A tall quantum tale

By Ir J.A.J. van Leunen

I state you a proposition

and that proposition indicates

how the world works

A group of elderly Magi sit in a circle and discuss what happens around them. That is not much. The youngest of them gets bored and starts considering their discussion. The chat appears regulated, because if they start from a false proposition they will be able to draw any inference, whether true or not true, and then the conversation ends only in balderdash ad infinitum. After some time, he has collected the rules. These rules prevent the conversations from getting out of control. He proposes these rules to his companion discussers. They are very pleased. From this moment on, every conversation runs fluently. The inventor writes his finding in a book and calls that book "Logic". However, in their environment still little occurs that is worth a proper discussion. Since the talks no longer get out of control, most of the time passes in silence. The inventor feels bored again and therefore he tries to invent something else. He realizes that if he changes the rules in his book a little, then as a result, the discussions could be become much more interesting. He writes a new book that contains the changed rules. Next he changes the forest that exists in their neighbourhood in order to reflect the discussion rules. After finishing this book and the forest, the situation has completely changed. Continuously, things appear in the forest around them that keep their conversations for ever alive. The writer calls the second book “Quantum Logic” and he renames his first book “Classical Logic”. The toolkit that he uses to create the new structure of the forest also has a name. It is called mathematics.

An old, very experienced senior meets a young curious guy, which is full of questions about the things that he has observed during his trip through his world. The youngster asks the elder whether he can ask him a few of his most urging questions. The senior reacts positively by nicking shortly. However, already the first question of the studious guy startles him:

**S**: Mister,
can you explain me how the world works?

The elder thinks a while very deeply and comes then with his answer:

**M:** That
would be a hell of a job, but I can at least give it a try. Please, sit down on
that stone, because this will take some time.

The lad sits down and looks expectantly to his narrator. The old man takes a breath and starts:

**M:** This can
be done in the form of a tale. It could be done better in the form of a truck
load of formulas, but I doubt that you would understand these formulas. Do you
accept that I pack the story in a tale?

**S:** Well I
like a tale much better than a truck load of formulas. I probably would not
understand one of them. So please start with your tale.

The elder takes a breath and starts his tale.

**M:** The world
is governed by a book of laws. It must conform to these laws. There is no
punishment in not following the laws, but the world cannot do anything else
then operate according to the rules that are written in the book of laws.

**S:** Where is
that book and how is it called?

**M:** It is in
the possession of the governor of Hilbert’s bush. The book’s name is “The rules
of quantum logic”.

**S:** What is
in that book?

**M:** The book
contains a small set of rules that regulate what the relations are between
propositions that can be made about things that live in our world.

**S:** What
things?

**M:** Well,
anything that has an identity and that stores the condition it is in. Let us
call such a thing an item or a particle and let us use the name state for the
condition it is in. Mostly the concerned things are very small. However, these
things can be very large.

**S:** What is
different with that logic? I know only one kind of logic.

**M:** You know
the kind of logic that humans base their reasoning on. They use the rules of
logic in their discussions when they start with truth and want to stay with
truth. Nature uses a kind of logic that has a much richer structure. However, in
that logic only one rule is different.

**S:** How many
rules contains the book and what do these rules mean?

**M:** The book
contains somewhat more than twenty rules and they specify the structure of the
relations between the allowable propositions.

**S:** There are
not much rules in the book! How can that book rule the world?

**M:** You are
right about this, but these rules are very powerful.

**S:** Please
explain that.

**M:** Well, the
structure of the propositions is reflected in the structure of Hilbert’s bush. Hilbert’s
bush is a huge and dense forest and is connected to our world. Via these
connections Hilbert’s bush controls how the world works.

**S:** Thus, if
I visit Hilbert’s bush, then I can see how the world works?

**M:** No, if
you visit Hilbert’s bush, then you can see how the world is controlled.

**S:** How, can
I visit Hilbert’s bush?

**M:** Well, you
can join me on a virtual trip to Hilbert’s bush. I will be your guide.

**S:** Fine. How
does Hilbert’s bush look?

The man describes a very strange environment. The chap follows the old man in his mind and shows astonished. However, in advance his guide warned that he would present a tale. So, he must belief what the man tells.

**M:** It is
like a huge forest of poles. All poles have the same length and the feet of all
poles are hooked at the same point in the centre of the bush. In this way the
poles form an enormous sphere.

** **

**S**: Where do
these poles stand for?

**M**: The poles
are the axes of a multidimensional cube that has an enormous dimension. First
think of a three dimensional cube. Take a corner of it and take the three axes
at that corner. You can identify the position of all points in the cube by
three positions on rulers that are taken along the three axes. Now, as in an
umbrella, fold these axes together, such that they form a small bundle. Now add
a large amount of axes to that bundle. Give every axis a unique label in the
form of one or more numbers. Add a ruler to each of these axes. You can still
define the position of each point in the multidimensional cube by stating the
corresponding positions on the rulers. Next increase the number of dimensions
until it reaches infinity. The axes now form a dense ball. Still they all are
mutually perpendicular and they all are numbered with a unique label. Finally
unfold in your imagination the “umbrella” again until all axes are again
perpendicular to each other. You can start counting the dimensions of the cube,
but you will never finish counting.

