LOCAL CAUSALITY AND THE FOUNDATIONS OF QUANTUM MECHANICS

Robert S. Goldstein

B. Sc., University of I I1 inois, Urbana Champaign, 1 985

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THE REQUiREflENTS FOR THE DEGREE OF

MASTER OF SCIENCE

in the Department

of

Physics

@ Robert S. Goldstein 1987

SIMON FRASER UNIVERSITY

July 1987

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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

for their contributions 'off the field'.

Approval . . . . . . . . . . . . . . . . . . . . . . . . . i i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . i i i

Dedication . . . . . . . . . . . . . . . . . . . . . v

. . . . . . . . . . . . . . . . . . . . . Acknowledgement v i

. . . . . . . . . . . . . . . . . . . . . List of Figures x

I. INTRODUCTION . . . . . . . . . . . . . . . . . . .

I I. MOTIVATION

2 . 1 Preliminaries . . . . . . . . . . . . . . . . .

2 . 2 Simple Locality Argument . . . . . . . .

2 . 2 . 1 Local hidden variables versus quantum

BACKGROUND

Einstein -Podolsky -Rosen

Bell Inequalities . .

. . . 8

predictions 16

. . . . . . . . Paper

. . . . . . . . . . . .

3 . 2 . I Insuff lclencies of local hidden variable theorles

CHSH Inequality . . . . . . . . . . . .

Experimental Results

I V . CAN HUMANS SEND SUPERLUMINAL SIGNALS (SLS) ?

4 . 1 Problem w i t h SLS . . . . . . . . . . . . . .

4 . 1 . 1 ' K i 11 ing your grandfather' paradox . . . .

4 . 2 A Gedankenexperiment . . . . . . . . . . .

4 . 2 . 1 Qualitative description . . . . . . . . .

4 . 2 . 2 Fi rs t order interaction picture . . . .

4 . 3 Useful Approximation Technique . . . . . . . .

4 . 4 General Proof Against Sending SLS . . . . . . .

V . RETARDED FIELDS AND THE CONSISTENCY OF QM

5 . I Can One Measure a System Wlthout Affecting I t ? . . 56

5 - 2 Experiment 1 . . . . . . . . . . . . . . 59

5 .3 Experiment 2 . . . . . . . . . . . . . . 65

5 . 3 . 1 Possible results . . . . . . . . 67

5 . 3 . 2 Standard measurement theory . . . . . 70

5 . 3 . 3 Pragmatic state reduction . . . . . 75

VI . CONCLUSION

6 . 1 Are Quantum Systems Sending SLS Amongst Themselves? 79

6 . 2 A Model for Non-Local Correlations . . . . . 80

6 . 3 A Generalization to Quantum Theory . . . . . . 87

APPENDIX A

APPENDIX B

REFERENCES

LIST OF FIGURES

Flgure 1 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . Figure 2

. . . . . . . . . . . . . . . . . . . . . . . Figure 3

Figure 4 . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . Figure 5

Figure 6 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . Figure 7

. . . . . . . . . . . . . . . . . . . . . . . Figure 8

. . . . . . . . . . . . . . . . . . . . . . . Figure 9

. . . . . . . . . . . . . . . . . . . . . . . Figure 10

. . . . . . . . . . . . . . . . . . . . . . . Figure 1 1

. . . . . . . . . . . . . . . . . . . . . . . Figure 12

. . . . . . . . . . . . . . . . . . . . . . . Figure 13

I. INTRODUCTION

The subtle relationship between the predictions of quantum mechanics and

the demands of local causal i ty i s investigated. Local causal i ty, the requirement

that one event cannot be affected by another event whlch dfd not occur wlthln

I t s light cone, appearsnecessay for special relativity to be self-consistent. .

We motivate the investigation by using a simple thought experiment which

considers the restrictions placed on correlations when the following type of

locall ty IS Imposed:

the setting of an apparatus does not affect the results

obtained at another remote apparatus.

We find that quantum predictions imply correlations which are stronger than

allowed by this locality constraint. We then make the argument quantitative by

reproducing the orlglnal work by t3el11 whlch demonstrates that no local theory

can reproduce quantum correlations. A generalization of this argument due t o

Clauser, Home, Shimony, and ~ o l t ~ i s then given which demonstrates that

quantum theory itself is in some sense non-local.

Slnce these non-local correlations have been shown to exlst experimentally,

we ask whether it i s possible to use them is such a manner which allows

humans to communicate faster than the speed of light. A thought experiment

which at f irst seems to allow superluminal communication is resolved by

demanding that the time development of quantum systems be described by

unitary operations. Indeed, this resolution points the way to an argument which

demonstrates in genera1 the impossibility of superluminal information

transmission within the quantum formalism, and suggests a simple model that

i s useful for studying quantum mechanics at i t s foundations.

Next, we question the consistency of quantum mechanics at the relativistic

level when one considers measurement of two non-commuting observables, one

of which i s accomplished by a retarded field measurement. Even if, under

further investigation, the quantum formalism proves to be consistent, 'state

reduct ionm in a practical sense can be said to have occurred.

Flnally, we attempt to create a model which explains why non-local

correlations seem to exist despite the impossibility of superluminal signalling.

The belief that the quantum systems are sending superluminal signals amongst

themselves is abandoned in favor of a model which takes the view of the

quantum particles. Such a view seems to imply that one should be able to

describe a quantum system and i ts development from the 'reference frame' of

another quantum system.

11. MOTIVATION

We investigate here the relationship between local causality and the

predictions of quantum mechanics. The crucial point made in the f i rs t few

chapters 1s that, although the quantum formalism affords non-local

correlations, it i s impossible for information to be sent faster than light

through use of these correlations. To a person who believes in causality (i.e.,

that for a particular system to be affected, some entity must physically

Interact wl th It at the spacetlme polnts at whtch this system Is located), this

must surely look as though the quantum particles are conspiring agalnst their

observers-- sending superlum inal signals amongst themselves, yet not a1 lowing

these signals to be used by their observers. This point w i l l be discussed later.

Here, we attempt to motlvate the reader by displaying that quantum

correlations are 'stronger' than allowed by any local model.

2. 1 Preliminaries -

We attempt to demonstrate the 'problem' here by somewhat classical

reasoning. This w i l l also allow us to introduce certain techniques and

terminology that w i l l be used throughout the paper. Consider a neutral

composite particle wl th spin zero. I t splits into two neutral spin 1 /2 particles

4

which f l y away from each other. See Fig. 1. From classical conservation laws,

t = 6: t 3

0

t + t = t,:

spin 4 D

spin f

Fig. 1

we expect that whatever angular momentum we find for particle 1 (moving to

the left), we shall find the opposite result for particle 2 (moving to the rlght).

