"Elements of Physical Reality"

                     ______________________________

 

                          Timothy Paul Smith

                          

                         Department of Physics

                      University of New Hampshire

                    Durham, New Hampshire 03824 USA



                           tim.smith@unh.edu



                       (received: July 19, 1995)

             (first printed - Vol. 2, No. 2 - August 1995)

 

 

I. What is so hard to believe in Quantum Mechanics?

___________________________________________________

 

	James Chen, my undergraduate advisor, in a lecture which 

introduced me to modern physics, to relativity and quantum mechanics, 

listed many of the peculiar of these two theories.  The dual nature 

of quanta (both wave-like and particle-like). The mixture of mass and 

energy. The constant speed of light in all reference frame.  The set 

orbits of atoms which can not degenerate and spiral in like planets.  He 

then asked, "Are all of these things too hard to believe?  Maybe not for 

you,  you are being raised with them.  But when they were new they 

seemed to defer reason to the scientist of the day.  So how come they 

are now all accepted?  We waited for the older scientist to die out."

 

	It is hard to believe that just being raised on a concept will 

mitigate all logically flaws of the system.  If the logical flaws do not 

alarm me, at lest I too should be able to see where quantum mechanics 

come in conflict with common sense and everyday experience.  I was 

indoctrinated into the non-quantum view of the world, having spent the 

first two decades of my life,  the most impressionable years,  essentially 

ignorant of quantum mechanics and its unusual properties.  



	So what are these properties? I will not pretend to be completely 

original or comprehensive.  One can cite wave-particle duality,  the 

uncertainty related to subsequent measurements, and so forth.  Perhaps 

these are all just different facets of the question "what in the world 

is this thing in quantum mechanics called a wavefunction?"

 

II. Einstein, Podolsky, Rosen (EPR) and Bohr

____________________________________________

 

	In 1935 Einstein, Podolsky and Rosen wrote their famous paper, 

usually called the `EPR' paper, where they turned the question around 

and said "We know what elements of physical reality are. But how does a 

wavefunction relate to that?"

 

	"*every element of the physical reality must have a

	counterpart in the physical theory*"

 

	"The elements of the physical reality cannot be

	determined by *a priori* philosophical considerations,

	but must be found by an appeal to results of experiments

	and measurements. A comprehensive definition of reality

	is,  however, unnecessary for our purpose. We shall be

	satisfied with the following criterion, which we regard

	as reasonable. *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 quantity.*" [1]

 

		(phrases in *asterisk* are in italic)

 

	What was it about quantum mechanics which bothered Einstein, 

Podolsky and Rosen to such a degree to drive them to write this paper

and try and define elements of physical reality?  When they looked at

the quantum mechanical wavefunction they saw its most famous

characteristic, the prediction of probability distributions.  A

wavefunction could tell you what the likelihood of the particles

position and momentum (for example),  but not exactly the position and

momentum.  EPR were of the opinion that there really was a definite

position and momentum of the particle,  despite the Heisenberg

uncertainty relationship (which EPR thought might only tell us about

what you could measure).  



	(uncertainty-in-position)*(uncertainty-in-momentum) >= h-bar



	Where does this uncertainty relationship come from?  It can be

derived from the commutation relationship of the position and momentum

operators.



	[X,P] = XP - PX = h-bar   



which means that if I take a system and measure [X] and then [P], I will

get a different answer then if I had measured [P] and the [X].  I get a

different answer for position and momentum depending on the order in which

I measure them,  *even if I disturb the system as little as I might

desire!*



	So here is the paradox which EPR proposed.  Imagine a system of 

two particle prepared in such a manner that their position and momentum 

are correlated.  Think of two particles starting back to back and moving 

out into a left and right detector. If I measure the momentum of the 

left particle, I can infer the momentum of right particle, because of 

the correlation.  Then what is to stop me from directly measuring the 

position of the right particle? Why should the measurement of momentum 

in the spatial removed left detector effect the results in my right 

detector?  Quantum mechanics says that for a system, be it one particle 

or two,  we can not measure position and momentum with precision, at the 

same time.  EPR concluded that of course you could do the measurement,  

if the two measurements are far enough apart. Therefore the quantum-mechanical 

two-particle wavefunction was in fact incomplete.  Each particle really 

did know its position and momentum.  These position and momentums which

are unmeasurable, but existed,  were later labeled `hidden variables'.

