Qualitative Quantum Mechanics


If we wish, we can take the single holistic state of the universe,
and cut it up conceptually into aspects, so that it can be described
by picking one value on each of the following lines:

Electron A has x coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3....
Electron A has y coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3....
Electron A has z coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3....
Electron A has spin:   up   down
Electron B has x coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3....
Electron B has y coordinate integer part: ...  -5  -4  -3  -2  -1  0  1  2  3  4  ...
Electron B has y coordinate fractional part:  0....0.2....0.35....0.64....0.79....0.99
Electron B has z momentum: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3....
Electron B has spin:   left   right

Then we can imagine a vertical squiggle going down through the lines
above, whose position at each line indicates what the value of that
aspect is -- such a squiggle yields a state of the universe.

There are obviously very many possible such squiggles.  Physicists call
these squiggles "base states".  Each is a possible state of the universe.

But there are more possibilities than just the squiggles!  Typically,
say for an electron in a hydrogen atom, the electron is not at a
specific point, but rather at a cloud of points, all around the nucleus.
Electron A might be at such a cloud, meaning that the current state of
the universe must somehow be a combination of many different squiggles,
going through different possibilities for A's coordinates.

So squiggles are possible states of the universe, but combinations of
squiggles are also possible states (and are much more common!).
(Physicists call such combinations of squiggles "superpositions of
base states".)

Above, we listed B's z momentum rather than its z coordinate.  Why not
list both?  Because they are two different ways of talking about the
same thing.  It turns out that the momentum and the position are the
Fourier transforms of each other, meaning that a specific value (or more
likely the "value cloud") for one lets you work out the value cloud for
the other, so you can't just arbitrarily specify both -- you only get
to arbitrarily pick one of them.

But clearly we have some freedom when picking what aspects to list,
for example we got to choose between position and momentum, and clearly
we could have picked a different xyz coordinate system for any of the
electrons.  Not so clearly, but just as truly, for measuring the electron's
spin, you can pick any pair of opposite directions that you want.  For
example, if you tell me how much an electron is spinning "up" and "down",
I can tell you how much it's spinning "left" and "right".  (Don't worry
about this too much, since nobody has a very good picture of what the
"spin" means anyway!)


If electron B is also at a cloud (like A is) rather than a single point,
then the current state is the combination of a whole lot of squiggles --
for each possible path through A's aspects and each possible path
through B's aspects, there is a contributing squiggle!  If it takes N
squiggles to represent a cloud, then it takes N^2 squiggles to represent
two clouds.  (Of course, if you believe in continuity, N is infinite.)

Now, let's consider a peculiar state of the universe:  Suppose electron A
is in its hydrogen atom cloud, and electron B is at the "opposite" point
in the cloud, so if A is one Angstrom to the right of the nucleus, then B
is one Angstrom to the left of the nucleus, and so on.  (Does this even
make any sense?  I mean, A's cloud is itself symmetric ball-shaped cloud
around the nucleus, so isn't the "opposite" cloud just the same thing?
Let's see.)  Well, let's see if we can represent this idea with squiggles.
For each squiggle contributing to this state, if it specifies that A's
x coordinate is 1.3, then it should specify that B's x coordinate be -1.3,
and so on.  That was easy!  We see how both A and B can be at the "same"
cloud, yet they are opposite each other.  We could also easily imagine
how they could be at the same cloud, at the same place as each other
(instead of opposite each other).  Or, they could be "either at the same
place or opposite each other" -- this would consist of both "same place"
squiggles and "opposite place" squiggles -- this too is one possible
state of the universe.

In a way, these peculiar relationships between A and B are even simpler
than when there is no relationship.  When there was no relationship, it
took N^2 squiggles to represent the state of the universe in terms of our
chosen aspects, but when they are related in one of these peculiar ways,
it only takes N squiggles (or 2N for the "same or opposite" one) to
represent the state.  Of course, if we had picked different aspects to
build up our base states, then the number of needed squiggles would be
different, so there's not much physical meaning to an analysis like this.


So anyway, the universe's state may be such that A and B are related,
or it may be such that they are not related.  If they are related,
then the way a squiggle goes through B's aspects will depend on how
it went through A's aspects.  But if they are not related, then we
can independently specify A's cloud and B's cloud, and this is enough
to tell us what squiggles are contributing.  For example, A and B could
both have the same cloud around a nucleus, but be unrelated.  This
would mean that knowing how a squiggle picked A's aspects doesn't tell
you anything about how that squiggle picks B's aspects.

