The Greek Atom and Quantum Mechanics
                    ____________________________________
 
                             Timothy Paul Smith
 
                             Department of Physics
                             University of New Hampshire
                             Durham, New Hampshire 03824
                             tps@fermi.unh.edu
 
Abstract:
_________
 
	The reasoning behind the Greek `Atomic Theory' is logical and 
rational.  It divides the world into atom and void.  This atomic theory 
went through a renaissance with the work of John Dalton and the rise of 
modern chemistry.  The Greek logic even hold up to most of the physical 
evidence of modern science.  But what of quantum mechanics?  Here, it 
can be argued that the most real part of the world is the wavefunction.  
Particles do not have a definite location,  momentum and (contrary to  
Democritus) there is not a definite bounds between the particle (or 
atom) and the void.
 
I. Introduction
_______________
 
	"Probing the Nature of Matter", proclaims CEBAF, "Discover the 
Nature of Nature", says FermiLab. What are the elements asked the 
Greeks.  I do not know if the question of "what is the world made of" is 
as old as mankind. We do know it took the contemplative age of the 
Greeks before people tried to answer it, or at least committed their 
answers to paper.
 
	The Greek atomic theory is logical, reasonable and was highly 
regarded even into our own century.  For most purposes it is still 
fruitful theory of matter.  We have adopted the name `atom' for the 
chemical indivisible and in everyday terms, `elementary' particles of 
our world.  We now measure a particle about
		     -10
		1x10         meters
across and call it an `atom'.  One of the most important aspect of the 
atomic theory was that between atoms was void.  Void allowed change,  
and change is what we see.
 
	In this century we thought we found the atom.  It might be the 
chemical atom at 10**(-10) meters in size,  or nucleons at 10**(-15) 
meters,  or quarks at 3x10**(-16) meters.  Or it maybe smaller,  or 
perhaps even a `Dirac' point-like particle.  All of these candidates in 
some sense would satisfy the Greeks.  But not the wavefunction of 
Quantum Mechanics.  The quantum mechanical wavefunction is continuous in 
spatial and temporal extent, as opposed to atoms and void.
 
	In this essay I will describe the atomic theory of the Greeks.  
I will try to put it in the context of the debates of its day.  I will 
then describe two interpretations of the quantum mechanical 
wavefunction.  In conclusions I will discuss the incompatibility of 
these two theories.
 
II. The Greek Theory of the Atom
________________________________
 
	
	When I was very small my brother once frustrated me for the 
length of a car trip by stating, "You can never have half of something".
	
	I retorted that of course, "If you cut an apple in half you 
have half an apple".
 
	He tauted me with, "No, you still have a whole".
 
	It took me years to realize what he meant was, "No, you still 
have a whole something - a whole object".  Weather I had a whole apple 
was not the question.
 
	The Greeks were caught up in a similar type of question.  Can I 
divide things forever?  Does a bit of apple remain sweet no matter how 
small? Anaxagoras believed "bones are made of very tiny little bonelets, 
and flesh of infinitesimal bites of flesh"[1],  but he was in a minority 
in classical Greece.
 
II.A  Parmenides and Zeno - The Eleatics
      __________________________________
 
	The language and reasoning of Parmenides seems very foreign to 
the modern scientist - something near to the mystics.  The Eleatics 
regarded reason as the ultimate  proof,  and sensory perceptions as 
potentially deceptive. Modern scientist may rely on mathematics and 
reason when trying to understand such murky issues as quarks and 
black-holes,  but we do not deny the senses when they are stubbornly 
irreconcilable with reason.  However the Eleatics did deny the senses 
when there was a conflict.
 
	Parmenides born in Elea in southern Italy and founded the 
Eleatics school of thought in the fifth century B.C.. In his major book 
"The Way of Truth" he explains that there are two possible things:
 
	"It is, or it is not" [2],
 
a Hamlet like status to everything:  "To be, or not to be".  From this 
humble beginnings he eventually concludes that the universe is one solid 
object that can not have parts which evolve and move. How did he arise 
at this conclusion?
 
	He considers the possibility of "at one time existed but now 
does not",  and "at one time did not exist,  but now does",  and 
concluded that something could not at one time be thinkable,  and one 
time not thinkable.  That is unthinkable,  therefore the objects which 
are created or perish  can not exist.  Therefore the world is made up 
only of what exist and what does not exist.  One can  not think of what 
is unthinkable and so therefore there is only what exist.
 