 

	I think that the real point of the paper was not the 

gedankenexperiment which they proposed,  but rather the paper was 

meant to send us on a quest to discover what the elements of physical 

reality are. (I have thought that we really should think of EPR = 

`Elements of Physical Reality',  perhaps a pun buried in the author 

list?). They didn't insist that position and momentum are `Elements 

of Physical Reality',  but used that as an example.

 

	Neil Bohr was ready to pounce on this (the EPR paper appeared 

in the May 15 issue of Physical Review,  Bohr's reply was received by 

Physical Review on July 13 [2]).  To be fair he had written his book 

"Atomic Theory and Description of Nature" the previous year,  and had 

already formulated his viewpoint as `complementarity'[3].  He claimed 

the the problem with the gedankenexperiment was a lack of understanding 

of experiments.  One can not separate the event and the apparatus or 

mechanism for measurement of the event.

 

	He described at great lengths how one measure position and 

momentum.  First we must remember that a particle is described by a 

wavefunction, or a DeBroglie wave-packet.  To measure its position we 

pass it through a slit.  If the slit is narrow,  the position is well 

defined,  but the wave's momentum is distributed.  If the slit is wide, 

the momentum is undistributed,  but the position is ill-defined.

 

 

	         |                            |

	         |

	~~~~~~~~> -------->         ~~~~~~~~~~~~~~~~~~~~>

	         |

	         |                            |

 

		(a)			     (b)

 

	Figure 1.  (a) A narrow slit gives a well defined

		position but disturbs the wavefunction.

		(b) A wide slit gives a poorly defined

		position but leaves the wavefunction

		undisturbs.

 

He then continues to tell us that a position measurement is made when 

the slit is solidly tied to some space-frame, let us say the top of the 

lab bench.  A momentum measurement is when the slit is free to absorbed 

momentum.  Now he reconstructs the EPR gedankenexperiment.  Two particles pass 

through a pair of slits, S(1,2).  This will give you their initial 

relative position.  To then measure the position of the second particle 

slit S(2) is ridgely mounted to the same space-frame (lab bench) as 

S(1,2).  Thus the position of particle 1 can be inferred.  To perform 

the momentum measurement on particle 2 the relative position of S(1,2) 

and S(1) is measured,  but S(1,2) is moved because of momentum absorbed 

from the measurement of position by S(2)! Thus a position measurement at 

S(2) as diagrammed below disturbs S(1,2) and precludes a momentum 

measurement at S(1)!

 

 

 

 

		|				| S(1)

		|				|

     1  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~------------>

		|				|

		|				|

		|

		|

		| S(1,2)			| S(2)

		|				|

     2  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~------------>

		|				|

		|===============================|

 

	Figure 2.  A measurement of position at S(2) disturbs

		S(1,2) and precludes a momentum measurement at S(1).

 

	Bohr's conclusion is that the gedankenexperiment of EPR was 

only a paradox because of the incomplete understanding of the role of 

the measurement apparatus.  Also without the paradox the need for hidden 

variables vanishes.

 

	Sometimes when I reread the Einstein-Bohr dialogue I think 

that they were talking right past each other.  I think Einstein believed 

that the particle must know its momentum and position without being 

measured,  but being practical, he cast the problem in terms of a 

gedankenexperiment.  Neils Bohr only saw the gedankenexperiment.  His 

reply is insightful,  and perhaps really meant to say was, `we could 

not know about the hidden variables which your talking'.  Instead, he 

comes across with a more powerful, `it is a question which we can not 

answer,  so we should not ask.  It is a non-question.'

 

 

III. David Bohm - A new paradigm[4]

___________________________________

 

	One of the problems with the gedankenexperiment we have used so 

far is that it is completely unmeasurable.  We can not do the experiments 

with the necessary precision.  David Bohm suggested a new experiment.  

He pointed out that Sx (the component of the spin-vector of an particle 

in the x-direction)  and Sy (component of spin in the y-direction) are 

non-commuting operators. That means that if we measure Sx with some 

precision, we have disturbed the system so much that we can not measure 

Sy with much precision.  Thus it qualifies as an EPR experiment, only

the uncertainties are potentially measurable.

 

	In fact the possibility of preparing a system of two particles 

with correlated spins is not hard,  three examples are;  1) photon 

cascade  (two step de-excitation) 2) electron-positron annihilation (two 

