 |
The Left Hand of the Electron |
1 - ODDS AND EVENS
I have just gone through a rather unsettling experience. Ordinarily
it is not very difficult to think up a topic for these chapters. Some interesting
point will occur to me, which will quickly lead my mind to a particular line
of development, beginning in one place and ending in another. Then, I get
started.
Today, however, having determined to deal with
asymmetry (in more than one chapter, very likely)
and to end with life and antilife, I found that two possible starting points
ocurred to me. Ordinarily, when this happens, one starting point seems so
much superior to me that I choose it over the other with a minimum of hesitation.
This time, however, the question was whether to start with even
numbers or with double refraction, and the arguments raging within my head
for each case were so equally balanced that I couldn't make up my mind. For
two hours I sat at my desk, pondering first one and then the other and growing
steadily unhappier.
Indeed, I became uncomfortably aware of the resemblance of my
case to that of 'Buridans ass'.
The reference, here, is to a fourteenth-century French philosopher, Jean
Buridan, who was supposed to have stated the following: 'If a hungry ass
were placed exactly between two haystacks in every respect equal, it would
starve to death, because there would be no motive why it should go to one
rather than to the other.'
Actually, of course, there's a fallacy here, since the statement
does not recognize the existence of the random
factor. The ass, no logician, is bound to turn his head randomly so that
one haystack comes into better view, shuffle his feet randomly so that one
haystack comes to be closer; and he would end at the haystack better seen
or more closely approached.
Which haystack that would be, one could not tell in advance.
If one had a thousand asses placed exactly between a thousand sets of haystack
pairs, one could confidently expect that about half would turn to the right
and half to the left. The individual ass, however, would remain unpredictable.
In the same way, it is impossible to predict whether
an honest coin, honestly thrown, will come down heads
or tails in any one particular case, but we can confidently predict that
a very large number of coins tossed simultaneously (or one coin tossed a
very large number of times) will show heads just about half the time and
tails the other half.
And so it happens that although the chance of the fall of heads
or tails is exactly even, just fifty-fifty, you can nevertheless call upon
the aid of randomness to help you make a decision by tossing one coin once.
At this point, I snapped out of my reverie and did what a lesser
mind would have done two hours before. I tossed a coin.
Shall we start with even numbers, Gentle Readers?
I suspect that some prehistoric philosopher must have decided
that there were two kinds of numbers: peaceful ones and warlike ones. The
peaceful numbers were those of the type 2, 4, 6, 8, while the numbers in
between were warlike.
If there were 8 stone axes and two individuals possessing equal
claim, it would be easy to hand 4 to each and make peace. If there were 7,
however, you would have to give 3 to each and then either toss away the
1 remaining (a clear loss of a valuable object) or let the two disputants
fight over it.
The fact that the original property that marked out the significance
of what we now call even and odd numbers was something like this is indicated
by the very names we give them.
The word 'even' means fundamentally, 'flat, smooth, without unusual depressions
or elevations'. We use the word in this sense when we say that a person says
something 'in an even tone of voice'. An even number of identical coins,
for instance, can be divided into two piles of exactly the same height. The
two piles are even in height and hence the number is called even. The even
number is the one with the property of 'equal shares'.
'Odd', on the other hand, is from an old Norse word meaning
'point' or 'tip'. If an odd number of coins is divided into two piles as
nearly equal as possible, one pile is higher by one coin and therefore rears
a point or tip into the air, as compared with the other. The odd number possesses
the property of 'unequal shares', and it is no accident that the expression
'odds' in betting implies the wagering of unequal amounts of money by the
two participants.
Since even numbers possess the property of equal shares, they
were said to have 'parity', from a Latin word meaning 'equal'. Originally,
this word applied (as logic demanded) to even
numbers only, but mathematicians found it convenient to say that if two numbers
were both even or both odd, they were, in each case 'of the same parity'.
An even number and an odd number, grouped together, were 'of different
parity'.
To see the convenience of this convention, consider the following:
If two even numbers are added, the sum is invariably even. (This can be expressed
mathematically by saying that two even numbers can be expressed as 2m
and 2n where m and n are whole numbers and that the
sum, 2m+2n, is still clearly divisible by two. However, we
are friends, you and I, and I'm sure we can dispense with mathematical reasoning
and that I will find you willing to accept my word of honor as a gentleman
in such matters. Besides, you are welcome to search for two even numbers
whose sum isn't even.)
