On the Problem of Hidden Variables in Algebraic Quantum ... - J-Stage

Annals of the Japan Association for Philosophy of Science

On the Problem

of Hidden

Quantum

25

Variables

Field

in Algebraic

Theory

Yuichiro KITAJIMA* Abstract

have

been many

in nonrelativistic

There

quantum

without

contradiction.

not

observational

all

interpreted operator field

theory.

algebras

about

mechanics

way. point

Then

can be taken

I point

be

Jauch

present

in order out

that

as definite,

paper, all

in addition

1.

propositions

as simultaneously

and Piron,

to apply not

all observational

taken

in nonrelativistic

In the view

whether

can

Von Neumann, propositions

in this algebraic

discussions

definite

and Bell have shown

quantum

I examine

mechanics

these

theorems

these theorems

to algebraic

observational

propositions

to nonrelativistic

quantum

that

can

be

from

an

quantum in local mechanics.

Introduction

In algebraic quantum field theory, some von Neumann algebra is associated with each bounded open set in Minkowski space-time. Any projection in that algebra is regarded as an observational proposition in the set in Minkowski space -time to which it is associated. Some physically proper conditions are imposed on that von Neumann algebra, which is called a local algebra. A state on a local algebra gives the probability of an observational proposition in the bounded open set in Minkowski space-time to which that local algebra is associated. There have been many discussions about whether there exist hidden variables that determine truth-values of all observational propositions about a given nonrelativistic quantum mechanical system, and whether the probability given by a quantum mechanical state can be regarded as that probability measure on the set of all hidden variables. For example, von Neumann (1955), and Jauch and Piron (1963) mathematically defined such hidden variables, and showed that there is none in quantum mechanics. But they imposed a condition for incompatible observa tional propositions on hidden variables. Bell (2004) argued that it is not physically proper to impose that condition on hidden variables although it is proper to impose it on quantum mechanical states (pp. 4-6). Then Bell (2004) defined a hidden variable which was not bound by the condition on incompatible observational propositions, and showed that no such hidden variable in nonrelativistic quantum * International

Buddhist

University -25-

26

Yuichiro

KITAJIMA

Vol. 15

mechanics exists (pp. 6-8). In sections 3, 4 and 5, we will see those arguments from an operator algebraic point of view and it will be seen that there is no hidden variable in either algebraic quantum field theory or nonrelativistic quantum mechanics. On the other hand, for a local algebra in algebraic quantum field theory as well as for the set of all bounded operators on a Hilbert space in quantum mechanics, some subset of the set of all observational propositions does admit hidden variables. Halvorson and Clifton (1999) defined these sets from an operator algebraic point of view (see also Clifton, 1999), and named them 'beable algebras' following Bell's terminology (Bell, 2004, chapters 5, 7 and 19). In Section 6, we will look at Clifton's interpretation of algebraic quantum field theory in terms of beable alge bras.

2. 2.1.

Operator In

of

all

let

present

bounded

paper, operators

we

use

the

following

on

a Hilbert

space

{A•¸B(H)|AB=BA

P(_??_)

_??_•¿_??_•L is

for

denote

the

set

called

the

center

Definition

2.1.

ƒÏ satisfies

A

A

if ƒÖ(A2)=[ƒÖ(A)]2

Definition if

(_??_) are

A

orthogonal

2.4.

Let _??_

called

projections

any I

a

Where _??_ denote

the

unital

is a subset

is denoted

Where_??_is

let „H(_??_)

on

If S

which

in _??_.

and

element

is

the

on

any

von

each

Definition

B•¸S}.

functional ƒÏ

state ƒÖ for

2.3.

all

notation. H

of

the

by B(H),

a von is a von center

set

letS•L

Neumann

algebra,

Neumann

algebra,

of _??_.

C*-algebra _??_

is called

a

state

if

conditions:

where

2.2.

all

of _??_

for

2. ƒÏ(I)=1

Definition

of

linear

following

1. ƒÏ(A*A)_??_0

algebra

preliminaries

algebras

the

denote

Mathematical

A•¸_??_;

identity

a

unital

element

equivalent

if

of

von

non-zero

Neumann

there

is a

is called

is

called

projections

algebra.

partial

a

dispersion-free

state

A•¸_??_.

algebra _??_

family

a

in _??_.

C*-algebra _??_

self-adjoint

Neumann

be

operator

in _??_

Two

isometry

a ƒÐ-finite

von is

countable.

projections

V in _??_ such

Neumann

P that

and

Q

in

P=VV*

P and

Q=V*V. If

Definition

two

projections

P

2.5.

be

Let _??_

and

a

Q

von

are

equivalent,

Neumann -26-

we

algebra.

shall

A

write

projection

P•`Q.