photons) 3) proton-proton scattering. In all three cases, the two 

photons or protons must have opposite polarization or spin to conserve 

the total spin of the system.  The essential point is that if a system 

of two particles are bound and in a singlet (spin-0 state) when they 

decay and fly off in opposite directions, they must conserve spin, so 

one will have spin +1/2 and the other spin -1/2.

 

	Conceptually we know how to measure Sx and Sy with with a 

`Stern-Gerlach' Device.  A Stern-Gerlach device is a set of magnets which 

separate spin +1/2 and spin -1/2 neutral particle.  Thus we need to 

build a Stern-Gerlach device on either side of the decaying bound-state,  

one with its axis pointed in the x-direction (to measure Sx), and the 

other with its axis pointed in the y-direction (to measure Sy). However, 

there was a problem.  There was no really `smoking gun' experiment.  

Quantum mechanics and hidden variables may still coincidently predict 

the same outcome.

 

IV. J. S. Bell and his `Inequality' [5]

_______________________________________

 

	Finally in 1964 John Stewart Bell moved the problem from 

gedanken- to a potential-experiment.  First he realized that what we can 

measure is the probability of finding a particle in a spin +1/2 (or 

-1/2) relative to the orientation of the axis of our Stern-Gerlach 

device.  The correlation which we want is the probability of measuring 

spin +1/2 in Stern-Gerlach-#1 which in oriented in direction [a], and 

measuring spin +1/2 in Stern-Gerlach-#2 which in oriented in direction 

[b].  We call this: P(a,b).  Now Bell showed that for any `local 

reality' theory, the following was true;

 

	|P(a,b)-P(a,c)| <= 1 + P(b,c)	.

 

This is commonly refereed to as `Bell Inequality'.

 

	So what is this `Local Reality' and what does it have to do 

with the EPR-paradox and test of quantum mechanics?  The details go 

something like this.  The probability P(a,b) can *always* be written as;

 

	P(a,b) = Integral[ AB(a,b,l) f(l) dl]  	.

				(this is always true)

 

that is, in general P(a,b) can be written as a function AB(a,b,l) of 

[a,b] (the orientation of the Stern-Gerlach device)  and any number of 

hidden-variables [l] times a distribution function  of the hidden-

variable [f(l)] , summed (integrated) over all [l].  This is compatible 

with quantum mechanics,  although it has a `hidden variable'.

 

	Now the `Locality Hypothesis'.  If the measurement at  

Stern-Gerlach-#1 only depends on the setting [a] and hidden variable(s) 

[l] (and likewise with #2),  then we could write:

 

	P(a,b) = Integral[ A(a,l) B(b,l) f(l) dl]	.

				(true for all `local reality' theory)

 

Given this, it is just algebra to derive `Bell Inequality'.

 

	Quantum Mechanics is not a local theory.  In quantum mechanics 

you can not separate AB(a,b,l) into A(a,l) B(b,l).  If the position and 

momentum are `encrypted' into the hidden-variable,  particle #1 does not 

have its own hidden-variable or its own position and momentum.

 

	Still, the quantum-mechanical prediction for David Bohm's 

experiment happens to satisfy the inequality *if a,b and b,c are 

separated by 90-degrees*  which is what Bohm had envisioned.  However if 

they are separated by (for example) 60-degrees,  the inequality is 

maximally violated.  This is the smoking gun experiment needed to 

test for `local realism'.

 

V Experiments

_____________

 

	Undoubtedly the most famous series of experiments to test for 

local realism was performed by Alain Aspect *et. al.* [6]. In it,  two 

photons with opposite spin are produced from a cascade decay of a 

calcium atom.  The photons fly off in opposite directions,  with a total 

polarization of zero.  Detectors (not Stern-Gerlach, rather polarization 

filters) measure the polarization of the two photons at macroscopic 

separated locations (see figure 3)

 

      detector Polarization    Calcium    Polarization    detector

        #1      filter #1       Source      filter #2       #2

 

         [ <---------|----------- Ca ---------|------------> ]

 

 

	Figure 3. Aspect's experiment.  Two correlated photons

		propagate in opposite directions.  They pass through,

		or are absorbed by the polarization filers, depending

		upon their polarization and the orientation of the

		filer (a,b or c).  They are then detected in detector

		#1 or #2.

 

	This experiment measures P(a,b) as the ratio of photon-pairs 

counted in #1-and-#2 with polarization filters set at a-and-b,  to all 

photon-pairs.  The experimental results violated Bell's Inequality by 

*five* standard deviations,  and agreed with quantum-mechanical 

predictions.

 

	One of the later improvements to this experiment was an ultra 

fast filter switch.  The problem is that one can imagine that the signal 

measured at detector-#2 is influenced by the settings of filter-#1 

