LOCAL CAUSALITY AND THE FOUNDATIONS OF QUANTUM MECHANICS Robert S. Goldstein B. Sc., University of I I1 inois, Urbana Champaign, 1 985 THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUiREflENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics @ Robert S. Goldstein 1987 SIMON FRASER UNIVERSITY July 1987 All rights reserved. This work may not be reproduced i n who1 e or in part, by photocopy or other means, without permission of the author.
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LOCAL CAUSALITY AND THE FOUNDATIONS OF QUANTUM MECHANICS
Robert S. Goldstein
B. Sc., University of I I1 inois, Urbana Champaign, 1 985
THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUiREflENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department
of
Physics
@ Robert S. Goldstein 1987
SIMON FRASER UNIVERSITY
July 1987
All r ights reserved. This work may not be reproduced in who1 e or in part, by photocopy
or other means, without permission of the author.
PARTIAL COPYRIGHT LICENSE
I hereby g ran t t o Simon Fraser Un ive rs i t y the r i g h t t o lend
my thes i s , p r o j e c t o r extended essay ( the t i t l e o f which i s shown below)
t o users o f the Simon Fraser U n i v e r s i t y L ibrary, and t o make p a r t i a l o r
s i n g l e copies on l y f o r such users o r i n response t o a request from t h e
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f o r m u l t i p l e copying o f t h i s work f o r scho lar ly purposes may be granted
by me o r the Dean o f Graduate Studies. I t i s understood t h a t copying
o r p u b l i c a t i o n o f t h i s work f o r f i n a n c l a l ga in s h a l l no t be a l lowed
w i thout my w r i t t e n permission.
T i t l e of Thesis/Project/Extended Essay
Local Causality and the Foundations
of Quantum Mechanics
Author:
(s ignature)
Robert S. Goldstein
(name
ABSTRACT
Quantum mechanics implies the existence of correlations which cannot be
described by any local theory. These non- local correlations possibly suggest
the existence of particles traveling faster than light (tachyons). But the
potential paradoxes associated with their existence imply that, even i f these
tachyons do exist, it should not be possible to control them in a manner
necessary for human superluminal (faster than light) communication. Indeed,
the unitary nature of the time development operator precludes the possibility
of using non-local correlations to communicate superluminally within the
quantum formalism. A simplified model which preserves unitary time
development i s shown to be useful in studying 'thought experiments' which
appear init ially to allow for superluminal communication.
The consistency of quantum mechanics is questioned when measurements of
retarded fields are considered. I f certain idealized measurements are possible
in principle, then the quantum formalism appears inconsistent. I f the
measurements are not possible in principle, then 'state reduct ion' (non-unitary
time evolution of the state operator) in a pragmatic sense can be said to have
occurred.
The suggestion that tachyons be used to explain the non-local correlations i s
abandoned in favor of another model whose philosophy is supported by the
tenets of relativity theory. Such a model may imply the existence of a more
general quantum theory where a quantum system itself can be taken as a
(generalized) frame of reference.
DEDICATION
To Ana. who was always with me when I needed her. and who shall always
remain with me, forever.
ACKNOWLEDGEMENTS
1 wish to express my gratitude to L. E. Ballentine for his help and support
through count less number of revisions. Thanks also goes to D. Wilson and M Gee
that the same result derived in the schematic argument was obtained, namely
that the spin part of the wavefunction looked like:
and hence again we find particle 1 to be described by a pure state T , as opposed Y
to the mixed state that described it before particle 2 traversed the apparatus.
Thus, uslng an 'acceptable' approxlmatlon, It lndeed appears that thls device can
affect what i s measured for the other particle.