P

in _??_

is

said

to

No. 1

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

be finite is

if

an

P•`Q_??_P

Abelian

Definition

2.6.

If Q

for

such

then _??_

is

If

for

such

that

of

type

to

be

_??_ is

said

all

can

is

of

there

type

is

I.

to

be

a

be

infinite.

If

P_??_P

Abelian.

non-zero

If _??_

orthogonal

P no

II

and

in „H(_??_)

non-zero

and

of

the

type

finite

algebra

there

is of

Abelian

type

I

equivalent

II

is a

Abelian

identity

projection

and

the

Abelian

and

non-zero

identity

projections,

finite

projection,

operator

projection

then _??_

is

in _??_ is finite,

the

identity

operator

is

to

an

Abelian

projection, _??_

of on

(Kadison

type

is

space

Ringrose,

there

expressed

I1

a Hilbert

and

Then

be

said to

Q

said

to

be

then _??_

is said

in _??_ is infinite,

then

type II•‡.

algebra.

and _??_

be

is

said

algebra.

in „H(_??_),

mutually

has

operators

2.1

to

it is

In.

type

non-zero

bounded

Neumann

if _??_

Neumann

Proposition

_??_

of no

P

projection

is of

P

Neumann

said

n

type

If _??_

be

has

von

is

of

of

and

II1.

to

von

of

be

If _??_

type

said

sum

non-zero

Q_??_P,

of

A

to

a

then

projection

then _??_

the

any

II.

If _??_

set

Q_??_P,

Otherwise,

algebra,

be

non-zero

in _??_ is

P=Q.

Neumann

Let _??_

any

that

operator

implies

von

27

are

said

is a von

1997,

projections

be

von

of

Neumann

Theorem

PI,

PII1,

type

III.

Neumann

algebra. algebra

6.5.2). PII•‡,

and

of

Let _??_ PIII

in

The type

be

a

I.

von

the

center

of

is of

type

II•‡

as

_??_ =_??_PI_??_PII1_??_PII•‡_??__PIII where _??_PI

is

or PII•‡=0

and _??_PIII

Definition algebra in _??_

of

type

2.7. if

such

A

for

any

that

I

or is

von

PI=0, _??_PII1 of

type

Neumann

non-zero

is of

III

or

algebra

projection

type

II1

or

PII1=0, _??_PII•‡

PIII=0.

P

is

called

in

the

a properly center

infinite

of _??_,

there

von is

a

Neumann

projection

Q

P•`Q•ƒP.

Any local algebra in algebraic quantum field theory is properly infinite (Baum gartel, 1995, Corollary 1.11.6). Definition there

2.8.

is

a

Definition

I-PƒÏ

density

2.9.

is

A

called

state ƒÏ

on

operator

Let ƒÏ

D

be

a

a such

is

the

of ƒÏ.

Neumann

algebra _??_

that ƒÏ(A)=tr(DA)

normal

PƒÏ:=sup{P|P support

von

a

state projection

-27-

on

is for

a

von

of _??_

Neumann such

called any

a

normal

operator

algebra _??_.

that ƒÏ(P)=0}

state

A•¸_??_.

Define

if

_??_ƒÏ _??_ƒÏ

28

Yuichiro

Definition

2.10.

Let ƒÏ

be

a

normal

KITAJIMA

state

on

a

von

,_??_:={A•¸_??_|ƒÏ(AB-BA)=0 ,_??_is

called

the

centralizer

of ƒÏ

Vol. 15 Neumann

for

all

algebra _??_.

Define

B•¸_??_}.

of _??_.

A centralizer of any normal state on a von Neumann algebra _??_is a von Neumann subalgebra of _??_ (Kitajima, 2004, Lemma 8). 2.2. Lattice theory Definition 2.11. A set _??_ is called a partially ordered set if there is a binary relation _??_ on _??_ which satisfies the following conditions: 1. •Ía•¸_??_ 2. •Ía,

b•¸_??_

3. •Ía,

b,

Definition b

in _??_ have

Definition the

and

[[a_??_b

b_??_c]•Ëa_??_c].

ordered

upper

bound

greatest

A

b_??_a]•Ëa=b]; and

partially

least the

2.13.

satisfies

a•Éb

element,

if

non-empty

following

set _??_ is and they

subset

a•¸I

and

b_??_a

imply

b•¸I.

2.

a•¸I

and

b•¸I

imply

a•Éb•¸I.

2.14.

mapping

A

a•¨a•Û

1.

a•Éa•Û=1

2.

a_??_b

3.

