Vol.

1,

No.

3

Using

(20) then

Then

using

(19) and 21)

Finally,

using

(18),

or

ON

THE

EINSTEIN

PODOLSKY

ROSEN

PARADOX

1

?c:;,

b)

-?c:;,

;;)1

.'.S

;:11

...

ru,)

[1 +

A"ci,

A)

8c;;,

A)]

+

/aAp(A)

[1 +

A(b,

A)

B(b,

A)]

I

:;

·

;;

-

:;

·

"b

I - 2

(E

+

o)

< 1 -

"b

·

;;

+ 2

(E

+ o)

~

~

~

7 7

~

4

(E

+

0)

2:

I a · c - a · b I + b • c - 1

Take

for example a ·

(;

=

0,

a · b = b ·

(;

= l /

y'2

Then

4

CE

+ o)

;::,

v 2 - 1

Therefore,

for

small

finite

o,

€

cannot

be

arbitrarily

small.

199

(22)

Thus,

the

quantum

mechanical

expectation

value

cannot

be

represented,

either

accurately

or arbitrar-

ily

closely,

in

the

form

(2).

V.

Generalization

The

example

considered

above

has

the

advantage

that

it

requires

little

imagination

to

envisage

the

measurements

involved

actually

being

made. In a more formal way,

assuming

[7]

that

any Hermitian oper-

ator

with a

complete

set

of

eigenstates

is

an

"observable",

the

result

is

easily

extended

to

other

systems.

If

the

two

systems

have

state

spaces

of

dimensionality

greater

than 2 we

can

always

consider

two dimen-

sional

subspaces

and

define,

in

their

direct

product,

operators

d

1

and d

2

formally

analogous

to

those

used

above

and which

are

zero

for

states

outside

the

product

subspace.

Then for

at

least

one

quantum

mechanical

state,

the

"singlet"

state

in

the

combined

subspaces,

the

statistical

predictions

of quantum

mechanics

are

incompatible

with

separable

predetermination.

VI.

Conclusion

In a theory

in

which

parameters

are

added

to quantum

mechanics

to

determine

the

results

of

individual

measurements,

without

changing

the

statistical

predictions,

there

must

be

a mechanism whereby

the

set-

ting

of

one

measuring

device

can

influence

the

reading

of

another

instrument, however remote. Moreover,

the

signal

involved

must

pr~pagate

instantaneously,

so

that

such

a theory

could

not

be

Lorentz

invariant.

Of

course,

the

situation

is

different

if

the

quantum

mechanical

predictions

are

of

limited

validity.

Conceivably

they might apply only to

experiments

in

which

the

settings

of

the

instruments

are

made

suffi-

ciently

in

advance

to

allow

them

to

reach

some mutual rapport by

exchange

of

signals

with

velocity

less

than or

equal

to

that

of

light.

In

that

connection,

experiments

of

the

type

proposed

by

Bohm

and

Aharonov

[6],

in

which

the

settings

are

changed

during

the

flight

of

the

particles,

are

crucial.

I

am

indebted

to Drs.

M.

Bander

and].

K . Perring for

very

useful

discussions

of

this

problem. The

first draft

of

the

paper

was written during a

stay

at

Brandeis University; I

am

indebted

to

colleagues

there

and

at

the University

of

Wisconsin

for

their

interest

and

hospitality.