We shall soon see the reason that we can apparently transmit SLS i s because
we have allowed non-unitary transformations to describe our state
development. One might be surprised t o find that the problem is not wi th the
spin flip, but wi th the recombination. The reason why the f i rs t approximation
to the interaction picture gives us incorrect results i s because the time
development operator i s non-unitary, as we now show. We have:
Now, for an arbitrary operator V t o be unltary:
v v t = v t v = I
But indeed we find:
( 1 + ( 1 / i f i ) ld ts WI (to) It ( 1 + I 1 / - i f i l ldt ' WI (t') 1 = 1 + (( 1 /fi)(dte WI (t') l2
f 1
We shall soon see that it is the unitarity of the time development operator
which prevents human transmission of SLS.
4 . 3 A Sim~le. Exactlu Soluble Model -
Before giving a general proof of the impossibility of superluminal
communication, l e t us demonstrate a simple way of clearly locatlng the
problem i n the above example. We have shown that we cannot use normal
perturbation methods to give a correct qualitative description of the above
example, since they approximate time evolution in a non-unitary fashion. So let
us t r y a dlfferent approach. We treat space as three valued, wl th the three
ei genvectors:
corresponding to the positions:
positive deflection no deflection negative deflection,
respectively. For example, l e t our particle start initially in the undeflected
posit ion. After traversing a SGA, the wavefunct ion has two components,
namely, positive and negative deflect ion in our coordinate space. This
separation wl l l allow us to act upon only one of the amplitudes with our spin
flipper. Then we shall recombine the two amplitudes. What i s important here
i s that we use only unitary operators to transform the state vectors. Finding
unitary operators that act on the state vectors in the desired manner i s simple
because of the small dlmenslonall tles of the state vectors used here.
With the above model, we see that the three relevant variables are:
1. spin of particle 1
2. spin of particle 2
3. posltlon of partlcle 2
We initially have the particles in a spin-singlet state, and the second particle
in a no-deflection position:
(from now on, we drop the z- subscript). First, we couple, by use of a SGA, the
position of particle 2 with i ts spin. This can be accomplished by the unitary
operator:
(The identity operator acting on variable I being understood) which gives for a
state vector:
Next, we flip the spin of the amplitude of, say, the bottom path. A suitable
unitary operator is:
Giving a resultant state vector:
We see that the spin 2 vector now comes off as a factor. Finally, we attempt to
recombine the two amplltudes, say, at the undellected position. We must do
this without affecting the spin variables, or else we would be back where we
started. Obviously, we must have this operator in the form:
(where a, b, and c, are lef t arbitrary for now), for only this matrix w i l l allow
both:
['I and
toreturn to thenondeflec ted state vector. But this matrix Is degenerate,
having two identical columns, and hence i t cannot be unitary for a values of a,
b, and c. We now see that al l we have effectively done thus far i s to transfer
the correlation from spin1 - spin2 to spin 1 - position2 We cannot recombine the
two paths, although in real coordinate space we can of course make the two
amplitudes overlap. That is, although we can, in real coordinate space, make
the two amplitudes condense into the same spacetime region, these two
amplitudes must necessarily be orthogonal, as we now show.
In real coordinate space, the singlet state for the three variables above
looks 1 ike:
There are two amplitudes associated wi th the second particie (for it i s in a
mixed state). These amplitudes start off orthogonal, because the two possible
spin values are orthogonal:
Now, any un i tay operator which acts upon these two amplitudes w i l l maintain
their orthogonality-- a well known property of un i tay operators. I f we
consider the SGA separatlon, spln flip, and SGA recombination as one large
unitary operation, we must necessari 1 y produce a wavef unct ion orthogonal to
the original (since now the spin part of the two amplitudes are colinear,
namely, both are spin 1,). In particular, i f we only act upon the bottom path
w i th the spin flipper (as in the above example), we get:
Hence, the two possible spin values of particle 1 are now correlated wi th
orthogonal wavefunct ions:
Y = I t l& - 1 , p 2 I @ 7Zz
and upon a partial trace over the variables of particle 2, particle 1 w i l l s t i l l be
described by the same mixed state that described it initially, having a
probability 0.5 of being found in any spin direction that we decide to measure.