To measure the angular momentum of one of these systems, we shall put in i t s

path an electromagnet wi th a highly inhomogeneous magnetic field, known as a

Stem-Gerlach maratus (SGA). The SGA has no 6-component along Oy-- the

dlrectlon which the lncomlng particle lnlt lally travels. I t s largest component

i s along 02, ard it varies strongly wi th z as demonstrated by the change of

. density of field lines as a function of z i n Fig. 2. Of course, since div B = 0, B

must also have a component along Ox. But near x - 0, the path actual 1 y taken, Bx

can be made arbitrarily small.

Fig. 2

The SGA w i l l produce a deflection of the particles which to some extent can

be understood classically. Since the particles are neutral, they w i l l not be

subjected t o a Lorentz force. But since they do carry angular momentum, they

w i l l posses a magnetic moment M, the two we assume proportional by:

where p i s called the gyromagnetic ratio. The resulting force can be derived

from a potential:

. giving:

F = V ( f l . B )

The SGA w i l l also produce a torque:

r = M x B

and hence, from classical mechanics:

aS/at = PS x B

The particle would thus behave like a gyroscope. The angular momentum of the

system would turn about the magnetic field, the angle Q between S and B

remaining constant (see Fig. 3). The components of M perpendicular to 6 w i 11

Fig. 3

thus t lme average to zero (asswnlng the angular frequency 1s great enough to

complete many cycles while in the SGA). Hence, i n calculating the deflection,

- we can ignore rl, and % Thus.

F - VM, 6, - MzVB,

But since aB,/ay = 0 (0 is independent of y by construction) and aBz/ax = 0 for

the path of interest (due to symmetry considerations), we finally obtain:

F = Fz = Mz aszm

and hence the force applied can deflect the particle in only two possible

directions-- either toward the north or south pole. We see that the SGA

effectively couples the z-component of angular momentum of the particle to i t s

position (see Fig. 41, and thus measuring the position of the particle after it has

passed through the SGA (by use of some type of detector) glves us the

corresponding z-component of the particle's angular momentum.

t

spin f

Fig. 4

22 u A physicist friend prepares an experiment for us to do. He has a box which

spews out these partlcle patrs whlch tntttally came from a composite system

wi th spin zero. The two particles of each pair separate from each other, and

each heads toward a SGA which can rotate about the axis of the in i t ia l

trajectory of the systems. He calls the pair a 'singlet state' system. He asks

us to construct a model that describes the slnglet state from the analysis of

results obtained for different orientations of the two SGA

First, we tabulate from, say, the le f t side, the position of particle 1 after i t

has traversed the lef t SGA Classically, i f we picture the in i t ia l composite

system as a bomb wi th zero angular momentum and the two components as

fragments flying away from each other, we would expect the fragments to

leave the explosion wi th any possible S (and thus, MI. But we are told that the

fragments are indeed spin 1 /2 systems. We therefore observe that after many

trials, only two possible deflections are observed, and they occur w i th equal

statistical frequency.

Now, let us look for correlations between particles 1 and 2. Let us place

both SGA magnets at R = 0 ((I corresponds to the angle between the z-axis and

the north pole of the magnets). We flnd over a number of tr ials that, for a given

side, either of the two possible results occur wi th equal frequency, but within

a single trial, we always obtain opposite results for particles 1 and 2. Let us

associate a value S, - + 1 wi th a deflection toward the north pole, and a value

SZ = -1 wi th a deflection toward the south pole. Recall both SGA have their

north pole polntlng ln the +z dlrectlon. Hence, for each trlal:

S z Z = + I - or (-I)(+!) = -1

Thus for N trials:

Ma tha t i ca l l y , we wri te this a s

' s,zs2z > = -1

Now, as stated above, for a given side, S, = r 1 occur wi th equal frequency.

Thus:

I n general, for arbitrary observables P and Q, when < PQ > * < P >< Q > for a

given preparat Ion procedure, we say that the results are correlated. (Note that

this does not imply the converse of this statement i s true-- we w i l l come back

to this later). Once again this correlation can be understood classically from

the exploding bomb model and conservation of angular momentum. Since what

we choose to call the z- axis i s arbitrary, it must be that for any 0 - Q2 (i.e.,

both SGA measire the same component of spin), we find:

= - 1

After repeating the above experiment numerous times, and always obtaining

opposite results for particles 1 and 2, we feel i t prudent to state that any pair

prepared in the singlet state i n the future would be found to have this quality i f

lndeed we declded t o measure It. Let us emphasize that these measurements

occur far from each other, and so are apparently independent. That is, the act

of measuring one particle would not seem to affect the result obtained for the

other particle. But since we can predict the result of any chosen spin

component of, say, particle 2 by f lrst measuring that component of part lcle 1,

it follows that if we do indeed assume that the measurement of particle 1 in no

way affects the result obtained for particle 2, then the result that we obtain

for particle 2 upon measurement must alreadu be determined before the

Let us now try to develop a model which wi l l predict results for a third

test, when we l e t 0, differ from Rp Assuming predetermined results and

quantization, a slmple model presents Itself. Consider two circles whlch are

perpendicular and concentric to the line joining the two particles, as in Fig. 5.

Every point on each circle corresponds to a direction in which the two SGA can

Fig. 5

be oriented. With each point we associate a red dot i f SI (i - 1, 2) would have

been + 1 upon measurement, and a blue dot i f Si would have been - 1. From the

results of the second test above we see that, for any Q - Q l - Q2 (measured

from the z-axis), i f on circle 1 we have a red dot, then the corresponding point

on circle 2 must be a blue dot, and vice versa (for then S,,S,, = - 1 1. Also,

prompted by our classical calculation of the deflection, we expect that i f we

would have flipped the SGA (i.e., i f R 4 R + fl), then we would have obtained the

opposite result. Thus, i f on circle 1 at an angle rb we have a red dot, then on

clrcle 1 at angle (Q, + n) we must have a blue dot. This assumption also

guarantees that < S1 > = < S2 > = 0, for now there are necessarily as many red

dots as blue dots.

We have seen that when 0, - Q,, we have < SIS2 > = -1. Now let us guess

what w i l l happen when R2 = 0 + SR, with SR (: a. With any feeling for physical

phenomena, one would guess that < S I S 1 > = -1. This implies that we expect, if

O , corresponds to a blue dot, say, then most surrounding dots w l l l be blue.

