Baggott J. The Meaning of Quantum Theory: A Guide for Students of Chemistry and Physics
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138
Putting
It
to
the
test
essentially
means
no
communication
faster thall
the
speed
of
light.
Recall once· again
that
Einstein suspected
that
the
Copenhagen
inter-
pretation
of
quantum
theory
might
necessarily lead
to
a
violation
of
the
postulates
of
special relativity.
Now let
us
take
a
look
at
some
specific
orientations
in
actual
experi-
ments.
Consider
the
four
experimental
arrangements
summarized
below:
Experiment
Photon
A
Photon
B Difference
PAl
orientation
PA
2
orientation
.---~----
~-----~~-
a
(0')
b
(225')
b - a = 22.5'
2 a
(0')
d
(67.5')
d - a =
67.5'
3
c
(45
')
b (22.5') b - c =
-22.5'
4
c
(45')
d
(67.5')
d - c =
22.5'
From
eqo (4.19), we
note
that
the
expectation
value
for
the
joint
mea-
surement
with
the vertical axes
of
the
polarization
analysers
at
an angle
(b
-
oj
is
cos2(b
.-
a).
Thus,
SOT =
IE(a,b)
-E(a,d)1
+
JE(c,b)
+
E(e,d)j
_ jcos(45°) -
cos(135°)1
+
Icos(
-.45°)
+
cos(45°)
I
1 1 1 I
-
172
-+
72
I +
171
+
71
1
= 2../2 = 2.828.
(4.38)
where SOT
denotes
the
quantum
theory
prediction
for
S.
Equation
(4.38)
is
in clear violation
of
inequality
(4.37). Readers
can
once
again
satisfy themselves
that
the
simple local
hidden
variable theory
described
in
Section
4.2,
which
prediclsE(a,b)
= 1 -
Ib
-
01/45°
for
0°
:I; I b - a I
,;;;
90°,
further
predicts
SHY
= 2, where
the
subscript HV
stands
for
'hidden
variables'.
This
exercise merely
confirms
once
morc
that
quantum
theory
is not
cOl)sistent
with
local reality.
Correlations
between
the
photons
can be
greater
than
is
possible
for
two
Einstein
separable
particles since the
reality
of
their
physical
properties
is
not
established until
II
measurement
is
made.
The
two
particles
are
in
'communication'
over
large distances
since their
behaviour
is
governed
by a
common
state
vector.
Quantum
theory
demands
a
'spooky
action
at
a
distance'
that
violates special
relativity.
The
Aspect
experimenfs
139
4.4
THE
ASPECT
EXPERIMENTS
It
is
probably
reasonable
to
suppose
that
the derivation, in the late
19605
and
early 1970s,
of
an equation which
is
demoMtrahly
violated by a
quantum
theory then over 40 years
old
should have settled
the
matter
one
way
or
the
other,
once and for all. Correlated
quantum
panicles
are
everywhere in physics
and
chemistry, the simplest
and
most obvious
example being the helium
atom,
an
understanding
of
the
spectroscopy
of
which
had
led
to
the introduction
of
the Pauli principle in the first place.
Blit
it
became
apparent
that the special circumstances
under
which Bell's.
inequality could be subjected
to
experimental test had never been realized
in the
laboratory.
Suddenly, the race was on
to
perfect
an
apparatus
that could
be
used
to
perform
the
necessary measurements
on
pairs
of
correlated
quantum
particles.
As early as 1946, the physicist
John
Wheeler,
then
at
Princeton
University,
had
proposed studies
on
correlated
photons
produced by
electron
-positron
annihilation. But the polarization correlations
of
two photOns emitted in rapid succession (in a 'cascade') from an excited
state
of
the calcium
alom
proved
to
be the most accessible to experi-
ment
and
ultimately closest to the ideal. Carl A. Kocher
and
Eugene
D.
Commins
at
the University
of
California
at
Berkeley used this source
in
1966
ill a
study
of
correlations between the linear polarization states
of
the
photons,
altbough they did
not
explicitly set
out
to test Bell's
inequality.
The
first such direct tests were performed in 1972,
by
Stuart J.