Since the same statistical operator s t i l l describes particle 1, someone
observing particle 1 w i l l not be able to distinguish i f particle 2 went through
the spin flipper or not, and thus no SLS can be sent in this manner.
4 . 4 General Proof Aaainst send in^ SLS -
Above we have shown a scheme whlch at rlrst sight apparently allowed one
to send SLS, but later was shown to be incorrect. Was the mistake an obvious
one, never to be made by advanced physicists? Let us just state here that after
the above problem was resolved, a similar suggestion was indeed found i n the
published ~t terature '~. T ~ U S 1t is forgivable not t o have found the mistake
instantly, and we see that the above simplified model may be a very useful tool
for checking future gedankenexperiments which may arise.
The question now becomes, can a general proof be given which demonstrates
the impossibility of sending SLS by means of non-local QM correlatlons? From
the above discussion, the most important fact seems to be that the time
development operator is unitary. We shall see that this fact alone, together
wi th use of the partial trace formalism, allows one to demonstrate the
impossibility of human transmission of SLS within the framework of QM.
Consider a three component system, w i th variables a, b, and c (which are
possibly correlated). The combined system in general can be described by a
state operator:
where 2' implies that the summation i s t o be taken over all six variables. We
can think of a and b as the two spin variables prepared in the singlet state, and
c as any apparatus that we let operate on b attempting to create an observable
change In a. We think of a and b as 'separating' in that they no longer Interact--
no term in the Hamiltonian couples them. We now allow for any interaction
between b and c that i s permissible within the quantum formalism. This
implies that the new combined state operator can be written as:
P, abc . (@c)t p abc Ubc i
where ubC is some unltary operator dependent only upon b and c. Using the
partial trace formalism, we find the state operator that describes variable a
init ial ly is:
= Z n.n0mp
W nmg; n'mg la"> <$'I
Let us compare this with the state operator that describes variable a after the
interaction between b and c:
= 2' < bm* ,P'I ubc ( ~ b c ) t W n.m,p; n",m',pg I bm cp > lan> can' I
With ubC unitary, we have ubC (ubc)t = 1, implying:
pp = z nn'mp
W n.ma n'mp Ian><<6'1
This simple proof demonstrates that the statistical averages associated
wl th a varlable 'a' can not be affected by anything done t o other systems,
whether or not the systems have interacted in the past.
This last statement deserves some attention I s this not in contradiction to
what was proved wi th the (generalized) Bell type arguments? Definitely not.
There we saw that, for individual systems described by a certain state
operator, there exist non-local correlations i n their observables. But what
this proof demonstrates i s that the averages of the observables for a large
ensemble of similarly prepared systems are i n no way affected, and therefore
(l)~ctually, one must be caeful when manipulating infinite dimamional spaces. Although for finite dimensional operators X and Y, Tr (XY - YX) messwily squab zero, we sea, far example whsn X - Q Y - P, ws ham the corntator[& PI = ih, md Uws the trace is not even defined. But since we c#.e dealing here with state operators, which bg definition have a finib indeed unit, value of trace. this problem does not seem to be nl-L han. How-. in Ua original proof'*. Ua authors showed kt, for any observAle Y of particle a:
' Ya , = Ya pr' '
We see that this proof can be criticized since there exists density operators p and operators Y such that : Tr W, pa I
is not defined. and h 8 ~ 0 the &ow assumption is invalid in general.
no information can be sent. We note here that a l l of this i s independent of
whether or not one be1 ieves in 'reduction of the wave packet'.
This also demonstrates why such a dif f icult technique was needed to show
that the r e W A is not indepenndent of the setting b . We noted earl l er that
usually correlations are looked for by comparing, in this example, < A b > w i th
< A >< b >. We see now that necessarily < A b > - < A >< b > must equal zero, or
else SLS could be sent, since we can control b (the orientation of our
Instrument). Thus although for arbitrary observables X and Y, < XI > r < X > < Y >
implies that correlations exist, the equality (XY > = < X > < Y > does not imply
that no correlation, in some sense of the word, exists.