Indeed, we see that f w small (Ql - 09, the stronaest correlation that can be

obtained is when circle 1 i s composed of a semicircle of blue dots and hence the

other semtcircle composed of red dots, wi th the opposite for circle 2. For now,

unless we are within 80 of one of the two transition points where red changes

to blue, we necessarily have opposite results (different colors) for particles 1

and 2 (and hence S ,Sq w i l l equal - 1 ). The values 0 and B + fl where the

transition from red to blue occurs i s le f t arbitrary.

With this model, It is easy to predict how (. S ,S2 > is related to !J - R2 =

AQ. Since clrcle 1 and circle 2 are just opposites, one of them carries no

additional information, and can theref ore be neglected. Thus consider one

circle composed of a red and blue semicircle. On this circle are two 'pointers',

each signalling the orientation of one of the SGA We associate a (- 1 ) for S ,S2

whenever the pointers point to the same color, and a + 1 whenever they point to

different colors. We sum over al l possibilities for a given U2, - S12) by

rotating the pointers through 2 ~ f , keeping ( Q l - a2) constant. Obviously for

(0, - Rp) = A n - 0, we w i l l always have both pointers situated on the same

color, and thus < S1S2 > = -1. Also note that the result obtained w i l l be

independent of the value of 6, 6 + rf where the color transition occurs.

Consider 'starting' D I at a transition point. See Fig. 6. Through a rotation of

Fig. 6

tn -, AS21 we have < S1S2) = -1, for the pointers are on the same color. For the

next rotation of AD, we have < SIS2 > - + 1, since the pointers are situated on

different colors. For the third rotation of - An) we have < S,S2 > = - 1

(same color). Finally, for the last rotation through An, we have ( SIS2 > = + 1

(different color). Thus for any AR (defined to be i n), we have:

< S > = ( 1 /2fi) {(- 1 )(rf - An) + (+ 1 )(An) + (- 1 N f l - An) + (+ 1 )(An))

Thus according to this model, < S ,S2 > approaches - 1 1 inearly as An approaches

zero.

We note the attributes of this red-blue circle model: We see that it

describes quantization (only two results, red - +I, blue - -1). It also has < Sl >

= < S2 > = 0, and for AQ = 0, < S I S 2 > = - 1 . Finally, for A n 4 0. < S 1 S 2 > 4 - 1 .

Let us state the fmplicit assumptions made here (which we know to be relevant

from hindsight):

1. We are assuming that our particles, after the explosion, leave wi th

definite values for al l possible spin measurements we may in the

future decide to measure. That is, once the particles have long

separated, the corresponding color of dots is fixed. Indeed, i f they

could change randomly, then for AQ = 0, we would find < S1S2 >

not equal to - 1, In conf i ic t wi th experiment.

2. We assume that since the measurements occur far from each other

that the measurement of system 1 has no effect on the result

obtained wi th system 2, and vice versa.

Assumptions 1 and 2 are similar in that they are based upon some type of

locality-- the separation between systems 1 and 2 makes the assumptions very

plausible. Again let us emphasize that this model gives us the strongest

correlations possible (for small AR, anyway) i f we assume the systems leave

with definite values of spin. f3y strongest correlation we mean that I < S ,S2 > I

1 - 1 + 2AR/n I i s the maximum possible value consistent wi th the above

asswnptlons. Hence we expect the actual measured correlations not t o exceed:

(We can describe any correlation weaker than the above by considering any

other conflguratlon of red and blue dots).

2 . 2 . 1 Local hidden variables versus auanturn re dictions

To our surprise, we find for the singlet state system, that for any An:

<s1s2> - -COS(AR) - I + ~ ( A R P

(which i s the quantum prediction, as shown in appendix A). The observed

correlations are thus stronger than allowed 'classically' and hence in conflict

wi th the assumptions made. Indeed, we see that for small AR, < S1S2 >

increases as f (AQ)~, as opposed to the linear increase predicted by our local

17

model. This i s shown graphically in figure 7. We see that our classical model

Fig. 7

can only explain the corretatlons for R = 0, fl12, and fl. Thus we conclude that,

for our singlet state system, at least one of the assumptions:

1. The particles leave with definite spin values just waiting to be

measured.

2. Orientation of one SGA has no effect on what result is obtained at

the other remote SGA.

appears to be incorrect. (Actually, there are other assumptions made here,

such as human free w i l l to orient the SGA as one desires, but these questions

w i l l be mainly glossed over throughout the paper. But see, for example, Peres

and Zurek3).

I n the quantum formalism, we can prepare a system so that at most only one

point on our circle (and i t s corresponding point on the opposite side) can be

attributed a predetermined value. Assuming that al l the other points have

delinlte results just waltlng to be measured is to believe, within the

framework of QM, in 'hidden variables'. Thus we have shown that any local

hidden variable theory (local meaning that events which occur in space-like

separated regions cannot affect each other) cannot explain quantum

correlations. TNS result was llrst obtained by ell'. Later, we shall show

that a more general argument can be given which situates QM itself against

some type of locality.

I1 I. BACKGROUND

3.1 EPR P a ~ e r -

The f i rs t to contrast the implications of QM and locallty were Einstein,

Podolsky, and i?osen4. I n their landmark paper, the authors attempted to show

that QM was incomplete. To demonstrate their point, they used the

noncommut ing observables of posit ion and momentum. We shall here continue

to use the singlet state example because of i t s simplfcity. This was f i rs t done

by ~ o h r n ~ .

The singlet state can be written (ignoring spatial variables):

or more symbolical 1 y:

We are using the spin-z representation. One can prove that the singlet state

has the same form in any basis, thus demonstrating that for any 0, = Q2 we

wlll obtain opposite results from a spin measurement. For example, in the

spin-x represent at ion:

where, for Instance:

The EPR argument, using the singlet state as an example, proceeds as

follows:

1. For a physical theoy to be considered comp/ete a necessary

requirement is that, "every element of physical reallty must have a

counterpart in the physical theoy".

Now, what one defines as an e/ement o f ,&pica/ rea/ity is somewhat

arbitray, but a condition sufficient enough for the present Purpose, and

extremely reasonable, Is that:

2. "If, without in any way disturbing a system, we can predict with

certainty (i.e., with probability equal to unity) the value of a

physical quantity, then there exists an element of physical reality

corresponding to this physical quan t l ty'.

