and sufficient that the sequence of
. expectations
I
converges for
C (-co, +00).
o
U E
In this case
E (u) ... E (u) .
n
(b)
If the convergence is proper
(*)
holds for
all
C(-oo, +00) .
U E
Here proper convergenc.!= means that
is a proper prObabi Li ty
The above theorem can be found in Feller (1966, VoL 2,
distribution.
p.
F
243).
No~
let
x
E (9
and
x
the associated multiplication
Opt";r'ctLH
x(~) = x ~.
(J
Recall that if
/.La
(E)
=
(Xgx,
a)
a
is a state and
where
=
x
/.La
is the measure defined by
a = fJ\.fd§, then
Jx
Thus if
/.La
(t)d]J. (t)
(J
Jtdj.La (x
a
-1
(J
(t)).
is the spectral measure of
x
in the state
~
95
1
01, (1) .
x- (E)
(x
(J"
(Observe that
/.l
x
01
has compact support since
x
is bounded. )
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 (/.l n} and [/.lOin} converges in distribution to
01
x
/.l0l
for every
01·
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
(f(x
Thus,
n
)01,
n
(1) ~
(f(X)OI, (1)
for each state
01'
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
ff(t)d~~n(t) ~ ff(t)~~(t)
x
and
~N
~
n
~ ~
x
~
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.
and also that of
x
~~
Let
I
be such an
We have that
x
fltd~Q'n(t) -+ flt~:(t)
(Here we use (a) of the Theorem cited from Feller and the fact that
x
the supports of the (~~n} lie in I.) It follows from the spectral
theorem that
-2
(x
Since
(x}
n
converge to
lim (
n
n
-2
~, ~) ~
(x a,
x
converges weakly to
-2
x
~)
(c£'. Naimark (1970), Chapter 7).
(xn -x) 2 a,
2
a) = lime (x
n
n
-
xx
n
- x x +
n
x2 )a,
Thus,
a)= 0
and
[;
2
limlll{'-n - x )all
o.
n
It follows that
x
n
~
-
x
strongly.
The theorem follows.
97
We have
is the same
strong
St~2n
a,~;
that considering convergence of distributions on
R
.Imposing a topology on the observables which is a
opc~l'atof'
topology.
Thus, convergence of distributions is
equivalent to the view that only a finite number of experiments with
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 fa.rnily of random variables which is parameterized by time
is called a random process.
given a finite set
Say
(h }
t
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
[h }
t
Thus, the interest of a study is to
from a finite dimensional distribu.·tion.
is clearly; in general, not possible.
asking what we may say about
dimensional distribution.
problem of van del' Vaart
(h }
t
This
We must content ourselves ''lith
when we know only some finite
This is really a rephrasing of the third
(1973).
That is, given a difference
equation find a differential equation whose solutions behave in a
simi lar manner.
We know of one result along these lines, which is round in
Gikhamn awl 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,
s(w):
Let
[a,b] -+ f?, ,
Then
st (w)
s(w)(t) ~
a sample function
(w
is fixed as
E: ('2
t
cbangeEj.
theoremlve will consider a special class
CSn(t)}
processes, where the sample functions are in
probability 1.
h~ife
We
x
(ulu(t)
~
C[u,b]
= St(w), w
E
o}
and furtLerwe have a natural measure for
!-L(A)
=
Q'((wlst(w)
=
u(t), u
X.
E
and
C[a,b]
X
now a space
For this
CSt}
of
with
where
Ii'
.A. eX,
A}) •
This is called the measure corresponding to the process.
By our discussion on page 97 , we are interested in the
convergence of finite dimensional distributions of
process of
(St}
(Sn(t)}
La the
(in the strong operator topology or equivalently as
convergence in distribution).
Theorem 3:
Suppose that the finite-dimensional distributions of the
processes
(s n (t)}
converge (as above) to the finite dimensional
distributions of a process
(~}
':>t
distributions of
to converge to the distrilmtion off(St)
f(Sn (t))
for all func ti onals
f
Then for the sequence of
that are conti nuous on
C[ a, b]
necessary and sufficient that
lim sup Q'[
sup
h~O n
It/-t / l l~h
for every
E
> 0 .
Is
n
(t') - S (VI)I > t}
n
0
J
it is
99
If
f
is a continuous functional on ; era, b]
the Riesz Representation Theorem to a function
it corresponds via
g, of bounded
variation (or as often stated, to the measure generated by
The functionals,
averages.
f , of
St
g).
correspond in some sense to time
One could presumably check these.
Thus,
100
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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
AppliGations, Vol. II. John Wiley and Sons, Inc., New York,
New York.
Gi~hman,
Skorokhod. 1969. Introduction To "The Theory of Random
Proces8'es. W. B. SaundeXS Co., Philadelphia, Pa.
Gudder, S. 196}. Spectral methods for a generalized probability
theory. Trans. American Math. Soc., 119:423-442.
Gudder, S. 1966.
observables.
Uniqueness and existence properties of bounded
Pacific J • . of Math., Vol.. 19, No.1, pp. '81-93.
R. 1,972. Quantum field theory.
pp. 1-14.
(R. Streater, Editor).
New York.
Haag,~
Math. of Cont. Physics,
Academic Press, New York,
Mackey, George W. 1963. ,Mathematical Foundations of Quantwn
Mechanics. Benjamin Publ., New York, New York.
Mackey, George W. 1968. Induced Representations of Groups and Quantum
Mechanics. Benjamin Publ., New York, New York.
M. A. 1970. Normed Rings.
Groningen, Holland.
N~mark,
Walters-Noodhoff Publ.,
Nelson, Edward. 1967. Dynamical Theories of BrownWn Motion.
Princeton Univ. Press, Princeton, New Jersey.
Nelson, Edward. 1969. Topics In Dynamics, I:
Dniv. Press, Princeton, New Jersey.
Flows.
F. and B. Sz.-Nagy. 1955. Functional Analysis.
Ungar Publ., New York, New York.
Ri~sz,
Sakai, 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. 195-211.
101
Varadarajan, N. S. 1962. Probability in physics and a theorem or.
simultaneous observabi l i ty. Comm. of P. Appl. Math., Vol. XV,
pp. 189-217·
von Neumann,
1953. Mathematical Foundations of Quantum Mechanics.
Princeton University Press, Princeto~ New Jersey.
Yosida, A. M. 1962.
West Germany.
Functional Analysis.
Springer-Verlag, Berlin,