V. RETARMD FIELDS AND THE FOUNDATIONS OF QM
5 . 1 Can One Measure a System Without Affectina It ? -
We have shown that in the non-relativistic framework, SLS are an
impossibility. What about within the relativistic framework? A1 though the
mathematics is more complex, relativistic QM is s t i l l based upon state vectors
and unitay time development, so the above proof would seem to be valid i n this
realm also. Still, there I s always the possibility of hidden assumptions in one's
proofs (as was the case with Von Neumann's proof of no hidden variables).
Furthermore, in relativistic quantum theory, there are st i I1 many unresolved
problems a t the foundational level. I f indeed a thought experiment was devised
that could apparently be used to send SLS, It would be evldence l o r the
inconsistency of the quantum formalism. Let us see i f there are any new
instruments in the relativistic l imit that may be of some use in possibly
sending SLS.
We shall now discuss a rather Involved gedankenexperiment which seems to
imply that either:
1. One can send SLS
or:
2. Quantum predictions cannot be satisfied for al l inertial
reference frames simultaneously.
But since the quantum formalism predicts that SLS transmission i s impossible,
I f the following argument i s valid, then either way quantum mechanics must be
incorrect.
Our main weapon w i l l be, ironically, the f ini te speeds at which fields
propagate. To motivate the following, le t us reproduce an often-quoted
statement used to describe quantum measurement theory:
to measure a property of a system inherent& affects the system.
Let us note that this statement does have some support from Newton's third
law:
For ever- action t . e Is an equa/ and opposite reaction
For, i f particle B acts upon particle A without A acting upon B, then in some
sense, A has measured some aspect of B without B being affected. Now in fact,
special relativity, w i th i t s f inite maximum speed of information transfer,
allows for such an occurrence. For example, consider a charged (classical)
particle at (x, y) = (-1, 0) traveling in the +X direction, approaching a target at
the origin. See Fig. 10. At (0, 10) i s an uncharged ferro- electric crystal
Fig. 10
which can be converted into an electric dipole upon command. Because of the
flnite speed at which the dipole's field propagates, It i s posslble to convert the
dipole at a time in which it can s t i l l be affected by the charged particle's field,
yet the charged particle cannot be affected by the dipole's field, since the
particle is already registered on the detector before the dipole's f ield can
reach It. Hence It seems plausible that an experiment could be posed that would
put quantum mechanics in conflict wi th special relativity. We attempt here to
do just that. First we shall introduce a new method for measuring spin. Then,
we shall show how such a measurement can be used to make quantum theoy
incompatible wi th special relativity.
Let us describe a new method for spin measurement. As usual, we f i rst
spatlally separate the two posslble paths 01 a neutral spin 1/2 partlcle by use
of a SGA But instead of measuring i t s posit ion (and hence spin), we measure
the direction of the magnetic field that i s created by the particle at a position
in between i t s two possible paths. (Since the particle has a dipole moment
H = pS, it creates a magnetic field). However, we do it in such a way that the
instrument measuring the field does not itself disturb the particle, which i s
possible because of the finite speed at which a field (of the instrument)
propagates.
We start with an uncharged particle prepared in a spin t, state traveling at
speed c/J2 in the -y direction (in the apparatus' frame). See Fig. 1 1. It travels
through an X- oriented SGA at R, which creates two possible paths which the
particle can take. Some bending apparatus is then used so that the particle
again travels in the -y direction. Call t = 0 the time when the particle's
amplitudes reach points S. At this time, a potential is turned on to recombine
the two paths at a point T. The particle then passes through another SGA
(orientation left variable for now) and i s finally registered at one of the two
Fig. 11
detectors positioned at points U.