Consider the singlet state experiment discussed earlier. As the two

particles separate, they no longer interact physical 1 y. We theref ore assume

that what we decide to measure. say, of system 1. whether it be Slxor S1,. in

no way affects the physical situation of system 8. We see though that i f we

decide to measure S then the z-component of spin for system 2 becomes an

element of physical reality, namely the opposite value that was found for

sys tern 1. On the other hand, i f we measure S , , , then the x-component of spin 2

becomes an element of reality. But, the quantum formalism cannot describe a

system which has definite values for two non-commuting operators, such as SZ

and S,. Thus, if we assume:

1. The direction in which we decide to measure system 1 in no way

affects what result we obtain for system 2.

then it follows that:

2. Quantum theory is incomplete.

When one reads the original article, one gets the feeling that assumption 1

was taken rather for granted. Indeed, even the rebuttal by ~ o h F did not contest

this assumption This assumption has support from classical causality-- there

exists no known mechanism to explain why what we measure on system 1 should

affect system 2. It also has support from special relativity, since the two

events (measurements) can be made to take place such that they are space-like

separated, and thus it would appear that a measurement of 1 affecting 2 would

imply su9erlumtnal communlcatlon-- that is, propagation of information faster

than light. We w i l l come back to this point later. Let us mention here that

Bohrms 'refutation' of the EPR paper was mostly irrelevant, and never really

addressed the EPR argument. He demonstrated the consistency of QM in the EPR

argument through use of the uncertalnty prlnctple, but hardly defended the

com~leteness of the theory, which was questioned by EPR.

3 . 2 BellIneaualities -

The questlon of whether 'hldden variables' m exist wlthin the framework

of OM i s as old as QM itself. Could it be that there exists other parameters not

included in the wavefunction which give purely determined results for any

experimental procedure, and it i s only our ignorance of the values of these

parameters which gives the appearance of non-causal l ty in the theory? That is,

could it be, for a given preparation procedure and measurement, that knowledge

of these hidden variables give definite results, and only when averaging over a l l

these possible values (with a suitable weighting factor), quantum predictions

are obtained? The f i rs t attempt to demonstrate the impossibility of hidden

variables was by Von ~eumann~. It appeared he had shown that any hidden

variable theory which gave definite results for a l l experiments could not be

compatible wi th the predtctions of QM But this 'proof' was later shown lacking

when 0ohm8 indeed created a hidden variable theory. Later ~ e 1 l9 showed where

Von Newnann's proof went wrong; he had assumed, for arbitrary observables P

and Q, that < P + Q > = < P > + < Q >. This assumption, which i s indeed true in

QM, was shown by Bell to be unduly restrictive for hldden variable theorles.

After criticizing a l l past arguments such as Von Neumann's, ell', using the EPR

paper as a basis, demonstrated the impossibility of any local hidden variable

theory which would reproduce quantum predict ions. (Bohm's model was

extremely non-local). By a local theory, we mean, for the example of our

singlet state system, that the setting of one SGA cannot affect the results

obtained at the other SGA Once again this seems very reasonable i n light of

special relativity. Actually, the 'proof' given earlier can be made as rigorous

as Be1 1's original, but for historical purposes, we reproduce Be1 1's argument

here. This w i l l also introduce mathematical arguments that w i l l be used when

we generalize this argument to conflict QM itself w i th locality.

3 .2. 1 Insufficiencies of local hidden variable theories

Let us assume, for the singlet state, that the orientation of the left SGA and

subsequent measurement of system 1 does not affect the results obtained at

another apparatus. Since we can predict in advance the result of any

measurement on system 2 by previously measuring the same component of spin

of system 1, it then follows that the results of any such measurement must

actually be predetermined.

Let X correspond to all possible local hidden variables, with @(XI the

corresponding (normalized) probability distribution:

lab pod- I

Let A and B correspond to the result up or down, respectively, for systems 1

and 2. A and B can only take on the values 21. Let L and R be the left and right

SGA, respectively. Finally, let a and b describe the orientation of L and R,

respectively. For any hidden variable theory, the expectation (mean) value of

the product of the two components a t . a and a l - b is:

But for the theory to be local, A must be independent of the setting of SGA 2,

thus independent of b, and B must be Independent of a:

The question is whether there exist variables X (and distribution ~ ( h ) ) such

that:

for all a and b, where - a - b i s the quantum prediction.

Now, P(b, b) must equal - 1 in order to reproduce quantum predictions. Hence,

we have:

A (b, A) = -B (b, X)

Thus:

P(a,b) = - J ~ x pod A(a,X) A(b,X)

Duplicating this equation for another orientation c, and then subtracting, we

get:

P(a,b) - P(a,c) = -121 p(N (A(~,x) A(b.1) - A(a,M ~ ( c , d

Since {A (b)} - t 1 , we have {A (b)j2 - 1. Thus:

because p(X) r 0 for all X, since it is a probability distribution.

Since A(d) = + I (for any dl, it follows that:

I P(a,b) - P(a,c) 1 r IdX pod { 1 - A(b,X) A(c,X) 1

Now, can the above inequality be compatible with P(d, e) = - d . e ? With a,

b, and c unit vectors, consider L ab = R, L ac = Q + 5Q, wi th 5Q 4 2fl. Then the

right hand side of the equation, for P(d, e) -- - d e, looks like:

1 cos R - cos (R + 6R) 1 = I cos R - (cos(R) cos(6R) - sin (R) sin (8R)) 1

"- I cos R - cos R + 6Sl. sfn Q I

"- 88a. sin D

That is, the RHS is linear in 6R for small 6R. But then, for the inequality t o

hold, P(b, c) must necessarily also be 1 inear for small 6R:

~ ( b , C) = - I + c l (sn) + c2(6ClF

But then P(b, c) cannot be the cosine function, since -cod L bc) i s stationary at

the minimum (C t - 0). Hence the inequality i s incompatible wi th local hidden

variables reproducing quantum predictions.

- - Bell then went on to show that neither could P(d, el, which equals P(d, e)

averaged over a small neighborhood about d and e, become arbitrarlly close to

quantum predictions. He showed that, for :

bounded by e, e must satisfy:

e 2 [ J 2 - 1 1 / 4

and thus e cannot be made arbitrarily small, implying local hidden variable

theories cannot reproduce quant um predict ions to an arbitrary accuracy.

3.. 3 CHSH Ineauali t y -

Guided by Bell's work, it was later shown that there was a far more general

conflict between quantum predictions and the principle of locality. We shall

derive here the CHSH~ inequality, which demonstrates that quantum predict ions

themselves violate the principle of locality. No mention of hidden variables

w i l l be made. The argument is somewhat similar to Bell's derivation.