Freedman
and
John
F, Clauser at the Lawrence Livermore Laboratory
in
California,
who also used the calcium
atom
source. These experiments
produced
the violations
of
Bell's inequality predicted by
quantum
theory
but,
because
of
some further 'auxiliary' assumptions
that
were necessary
in
order
to
extrapolate the
data,
only a weaker form
of
the inequality was
tested. These auxiliary assumptions left unsatisfactory loopholes for the
ardent
supporters
of
local hidden variables
to
exploit.
It
could still be
argued
then
that the evidence against such hidden variables was only
circumstantiaL
To
date-;-lhe best, most comprehensive experiments designed speci-
fically to test the general form
of
Bell's inequality were those performed
by Alain Aspect
and
his colleagues Philippe Grangier,
Gerard
Roger
and
Jean
Dalibard,
at
the Institut d'Optique Theoretique et Appliquee,
Universite Paris-Sud in Orsay, in
1981
and 1982. These scientists also
made
use
of
cascade emission from excited calcium
atoms
as a source
of
correlated
photons.
We
will
!lOW
examine the physics
of
this emission
process in detaiL
140
Putting It to the
test
Cascade
emission
In
the
lowest energy
(ground)
state
of
the
calcium
atom,
the
outermost
4s
orbital
is
filled with
two
spin-paired
electrons.
The
vector sum
of
the
spin
angular
momenta
of
these
electrons
is
therefore
zero, and
the
state
is
characterized
by
a
total
spin
quantum
number
S
'"
O.
The
spin multiplicity (2S +
I)
is
unity
and
so
the
stale
is
called a singlet
state.
The
total
angular
momentum
of
the
atom
is
II
combination
of
the
intrinsic
angular
momentum
that
the
electrons
possess
by
virtue
of
their
spins
and
the
angular
momentum
they possess
by
virtue
of
their orbital
motion.
We
can
combine
these
two
kinds
of
angular
momentum
in dif·
ferent ways.
In
the
first, we determine
separately
the
total
spin angular
momentum
(characterized
by
the
quantum
number
S)
and
the
total
orbital
angular
momentum
(quantum
number
L)
and
combine
these to
give
the
overall
momentum
(quantum
number
J).
In
the
second we com·
bine the spin
and
orbital
angular
momenta
of
each
individual electron
(quantum
number
j)
and
combine
these
to
give
the
overall
lotal.
The
former
method
is
appropriate
for
atoms
with
light
nuclei and
we
will use
it here.
In
fact,
for
the
ground
state
of
the
calcium
atom,
the
outermost
elec·
trons
are
both
present in a spherically
symmetric
s
orbital
and
therefore
possess
no
orbital
angular
momentum:
L
'"
I), S = 0
and
so J =
O.
The
state
is
labelled 4s' 'So, where
the
superscript
I indicates
that
it
is a
singlet
state,
the S indicates
that
L = 0 (S
corresponds
to
L
'"
0,
P cor·
responds
to
L = I, D
corresponds
to
L = 2,
etc)
and
the
SUbscript 0
indicates
that
J =
O.
If
we
use light
to
excite
the
ground
state
of
a calcium
atom,
the
photon
that
is
absorbed
imparts
a
quantum
of
angular
momentum
to
the
atom.
This
extra
angular
momentum
cannot
appear
as
electron spin, since
that
is
fixed
at
t
II.
Thus,
the
angular
momentum
must
appear
in
the
excited
electron's
orbital
motion,
and
so
the
value
of
L
must
increase
by
one
unit,
Promoting
one
electron
from
the
4s orbital
to
the
4p
orbital satisfies this
selection rule.
If
there is
no
change
in
the
spin
orientations
of
the
elec-
trons,
the excited
state
is still a singlet
state,
S = 0
and,
since L = I, there
is
only
one
possible
value
for
J, J =
I.
This
excited
state is labelled
4s4p
'P"
Now
imagine
that
we
could
somehow
excite a
second
electron (the one
left
behind
in
the
4s
orbital)
also
into
the
4p
orbital,
but
stili maintaining
the
alignment
of
the
electron spins.
The
configuration
would
then
be
4p',
which can give rise
to
three different electronic states correspon-
ding
to
the
three different ways
of
combining
the
twO
orbital
angular
momentum
vectors. In
one
of
these states
the
orbital
angular
momentum
The
Aspect
experiments
141
vectors
of
the
individual
electrons
cancel, L = 0
and,
since S = 0
we
have
J =
O.