Let us use numbers here to facilitate comprehension. We take the radius 'r'
and the speed of light c to be unity. With the particle reaching S at t = 0 and
traveling at speed v - 1 /J2, we see that it arrives at T at time:
t, = 4211
Since the distance TU i s equal to .394 we find that the particle i s registered on
. one of the detectors at U at time:
s, = 42 fi + 42 (.394) = 5.000
At point V, a person controls whether or not a neutral macroscopic particle
dissociates into two highly charged (oppositely, of course) test bodies,
separating in the z- direction. The controller decides this at a time just before
t - 242. Note that t - 242 i s exactly the time which, i f the recombining
potential was not turned on at polnts S, the two possible positions of the
particle would have been collinear wi th V, namely, at points W. Also note that
wi th distance ST = 2, and TV - 2, we have SV = 242. Therefore t = 242 i s also
the time which a light signal originating from S at t - 0 would reach V. Hence
from classical electrodynamics, for times less than t = 242, the B-field
detected at V would be as i f the particle was not deflected at S. (If this was
not so, someone at V would know that the combining potential was turned on at
S faster than allowed by SR, and hence a SLS would be sent). Thus we see that
the possible flelds detected around the time t = 242 would be the strongest
obtainable, and i n opposite directions. See Fig. 12.
With the charged test particles separating in the ?z directions at great
velocity, we see that a (classical) force would deflect them in the ?y
dlrectlons. That is, from:
F = qv x B = (pO/4d) qv x pS/ 3
position \ 9 c$ "w"
I( position
Fig. 12
we see that, for the positive test charge initially traveling in the +z- direction,
say, a deflection in the +y direction would correspond to a 8-field pointing in
the +x direction. I f p is positive, this would imply that the particle took the
upward path. and thus has a spin value S, = + 1 /2. Hence by observing the
deflection of the charged test bodies in the y- direction, one can infer the value
of Sx for the particle in question
Let us note here that, even i f the controller at V creates the charged test
bodies, their fields cannot affect the result obtained at U, since their fields
propagate a t speed c. Indeed, with the distance VU equal to 2.394, the fields
would not reach U until:
t, = 242 + 2.394 - 5.222
and thus cannot affect the result obtained there occurring at t = 5.000.
Let us consider possible results for di f f e m t orientat ions of the second
SGA. Flrst, consider I f the second SGA 1s oriented in the z- direction. We have
earlier discussed the abi 1 i ty to split and recombine amplitudes coherently, and
thus we predlct that the partlcle would lndeed be found ln the t, state at U,
since It was prepared so inltially. Now, is it possible for the actions a t V
(whether or not the controller created the test charges) to affect this result?
We see that if it did, then one could send SLS, for an observer at U, knowing
that a particle prepared in the tz state must be found at U to s t i l l be in the tZ
state unless something interacted with it, would then conclude that the
controller at V must have indeed elected to produce the test bodies. But the
observer a t U would know this at time t = 5.00 (since this is the time when he
obtains the result for S,), which is faster than a light signal from V could
reach U (namely, t = 5.222) Therefore, for SLS not to be sent, the particle must
always be found upon measurement to be in the same state as it was in initially.
Now le t us conslder the posslble results i f the second SGA 1s orlented In the
x- direction. Note then that 5, i s being measured at both U and V. I n what
follows, we shall assume that the result obtained at U w i l l be consistent wi th
the result obtained at V ( i f the controller at V created the test charges, that
is). I n other words, we assume that the measurement carried out at V i s a
valid measuring process which gives reproducible results for spin observables.