Again consider the singlet state preparation. Both SGA L and R have a knob a

and b, respectively, each which can only be set i n 2 positions; a', a2, or bl, b2.

The events:

1. Setting of knob a and subsequent measurement

2. Setting of knob b and subsequent measurement

w i l l be spacelike separated. Thus it appears that special relat ivi ty would

impose the requirement that the result of measurement at L must be

independent of the setting of knob b, and vice versa.

Agatn, the results A and B at L and R, respecttvely, can be only 21. Let us

define:

< A ~ ~ B ~ ) = LimW-,,) ( / N ) A ~ ~ 6f I

where i, k (i, k = 1, 2) describe which setting at L, R was chosen, respectively.

Obviously, there are four possibilities. Now, for the ensuing derivation to

follow logically, one must assume that ~t makes sense to consider the results

of a future experiment for several different settings of knobs a and b, although

obviously only one setting (at most) can actually be chosen. That is, we assume

that any apparatus setting that we could have chosen would have produced some

result, regardless of what that result may have been. Now, this would not be

acceptable i f one believed that there was no free w i l l for the experimenter to

choose which apparatus setting he desired. But except for this possibi 1 i ty, the

assumption appears to be quite reasonable, and we shall therefore accept it.

We can now show that quantum predictions are incompatible with the

stat emen t, ' the result A is independent of the setting of knob d'.

Define for trial j:

(Note that the assumption discussed above is used here -- V j is composed of

four terms relating to the four different experimental arrangements). In

general, A = A (a, b). That is, our result at t, namely, A, could depend upon I f

both the settings of a and b. But using our locality assumption, we get A = j

~ ~ ( a ' ) . With this, we see the fourth t e n is just a product of the f irst 3 terms:

Using A*', B -k = 2 1: 1 1

Thus, in general Vj = + 4 +2, 0, -2, -4. But wi th the locality requirement, we

see that V1 = 24 i s impossible. (For Vj to equal +4, al l four terms must equal

+ 1. But i f the f i rs t three terms equal + 1, the fourth term, as shown above,

equals - 1 ). Therefore, we have:

It follows that:

For quantum predictions not to violate the locality assumption, the inequality

must be satisfied for a l l conceivable ale2, b1e2. However, if, for example:

a' = 45'. a2 = O', b' = 22.5', b2 - 67.5'

we obtain the contradiction:

242 < 2

Thus, quantum predictions are incompatible wi th our locality assumption.

The f i rst experiments to test local hidden variable theories versus OM were

done on photons prepared wi th correlated polarl~atlons'~. The preparation i s

accomplished by a two-photon cascade. A particle excited in a spin J = 0 state

'falls' to a lower energy J = 1 state by emission of a photon, and then 'fallse once

more to an even lower J = 0 state by emlsslon of a second photon. Since the

particle began and ended with zero angular momentum, conservation of angular

momentum necessitates (assuming no other interactions took place) that, i f one

photon traveling in, say, the y- direction, i s found to be polarized i n the 9-

direction, then the other photon, i f traveling in the y- direction, must be found

upon measurement to be polarized i n the (8 + 71/21 direction. The experiments

measured the pairs' polarizations at certain angles which theoretically would

produce the greatest difference between the OM and local hidden variable

theorles. Experiments using protons prepared in the slnglet state were also

done1 I . I t should be mentioned that additional assumptions are needed at the

experimental level to test Bell-type locality, such as assuming that the

efficiency of detectors used in the experiment i s independent of whether the

incoming particles have traversed a f i 1 te r or not. With these assumptions,

most experiments contradicted hidden variable predict ions in favor of the

quantum predictions, although some conflicting results did occur at f i r s t 12.

One simplification made i n these original experiments was that the SGAs or

polarizers were oriented from the start of each trial, so it was possible, at

least i n theory, that the remote apparatuses could communicate wi th each

other sublwninally, and thus reproduce the quantum correlations. Hence, a

superior set up was needed where the orientation of the analyzers or SGA

would change while the correlated pair was in flight, thus removing this

possibil ityl? With improvements in the design of these experiments over the

past decade, t t ts safe to say that a i l now agree that quantum predictions have

been found correct over the local hidden variable theory predictions.

1V. CAN HUMANS SEND SUPERLUMINAL SIGNALS (SLS)

4 1 ProbtemsWithSLS L

We have shown that In some sense quantum mechanlcs Is now local In that

the setting of an apparatus seems to affect what i s measured at another

remote apparatus. These correlations give the appearance that the particles

composing the singlet state, or the two SGA's (or both), are in communication

wl th each other, w l th lnformatlon belng sent at effectively lnflnlte speeds. We

call signals traveling faster than c 'superluminal signals (SLS), and particles

which have a velocity greater than c 'tachyons'. Two quest ions come to mind:

1. Are the particles (or apparatuses) sending SLS amongst

themselves?

2. Can humans somehow use these non-local correlations to

communicate superluminally wi th other humans?

It appears that i f *2 i s answered in the negative, then * 1 becomes mainly a

questlon of lnterpretatlon and phllosophy, slnce It seems that # I would then

have no physical consequences. We shall come back to # 1 much later-- le t us

now focus on '2. But before we attempt to answer it, let us look at the

consequences involved i f indeed it was possible for humans to communicate

superluminally '?

4 . 1 . 1 'Kill h a uour orandfather' D ~ ~ O X

Again we have our singlet state preparation machine. At time t = 0,

observers L and R leave the machine, traveling at u = 2 (113) c (with respect to

the preparation machine). From t = 0 to t 4 1 (arbitrary units), the machine

spews out several singlet state systems, a l l nicely aligned so that the particle

furthest to the right is correlated wi th the particle furthest t o the left, the

second to the right correlated wi th the second to the left, etc.. These two

pulses of particles are sent out at a speed which i s infinitesimally greater than

that of L and R, so they for the most part travel with L and R, and thus L and R

see a pulse slowly pass themselves. Now, we are assuming here that there

exists some type of apparatus that uses the non-local correlations to send SLS.

This apparatus has the ability to change what the other observer detects, and

hence this observer would know whether the SLS transmitter was turned on or

not. We give to L and R both a transmitter and a SGA to detect whether the

other's transmitter was used.