This
particular
doubly
excited
state
is
labelled
4p'
'So'
If
this
doubly
excited
state
is
produced
in
the
laboratory,
it undergoes
a
rapid
cascade
emission
through
the 4s4p
'P,
state
to
return
to
the
ground
state
(see Fig,
4.8).
Two
photons
are
emitted.
Because
the
quan-
tum
number
J changes
from
0
->
I
....
0 in
the
cascade,
the
net
angular
momentum
of
the
photon
pair
must
be zero.
The
photons
are
there-
fore
emitted
in
opposite
states
of
circular
polarization.
In
fact, the
photons
have
wavelengths
in
the
visible region,
Photon
A,
from
the
4p'
'So
- 4s4p
'P,
transition,
bas
a wavelength
of
551.3
nm
(green)
and
photon
B,
from
the
4s4p
'P,
->
4s'
'So
transition.
has a wavelength
of
422.7
nm
(blue).
Experimental
details
In
the
experiments
conducted
by
Aspect
and
his colleagues,
the
4p'
'So
state
was
not
produced
by
the
further
excitation
of
the
4s4p 'P, state,
since
that
would
have
required light
of
the
same
wavelength as
photon
B.
making
isolation and detection
of
the
subsequently
emitted
light very
difficult.
Instead,
the scientists used two high-power
lasers.
with wave·
lengths
of
406
and
581
nm,
to
excite
the
calcium
atoms.
The
very high
intensities
of
lasers
make
possible otherwise very low
probability
multi-
photon
excitation.
In this case
two
photons,
one
of
each
colour,
were
absorbed
simultaneously
by a calcium
atom
to
produce
the
doubly
excited
state
(see Fig.
4.8).
Aspect,
Grangier
and
Roger
actually used a calcium
atomic
beam.
This
was
produced
by
passing gaseous calcium from a high
temperature
oven
through
a
tiny
hole
into
a
vacuum
chamber.
Subsequent
collima-
tion
of
the
atoms
entering
the
sample
chamber
provided
a well defined
beam
of
a!Oms with a density
of
about
3 X 10'°
aloms
cm-'
in
the
Fig.
4.8
Electronic states of the calcium atom inVOlved in
two~photon
cascade
emission process.
142
Putling it to the
test
region where the atomic beam intersected the laser beams. This low
density (atmospheric pressure corres
pond
s to
about
2 x
10
" molecules
cm -
J
)
ensured
that
the
calcium atoms did not collide with each
other
or
with
the
walls
of
the chamber
before
ab
s
orbing
and
sub
sequently emit-
ting light. It also removed the possibility
that
the
emitted
422.7 nm light
would be
reabsorbed
by
ground
state calcium
atoms.
Figure
4.9
is a schematic
diagram
of
the
apparatus
used by Aspect
and
hi
s colleagues.
They
monitored light emitted
in
opposite
directions from
the
atomic
beam
source
, using filters
to
isolate
the
green
photons
(A)
on
the
left
and
the blue
photons
(B) on the right.
They
used two polariza-
tion
analy
sers,.
four
photomultipliers
and
electronic devices designed
to detect
and
record coincident signals from
the
photomultiplier
s.
The
polarization
analysers were actually polarizing cubes, each
made
by
gluing
together
two prisms with dielectric coatings
on
those
faces in con-
tact.
The
se cubes
tran
smitted light polarized parallel (vertical)
to
the
plane
of
incidence,
and
reflected light polarized
perpendicular
(horizon-
tal)
to
this
plane
.
Thus,
detection
of
a
transmitted
photon
corre
sponds
in
our
earlier discussion
to
a + resuit, while detection
of
a reflected
photon
corresponds
to
a - result.
The
polarizing cubes were neither quite perfectly
tran
smitting
for
pure
verti
ca
lly polarized light
nor
perfectly reflecting
for
pure
horizontally
polarized light.
The
physicists measured the
transmittance
of
PA, for I
r-1
0
PMT
"'1(
----
---13
m--
--
---~)o!
®
v J / 1
PA
,
1/
I
h
r
e
PMT
e
0
n'
'n
fi
Uers
/ "
•
U
Sou
r
ce
U
lo
u
r·
laid
coincidenc
es
e
e
CO
in
CI
dence c
ounl
er
®
,
~
v'
PA
L
~
h'
PMT
e
r
o
r-
PMT
Fig.4
.9 Sc
hematic
diagram
of
the experimental apparatus used
by
Aspect,
Grangier and Roger
(PMT
is
a photomultiplier).