One may be w a y of the measurement procedure used at V. Obviously, the
0-field created by an actual spin 1/2 particle would be extremely small. But
we see from the derivation of B from a classical dipole M that for large p and
small r, this 0-field can be made arbitrarily large. Also, note that we do not
have to measure B accurately for a successful measurement. Indeed, a l l we
need to measure i s the direction (sign) of B to calculate whether S, equals
+ 112. Furthermore, we can consider more involved experimental arrangements
where an incoming spin 1/2 ion has an arbitrarily large charge (one cannot use
a SGA on an electron for spin measurements due to i ts l ight mass1g). We can
measure the electric f ield that it creates, which can be made arbitrari ly large
due to i t s large charge. Although this set up also has drawbacks (e.g.,
acceleration of charged particles would produce radiation, which would carry
away angular momentum) we see that it i s the Question of measurabilitu of
retarded fields that i s relevant here, and not the problems inherent i n any
particular set up. Indeed, we can create fictit ious fields which act in the
manner desired, for the quantum formalism i s believed to be independent of the
actual forces of nature (e.g., in elementary QM we consider square well
potentials, 6-function potentials, etc, without questioning their reality, and we
s t i l l expect the quantum formalism to 'work' regardless of their reality).
Indeed, i f the only resolution of the forthcoming paradox i s obtained by arguing
that we should only conslder the actual forces in nature, then we w i l l have
found a new relationship between QM and conceivable worlds that has not
previously been discovered.
5 2 oeriment 2
Now wi th the groundwork laid, we get to the paradox. Assume we have two
neutral spin 1/2 particles prepared in a spin singlet state (see fig. 13). They
dissociate at t - 0, and travel at speed c/?2 with respect to the rest frame.
Particle 2 goes through a similar set up as in experiment 1, now wi th the
second SGA oriented in the I-direction. We wait, say, 100 time units as the
S W
FIG. 13 -- - -- --
particles separate. The times of the events are now:
Particle 2 reaches S
V decides whether to create test charges
(and hence, measurement at V ) t = 102.828
Measurement at U
Result at V to reach U
While al l of this i s going on, a human at L observes the result obtained for
particle 1 with a SGA oriented i n the z- direction at time, say, t = 102 .5. We
note that since U is closer to L than i s V, that L w i l l receive the result obtained
at U (sent by a light signal) before he receives the result obtained at V. I n
particular, we assume the apparatus is set up so that L receives the results
obtained at U and V at times (respectively):
5 . 3 . 1 Possibleresults
Let us consider the m u 1 ts i f V decides not to create the test charges. We
see then that partlcle 2 wl l l recornblne coherently a t T, slnce nothlng
interacted with it between R and T. Because both L and U measure the z-
component of spin (on part ides 1 and 2, respect iveiy), and since the part ides
were prepared in the singlet state, L and U wi l l always obtain opposite results.
NOW l e t us consider the posstble results II v decides to create the test
charges. First, consider the scenario that L and U do not obtain opposite
results. How does L view the situation? At t = 102.5, he measures 5 ,, and
obtains, say t,,. At t - 205.000, he is informed (by light signal) that U
obtained t2z. Since L knows that, i f V does not create the test charges, then L
and U must obtaln opposite results, L can only conclude that V must have indeed
created the charges. He concludes this at t = 205.000, but a 1 ight signal from V
tell ing of V's actions does not reach L unti l t = 205.222. Hence, a SLS has been
sent.