Experiment 1: With u = + ( 1 /3)c, we find by the law of addition of velocities:

Vm,=KI/3)c+ (1/3)cJ/(l +[(1/3)c.(1/3)cl/c2) = (3/5)c

where V,, i s the velocity of R wi th respect to L. We tel l L to 'transmit* at time

t = 20 (wi th respect to himself). In his own reference frame, he sees the signal

sent wi th infinite velocity (which i s necessary for quantum predictions to be

satisfied in his frame). That Is, whatever he dld to one particle In the singlet

state, i t s correlated particle i s immediately affected. Thus for him:

event 1 = system 1 acted upon by him -- t 1 = 2 0 X1=O

We use a Lorentz transformation to find how R observes these events:

ti' = 'l [t, + V,, . xi/c21 x i = a f xi + v , . ti 1

where 3 - 1 - (v,,/c~ 1-4, which in our case equals 5/4. Thus:

event 1 ' -- t lo 25 X i = - 1 5 ~

event 2' - - t2' - 16 X2' - 0 Thus R 'sees0 event 1, the cause of event 2, occur at time t' = [25 - 16 ] = 9 time

units after event 2. Although unusual, no paradox occurs here.

Experiment 11. We tel l R that, i f indeed he receives a slgna? at t' = 16 (wi th

respect to his reference frame), then he should send his own signal at t' = 20.

This command creates a symmetrical set up with respect to R and L. Even

though R and L disagree about the order of events, s t i l l no paradox exists.

Experiment 111. Now, we tell L to transmit a signal (at t = 20) i f and only if

he receives no signal at t = 16. Let us assume that no signal was present. Then

L at t = 20 transmits. R receives this at t' = 16, and transmits his own signal at

to = 20. But then L indeed receives a signal at t = 16, contrary to assumption.

Conversely, assume L does receive a signal at t - 16. Then he does not transmit

at t = 20. But then R does not recelve at t' = 16, and thus, as Instructed, does

not transmit himself. But then L does not receive a signal, contrary to

assumption. Hence either scenario is inconsistent -- a paradox is apparent 1 y

created i f it is possible for humans to communicate superluminally as above

and special relatlvtty predictions are correct. ~ h t s *ktlltng your grandfather as

a boy' paradox makes it very difficult to believe in superluminal signaling,

which thus illustrates the significance of Bell-type arguments.

Later we shall show that human superluminal signaling is impossible,

despite the existence of non-local correlations.

4 . 2 - A Gedankenex~eri ment

Let us now look at a particular arrangement which does seem to permit SLS.

The exposure of the reason for i t s failure w i l l point the way to a general

argument demonstrating the impossibility of using non-local correlations to

send SLS.

Consider a neutral particle prepared in the aZ - + I state, and let i t pass

through a SGA oriented i n the X-direction. There are two possible paths

(associated wi th the two posslble eigenvalues ax = 21) which the particle can

take. Note that so far no measurement has taken place. Indeed, it can be

shown15 rather easily using propagators that these two amplitudes can be

recombined to reproduce the init ial oZ = 4 state. I n fact, similar processes

have actually been done experimentally using neutron interferometryi6, where

two amplitudes have been separated by a few centimeters and recombined

coherently. Can we somehow use this to send SLS?

4 . 2 . 1 Qualitative descri~t ion

Let us prepare two particles in a spin singlet state. The combined state

vector has the form:

*trlJrzJoJt=o) = s,trl) ~,tr,) h 1 r 2 - rlt21

where the spatial wavefunctions can be thought of as gaussians symmetric

about r,, r2. The particles separate in the Y-direction (at speeds small

compared wlth c). Much later, particle 2 interacts with a 2-oriented SGA I f

we neglect the 'spreading of the wavepacket*, we have, approximately:

Note that the z-coordinate of particle 2 is now correlated wlth i ts spin. Before

we recombine these two amplitudes, l e t us put a spin flipping device in one of

the two possible paths of particle 2 (see Fig. 8). Because the two paths can be

separated as much as desired, the spin flipper can be made to have no effect on

spin f l ip

Fig. 8

the other path. For example, a suitable potential energy term that produces the

spin f l ip in the desired manner is:

Win, = - (1/2) gBazxA for v t ' i y i vt': 2 =: - 6~

= 0 otherwise

Then in the impulse approximation (thus neglecting the kinetic energy term):

*(r rib a, t= t9 = exp{(-i/N Iwlnt- dt ) *(r,, r,, a, t= t')

We shall set the apparatus so that pB(t" - t*) = n. We see that:

and thus we find:

- @, (xl ,yl -vt.,zl )[ iQ 2 ( ~ 2 , y 2 - ~ t a ; ~ 2 + ~ ~ ) t , - @ 2 ( ~ 2 , Y 2 - v t ~ : ~ 2 - ~ ~ ) ~ , I B t

Note that the second spin state vector now comes off as a common factor. Next,

we recombine the beams from the two paths:

And upon a partial trace over the variables of particle 2, we find that particle I

i s described by a pure state ty , as opposed to the mixed state that described i t

initially, where it had a 0.5 probability of being found in the ty state. Thus

someone measuring the spin of particle 1 In the y- dlrectlon can te l l whether

the spin flipping device was applied to particle 2 by whether he finds (for an

ensemble of singlet state systems) < S , > to be + 1 (spin flipper was applied to

partlcle 2) or zero (spln fllpper was not applied to particle 2). Actually,

instead of comparing whether the spin flipping device was turned on or not, a

better experiment is to te l l the controller of the spin flipper to use the

apparatus, but decide whether to use a positive or negative magnetic field. (For

a negative magnetic field, we have pB (t" - t') = -n, hence $(o) = I-it - r> e It>,

and thus @,(dl = ly 1. Then the probabilities of finding spin t for particle 1 are Y

unity and zero, instead of unity and 0.5 above. (And hence the two possible

values of < S > are + 1 and - 1, instead of + 1 and 0 as above). Thus it appears

that such a device would even allow humans to communicate superluminally

using just one spin singlet pair. For, i f someone measures S and finds ty, 1 Y

then he knows the controller of the magnetic f ield (who i s situated far away,

near particle 2) used a positive B-f ield, and vice versa i f he finds ly.

4 . 2 . 2 Interaction ~ i c t u r e

The argument glven above was rather schematlc, so l e t us lmprove It here.