(a•Û)•Û=a

When

Definition

lattice _??_

from _??_ and

implies

a_??_b•Û,

we

2.15.

for

Definition

is

2.16.

a

greatest

exist,

I

of

a

are

a

lattice lower

denoted

lattice _??_

is

if

any

bound

elements a•Èb.

by 0

and

called

an

1

a The

and least

respectively.

ideal

of _??_ if

An

An

0

and

itself

1 is

called

satisfying

the

orthocomplemented following

when

there

conditions:

a•Û_??_b•Û; any

element

a•¸_??_.

a•Ûb.

orthocomplemented for

a von

with

into

a•Èa•Û=0;

write

b=a•É(b•Èa•Û) When _??_

called

I

conditions:

1.

Definition

when

A

a

and

[[a_??_b

c•¸_??_

2.12.

element

a

[a_??_a];

any

Neumann

elements

lattice _??_ a,

algebra,

orthocomplemented

is

b•¸_??_

such

P(_??_)

is

called that

an

(a•Èb)•Éc=(a•Éc)•È(b•Éc)

-28-

orthomodular

orthomodular

lattice _??_ is called

(a•Éb)•Èc=(a•Èc)•É(b•Èc)

an

lattice

a_??_b.

a Boolean

lattice.

lattice

when

is

No. 1

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

29

for any elementsa, b and c in _??_. As easily seen, a Boolean lattice is an orthomodular lattice.

3.

von

Neumann

In this section we see von Neumann's 'impossibility algebraic point of view (Proposition 3.1).

proof' from

an operator

Definition 3.1. Let _??_ be a von Neumann algebra. A mapping Exp from the set of all self-adjoint operators in _??_to R is called a dispersion-free expectation value assignment on N if it satisfiesfollowing conditions. 1. _??_

and

Exp for

(aR+bS)=aExp(R)+bExp(S)for any

a,

any

self-adjoint

operators

R,

S

in

b•¸R.

2.

Exp(R2)=[Exp(R)]2

3.

Exp(I)=1.

for

any

self-adjoint

operator

R

in _??_.

Von Neumann defined a dispersion-free expectation value assignment on the set (BH) of all bounded operators on an infinite dimensional Hilbert space H (von Neumann, 1955, pp. 311-312) and showed there is no dispersion-free expectation value assignment on B(H) (von Neumann, 1955, p. 321). In Proposition 3.1 we state this fact more generally.

Lemma 3.1. Let Exp be a dispersion-freeexpectationvalue assignment on a von Neumann algebra _??_.Then Exp can be extendedto a dispersion-freestate on _??_. Proof.

Define

a mapping ƒÖ

from _??_

to C

as

ƒÖ (A):=Exp(A*+A/2)+iExp(i(A*-A)/2) Let

A

expressed

and

B as

be a+ib

any

operators (a,

b•¸R).

in _??_

and

let

(•ÍA•¸_??_). c be

Since

ƒÖ (A*A)=Exp(A*A)=Exp(|A|2)=[Exp(|A|)]2_??_0,

-29-

any

complex

number.

c can

be

30

Yuichiro

KITAJIMA

Vol. 15

and

ƒÖ is

a

state

on _??_.

(R)=Exp(R). We Doring

For

any

self-adjoint

Therefore ƒÖ prove

the

is a

following

operator

R, ƒÖ(R2)=[ƒÖ(R)]2

dispersion-free

theorem,

state

making

because ƒÖ

on _??_.

reference

to

the

proof

of

Lemma

19

of

(2005).

Proposition 3.1. There is no dispersion-freeexpectationvalue assignment on a von Neumann algebra without direct summand of type I1. Proof.

1. that set

Let _??_ there

{P1,

tions

be

is P2,

...,

Pn}

in _??_ such

ƒÖ on _??_.

a von

(2_??_n•ƒ•‡)

There

expectation

of

mutually By

only are

Lemma

2

(V*)ƒÖ(V)=ƒÖ(V*V)=ƒÖ(Pk). sion-free

of

Pi=I.

Since ƒ°ni=1ƒÖ(Pi)=1,

By

algebra expectation

that ƒ°ni=1

(Pk)=0(k•‚j). Pk=V*V.

Neumann

a dispersion-free

a partial of

Misra This

value

type

In, where

value

assignment

orthogonal

Lemma one

3.1

there

of ƒÖ(Pk)s,

isometry

2_??_n•ƒ•‡.

Suppose

on _??_. and

There

equivalent

projec

is a dispersion-free say ƒÖ(Pj),

V in _??_ such

is that

is

unity, Pj=VV*

state and ƒÖ and

(1967) ƒÖ(Pj)=ƒÖ(VV*)=ƒÖ(V)ƒÖ(V*)=ƒÖ is a

assignment

contradiction.

on _??_. - 30-

So

there

is no

disper

a

No. 1

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

2.

Let _??_

algebra

of

Ringrose

either

a properly

type

II1.

(1997)

there

that

similar 3.

is no

manner By

to

a von

Lemma

2.1

Neumann

P

algebra

is no

that

is

P=VV*

of

type

Jauch

and

and

In

direct

Neumann

Kadison

.