Now consider i f L and U do always find opposite results. How does V view
the sltuation? At t = 102.828, V measures the x- component of spin of particle
2. Let us say that he finds spin 12*. Familiar with the experimental set-up. V
knows that (in his reference frame, anyway) he i s the f i r s t to perform a
measurement on particle 2 from the time that It was prepared In the slnglet
state. Conservation of angular momentum (and quantum predict ions) demand
therefore that, i f L were to measure S , , then he would necessarily f ind r l ,
Hence from V's measurement of S2,, he concludes that particle 1 i s i n a pure
state qx. NOW, it can be shown that i f a system i s in a pure state, then
uncorrelated wl th any other system. (See a~pendlx 8). That is, consider
particles. They can in general be described by a state operator :
P ' ~ = Wnm; n*, lan bm > < a"' bm' I
I f we are told that particle 1 i s in a pure state:
p1 = Tr, p12 = I Y > c Y 1
then it i s always possible to wr i te pl* i n a factorized form:
it i s
two
Now, If indeed pI2 = p1 s d, then for any observables XI and Y2 acting on
particles 1 and 2, respectively, we find:
= < X , > < Y 2 >
Thus V concludes that any measurement performed on particle 1 w i l l be
uncorrelated wi th any measurement performed on any other particle, i n
particular, any measurement performed on particle 2 which occurs after V's
measurement. For example, for XI = S1,, Y2 = Sa, V expects:
< SlZ s2z > - < SIZ > < Sa > - 0
But this i s contrary to our ini t ial assumption, where:
and hence this scenario i s inconsistent wi th quantum predictions from V's
frame of reference.
One can alternatively demonstrate the inconsistency of this scenario wi th
QM in the following way. Since the measurement at U and V are spacelike
separated, the time ordering of these two events dif fer in certain reference
frames. To satisfy the quantum correlations in both types of frames, particle 1
must simultaneously have probability equal to unity for a certain result for Sx
and S, (namely, the opposite results that were obtained at U and V). Hence, If
the perfect correlation 1s always kept, then this set up can be used as a state
preparation procedure for particle 1 for preparing definite values for two
non-commuting observables, which i s impossible within the quantum
formalism. Note that this i s not an EPR argument which attempts to show QM i s
incomplete. Rather, t h ~ s attempts to show QM 1s inconststent wi th Itself.
5 . 3 . 2 Standard measurement theorq
To demonstrate the apparent quantum predictions more clearly, we look at
the possible scenarios using standard measurement theory, where both the
quantum system and apparatus are included into the state vector. We shall use
the simplified model introduced previously. Both the position of particle 2 and
the deflection of the charged particle w i l l be treated as three valued. We see
that we now have four relevant variables:
1. Spin of particle 1
2. Spin of particle 2
3. Positionof particle 2
4. Position of test charge
First, le t us consider the case i n which the contoller at V decides not t o
create the test charges. Initially, we have:
that is, spin 1 and 2 are prepared in a singlet state, particle 2 i s undeflected,
and the positive test charge has not been created
Next, we use a SGA on particle 2 at R. We omit the unitary operator that
performs the transformation, and just write the new state operator:
and thus the position and spin of particle 2 are now correlated. Since the
controller at V does not create the test charges, we simply have recombination
at point T:
As stated earlier, the singlet state has the same form
tr iv ia l exercise to demonstrate this by expanding the
in any basis. It i s a
above in terms of
eigenvectors of S,, and S,. Hence this state vector i s equivalent to:
In this form, it Is easy to see that the quantum formalism predicts:
' SIz S2z > - f {(+
A more lnterestlng case occurs lr
st i l l have the same splitting at R:
1 - 1 + - l + l - - 1
v decides to create tne test cnarges. we
- - 0
as stated above.
This approach gives correct qualitative descriptions for non-relativistic
examples in QM However, it treats a l l interactions as instantaneous, and so
does not account for retardatlon affects, whlcb are so cruclal to the paradox
above. Indeed, if one were to consider a reference frame where the
measurement a t U occurs before the test charge creation at V, we would obtain
different predictions. (The above analysis for such a frame predicts that
Sl& ) = - 1 , and that the test charges would not even be deflected!). Thus
such an approach i s frame-dependent, and hence gives no additional support to
the accurateness of the paradox presented above.
This gedankenexperiment i s not necessarily meant to be experimentally
reallzable; It Is slmply questlonlng a prlnclple. Clearly I t s weakness 1s that It
assumes the fields created by quantum system can be measured to a certain
accuracy. Although classically there i s no l imi t to this degree of accuracy,
quantum mechanically there exist uncertainty relations similar to the ones
placed on material bodies.". Can the paradox be resolved by imposing the
restrictions of these uncertainty relations?