The Aspect experiments
143
vertically
polarized
light,
T;,
and
the
reflectance
of
PAl
for
horizon-
rally
polarized light,
R?
for
light with a wavelength
of
551.3nm.
They
obtained
T;
=
R;
= 0.950.
They
also
measured
n =
R;
= 0.007.
These
latter
figures
represent
a small
amount
of
'leakage'
through
the
analyser. Similarly,
for
light with a wavelength
of
442.7
nm,
they mea-
sured
T;
=
R~
= 0.930
and
n =
R;
= 0.007.
Each
polarization
analyser
was
mounted
on
a
platform
which allowed
it
to
be
rotated
about
its optical axis. Experiments
could
therefore
be
performed
for
different
relative
orientations
of
the
two
analysers.
The
analysers were placed
about
13
m
apart.
The
electronics were set
to
look
for
coincidenc'es in
the
arrival
and
detection
of
the
photons
A
and
B
within a
20
nanosecond
(ns) time
window.
This
is
large
compared
with
the
time
taken
for the
intermediate
4s4p 'P,
state
to
decay
(about
5 ns),
and
so
all
true
coincidences were
counted.
Note
that
to
be
counted
as a coincidence,
the
photons
had
to
be
detected within 20 os
of
each
other.
Any
kind
of
signal passed between
the
photons,
'informing'
photon
B
of
the
fate
of
photon
A,
for
example,
must
therefore
have travelled
the
j 3 m between the
analysers
and
detec-
tors within
20
ns. In fact,
it
would
take
about
40 ns
for
a signal
moving
at
the speed
of
light
to
travel this distance.
The
two
analysers were
therefore
space-like
separated.
The
results
Aspect,
Grangier
and
Roger
actually
measured coincidence rales (coin-
cidences
per
unit
time).
For
the specific
arrangement
in which
PA,
has
orientation
a
and
PA,
has
orientation
b, we write
these
coincidence
rates
as R
..
(a,b)'
R,_
(a,b),
R_,
(a,b)
and
R
__
(a,b).
After
correction
for
accidental coincidences,
the
physicists
obtained
results
which
varied
in
the
range
0-40
coincidences
S-I
depending
on
the
angle
between the
vertical axes
of
the
polarisers
(b
-
oJ.
They
then
used these
results
to
derive
an
experimental
expectation
value, E
(a,
bJ"."
for
comparison
with theory (cf.
eqn
(4.18));
E(a
b)
=1!~Ja,
b)
-
R,jcL.,tJL
- R_+ (a,
bj
+
R._
(""-~
, "P'
R.,
(a,b)
+
R,_
(a,b)
+
R_,
(a,b)
+R
__
(a,b)
(4.39)
Dividing by
the
sum
of
the
coincidence
rates
normalizes
the
expectation
value (it is equivalent
to
dividing
by
the
tOlal
number
of
photon
pairs
detected).
The
physicists
measured
the
expectation value for seven different sels
of
analyser
orientations,
and
the
results they
obtained
are
shown
in
Fig.
4.10.
From
eqn
(4.19), we know
that
the
quantum
theory
prediction
144
Putting
it
to
the
tos
t
T~
05
E(a,b)
-1
,
(b-a)
Fig.
4.10
Results of measurements
of
the expectation value
Eta,
b J for seven
different
relative orientations of
the
polarization analysers. The continuolJs line
represents
the
quantum
theory predictions modified
to
take
account
of
;nstru~
mental
-factors (see
text).
Reprinted
with
permission from Aspect er al.
{i
982;.
Physical
Review
Letters,
49,
91.
for
E(a.
b)
is
cos2(b
-
a).
However, (he
extent
of
the
correlation
obsdrved
experimentally
was
dampened
by
limitations
in
the
apparatus,
as
described
above.
The
physicists
therefore
derived
a slightly
modified
form
of
the
quantum
theory
prediction
that
takes
these limiting factors 1
into
account.
They
obtained:
.
(T;
-
Tn
(T;
- n 1
E(a,
b)
~
F
1h
+
T
;)
(1";
+
Tn
cos2(b
-
oj.