It

and

follows

P•Û=V*V

.

assignment

that We

on _??_

can in

a

(2_??_n•ƒ1). expectation

I1

a von of

P•`P•Û

value

type

or

6.5.6

that

dispersion-free

without

4.

Lemma

expectation

where _??_

there

and

algebra1

in _??_ such

V in _??_ such

case

Neumann

6.3.3

dispersion-free

the

Proposition

von

is a projection

isometry

there

infinite

By

there

is a partial

prove

on

be

31

value

assignment

summand.•

Piron

Jauch and Piron (1963) evaluated von Neumann's argument (especially the condition 1 in Definition 3.1) as follows (see also Bell, 2004, pp. 4-5). Concerning states, it is difficult to justify the additivity of the expectation values on non-compatible observables. (p. 828) Then they defined the following truth-value Definition

4.1

omodular

(Jauch-Piron

lattice.2

We

if ƒÊ

satisfies

assignment

truth-value

call

a

assignment).

mapping ƒÊ

following

assignment (p. 833).

from _??_

to {0,

Let _??_ 1} a

be

Jauch-Piron

an

orth

truth-value

conditions:

1. ƒÊ(1)=1; 2. ƒÊ(a•Éb)=ƒÊ(a)+ƒÊ(b) 3.

for

If ƒÊ(a)=ƒÊ(b)=1,

any

elements

a,

b•¸_??_ such

that

a•Ûb;

then ƒÊ(a•Èb)=1.

Jauch and Piron proved that there is no Jauch-Piron truth-value assignment in quantum mechanics (Jauch and Piron, 1963, Theorem II). We prove Theorem II of Jauch and Piron (1963) in another way (Proposition 4.1). Lemma

4.1.

lattice _??_.

1 A

Let

For

type

I•‡

Neumann 2 Jauch following

1

any

von

a

Neumann

algebra and

be

Jauch-Piron

elements

Piron condition

a,

algebra,

are

properly

(1963) (p.

(P) •Ía,

truth-value

assignment

on

an

orhomodular

b•¸_??_,

a type

II•‡

von

Neumann

algebra

and

a type

III

von

infinite.

defined

an

orthocomplemented

lattice _??_

that

satisfies

the

831).

b•¸_??_[a_??_b•¨(a•Èb•Û)•Éb=(b•Èa•Û)•Éa]

As is easily seen, an orthocomplemented orthomodular lattice. - 31-

lattice that

satisfies condition

(P) is an

32

Yuichiro

KITAJIMA

Vol. 15

1. ƒÊ(a•Û)=1-ƒÊ(a); 2.

a_??_b•¨ƒÊ(a)_??_ƒÊ(b);

3. ƒÊ(a•Èb)=ƒÊ(a)ƒÊ(b). Proof. 2.

1.

Since

1=ƒÊ(a•Éa•Û)=ƒÊ(a)+ƒÊ(a•Û), ƒÊ(a•Û)=1-ƒÊ(a).

If

a_??_b,

then

(b•Û)

a_??_(b•Û)•Û,

that

is, a•Ûb•Û.

Since

1_??_ƒÊ(a•Éb•Û)=ƒÊ(a)+ƒÊ

=1+ƒÊ(a)-ƒÊ(b), ƒÊ(a)_??_ƒÊ(b). 3.

If

either ƒÊ(a)

or ƒÊ(b)

is

0, ƒÊ(a•Èb)=0

since

a•Èb_??_a

and

a•Èb_??_b.

Therefore ƒÊ(a•Èb)=ƒÊ(a)ƒÊ(b).

Proposition 4.1. Let _??_ be an orthomodular lattice. Thefollowing conditions are equivalent. 1.

For

ment ƒÊ

any

such

2. _??_ Proof.

non-zero

element

a•¸_??_

there

is

a

Jauch-Piron

truth-value

assign

that ƒÊ(a)=1. is

a

1•Ë2

Boolean

Let

lattice.

a and

b be

•È b)•É(a•Èb•Û)_??_a.

any

elements

Because _??_

is

in _??_. an

Since

a•Èb_??_a a

orthomodular

and

a•Èb•Û_??_a,

(a

lattice,

a=(a•Èb)•É(a•Èb•Û)•É(a•È((a•Èb)•É(a•Èb•Û))•Û) Suppose

that

- Piron

a•È((a•Èb)•É(a•Èb•Û))•Û•‚0.

truth-value

Lemma

assignment ƒÊ

4.1 ƒÊ(a)=1

since

It such

follows

that

there

is

a

Jauch

that ƒÊ(a•È((a•Èb)•É(a•Èb•Û))•Û)=1.

a•È((a•Èb)•É(a•Èb•Û))•Û_??_a.