(no particular mechanism is proposed,  just the general principle that 

they could be connected),  and quantum mechanics is mimicked.  To remove 

this possibility,  Aspect's group switched the orientation of filter-#1 

after the photons are in flight,  therefore photon #2 is detected before 

it could know how filter-1 is set (information from filter-1 could not 

catch-up with the photon).[7]

 

                               * * *

 

	A less-know experiment was performed at Saclay,  near Paris, by 

Lamehi-Rachti and Mittig [8].  I will describe this experiment in detail 

since it contains some interesting and subtle features.

	

	Lamehi-Rachti and Mittig took a beam of protons (13.2 and 13.7 

MeV) and scattered them off a hydrogen target.  Since it was a few MeV 

beam,  the incident protons only saw the nucleus of the hydrogen,  ie.  

just a proton target.  The incident and target protons are momentarily 

bound in a state with total spin zero (which correlates their spin),  

and then they fly off to the detectors.  Each detector was to measure a 

component of the spin of the proton.  Lamehi-Rachti and Mittig could not 

use a Stern-Gerlach device since a Stern-Gerlach device will only work 

for neutral charged particles, (also Stern-Gerlach did their experiment 

with thermal silver-atoms, ie. *much* slower. The necessary magnets for an

MeV proton would be monstrous). They could not use a polarization filter 

(like Aspect) since filter are typically designed for light,  or at lest 

something with angstrom or larger wavelengths.

 

	Nuclear physicist have been using something called a "recoil 

polarimeter" for a number of years.  This device usually consist of a 

carbon slab which re-scatters the protons,  and then two detectors.  

Protons will scatter both left and right from the carbon,  but protons 

with spin up will tend to scatter more often to the left,  and protons 

with spin down will tend to scatter more often to the right.  That is 

enough to test for Local Realism. The orientation of the divide between 

left and right is exactly analogous to the orientation of the 

Stern-Gerlach [a,b,c] device.

 

	Let us first look at what is happening at the microscopic 

level.  As a proton approaches a carbon nucleus it experiences two types 

of forces, one is proportional to the distance between the proton and 

the carbon nucleus [ f(r) ] and the other is proportional again to the 

distance [ g(r) ] and the amount of angular momentum [ L ] oriented in 

the same direction as the spin  [ s ]:

 

	V(r) = f(r) + (s.L) g(r)

 

	The effect of the (s.L) term is that if [s] is up compared to 

the scattering plain,  then [ (s.L) g(r) ] is positive and repulsive.  

If [s] is down, then [ (s.L) g(r) ] is negative and attractive.  { The 

scattering plane is defined by the vector (or line) of the incident 

beam,  and the vector of the scatters proton}.  Fine, this yields the 

same trends for scattering which where mentioned before.

 

	But what about the scattering from the [ f(r) ] term?  It 

scatters right if the proton passes to the right of the carbon,  and 

left if the proton passes to the left of the carbon, independent of the 

spin.  This is where one has to be cleaver about measuring an asymmetry. 

I must not just look at the number which went left or right,  but 

rather at the difference.  When you subtract number-right from 

number-left you cancel out the contribution from the [ f(r) ] term.

 

	One of the  problems which Lamehi-Rachti and Mittig reported 

was that they could not create ultra-fast orientation switched like 

Aspect.  However on the microscopic scale the orientation is set by 

which side of the carbon the protons passes.  So detector-2 can only 

know what the `orientation' of detector-1 is when the proton is close 

enough to the carbon that it's trajectory is well determined.  Using the 

Heisenberg uncertainty relationship we can calculate that the detectors

must be separated by a distance greater then 0.03 mm,  a condition the

Lamehi-Rachti and Mittig experiment satisfied.

 

	The other interesting thing to look at is a `triple-scattering 

experiment' (the first scattering is proton-proton, the second and third 

scattering are on different carbon blocks).  In the first 

carbon-detector apparatus you can measure protons with a certain 

component of spin. By orienting the second carbon-detector apparatus in 

the scattering plane you can measure if the spin has changed.  If the 

spin was perpendicular to the first scattering plane it will not change. 

Thus if the proton is detected in the second detector you have selected 

protons with a component of spin perpendicular to the scattering plane,  

and all other components are zero.



                                 ] detector- #1

                               / 