Look at how we described B's y coordinate above.  We used two different
aspects: the integer part, and the fractional part.  Suppose that B's
cloud has sharp edges, and goes from y=-2.1 to y=2.1.  Then if a squiggle
picks B's y coordinate's integer part to be 2, then it has to pick a
value between 0 and .1 for the fractional part.  So in this case, the
integer part and fractional part are related aspects.  It seems like
these two aspects almost have to be related!  Can we imagine a situation
where they are not related?  Sure, say the y coordinate is known to be
exactly an integer -- the "y coordinate cloud" is all bunched up at a
bunch of separate points corresponding to the integers on a number line.
Then the fractional part is always 0, so there is no relation between
the fractional part and the integer part.  Put another way, we can give
a "cloud" for the integer aspect and another cloud for the fractional
aspect (this cloud is all bunched up at 0).  As another example, say
the y coordinate cloud just goes from 3.4 to 3.7.  Then the integer part
must always be 3, so again there is no relation between the integer and
fractional aspects.  Put another way, it makes sense to talk about
separate clouds for the integer aspect (all bunched up at 3) and for
the fractional aspect (going from .4 to .7).  But of course, usually,
these two aspects are related, and one cannot in general cut up the
"y coordinate cloud" into a separate cloud for the integer aspect and
an unrelated cloud for the fractional aspect.  (Similarly, one usually
cannot even separate B's y coordinate cloud out from its whole 3D cloud.
But it is quite normal to be able to separate B's cloud from A's cloud.)

Mathematicians, who like to keep things simple, would say that two
unrelated things are "independent".  Physicists, who like to keep
things confused, would say that two related things are "entangled".
Both of these terms are somewhat unfortunate.

Conceptually, when aspects are related, we think of them as really
describing just one single thing.  For example, we would usually think
of the "integer" and "fractional" aspects as really just describing
the y coordinate.  And we would probably think of A's 3 coordinates
as together describing its position.  Instead of the three coordinate
aspects, we could have just listed one aspect, the "3D position",
with each squiggle picking a value (a point in space) for that aspect.

When aspects are unrelated, we think of them as describing two separate
things.  For example, if A's cloud and B's cloud are not related, then
we can think of two separate clouds (perhaps at the same place), one
for A and one for B.  In real life, A and B will only be unrelated if
they are not near each other.  This is why we think of them as two
separate particles in the first place.  If they are near each other,
then they will become related to each other.  (Chemical bonding works
like this.)


We keep talking about "states of the universe".  A state of the universe
is what the universe is like at a particular time.  If we want to make
any progress, we'll need to talk about how the universe evolves with time.
That is, how does the state of the universe change over time?

In this section, we're going to use some mathematical notation.  Maybe
I'll come back and rewrite it after I understand things better, but for
now, we're going to have to use complicated numbers.  (Mathematicians
call them "complex numbers".)

A complicated number is like an ordinary number, except that it has a
direction (like north, or east, or south-by-southwest).  So -3 north is
the same thing as 3 south.  If you add 12 north plus 12 east, you get
17 northeast.  For 0, the direction doesn't matter.  Pretty simple,
really -- instead of a number line, we have a number area.

Now we have been saying that a state of the universe is a combination
of contributing squiggles, without discussing exactly how squiggles are
actually combined.  Well, combining them is easy -- you just list them,
and each one indicates its contribution with a complicated number.

Squiggles that don't contribute indicate this with the number zero.
Squiggles that contribute more have larger numbers.  And the directions?
Well, we'll talk about them in a minute.

Anyway, over time, a squiggle "leaks" into "neighboring" squiggles
according to various laws of physics.  "Leaking" means that it adds
a little bit of its complicated number to its neighbors' complicated
numbers (perhaps with a little twist prescribed by the physics).
But what are its neighbors?  Its neighbors are squiggles that are
"close" in a qualitative sense, in that the various aspects are
what we would call close to each other.  For example, if two squiggles
are the same except that one picks A's x coordinate to be 3.72 while
the other picks it to be 3.73, then those two squiggles are pretty
darn close.  For aspects like spin with just a couple of possible
values, those values are considered close.

"Close" can be considered the word we use to describe squiggles that
we feel to be qualitatively "near" each other.  So what does near
mean?  It means they're "almost the same".  What does that mean?
Well, in the end it just means that the physics treats them very
similarly, and they leak into each other.