	"One way only is left to be spoken of, that it is;  and on this 
way are full many signs that what is is uncreated and imperishable, for 
it is entire, immovable and without end. ...  Nor is it divisible since 
it is all alike;  nor is there more here and less there, which would 
prevent it from cleaving together, but it is full of what is. So it is 
all continuous; for what is clings close to what is. But motionless 
within the limits of mighty bonds, ... " [3]
	
(note: one of the important things which is therefore rejected in this 
analysis is `void'. )
 
	That is it,  there is just one existence which did not start 
and will not end.  It can not change.  Therefore all things what appear 
to move, evolve, are born, die etc. are just deceptions of the senses.
 
	Even in its day the ideas of Parmenides were not easyly 
understood and were heavely criticized.  Plato describes in detail a 
debate between the Eleatics ( Parmenides and his student Zeno), Aristole 
and Socrates [4].  It was left to Zeno to defend the Eleatics,  the one 
and wholeness of the world.  He did so with a series of paradox,  
demonstrating that motion was unthinkable.  For example the paradox of 
`Achilles and the Tortoise' [5].
 
	If the Tortoise is given a lead in its race with Achilles he 
can not lose,  since by the time Achille has moved to the starting place 
of the Tortoise the Tortoise will have been able to advance some more.  
When Achille has moved to this second point the Tortoise will still be 
ahead and advanced to a third point.  Although the distance between them 
shrinks Achille can not catch the Tortoise since he will have to move 
through an infinite number of points to do so,  which will take an 
infinite amount of time.  Motion, Zeno argues is therefore 
incomprehensible,  therefore what every we see is merely a deception.  
Of his defense of the views of Parmenides Zeno said,
 
	"This book, then, is a retort against those who assert a 
plurality.  It pays them back in the same coin with something to spare, 
and aims at showing that, on a thorough examination, their own 
supposition that there is a plurality leads to even more absurd 
consequences than the hypothesis of the One."[6]
 
 
 
II.B Leucippus and Democritus - the Atomic Theory
     ____________________________________________
 
 
	It was in reply to these continuous and unchangeable world 
theories that the atomic theory was born.  The theory was developed 
first by Leucippus,  (although there were other older `atomic' 
theories),  but it was his student Democritus (born c. 460 B.C.) who 
developed,  expanded and wrote about it.  All of Democritus' writings 
have been lost (except possible some debatable fragment,  quotations in 
other writings) and what we know of him comes by reputation from other 
authors.  Aristole provides one of the most extensive description of the 
atomic theory.  In his introduction to "Metaphysica" he surveies his 
predisors and writes:
 
	"Leucippus, however, and his disciple Democritus hold that the 
elements are the Full and the Void - calling one "what is" and the other 
"what is not".  Of these they identify the full or solid with "what is", 
and the void or rare with "what is not" (hence they hold that what is 
not is no less real than what is, because Void is as real as Body); and 
they say that these are the material causes of things.  And just as 
those who make the underlying substance a unity generate all other 
things by means of its modifications, assuming rarity and density as 
first principles of these modifications, so these thinkers hold that the 
"differences" are the causes of everything else. These differences, they 
say, are three: shape, arrangement , and position; because they hold 
that what is differs only in `contour', `inter-contact',  and 
`inclination'. (Of these contour means shape, inter-contact arrangement, 
and inclination position.)  Thus, e. g., A differs from N in shape, AN 
from NA in arrangement, and Z from N in position.  As for motion,  
whence and how it arises in things, they casually ignored this point, 
very much as the other thinkers did."[7]
 
 
	Following Democritus there was a great deal of debate about did 
this model answer all the problem posed by the Eleatics.  Was motion to 
be understood in terms of atoms?  Perhaps space is not infinitely 
divisible.  Could atoms be of any size? Perhaps there is an atom the 
size of the universe - ie. the Eleatics solution again.
 
	In the first century B.C. Titus Lucretius Carus, a Roman, 
traveled to Greece to study,  and upon returning to Rome he wrote a near 
encyclopedic account of knowledge in Greece (at that time still the best 
graduate education).
 
	"When human life, all too conspicuous,
	Lay foully groveling on the earth, weight down
	by grim Religion looming from the skies,
	Horribly threatening mortal men, a man,
	A Greek, first raised his mortal eyes
	Bravely against this menace. No report
	Of gods, no lighting-flash, no thunder-peal
	Made this man cower, but drove him all the more" [8]
 
	Lucretius then details the Atomic theory as in stood in the 
first century,  and remain until perhaps John Dalton and the birth of 
modern chemistry.
 