As stated above, there are reasons to believe such a measurement i s
possible:
1. The fields created can be made large for short path separations
2. We need only to measure the direction (sign) of the f ield for a
successful measurement, hence large errors in the magnitude of
the f leld can be accommodated.
Still, because of the implications involved, certainly a closer investigation
i s called for. Obviously the best thing to do i s to idealize the set up above and
see what QM predicts. Unfortunately, relativistic quantum mechanics involving
retarded llelds ts wel l beyond the ablllties of the author. Therefore the next
best alternative is to study more in depth the new measurement procedure and
try to find the most l ikely resolution. (If indeed a resolution exists!)
5 . 3 . 3 Pragmatic State Reduction
Although many assumptions may have been made which were not explicit in
the above analysis, it seems safe to state that anu action of V cannot affect the
results obtained at U and L without orovidina a method to send SLS.
Mathematically, this implies that, i f we associate a value X = 1 w i th V's
decision to create the test charges and X = - 1 wi th V'S decision not to, then it
seems necessary for SLS not to be sent that:
' (SlZ S2,) X ' - ' (SIZ S2z)' ' X > = 0
where S,,, 52, refer to the results obtained at L and U, respectively. That is,
the correlation between measurements S,, and S2, must not be affected by the
actions of V, since -V's actions occur at a spacelike separation from the
measurements of S,, and S2,. Note that although the particles are prepared in
the singlet state, we do not presume here that (S, , S2z> - - 1 as before. That i s
because we are now taking into account the possibility that coherence i s lost
upon the splitting and recombination of the amplitudes of particle 2, even i f no
material body interacts wi th it. This might be possible if one accounts for
interact ions wi th the background electromagnetic f ield that exists in vacuum.
A possible resolution of the paradox i s now offered. Perhaps under further
investligation it can be shown that, for the fields created by particle 2 to be
large enough so that the value S2x can be inferred from the measurement at V,
the coherence between the two amplitudes of particle 2 is mostly lost due to
interaction wi th background fields. Hence, upon recombination of amplitudes at
point T (see Fig. 13), particles 1 and 2 are no longer described by a singlet state.
Then the measurement at U becomes weakly correlated wi th the result at L, and
then no paradox exists. (We note here that if indeed coherence Is lost by the act
of path separation, then it i s lost reaardless of the actions of V i f SLS are not
to be sent. See above).
I f indeed the above resolution i s found to be valid, then it would seem
possfble to establish numerlcal values where 'State reductton tn the practlcal
sense' occurs. I n Von ~ e u m a n n ' p postulates of OM, he speaks of state
reduction where a pure state 'reduces' to a mixed state upon measurement.
Here we see that i f the coherence i s lost upon amplitude splitting, then for a l l
practical purposes, the system can be descrlbed by a m lxed state, and reduct ion
can be said to have occured. Note that this does not imply that 'state reduction'
is an actuality, for i f one considers the background electromagnetic f ield in
vacuum as a quantum system and includes it in one's calculations, then the time
develoDment of the total system of singlet state plus apparatus plus background
field can s t i l l be described by unitary operations, which transform pure states
into pure states. But since in practice we could not then retrieve the coherence
and 'undo' the splitting, practical state reduction can be said to have occured.
Let us note here that "pragmatic state reduction' i s an old concept when
dealing wi th measurement processes. Since we in general cannot undo a
rnacroscoplc measurement procedure, then once a quantum system has
interacted with an apparatus, pragmatic state reduction can be said to have
occured. But here the situation i s different- usually a SGA can be considered
as an 'external apparatuse as displayed by the ability to use a second SGA for
coherent recombtnation. Thus the state reduction spoken of above Is more
general in that it i s seen to occur when nothing interacts wi th the quantum
system (save the background electromagnetic field).