As i s well known, few problems in quantum mechanics can be solved exactly,

and this one i s no exception. We must resort to some type of perturbation

technique. Since we are most interested i n when the particle interacts wi th

the additional apparatus (composed of the lni t fa1 SGA, spin-f l ipper, and

recombining SGA), le t us use the interact ion picture. We have:

H = Ho + W, where H, = p/2m

i f i d l d t {Uo) = HoUo Uo=exp{-iH,(t-t,)/(i)

I Y1 (t), = u0' (t, to) I PS (t),

where PI refers to the interaction picture, qS refers to the Schrodinger

picture. We get:

where:

WI (t) = u,t (t, to) W$t) Uo (t, to)

The relevant potentials are:

WS (SGA) = z 8' oz WI (SGA) = exp{ ip2 t/2m5) zB1aZ exp {-ip2 t/2mfi)

- whereBedescribesthegradientofthemagneticfieldinsidetheSGA,and:

WS (Spin flipper ) = Bo a, + WI (Spin flipper ) = Bo a,

To simplify the mathematics, we only used the f i r s t term of the expansion:

191 (tb = ( I + ( l / i % ) 1 dt' w1 (t.1 1 1 9 ~ (to)>

To get the desired result, one uses four different field gradients to separate

and recombine. See Fig. 9. For some distance d we use a splitting potential; for

I - I I - - 1 1 1 - I v - v B- 8, B=%, flip? B=%, B- Bo

Fig. 9

the next distance d we use the same magnitude but opposite combining gradient.

We then use our spin flipper. A combining potential followed by a repulsive

potential brings the paths back together. We omit the calculation and just state

herethatthe(average)pathsofthesystemcouldbeunderstoodclasslcally,and

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).

VI. CONCLUSION

6 . 1 Are Quantum Sustems Sendina SLS amonast Themselves? -

We exclude from further investigation the relativistic paradox just

presented. The rest of the paper can be summed up in one sentence: we have

shown that non-local correlations exist, although these correlations cannot be

used for superluminal signalling. Now, should we picture the particles of the

singlet state as sending SLS amongst themselves? There are (at least) two

difficulties with this approach:

1. Such a model would imply a 'conspiracye among the quantum

systems-- they can send informat ion superluminal ly amongst

themselves, but do not allow their observers to do the same.

Such an argument against a superluminal model is hardly physical, but the

'beauty' (or lack, of beauty) of a model has often been used to question i t s

validity.

2. Even i f we accept the conspiracy, for a theory involving SLS

and QM to be Lorentz covariant, an infinite number of signals would

need to be present so that in every reference frame there would at

least one signal traveling with infinite velocity (as required t o

explain the non-local corre1ations in al l inertial reference

frames).

Because of such problems, such an approach does not appear to be fruitful.

But i f we reject the possibility that the particles are communicating

superluminally, how do we explain the non-local correlations? Of course we

can ignore the problem, and be satisf led that the foMnallSmS of both QM and SR

are consistent, since SLS cannot be sent. (Indeed this approach seems to be the

one preferred by most physicists). Rather, let us look for a model that would

explain how non-local correlations can exist without SLS being transmitted,

and yet seems 'beautiful enough' t o be a possible solution. There does seem t o be

one model available, and i t s foundations have support from the tenets of

relativity theory.

6 2 A Model For Non I ocal ~o r re l a t lons -

We assume two postulates:

1. To any particle of non-zero rest mass there corresponds a

'reference frame' whose origin of phase space is defined by the

particle.

Simply stated, the particle 'sees' itself as having zero velocity and a t the

origin of the universe. This reference frame allows the particle to quantify

spatial and temporal relationships between itself and other particles (e.g.,

system 2 i s ten meters in the +X direction with respect to the particle). This

postulate has strong support from relativity theory, which further implies that

all such reference frames are 'as good as' any other (although particles acted

upon by forces may necessitate a more complex mathematics to describe events

wi th respect to their reference frame).

2. A wavefunction describes spatial and temporal relationships of an

individual microscopic particle with respect to a particular

reference frame. (As opposed to describing an ensemble of /

similarly prepared systems23. 1

This postulate has i t s origins In the Copenhagen Interpretation of quantum

mechanics, but it is free from any subjective qualities that the originators may

have proposed. We see that the human consciousness plays no role here. Indeed,

the wavefunction is regarded as an objective reality which describes

relationships wlth respect to any (classical) frame of reference.

We attempt to demonstrate the consequences of these two postulates

without invoking any other assumptions. For now, we are wil l ing to renounce

previous notions of the structure of spacetime as much as necessary for the

two assumptions to be consistent, and then we shall study these consequences.

Let us consider the reference frame of some macroscopic particle (e.g., a

spec of dust). We call it particle 1 (or just '1'). For simplicity, we only

consider one dimension. Consider a quantum system (particle 2). With respect

to particle 1, particle 2 is (approximately) described by the wavefunction:

Wx) = exp ( [x - v t - d?/b2 ) + exp { [x + v t - d?/b2 1 , d > 0

that is, two gaussians traveling in opposite directions. We let b be small i n

some sense so that the two amplitudes are extremely well localized. We see

that just before time t = d/v, particle 1 w i l l associate two amplitudes wi th the

quantum part lcle; one near Itself, and one near x = 2d.

Now we ask ourselves, how does the quantum particle 2 view the situation?

From postulate * 1, we see that it sees itself defining the origin of i t s phase

space. (Thus it cannot associate two amplitudes with itself). But then how does

the quantum partlcle vlew the spec of dust? A l l t t l e conternplatlon shows that

the only possible way the two postulates may not be contradictory i s i f the

quantum particle associates with particle 1 two amplitudes; one near the

origin, and the other near x = -2d. Clearly this i s the only interpretation which

Is covarlant wl th respect t o the two vlews.

Now, any 'classical' reference frame (i.e., the ones envisioned by SR) would

associate with the macroscopic particle 1 a unique position andvelocity. Hence

the view attributed to the quantum system describes a new type of reference

frame (hence the term 'generalized' reference frame) which can be seen to be a

1 inear combinat ion of 'classical' reference frames.

Thus, macroscopic systems describe microscopic systems in terms of

amp1 i tudes, and microscopic systems describe macroscopic systems i n terms

of amplitudes. The reciprocity needed for consistency i s clear, but one may ask

the usefulness of such an approach. First let us emphasize that such a

concluslon seems t o follow necessarily from the postulates. Thus t o crit lclze

the conclusion i s to criticize the postulates. But before we crucify the

approach, let us see i f i t has any useful insight.

We return to the example given above. From particle 1 's view, at time

t = d/v, one of the amplitudes of particle 2 w i l l be colncident wi th Itself. At

this time either particle 2 w i l l interact wi th 1 or it w i l l not. I f 2 does

interact wi th 1, then instantly, from 1's point of view, the remote amplitude of

2 near x - 2d disappears. The quantum system is now described by a

wavefunction which has nm-zero values only near the origin (with respect t o

1 1. Thus the macroscopic particle 1 observes non-local correlations between:

1. the interaction of itself wi th one amplitude of particle 2,

and:

2. the existence of the remote amplitude of particle 2.