                              carbon-2

                             / \

                            carbon-1

                           / \

                          /

                   ------>Hydrogen

                          \

                           \ /

                            carbon

                             \ /            

                              carbon

                               \

                                 ] detector- #2





	Figure 4. A triple-scattering experiment.  If a proton 

		scatters from the Hydrogen,  through carbon-1, 

		through carbon-2 and into the detector- #1,  

		then the component of spin in the 

		Hydrogen-carbon1-carbon2 plane in zero. 

 

	On the surface it appears that you are just measuring extra 

spin-information,  but quantum mechanically you measure:

 

	P(a,b) = <wave| (s.a)(s.a) (s.b)(s.b) |wave> = 1

 

or you can show that the quantum mechanical operator which corresponds 

to this measurement commute

 

	[(s.a)(s.a),(s.b)(s.b)] = 0

 

	The possibility of using a triple scattering experiment to test 

for Local Realism vanishes.

 

 

 

VI. Limits of the Experiments

_____________________________

 

	In light of the above discussion it is worthwhile to note what 

are the important points in designing these experiments.

 

	1) a correlated system

	2) the measurement of non-commuting observables

	   -> this means the whole experiment!

	3) Local-Realism will always satisfy Bell's Inequality,

	    Quantum-Mechanics will sometime (such as at 90-degrees)

	    satisfy Bell's Inequality.

 

 

V. Conclusion

_____________

 

	One of the interesting things in this EPR dialogue is that it 

has captured the imagination of a number of outstanding physicist over 

such a long period of time (1935-present).  But it is a dynamic problem 

which, by small steps has evolved from gedankenexperiment to real 

experiments.  EPR and Bohr cast the problem into an experiment,  albeit 

beyond the resolution of any real apparatus.  David Bohm shift the 

experiment to something feasible.  J. S. Bell then design the definitive 

statistical test. Finally experiments like Aspect and Lamehi-Rachti/

Mittig performed the test.

 

	Another evolution which has also taken place concerns what it 

is which we end up measuring.  Einstein-Podolsky-Rosen tell us



	"We shall be satisfied with the following criterion, which 

	we regard as reasonable. *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 quantity." 



They are suspicious that position and momentum qualify.  Even if we 

can not measure the position and momentum of a particle simultaneously, 

the position and momentum do exist, ie. they exist as hidden variables.  

But what we test for is called `Local Realism',  which means `all the 

information necessary to determine the outcome of the experiment is 

available locally (presumablely the information is contained in a hidden 

variable).  But the statistics and correlations measure tell us that

information is NOT all local.  All along the question has been, "does 

a proton have a momentum if you are in the act of performing a precise 

position measurement?".  I guess the answer is no.

 

	This leaves me with not with the question posed in the title of

the EPR paper ("Can Quantum-Mechanical Description of Physical Reality 

Be Considered Complete?"), but rather, since by the EPR definition, "



	"The elements of the physical reality cannot be

	determined by *a priori* philosophical considerations,

	but must be found by an appeal to results of experiments

	and measurements."

	

This leaves me wondering if they is any quantity which satisfies this

criteria for an element of physical reality?

 

 

References

__________

 

[1] A. Einstein, B. Podolsky and N. Rosen, "Can Quantum-Mechanical

	Description of Physical Reality Be Considered Complete?",

	Physical Review vol. 47,  p 777 (1935).

 

[2] N. Bohr, "Can Quantum-Mechanical Description of Physical Reality

	Be Considered Complete?",Physical Review vol. 48,

	p 696 (1935).

 

[3] J\/orgen Krogh, "The Epistemology of Neils Bohr & Albert Einstein:

	Two forms of realism", Metaphysical Review vol 1 no 9

	(April 1995).

 

[4] David Bohm, "Quantum Theory", ch. 22 sec. 16, p 614-5, (1951).

 

[5] J. S. Bell, Physics 1, p 195-200 (1964).

 

[6] Alain Aspect, "Proposed experiment to test the nonseperability 

	of quantum mechanics", Phys. Rev. D 14, p 1944 (1976).

 

[7] Alain Aspect, Jean Dalibard and Gerard Roger, "Experimental Test 

	of Bell's Inequalities Using Time-Varying Analyzers", 

	Phys. Rev. Lett. 49, p 91 (1982). 



[8] M. Lamehi-Rachti and W. Mittig, "Quantum mechanics and hidden 

	variables: A test of Bell's inequality by the measurement of

	the spin correlation in low-energy proton-proton scattering",

	Phys. Rev. D 14, p 2543 (1976).