This leaking over time is how the universe progresses.  If an electron
is moving to the right, that means that the complicated numbers for
the parts of its cloud are set up in such a way that when they leak,
the net result will be a rightward leakage, i.e. a rightward movement
of the cloud as a whole.


Sometimes, when we do an "experiment", we take a "measurement".  The
universe doesn't know we're doing this, and doesn't care.  It doesn't
have to watch a young experimenter and "judge" whether what they did
was really a valid experiment or not.  The universe just chugs along,
doing what it always does.

If we have chosen our aspects so that we regard the universe's state
as the combination of two squiggles, then we might be able to set things
up so that one of the two squiggles evolves very differently from the
other.  For example, maybe a photon is either "over here" or "over there",
and we have placed a delicate detector at one of the positions.  What
is a detector?  It's a device where, when a photon enters its detection
receptacle, the photon interacts with some other particles, which interact
with more particles, and the chain reaction eventually consists of human-
made gears cranking about and light bulbs being turned on and off, i.e.
large-scale high level things happening.

Now, whereas our two original squiggles may have been able to become
neighbors again and interact somehow (say if mirrors were to recombine
the two possible positions for the photon so that they would "interfere"
with each other), the detector really changes this, so after the
detector has cranked its gears about (following only one of the
squiggles), the two squiggles' offspring are hopelessly different --
so many aspects are completely different between the two clans that
it would be practically impossible for any further offspring to ever
coincide.  Maybe if the whole experiment falls into a black hole...

But what if we watch the whole same procedure but using a different
set of aspects, so that at the beginning we saw just one squiggle
instead of two?  Well, the physicists haven't figured everything out
yet -- in fact they have figured out very little, and they only know
how to analyze or predict the leakage formulas for certain limited
situations.  In particular, they are only able to work with certain
sets of aspects, but they will hasten to point out that this doesn't
matter, since the universe will do the same thing no matter what
aspects you choose to view it in terms of, so it really doesn't matter
what aspects you pick.  Well, this leaves us not really knowing how
to think about the measurement except for the two-squiggle, two-clan
point of view.  Oh well.

Anyway, at some point we observe the measurement -- we look at the
detector's lights, and jot something down in our notebook.  At this
point, one of the two clans is representing us jotting down a "no",
while the other clan is representing us jotting down a "yes".  Now
we are quite familiar with jotting things down, and we know that the
experience is of only doing one thing, not of doing both.  But this
makes sense, since the two clans are nowhere near each other, and
thus they aren't interacting at all, and each clan is in effect totally
unaware of the other clan.

Now some people (e.g. Everett) say that there is a you that jots down
"no" and a you that jots down "yes", and that the two of you just
never know about each other.  Most people, though, say that the you
you don't know about doesn't exist.  As we are limited to observing
what we can experience, the debate is purely philosophical.

Anyway, if you do the experiment a large number of times, you can
compare how many times you jotted down "no" with how many times you
jotted down "yes", and you can ask the statistical question, "What
is the chance that the next one will be a yes?"  The answer is that
the chance of becoming purely in the clan of descendents of a certain
squiggle turns out to be exactly proportional to the size of its
complicated number!  (Actually, to the size of a circle centered at
zero and containing the complicated number -- but this only matters
when doing exact calculations, which only physicists know how to do.)

And nobody has a good explanation for why this statistical observation
is what it is.  It is just a fact of nature, according to the physicists.


I'd like to get an intuitive feel for "quantum logic"...  I still have
several papers to read on that front yet though.

Obviously it will work along the lines of having quantum boolean variables:
v1: true  false
v2: true  false
v3: true  false
and then if the variables start out being unrelated, then each one can
start out as either true, false, or wishy-washy (combination of true
and false), and we'll naturally start out with 2^n squiggles where n is
the number of "wishy washy" variables.  This lets us do an exponential
number of calculations at once, in effect trying all possibilities at
the same time.  This could sure speed up a lot of searches!

But there are some outstanding issues:  How do we generate output?
How do we keep the variables in line during the calculation (normally
they would slowly leak between their two values)?


I'd really like to get an understanding of how relativity fits into
all this.  For one thing, we'll have to abandon the notion of "a state
of the universe", and replace it with something very local.

But I haven't been able to find any qualitative descriptions of
how these fit together.  Maybe someday...

All heckling, questions, insights, and other comments can be sent to Matthew Cook.