( 4.40)
The
factor
Fallows
for
the finite solid angles
for
detection
of
the
photons
(not all
photons
could
be
physically
'gathered'
into
the
detection
system).
They
found
F =
0.984
for
their experimental
arrangement.
The
term
involving
the
analyser
transmittances
accounts
for
the
small
amount
of
leakage
and
for
the
fact
tnat
not
all
photons
incident
on
the
analysers
were ultimately detected,
The
predictions
of
quantum
theory,
corrected
for
these
instrumental
deficiencies,
are
shown
in
Fig. 4.10 as
the
conti-
nuous
line. As expected,
the
predictions
demonstrate
Ihat
perfect
corre-
lation,
(b
-
oj
=
0°,
and
perfect
anticorrelation,
(b
-
0)
'"
90°,
were
not
quite
realized in these experiments.
Aspect
and
his colleagues
then
perfonned
four
sets
of
measurements
with
analyser
orientations
as described
on
p. 138.
Defining
the
quantity
S,," as
[IE(a,b)
-E(a,d)j
+
IE(c,b)
+E(c,d)ILp>(cLeqn(4.38»,
from
their
measurements
they
obtained
S"P'
= 2.697 ± 0.015
(4.41 )
a violation
of
Bell's inequality,
eqn
(4,37)),
by
83"70
of
the
maximum
The
Aspect
experiments
145
possible predicted by
quantum
theory (Le .
.J2,
see eqn (4.38)).
By
taking
the
mstrumentallimitations
into account, the physicists obtained a modi-
fied
quantum
theory prediction for this quantity
of
SOT = 2.70 ± 0.05,
in excellent agreement with experiment.
These results provide almost overwhelming evidence in favour
of
quantum
theory against all classes
of
locally realistic theories. One loop-
hole remained, however. The polarization analysers were set in position
before
the
experiments were initiated (i.e. before the calcium atoms
were
excited
and,
most importantly, before the correlated
photons
were
emitted).
Could
it
not be
that
the photons were
somehow
influenced in
advance
by
the
way the
apparatus
was set up?
If
so,
is
it possible that the
photons
could
have been emitted with
just
the right physical characteris-
tics (governed,
of
course, by local hidden variables)
to
reproduce the
quantum
theory correlations? Although this
is
beginning
to
look like
some kind
of
grand conspiracy on
the
part
of
the
photons,
it
is
not a
possibility
that
can be excluded by
the
experiments
just
described.
Closing
the
lasl
loophole
To
close this last remaining loophole, Aspect, Dalibard
and
Roger modi-
fied
the
experimental set-up to include two acousto-optical switching
devices (see Fig.
4.11). Each device was designed to switch the incoming
photons
rapidly
between
two
different optical paths,
and
each was
activated by passing standing ultrasonic waves through
a small volume
of
water
held
in
a
transparent
container.
The
ultrasonic waves, which
change
the
refractive index
of
the water and hence change
the
path
of
light passing
through
it, were driven at frequencies designed to switch·
between
the
two
paths every IOns. At the end
of
each
path
was placed
a polarization analyser (which could
be
oriented independently
of
the
fig.
4.11
Schematic diagram
of
the experimental apparatus used
by:
Aspect.
Dalibard and Roger.
146
Putting
it
to
the
test
rest
of
the
apparatus)
and
a
photomultiplier.
The
vertical axes
of
each
of
(he
four analysers were
oriented
in
different
direc!lons.
By this
arrangement,
the physicists prevented the
photons
from 'know-
ing' in
advance
along
which optical
path
they
would
be travelling,
and
hence
through
which analyser they would eventually pass.
The
end
result was
equivalent
to
changing
the
relative
orientations
of
the
two
analysers while the
photons
were in flight.
Any
communication
between the
photons
regarding the way the
apparatus
was set up was
therefore
restricted
to
the
moment
of
measurement,
in principle requir-
ing faster
than
Iigh! signalling between the
photons
to
establish the
correlation.
The
switching
arrangement
shown
in Fig. 4.11
was
difficult
to
operate
and
run
successfully. Aspect
and
his colleagues could
not
add
to
Ihis
difficulty by trying
to
detect
photons
both
transmitted
and
reflected
by
the
polarization
analysers (such
an
experiment would have required
eight
photomultipliers
and
the
necessary coincidence detection!).