If two odd numbers are added, the sum is also invariably even.
If an odd number and an even number are added, however, the sum is invariably
odd.
We can express this more succinctly in symbols, with E standing for even
and O standing for odd:
E + E = E |
O + O = E |
E + O = O |
O + E = O |
Or, if we are dealing with pairs of numbers only, the concept
of parity enables us to say it in two statements, rather than four:
(1) Same parities add to even.
(2) Different parities add to odd.
A very similar state of affairs exists with reference to multiplication, if we divide numbers into two classes: positive numbers (+) and negative numbers (-). The product of two positive numbers is invariably positive. The product of two negative numbers is invariably positive. The product of a positive and a negative number is invariably negative. Using symbols:
+ x + = + |
- x - = + |
+ x - = - |
- x - = - |
Or, if we consider all positive numbers as having one kind of
parity and all negative numbers as another, we can say, in connection with
the multiplication of two numbers:
(1) Same parities multiply to positive.
(2) Different parities multiply to negative.
The concept of parity - that is, the assignment of all objects of a particular
class to one of two subclasses and then finding two opposing results when
objects of the same or of different subclasses are manipulated - can be applied
to physical phenomena.
For instance, all electrically charged particles can be divided
into two classes: positively charged and negatively charged. Again, all magnets
possess two points of concentrated magnetism of opposite properties: a north
pole and a south pole. Let's symbolize these as +, -, N and S.
It turns out that:
+ and + or N and N = repulsion |
- and - or S and S = repulsion |
+ and - or N and S = attraction |
- and + or S and N = attraction |
Again, we can make two statements:
(1) Like electric charges, or magnetic poles, repel each other.
(2) Opposite electric charges, or magnetic poles, attract each other.
The similarity in form to the situation with respect to the
summing of odd and even, or the multiplying of positive and negative, is
obvious.
When, in any situation, same parities always yield one result and different
parities always yield the opposite result, we say that 'parity is conserved'.
If two even numbers sometimes added up to an odd number; or if a positive
number multiplied by a negative one sometimes yielded a positive product;
or if two positively charged objects sometimes attracted each other; or if
a north magnetic pole sometimes repelled a south magnetic pole, we would
say that, 'The law of conservation of parity is violated.'
Certainly in connection with numbers and with electromagnetic
phenomena, no one has ever observed the law of conservation of parity to
have been violated, and no one seriously expects to observe a case in the
future.
What about other cases?
Well, electromagnetism involves a field. That is, any electrically
charged particle, or any magnet, is surrounded by a volume of space within
which its properties are made manifest on other objects of the same sort.
The other objects are also surrounded by a volume of space within which their
properties are made rnanifest on the original object. One speaks, therefore,
of an 'electromagnetic interaction' involving pairs of objects carrying electric
charge or magnetic poles.
Up through the first years of the twentieth century, the only
other kind of interaction known was the gravitational. At first blush, there
seems no easy way of involving gravitation with parity. There is no way of
dividing objects into two groups, one with one kind of gravitational property
and the other with the opposite kind.
All objects of a given mass possess the same intensity of
gravitational interaction of the same sort. Any two objects with mass attract
each other. There seems no such thing as 'gravitational repulsion' (and,
according to Einstein's General
Theory of Relativity there can't be such a thing). It is as though,
in gravity, we can say only that E+ E = E or +x+ =+.
To be sure, there is a chance that in the field of
subatomic physics
there might be some objects with mass that possess the usual gravitational
properties and other objects with mass that possess gravitational properties
of the opposite kind ('antigravity'). In that case, the chances are that
it would turn out that two antigravitational objects attract each other just
as two gravitational objects do; but that an antigravitational and a
gravitational object would repel each other. The situation with respect to
the gravitational interaction would be the reverse of the electromagnetic
one (like gravities would attract and unlike gravities would repel) but,
allowing for that reversal, parity would still be conserved.