By

Further,

1=ƒÊ(a•È((a•Èb)•É(a•Èb•Û))•Û) =ƒÊ(a•È((a•Èb)•Û•È(a•Èb•Û)•Û) =ƒÊ(a)(1-ƒÊ(a)ƒÊ(b))(1-ƒÊ(a)(1-ƒÊ(b))) =(1-ƒÊ(b))ƒÊ(b) =0(•æƒÊ(b)=0 This

2•Ë1

or

is a contradiction.

b)•É(a•Èb•Û). This Let

a

_??_.

prove,

be

any

Theorem

making non-zero

Define _??_a

Lemma

4.1)

1).

Therefore

By

we

(•æ

(•æƒÊ(a)=1)

36.7 reference

element

a•È((a•Èb)•É(a•Èb•Û))•Û=0, of

Maeda to

(1970), _??_

Lemma

in _??_ and

that is

11.3

of

let _??_ be

the

a

Boolean

Maeda set

is,

a=(a•È

lattice.

(1980). of

all

proper

ideals

of

as

_??_a :={J•¸_??_|a_??_}. _??_ais partially has is

an not

ordered

upper

by

bound

empty

set

inclusion.

in _??_a because •¾_??_a

because

{0}•¸_??_a.

Zorn's

Every

linearly

is a proper lemma

ordered

ideal

such

implies

subset _??_a that

that _??_a

of _??_a

a_??_•¾_??_a. _??_a has

a

maximal

element _??_a. Let

b and

c _??_Ja.

c be Define

any

elements

in _??_ such

J•L as -32-

that

b•Èc•¸Ja

and

b_??_Ja.

Suppose

that

No. 1

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

33

J•L:={y•¸_??_|•Îx•¸Ja[y_??_b•Éx]}. J•L is an

ideal.

a•¸J•L.

Then

is an

Ja_??_J•L there

element

x2

in

since

is

an

Ja

such

b•¸J•L.

element

Because x1 in

that

Ja

Ja

such

a_??_c•Éx2

.

is

a maximal

that

element

a_??_b•Éx1.

Because _??_

in _??_a,

Similarly

is a

Boolean

there

lattice,

a_??_(b•Éx1•Éx2)•È(c•Éx1•Éx2)=(b•Èc)•É(x1•Éx2)•¸Ja. This

contradicts

that

a_??_Ja.

Therefore

c•¸Ja.

Therefore

any

elements

b,

c•¸_??_

, b•Èc•¸Ja•Ìb•¸Ja If

b_??_Ja, such

Define

a

then

b•Û•¸Ja

that

d•Ûe

since and

mapping ƒÊa

or

b•¸Ja

e_??_Ja,

from _??_

or

b•Û•¸Ja.

d•¸Ja to

{0,

c•¸Ja. Therefore

because 1}

e•Û•¸Ja

for and

any

elements

d,

e•¸_??_

d_??_e•Û.

as

1(x_??_Ja) ƒÊ a(x)=

Then ƒÊa

is

a Jauch-Piron

In quantum lattice.

theory,

Therefore

0(x•¸Ja)

truth-value

assignment,

and ƒÊa(a)=1.•

the set of all observational

there

is no

Jauch-Piron

propositions

truth-value

is not

assignment

a Boolean

in quantum

theory.

5.

Gleason

Bell (2004) stated that the condition 3 of Definition 4.1 is a' quite peculiar property' of quantum mechanical states (p. 6). If this condition cannot be justified, the argument of Jauch and Piron does not hold. Then we define a truth-value assignment which is not bounded the condition on incompatible observational propositions as follows. Definition the

set

additive

5.1 of

all

(A

finitely

additive

projections

truth-value

in assignment

a

truth-value von

assignment).

Neumann

on _??_

if ƒÊ

algebra _??_ satisfies

to

{0,

following

1}

A

mapping ƒÊfrom

is

called

a finitely

conditions:

1. ƒÊ(I)=1; 2.

For

Hamhalter

any

mutually (1993)

orthogonal proved

the

projections following

P, Q•¸_??_, ƒÊ(P•ÉQ)=ƒÊ(P)+ƒÊ(Q). lemma

using

a

generalised

Gleason's

theorem (see, e.g. Maeda, 1990, Theorem 12.1 or Hamhalter, 2003, Chapter 5). Lemma

5.1 (Hamhalter,

1993, Lemma 5.1). -33-

Let _??_be a von Neumann algebra

34

Yuichiro

without

direct

assignment

on _??_.

The prove

it

in

let ƒÊ

be

Let ƒÊ

It

be By

be

to

be

to

in

von

Neumann

a

a

finitely

additive

dispersion-free

Corollary

5.2

algebra

truth-value

truth-value

state

of

on _??_.

Hamhalter

(1993).