The
physicists could
therefore
only detect
those
photons
transmined
by
the
analysers: they could observe only + results.
Fortunately,
a version
of
Bell's inequality
had
been derived by
John
F.
Clauser, Michael
A.
Horne,
Abner
Shimony
and
Richard A. Holt in 1969 for Just this kind
of
experiment.
We
will derive this inequality
here.
Let us
return
for a
moment
to
the
expression for the expectation value I
E(a,
b)
given in
eqn
(4.18):
E
(c,
b)
=
PH
(a,
b)
- P + _
(c,
b)
- P _ + (a,
b)
+ P
~
_ (c,
b)
.
(4.18)
Our
difficulty arises because
if
only
transmitted
photons
are
detected,
then
only quantities related
to
PH
(a,
b)
can
be
measured.
However,
consider
the
equivalent
experiment
performed
with
PA,
removed com-
pletely. We
llse
the
symbol
00
instead
of
b
to
define a probability for
joint
detection,
P,
+
(a,
00),
in these circumstances.
Provided
the removal
of
PA,
in
no
way
affects
the
behaviour
of
either photDn A
or
B,
then
P"
(a,
(0)
should
include the probabilities
of
all possible joilll results
in which
photon
A
is
detected, I.e. it includes the possible
joint
reslIlts
in which
photon
B is detected
(+)
and
not
detected
(-):
PH
(a,
()OJ
~
PH
(a,b)
+
p~_
(a,b).
(4.42)
Similarly,
(4.43)
and
PH
(00,
00)
= P
++
(a,
b)
+ P + _ (a,
b)
+ P
~
+ (a,
b)
+ p
__
(a,
b).
(4.44)
The
Aspect
experiments
147
We
can
now
combine
these expressions to give
£(a,b)
=
4P
..
(o,b)
-
2P
q
(a,oo)
-
2P
..
(00,11) +
PH
(00,00).
(4.45)
Equation
(4.45) allows
us
to
calculate the expectation value using only
the
probabilities
of
joint
+ + results, for which related quantities can
be
obtained
from
experiment. •
From
eqn
(4.45) it follows
that
£ (0,
b)
- E
«(J,
d) = 4 P ++
«(J,
b)
- 4 p
..
(a,
d)
-
2P"
(00,
0)
+
2P
..
(00,
d)
(4.46)
and
E(c,
b)
+
E(c,
d)
=
4P
++
(c,
0)
+
4P
H (c,
d)
-
4P
..
(c, co)
- 2 P + +
(00,
b)
- 2 P
++
(00,
d)
+ 2 P H (
co,
00
) .
(4.47)
Equations
(4.46)
and
(4,47) can
now
be combined
to
give an expression
for S
in
terms
of
the probabilities
for
joint
-f. + results.
In
the
experiments, coincidence rates were actually measured. When
normalized,
these rates are related to
the
joint
detection probabilities via
relations such as
P
(
b)
=
R"
(G,
b)
.
• +
G,
R ( ) .
t+
00,00
(4.48)
All
the
quan!tt.es needed to
obtain
S from
the
experiments with
switched optical
paths
were measured by Aspect
and
his colleagues.
For
(J
=
0°,
b =
22.5°,
C = 45°
and
d =
67.5°,
they obtained S"" '"
2.404 ± 0.080, once again in clear violation
of
inequality (4.37). Taking
account
of
inefficiencies in
the
polarization analysers
and
the finite solid
angles
for
detection allowed them to obtain a modified
quantum
theory
prediction
SQT
=
2.448,
in excellent agreement with
experiment.
So,
where
does all this leave local reality?
For
the
purist,
the
last
loophole
is still not completely closed by these experiments.
The
standing
ultrasonic
waves used
to
drive
the
acousto-optical switches did not
provide completely
random
switching, although the
two
switches were
driven
at
different frequencies. However,
we
would need 10 invoke a
very
grand
conspiracy indeed
to
salvage local hidden variables
from
these
experimental
results. This immediately brings
to
mind
another
of
Einstein's
famous
quotes
(made
in
a
rather
different context):
'The
Lord
is
subtle,
but
he
is
not malicious.'
.
The
majority
of
physicists, including those like David Bohm
and
John
Sell
who
have rejected
the
Copenhagen view, have accepted
that
the