The trouble is, though, that the gravitational interaction is
so much more feeble than the electromagnetic interaction that gravitational
interactions of subatomic particles are as yet impossible to measure and
a sub-tiny attraction can't be differentiated from a sub-tiny repulsion.
So the question of parity and the gravitational field remains in abeyance.
As the twentieth century wore on, it came to be recognized that
the gravitational and electromagnetic interactions were not the only ones
that existed. Subatomic particles involved something else. To be sure, electrons
had negative charges and protons had positive charges and with respect to
this, they behaved in accordance with the rules of electromagnetic interaction.
There were other events in the subatomic world, however, that had nothing
to do with electromagnetism There was, for instance, some sort of interaction
involving particles, whether with or without electric charge, that showed
itself only in the super-close quarters to be found within the atomic nucleus.
Did this 'nuclear interaction' involve parity?
Every subatomic particle has a certain quantum-mechanical property
which can be expressed in a form involving three quantities, x, y, and z.
In some cases, it is possible to change the sign of all three quantities
from positive to negative without changing the sign of the expression as
a whole. Particles in which this is true are said to have 'even parity'.
In other cases, changing the signs of the three quantities does change the
sign of the entire expression and a particle of which this is true is said
to have 'odd parity'.
Why even and odd? Well, an even-parity particle can break up
into two even-parity particles or two odd-parity particles, but never into
one even-parity plus one odd-parity. An odd-parity particle, on the other
hand, can break up into an odd-parity particle plus an even-parity one, but
never into two odd-parity particles or two even-parity particles. This is
analogous to the way in which an even number can be the sum of two even numbers
or of two odd numbers, but never the sum of an even number and an odd number,
while an odd number can be the sum of an even number and an odd number, but
can never be the sum of two even numbers or of two odd numbers.
But then a particle called the
'K-meson' was discovered. It was unstable
and quickly broke down into 'pi-mesons'. Some K-mesons gave off two pi-mesons
in breaking down and some gave off three pi-mesons and that was instantly
disturbing. If a K-meson did one, it ought not to be able to do the other.
Thus an even number can be the sum of two odd numbers (10 = 3+7) and
an odd number can be the sum of three odd numbers (11 = 3+7+1), but no number
can be the sum of two odd numbers in one case and three odd numbers in another.
It would be like expecting a number to be both odd and even. It would, in
short, represent a violation of the law of conservation of parity.
Physicists therefore reasoned there must be two kinds of K-meson;
an even-parity variety ('theta-meson') that broke down to two pi-mesons,
and an odd-parity variety ('tau- meson') that broke down to three pi-mesons.
This did not turn out to be an altogether satisfactory solution, since there
seemed to be no possible distinction one could make between the theta-meson
and the tau-meson except for the number of pi-mesons it broke down into.
To invent a difference in parity for two particles identical in every other
respect seemed poor practice.
By 1956, a few physicists had begun to wonder if it weren't possible that
the law of conservation of parity might not be broken in some cases. If that
were so, maybe it wouldn't be necessary to try to make a distinction between
the theta- meson and the tau-meson.
The suggestion roused the interest of two young Chinese- American
physicists at Columbia, Chen Ning Yang and Tsung Dao Lee, who took into
consideration the following- There is, as a matter of fact, not one nuclear
interaction, but two. The one that holds protons and neutrons together within
the nucleus is an extremely strong one, about 130 times as strong as the
electromagnetic interaction, so it is called the 'strong nuclear interaction.
There is a second, 'weak nuclear interaction' which is only
about a hundred-trillionth the intensity of the strong nuclear interaction
(but still some trillion-trillion times as intense as the unimaginably weak
gravitational interaction).
This meant that there were four types of interaction in the
universe (and there is some theoretical reason for arguing that a fifth of
any sort cannot exist, but I would hate to commit myself to that):
(1) strong nuclear,
(2) electromagnetic,
(3) weak nuclear, and
(4) gravitational.
We can forget about the gravitational interaction for reasons
I mentioned earlier in the article. Of the other three, it had been well
established by 1956 that parity was conserved in the strong nuclear interaction
and in the electromagnetic interaction. Numerous cases of such conservation
were known and the matter was considered settled.
No one, however, had ever systematically checked the weak nuclear
interaction with respect to parity, and the breakdown of the K-meson involved
a weak nuclear interaction. To be sure, all physicists assumed that parity
was conserved in the weak nuclear interaction but that was only an assumption.