We

there .

summand

on _??_

truth-value

and

assignment

state

Kadison

is a

direct

let

P,

of

type I2,

Q

be

any

then ƒÊ(P•ÈQ)=1.

additive

of

without

assignment

a dispersion-free

6.1.7

that

a

a finitely

Theorem

follows

let ƒÊ

extended

If ƒÊ(P)=ƒÊ(Q)=1,

extended

-P•ÈQ=V*V

and

proved

additive

in _??_.

Proof. ƒÊ can

be

was

be

finitely

projections

I2

Vol. 15

way.

Let _??_

a

type can

lemma

another

5.2.

of

Then ƒÊ

following

Lemma

.

summand

KITAJIMA

and

partial

on _??_.

Ringrose

isometry

on _??_.

Let

P,

(1997),

V

Q be

By any

Lemma

5.1,

projections

in _??_

(P•ÉQ-Q)•`(P-P•ÈQ).

in _??_ such

that

P•ÉQ-Q=VV*,

P

Because

ƒÖ (VV*)=ƒÖ(V)ƒÖ(V*)=ƒÖ(V*)ƒÖ(V)=ƒÖ(V*V)

by Lemma 2 of Misra (1967), ƒÖ (P•ÈQ)=ƒÖ(P-V*V)=ƒÖ(P)-ƒÖ(V*V) =ƒÖ(P)-ƒÖ(VV*)=ƒÖ(P)-ƒÖ(P•ÉQ-Q) =ƒÖ(P)+ƒÖ(Q)-ƒÖ(P•ÉQ). Therefore

if ƒÖ(P)=ƒÖ(Q)=1,

then ƒÖ(P•ÈQ)=1

(cf.

Hamhalter,

2003,

Example

10.

1.1). •

We

can

prove

Theorem I2.

5.1. Then

Proof.

For

any

is

1•Ë2

Let

Abelian

By

Lemma

lattice tion (P)=1 We

in _??_

any

the

in _??_

finitely

there

is

additive

direct

summand

a finitely

additive

a

truth-value

on

P(_??_).

P=(P•ÈQ)•É(P•ÈQ•Û)

of

type

truth-value

of finitely

an

Since

and Maeda

assignment

Since P(_??_) for

is

in _??_.

Kadison

is

without

algebra.

Therefore _??_

36.7

Proposition

prove

P

assignment 4.1,

of

there

algebra

4.1.

equivalent.

Neumann

5.2

Theorem

by can

von

projections

2.5.3 by

P

any

Proposition

that ƒÊ(P)=1.

PQ=QP.

Proposition

Neumann

projection

Proposition

Q be

using

are

truth-value

Then P,

von

such

an

by

(_??_). 2•Ë1

a

theorem

conditions

non-zero

Jauch-Piron lattice

be

following

assignment ƒÊ 2. _??_

following

Let _??_

the

1.

the

Abelian

any

von

PQ=QP,

Neumann

Boolean P,

Q•¸P

algebra.

P=(P•ÈQ)•É(P•ÈQ•Û)

(1997).

Then

(1970).

Therefore

for

truth-value

a

projections

Ringrose

additive

on _??_ is a is

P(_??_) any

by is

a

non-zero

assignment

Boolean projec

1 such

that ƒÊ

4.1. • following

theorem -34-

using

Proposition

3.1

(cf.

Hamhalter,

No. 1

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

35

2003, Theorem 7.3.1).

Theorem 5.2. Let _??_ be a von Neumann algebra that has neither direct summand of type I1 nor direct summand of type I2. Then there is no finitely additive truth - value assignment on _??_. Proof. Suppose that there is a finitely additive truth-value assignment on a von Neumann algebra _??_ which has neither direct summand of type I1 nor direct sum mand of type I2. It followsthat there is a dispersion-free state on _??_ by Lemma 5.1. This contradicts Proposition 3.1, since a dispersion-free state on _??_ is a dispersion -free expectation value assignment on _??_.Therefore there is no finitely additive truth-value assignment on _??_. By Theorem 5.2 no finitely additive truth-value can be assignedsimultaneously to all projections on the Hilbert space whose dimensionis greater than 2. Moreover, no finitely additive truth-value can be assigned to all projections which belong to any local algebra in algebraic quantum field theory because any local algebra is a properly infinite von Neumann algebra (Baumgartel, 1995,Corollary 1.11.6). 6.

Clifton's

interpretation

of algebraic

quantum

field

theory

Theorem 5.2 shows that there is no finitely additive truth-value assignment on any local algebra. But some subalgebra of a local algebra can be assigned a finitely additive truthvalue and a state on that subalgebra can be regarded as a mixture of finitely additive truth-value assignments. Halvorson and Clifton (1999) called such a subalgebra a beable algebra, following Bell's terminology (Bell, 2004, chap ters 5, 7 and 19). For example, Bell (2004) said the following: We will exclude the notion of 'observable' in favour of that of 'beable'. The beables of the theory are those elements which might correspond to elements of reality, to things which exist. Their existence does not depend on 'observation'. (p. 174) A beable

Definition

6.1

- algebra, beable is,

if

states

algebra

(Halvorson

let _??_ algebra and on _??_

is defined

be

a

for ƒÏ

only such

if

and

unital if there

the following.