	"Our starting point shall be this principle:
	Nothing at all is ever born from nothing
	________________________________________
	By the god's will. ... " [9]
	__________________
 
Lucretius's arguments are truly poetic:
 
	"Now, if things come from nothing, all things could
	Produce all kinds of things;  nothing would need
	Seed of its own.  Men would burst out of the sea.
	...
	Thirdly, if things could come from nothing, time
	Would not be of the essences, for their growth,
	... And full-grown forests come leaping out of the ground.
	Ridiculous! We know that all things grow
	Little by little, as indeed they must
	From their essential nature." [10]	
 
The second axiom
 
	"Our second axiom is this, that nature
	Resolves each object to its basic atoms
	But does not ever utterly destroy it.
	If anything could perish absolutely,
	It might be suddenly taken from our sight,
	...
		Nature permits no visible destruction
	of anything.
	...
	By rights, if things can perish, infinite time
	And ages past should have consumed them all,
	...
		Nothing can
	Disintegrate entirely into nothing,
	...
	Matter is indestructible" [11]
	______________________
 
Finally void
 
	"By void I mean vacant and empty space,
	    ____
	Something you cannot touch. Were this not so,
	Things could not move." [12]
 
	So does this atomic theory overcome the difficulties posed by 
Parmenides and Zeno?  First and foremost, it allows motion. If an object 
moves the atoms of air can compress  by moving into the void between the 
atoms,  and so the atoms of air can move around the object.  Secondly 
the theory explains the variety of materials we observe by the motion, 
orientation and combination of atoms.
 
	In summary what drove the development of the atomic theory of 
the Greeks was the explanation of motion and variety of materials.  The 
theory required the indestructible atom, and the (unthinkable?) void.  
As a side benefit we have a proof, by definition (given the atomic 
theory), of the conservation of matter.  We also have the concept of 
vacuum which preceded its experimental observation by two millennium.
 
III. Quantum Mechanics
______________________
 
	The are a great many new developments in this century 
concerning the question of the nature of matter.  We lost the term 
`atom' to chemistry and perhaps now might only ask the question in terms 
of `fundamental particle'.  It is not clear if I should be talking about 
quarks and leptons, flux-tubs (string theory), space-time form,  an 
object on the Plank-length scale or a true Dirac particle.  What is 
clear is that we should be talking about some type of quantum mechanical 
wavefunction.
 
III.A The EPR-Paradox (Briefly) and the reality of the Wavefunction
      _____________________________________________________________
	
	I will not go into the Einstein-Podolsky-Rosen paradox ("Can 
Quantum-Mechanical Description of Physical Reality be Considered 
Complete?" [13]) in great detail (I invite a reader out there to take on 
this task),  suffice it to say that Einstein et al. had a hard time 
with:
 
	"the doctrine that physical properties have in general no 
objective reality independent of the act of observation." [14]
 
I will try to condense the EPR debate to this:
 
	Einstein (1935): I think that a particle must have a
		separate reality independent of the measurements.
		That is an electron has spin, location and so
		forth even when it is not being measured. "I like
		to think that the moon is there even if I am not
		looking at it." [15]
 
	
	N. Bohr ("Quantum-Mechanical Description of Physical Reality")
		(1935): [16] The measurement is all there is.  If
		a tree falls in the forest and no one is there
		to hear it, I will not ask the question, "did
		it make any noise?"
 
	J. S. Bell (1964): [17] In a certain class of experiments
		are distinct results if the particle has a
		`hidden variable',  that is, the particle has
		a unique spin, location, etc. as opposed to only
		the quantum mechanical numbers associated with
		the wavefunction.
 
	A. Aspect (1981): [18] We have performed Bell's experiment
		and the results exclude  `hidden variables'.  The
		results are consistent with only the quantum
		numbers of the wavefunction.
 
	T. P. Smith (1994): An `independent reality' means
		something which has its properties independent
		of a measurement.  Only the wavefunction has
		properties which are independent of a measurement.
		What is really the "Elements of physical reality"
		is the wavefunction.
 
 
 
III.B Continuous Particles?
      ____________________
 
	The simplest interpretation of the quantum mechanical 
wavefunction is that it is essentially a complex probability function.  
It tells me what is the probability that I will find a particle with a 
certain spin at a certain location.  If that is true then a particle 
need not be any different from the particles described by Leucippus and 
Democritus,  little hard spheres.  However given the results of Aspect's 
experiments one may be driven (as is the author) to say that a particle 
is really its wavefunction.  This then poses the interesting problem of 
where are the bounds of the particle?
 