Yang and Lee published a paper pointing this out - and suggested
experiments that might be performed to check whether the weak nuclear
interactions conserved parity or not. Those experiments were quickly carried
out and the Yang-Lee suspicion that parity would not be conserved was shown
to be correct. There was very little delay in awarding them shares in the
Nobel prize in physics in 1957, at which time Yang was thirty-four and Lee,
thirty-one.
You might ask, of course, why parity should be conserved in
some interactions and not in others - and might not be satisfied with the
answer 'Because that's the way the universe is.'
Indeed, by concentrating too hard on those cases where parity
is conserved, you might get the notion that it is impossible,
inconceivable, unthinkable to deal with a case where it isn't conserved.
If the conservation of parity is then shown not to hold in some cases, the
notion arises that this is a tremendous revolution that throws the entire
structure of science into a state of collapse.
None of that is so.
Parity is not so essential a part of everything that exists
that it must be conserved in all places, at all times, and under all conditions.
Why shouldn't there be conditions where it isn't conserved or, as in the
case of gravitational interaction, where it might not even apply?
It is also important to understand that the discovery of the
fact that parity was not conserved in weak nuclear interctions did not
'overthrow' the law of conservation of parity, even though that was certainly
the way in which it was presented in the newspapers and even by scientists
themselves. The law of conservation of parity, in those cases in which its
validity had been tested by experiment, remained and is still as much
in force as ever.
It was only in connection with the weak nuclear interactions,
where the validity of the law of conservation of parity had never been
tested prior to 1956 and where it had merely been rather carelessly assumed
that it applied, that there came the change. The final experiment merely
showed that physicists had made an assumption they had no real right to make
and the law of conservation of parity was 'overthrown' only where it had
never been shown to exist in the first place.
It might help if we look at some familiar, everyday case where
a law of conservation of parity holds, and then on another where it is merely
assumed to hold by analogy, but doesn't really. We can then see what happened
in physics, and why an overthrow of something that really isn't there
to begin with, improves the structure of science and does not damage it.
Human beings can be divided into two classes: male (M) and female
(F). Neither two males by themselves nor two females by themselves can have
children (no C). A male and a female together, however, can have children
(C). So we can write:
M and M = no C |
F and F = no C |
M and F = C |
F and M = C |
There is thus the familiar parity situation:
(1) Like sexes cannot have children.
(2) Opposite sexes can have children.
To be sure, there are sexually immature individuals, barren
females, sterile or impotent men, and so on, but these matters are details
that don't affect the broad situation. As far as the sexes and children are
concerned, we can say that the human species (and, indeed, many other species)
conserves parity.
Because the human species conserves sexual parity with respect
to childbirth, it is easy to assume it conserves it with respect to
love and affection as well, so that the feeling arises that sexual love ought
to exist only between men and women. The fact is, though, that parity is
not conserved in that respect and that both male homosexuality and
female homosexuality do exist and have always existed. The assumption that
parity ought to be conserved where, in actual fact, it isn't, has
caused many people to find homosexuality immoral, perverse, abhorrent, and
has created oceans of woe throughout history.
Again, in Judeo-Christian culture, the institution of marriage
is closely associated with childbirth and therefore strictly observes the
law of conservation of parity that holds for childbirth. A marriage can take
place only between one man and one woman because, ideally, that is the simplest
system that makes childbirth possible.
Now, however, there is an increasing understanding that parity,
which is rigidly conserved with respect to childbirth, is not necessarily
conserved with respect to sexual relations. Increasingly, homosexuality is
treated not as a sin or a crime, but as, at most, a misfortune (if that).
There is the further attitude, slowly growing in our society,
that there is no need to force the institution of marriage into the tight
grip of parity conservation. We hear, more and more frequently, of homosexual
marriages and of group marriages. (The old-fashioned institution of polygamy
is an example of one kind of marriage, enjoyed by many of the esteemed men
of the Old Testament, in which sexual parity was not conserved.)
In the next chapter, then, we'll go on with the nature of the
experiment that established the non-conservation of parity in the weak nuclear
interaction and consider what happened afterward.