Clifton,

C*-subalgebra and is

only a

if ƒÏ|_??_

probability

1999,

p.

of _??_

and

is

a

measure

that

-35-

2447).

mixture 1 on

Let _??_

let ƒÏ of the

be

a

be

state

dispersion-free space

S

of

a

unital

on _??_. _??_ states, dispersion-free

C* is that

a

36

Yuichiro

KITAJIMA

ƒÏ(A)=•çsƒÖs(A)dƒÊ(s)

Vol. 15

(•ÍA•¸_??_).

Clifton (2000) determined the maximal beable algebra for each faithful normal state in a von Neumann algebra under the condition that the beable algebra is determined solely in terms of the faithful normal state and the algebraic structure of the von Neumann algebra (Clifton, 2000, Proposition 1). The following theorem is a generalaized Clifton's theorem. Theorem

6.1

algebra3

on

of ƒÏ.

a

2. ƒÐ(_??_)=_??_

for

is be

uniquely

set

of

state ƒÏ

a separable with

the

ously

set

mechanics

normal

are states

hyperfinite

for

such

have

no

to

algebraic

a

faithful

Another

states

3 Since

local

(Bratteli 4

To

Clifton 5 In

and

represent assumed

physically

(Haag,

1996,

the

S

Neumann

be

the

following

support

conditions.

that ƒÏ•KƒÐ=ƒÏ.4 and

in

of

2. the

center

of

propositions.

concerns

Corollary

are

Robinson,

this

1987,

condition

reasonable

models,

in

Proposition any

local

V. 6).

-36-

of the

operators

on

have

simultane

quantum of

2004,

field

theory

nonrelativistic

quan

problem

a von

faithful

determined

states

maximal

solely

in

assignment.

truth-value

assignments, by

the

faithful algebra _??_

normal The

truth-value

vectors

concerns

Neumann

of

which

beable terms

of

If

algebra that

we

state

regard

it is natural

Reeh

-Schlieder

theorem

von

Neumann

algebras

are ƒÐ-finite

is

2.5.6).

solely in

set of

(Kitajima,

algebraic

3).

algebras

Proposition

is determined

the

terms

bounded

which

One

many

is

these

all

interpretation

When

separating

1.3.3),

of

in

theory.

invalid

and

as

solely

interpretation

on S•Û_??_S•Û_??_„H(_??_ƒÏ, _??_)Sas cyclic

regarded

projections

Proposition

which

an

set

of

trivial.

there 2000,

state

that _??_

all

interpretation. are

be

in S•Û_??_S•Û_??_„H(_??_ƒÏ, _??_)S coincides

modal

field

this

factor,5

have

is the

projections

modal

definite

section

von

let

determined

interpretation

normal

1995,

such

are

Kochen-Dieks

(Clifton,

algebras

(Baumgartel,

the

all

(1995))

centralizers

problem

dispersionfree

of

quantum

III1

nontrivial

and

satisfy

1

When _??_

set

Clifton

problems

trivial

which

Kochen-Dieks

which

are

a ƒÐ-finite

S•Û_??_S•Û_??_„H(_??_ƒÏ,_??_)S, where„H(_??_ƒÏ,_??_) is

Clifton's

the

type

centralizers

on _??_

algebra.

in

of

of

let _??_

Conditions

algebra

the

under

two

be

on _??_

in S•Û_??_S•Û_??_„H(_??_ƒÏ, _??_)S can

local

Therefore

extension

and

to as

local

space,

values 12).

There

a

(DefKD(W)

definite

Corollary

tum

that

Hilbert

state

of _??_.

in

on

respect

projections

propositions

normal

an

of ƒÏ

Let _??_

normal

automorphism ƒÐ

expressed

all

a

for ƒÏ.

with

centralizer _??_ƒÏ, _??_

definite

any

11).

be

of _??_

algebra

maximal

can

Theorem

let ƒÏ

C*-subalgebra

beable

The

a

space,

a

Then _??_ the

be

is

3. _??_

2004,

Hilbert

Let _??_ 1. _??_

is

(Kitajima,

a

terms

of ƒÏ 1 of

algebra

and

Clifton is

the

algebraic

structure

of _??_,

(2000). a hyperfinite

type

III1

factor

to

No. 1 think all

On the Problem of Hidden Variables in Algebraic Quantum Field Theory that

for

any

projections

dispersion-free

in

(_??_ƒÏ, _??_)S are when „H(_??_ƒÏ

a set

false

{Pi|i•¸N}

(ƒÖ(Pi)=0 a set

projections,

there

that ƒÖ•L(•Éi•¸NPi)=1 Then •Éi•¸NPi regarded

any

is

a

a valid

and

Pi

for

is false

section

nonrelativistic

5 we pointed

which

out

other

examined

than

any

this

S•Û_??_S•Û_??_„H

holds.

countably

But infinite

on S•Û_??_S•Û_??_„H(_??_ƒÏ, _??_)S such

(Kitajima,

i•¸N.