	In general wavefunction extend to infinite (although from the 
particle's most likely position the wavefunction soon decays to an 
infinitesimal amplitude).  This leads to some type of continuous 
particle,  or the most real thing is the wavefunction and the 
wavefunction is continuous and extends to all space, so that which is 
more real then extends to all space.
 
III.C The Greek and Quantum Mechanics
      _______________________________
 
	So what is the problem with the Greek Atomic Theory and Quantum 
Mechanics?  It is simple "where did the void go?"  This "void" is 
distroyed in several ways.  First,  since a particle's wavefunction 
extends to infinite a particle in some sense is everywhere.  Perhaps 
that is not so bad.  If one thinks of wavefunctions in their simplicity 
interpretation, as probability distributions, you can think about there 
being a very large probability that the particle is not at infinite.
 
	Let me now return to the reasons that Leucippus and Democritus 
postulated their atomic theory.  It was in reply to Parmenides and Zeno 
continuous world which could not change.  Motion was seen as being 
impossible without void.  It as also in a quest to explain the variety 
of materials which are observed in our world.  There theory was not 
unique in describing the variety of materials, however it managed to do 
it in a most economical way,  without a myriad of elements.
 
	So what does quantum mechanics say about motion?  Not only for 
two particles to move, but even exist we have a problem.  That particles 
extend on to infinite. Does quantum mechanics allow two particle to 
exist at the same location in space? Yes, except if it violates the 
Pauli Exclusion Principle.  So if particles are not identical they are 
their wavefunctions can occupy the same space.  What if they are 
identical?  Can two identical particle exist in the same universe?  No 
they can not be identical; they can not have all the same quantum 
numbers.  If two electrons have the can lepton number, spin, 
angular-momentum,  there must be something different.  Perhaps which 
hydrogen-atom they are bound in?
 
	If the wavefunctions of two particles can occupy the same space 
do we allow motion?  Yes!  For a particle to move it is merely changing 
the amplitude of the wavefunction at the point under consideration.
 
	The present `Standard Model' of particle physics gives us a 
small number of fundamental particles,  not quite as economic as 
Democritus, but they also need only "shape, arrangement , and position" 
to give rise to all observed materials.
 
IV. Conclusion
______________
 
	The Greeks; Parmenides, Zeno, Leucippus, Democritus and many 
others, where asking good question.  They make us think even today about 
the consequences of our modern theories of matter.  Our modern theories 
must still explain motion and the evolution of matter.  It must avoid 
the problems which beset Parmenides and Zeno's continuous theory.  It 
must explain the varieties of materials we observe around us.
 
	The question of what is the nature of matter came along ways in 
the fourth century B.C..  In the century we often think we are probing 
the same question (i.e. recent `top-quark' announcements) when we look 
at smaller and smaller particles.  But to answer the Greeks we must 
answer what might matter be made of.  This question is also actively 
presured in modern laboratories,  but its interpretation is very 
difficult.
 
References
__________
 
[1] Lucretius, "The Way Things Are", book 1, line 833.
 
[2] Parmenides, "The Way of Truth", fragment 2.
 
[3] Parmenides, "The Way of Truth", fragment 8.
 
[4] Plato, "Parmenides".
 
[5] Aristotle, "Physics" 239b.
 
[6] Plato, "Parmenides", 128.
 
[7] Aristotle, "Metaphysics" 985 B.
 
[8] Lucretius, "The Way Things Are", book 1, line 66.
 
[9] Lucretius, "The Way Things Are", book 1, line 151.
 
[10] Lucretius, "The Way Things Are", book 1, line 163.
 
[11] Lucretius, "The Way Things Are", book 1, line 219.
 
[12] Lucretius, "The Way Things Are", book 1, line 335.
 
[13] A. Einstein, B. Podolsky and N. Rosen, 
	"Can Quantum-Mechanical Description of Physical Reality 
	be Considered Complete?", Phys. Rev. v47, p777 (1935).
 
[14] N. David Mermin, 
	"Is the moon there when nobody looks? Reality and 
	the quantum theory", Phys. Today (April 1985).
 
[15] A. Einstein, reported in, A. Pais., 
	Rev. Mod. Phys. v51, p 863 (1970).
 
[16] N. Bohr, 
	"Quantum-Mechanical Description of Physical Reality", 
	Phys. Rev., v.48, p.696 (1935).
 
[17] J. S. Bell, physics v.1, p.195 (1964).