One can see how I may conclude that the quantum system is sending a SLS

between itself.

Let us now consider the view of the quantum system 2. A symmetrical

situation i s seen to occur. Just before time t' = d/v, the quantum system 2

associates wi th particle 1 amplitudes about x' = 0 and x' - -2d. I f at t' - d/v the

systems interact, then the quantum system sees the amplitude of particle 1 a t

xe = -2d suddenly disappear. Thus the quantum system views the amplitudes of

macroscopic particle 1 as demonstrating non local correlations, and considers

the possibility that the particle 1 may be sending SLS between itself.

The question now becomes, which particle is the a source of the non-local

correlations, and which ( i f any) i s signalling superluminally? The answer to

this i s found in a simple analogy with a thought experiment from special

relativity.

From relativity theory, we have learned to distinguish between, 'what is'

and, 'what i s wi th respect to a certain view of the situation'. For example,

consider two persons moving with respect to each other, and whose hearts beat

at the rate of 1 beat/sec with respect to themselves. From relativity theory,

we know that person 1 w i 11 see person 2 as aging slower, and that person 2 w i 11

see person 1 as aging slower. Such statements would have been contradictory

before the advent of relativity, since the time axis was considered to be the

same for all inertial frames. But to explain certain physical phenomena, we

have learned to accept a model of spacetime which differs from the one we

seem to experience.

A similar approach may be used here. We see that the question, 'who is

reallu causing the non-local correlations is meaningless. Such a question only

can be answered wi th respect to a particular frame of reference. But as we

see, no one is sendina SLS. Indeed, each particle sees itself as having a unique

posl tion In i ts phase space (namely, the orlgin), and thus each particle cannot be

sending SLS between itself since, with respect to itself, there is no second

amplitude to communicate with!

Compare the above dnalysis with how an observer sees one particle of a spin

slnglet palr when the particle's partner has i t s spin measured ln a direction

chosen by a second person. Before measurement of S2, (D i s the orientation

chosen by person 2) upon i t s partner, a human observer associates with particle

1 two possible values of spin i n the 0- direction. But after measurement, only

one value remains. Thus, how particle 1 i s perceived by the observer i s

immediately affected by a measurement at a distant location.

But let us now take the particle's view. It sees i tself as always having a

spin opposite to that of i t s ~ar tne?~, and hence nothing has changed upon

measurement. That is, according to particle 1, no information i s sent between

himself and particle 2 during the measurement process-- they le f t pointing i n

opposite directions, and that his how they remained.

But now one may ask, 'which direction was it that they were pointing to?'

Quickly one can convince oneself that i t cannot be any 'direction' which

corresponds to a direction (i.e., a unit vector) on a macroscopic view, for then

this would correspond t o a local hidden varlable model, which has been shown

to be incapable of reproducing quantum predict ions. Rather, such a 'direct ion'

must look to the macroscopic observer as an image of many directions, and only

upon measurement does a single direction remain.

Hence we see that we do not have to assume the existence of tachyons to

explain non-local correlations. Rat her, it seems preferable to view our

description of quantum systems, not as absolute, but only relative to our frame

of reference. Upon measurement of the distant particle, no tachyon i s sent

between quantum particles. Rather, the relative orientation between the

observer and the quantum system i s changed.

6 . 3 A Generalization of the Quantum Theory -

Does this model have any implications for the advancement of of the field, or

i s it just a cute way to visualize the problem? There does seem to be one

implication-- i f one can 'legally' consider the view of the quantum system, then

it should in general be possible to describe the state of one quantum system

(and i t s time development) from the frame of reference of another quantum

system. The mathematics involved to take such a view may be more dif f icult

(just as in relativity an accelerated frame of reference i s much more dif f icult

mathematically to use that of an inertial frame). But such an approach may

greatly generalize the quantum formal ism. Indeed, such an approach was

attempted by the author. We tried to describe one spin zero quantum system

with respect to another by use of wavefunctions, but the model was

inconsistent. (We found W x ) not to equal the Fourier transform of Wp)). Indeed

such an approach was rather naive; there is no reason to believe that such a

generalization to the quantum formalism should f i t neatly into the present

88

formalism. But at least here is a path which may be built upon.

APPENDIX A

We demonstrate here that, for two particles prepared in the singlet state

with their respective SGA's oriented with an angle AD between them, the

quantum prediction for < S1 S2) is:

This i s most easily done by using the direct product formalism. There, the

singlet state takes the form:

For simplicity, we shall define the axes so that the SGA acting upon particle

1 i s oriented in the z- direction (9 = 01, and the 5GA acting upon particle 2 is

oriented in the Oxz plane (@ = 0). We therefore are measuring S , zand Span.

Now, in general:

Therefore, we have:

Thus:

Which implies that:

as stated in the text.

APPENDIX B

I f a combined two- particle system is in a pure state, then i t can be writ ten

in the fwm:

where the ai and the b' each independently form an orthonormal set. I f Cil can

be written as y x v, . then the two particles are uncorrelated. By uncorrelated,

we mean that ph = I qab> < qrbl = pa @ pb. Let us demonstrate this. For the

@ glven above, we have for p*:

p* = Z Z uiv I urnvI.' I aJ e bJ ) < a j 'e bl'I i.1 i9,l'

Now, pa is defined by:

but 2, v, v,' must equal unity for !Pb to be normalized. Thus:

pa = 2 .U iU i . * I ai > < ai' 1 i, I

By a similar argument, we find:

pb = 2 v v * l b ' > < b r l 1.r 1

and thus we see that:

as promised.

Now, in general, two partlcle system can be descrlbed by the state

operat or:

as is necessary for pab to be a state operator. Now we obtain for pa:

Now, i f we assume that particle a is in a pure state, then we can choose the

basls I a > such that pa has only one non-zero term, equal to unity, and on the

diagonal:

pa l1 = 1 , all other matrix elements equal zero

Since, in general:

we have here:

But with pab a state operator, i ts trace equals unity:

Tr pb = 1 implles 2 2 P I C, Ik2 = I k n.m k

With ICm12 non-negative definite, we see from comparing the last two

equations that necessarily C, = 0 I f n 1 . But then all the Cm's can be wrltten

as u, x v, or zero. Thus, knowing that particle a is in a pure state necessitates

that it be uncorrelated with all other systems.

95

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