2005,

Therefore

Proposition

this

state

3.1). cannot

be

that

Remarks there

is no

hidden

quantum quantum

interpretation

of the Kochen-Dieks

the

whether

i•¸N

in

variable

in algebraic

mechanics. field theory

Therefore as well as

mechanics.

quantum mechanics to algebraic interpretation. In nonrelativistic tions

for

state,

orthogonal

state ƒÖ•L

any

Concluding

6 we saw Clifton's

is an extension

projections

is a normal

as well as nonrelativistic problems in algebraic

quantum

In section

If ƒÖ mutually

whenever

assignment.

7. In

i•¸N). of

(ƒÖ(•Éi•¸NPi)=0)

orthogonal

dispersion-free

truth-value

quantum field theory there are interpretation

is false

mutually

{Pi|i•¸N}

and ƒÖ•L(Pi)=0 is true

as

of

for

, _??_)S contains

non-zero

state ƒÖ, •Éi•¸NPi

37

quantum

modal

field theory

of nonrelativistic

and saw problems with this there are some interpreta

interpretation.

field theory

quantum

interpretation

quantum field theory, quantum mechanics

Kochen-Dieks

algebraic

of algebraic

modal

It therefore

can be interpreted

needs

to be

in another

way.

Acknowledgements The version

author of this

would

like

to thank

a referee

for valuable

comments

on an earlier

paper.

References

[1]

Baumgartel, H. (1995). Operatoralgebraicmethods in quantum field theory, Akademic Verlag, Berlin.

[2]

Bell, J. S. (2004). Speakable and unspeakable in quantum mechanics (Secondedition), Cambridge University Press, Cambridge.

[3]

Clifton, R. (1995). Independently motivating the Kochen-Dieks modal interpreta tion of quantum mechanics. British Journal for Philosophy of Science,46, 33-57.

[4]

Clifton, R. (1999). Beables in algebraic quantum theory, in J. Butterfield and C. Pagonis (eds.), From physics to philosophy, Cambridge University Press, pp. 12-44.

[5]

Clifton, R. (2000). The modal interpretation of algebraic quantum field theory. Physics Letters A, 271, 167-177.

[6]

Doring, A. (2005). Kochen-Specker theorem for von Neumann algebras. Interna tional Journal of Theoretical Physics, 44, 139-160. -37-

38

Yuichiro

KITAJIMA

Vol. 15

[7] [8]

Haag, R. (1996). Local quantum physics (Second edition), Springer, Berlin. Halvorson, H. and Clifton, R. (1999). Maximal beables subalgebras of quantum mechanical observables. International Journal of TheoreticalPhysics, 38, 2441-2484.

[9]

Hamhalter, J. (1993). Pure Jauch-Piron States on von Neumann Algebras. Annales de l'Institut Poincare, 58, 173-187. Hamhalter, J. (2003). Quantum measure theory, Kluwer, Dordrecht. Jauch, J. M. and Piron, C. (1963). Can hidden variables be excluded in quantum mechanics?. Helvetica Physica Acta, 36, 827-837. Kadison, R.V. and Ringrose, J. R. (1997). Fundamentals of the theory of operator algebras, American Mathematical Society, Providence, Rhode island. Kitajima, Y. (2004). A remark on the modal interpretation of algebraic quantum field theory. Physics Letters A, 331, 181-186. Kitajima, Y. (2005). Interpretations of quantum mechanics in terms of beable alge bras. International Journal of Theoretical Physics, 44, 1141-1156. Maeda, F. and Maeda, S. (1970). Theory of symmetric lattices, Springer, Berlin. Maeda, S. (1980). Lattice theory and quantum logic, Maki-Shoten, Tokyo (In Japanese). Maeda, S. (1990). Probability measures on projections in von Neumann algebras. Reviews in Mathematical Physics 1, 235-290.

[10] [11] [12] [13] [14] [15] [16] [17] [18]

Misra, B. (1967). When can hidden variables be excluded in quantum mechanics?. Nuovo Cimento, 47A, 841-859.

[19]

von Neumann, J. (1955). Mathematical foundations of quantum mechanics, Princeton University Press, New Jersy.

-38-