**S**: Thus the
poles are a plain set of axes.

**M**: Yes, but
the space between the perpendicular axes can also be filled with poles. In this
way several sets of mutually perpendicular axis poles can be found.

**S**:What is
the function of these axis poles?

**M**: The axis poles
have colours. Some axis poles are green poles. Together they form a base in
which the position of all other poles can be expressed. Another set of axis
poles are red. Also they form a base. Some of the poles are silver white. They
are not necessarily axis poles. The silver white poles appear in bundles.

**S:** That is a
strange kind of forest!

**M:** Indeed,
but it is not the only thing that is strange about Hilbert’s bush. Let me tell
more about the silver white poles. The bundles of white poles represent and at
the same time control the items in our world.

**S:** How is
that arranged?

**M:** The items
in our world are reflections of the bundles of white poles in Hilbert bush.
What happens to the bundles will happen to the items.

The student tries to imagine the strange situation. Apparently two worlds exist. One in which he lives and one from where his live is controlled. He visualizes the forest in his brain.

**S:** What is
the function of the green and red poles?

**M:** At their
top these other poles contain a data store in the form of a label. The data
stores of the green poles contain position data. They are a kind of kilometre
indications that you find along our roads. Instead of a single number the
stores contain all three coordinates. It works like a kind of primitive GPS
system.

**S:** With some
trouble I can understand what you paint for me.

**M:** The data
stores of the red poles contain speed data, or better said momentum data. In
this way a bundle of silver white poles can determine the current position and the
momentum of the moves of its pupil in the real world.

**S:** Why are
there two types of data poles?

**M:** The
governor arranged it that way. In this way the bundle cannot determine both
types of data at the same time. It is another detail of how the governor models
our world. The stores of the poles contain the values of the properties of the
type observation to which the pole belongs. Mathematicians call these values eigenvalues
and the corresponding poles eigenvectors. With this trick the governor leaves
us uncertain about our exact condition.

**S: **What are
mathematicians?

**M: **Mathematicians
are scientists that amongst other things study the mechanisms, which determine
the structure and behaviour of Hilbert’s forest. The creator of the forest used
mathematics to give it its functionality.

**S:** Can white
poles read data?

**M:** No, in
fact a shepherd that takes care of the silver white bundle does that. The
forest is very dense. So, the shepherd can walk on top of these poles and guard
his herd of sheep. From now on, I will call the silver white poles the
shepherd’s sheep.

**S:** How does
the shepherd read the data?

**M:** The
shepherd must turn to the data pole in order to read its data. If he is close
to a green pole, then he is rather far from a red pole. In fact he may be at
nearly the same distance from a series of red poles. He will usually read the
nearest data pole. The same holds when colours are exchanged. Thus, the governor
plays a strange trick with our world.

*For the insiders: This is the source for the existence
of Heisenberg’s uncertainty principle. It is the cause of the quantum behaviour
of small particles. *

**S:** I must
say, that is a strange situation!

**M:** Yes, let
me proceed. It will become even much stranger.

**S:** Please,
go on.

**M:** The
shepherd drives his sheep through Hilbert’s bush. He does that guided by the smells
that he receives from other silver white bundles. The smells are mixtures of perfumes
that are attractive and perfumes that are repellent. The shepherd reacts on
these smells.

**S:** What is
causing these smells?

**M:** These
smells are caused by the properties of the sheep. They hang as a blurring mist
around each white pole, thus around each individual sheep. The sheep may move
inside the herd. That movement may also be caused by the influence of the
emitted smells.

**S**: How does
the shepherd keep his sheep together?

**M**: Well,
that happens in a particular way. The bush is so dense, that it is impossible
to let the poles move. Instead at each of his steps the shepherd redefines the
poles that belong to his herd. These poles turn silver white. The poles that
get outside of the herd obtain their original green or red colour. Further
there exists another mechanism, which is called inertia.

**S**: What is
inertia?

**M**: Inertia
represents the combined influence of all other herds. The most distant herds
together form the largest part of the set of herds. So, they have the largest effect.
The influence of each individual herd decreases with distance. However, the
number of herds increases faster with distance. The difference between the
distant herds averages away. As a consequence the distant herds form a uniform
background influence.

**S**:What is
the effect of inertia on a herd?

**M**: Locally
the inertia produces an enormous smell pressure. A smooth uniform movement does
not disturb this potential. When the herd accelerates it stirs the perfumes and
in this way the inertia produces the smell that goes together with this
movement.

**S:** I
understand now how position is treated. What about time?

**M:** The
shepherd owns a simple clock. That clock counts his steps. His steps are all
the same size. When he drives his sheep around, he follows a track in Hilbert’s
bush. All shepherds take their steps in synchrony. In facts at each of their
steps the forest is redefined. In this process the smells act as a guide. They
store the current condition of the forest and these represent the preconditions
for the new version of the forest. You can say that the smells represent
potential versions of the forest. This includes potential versions of sheep.
These potential sheep are virtual sheep.

**S:** So, compared
to space, time is handled quite differently.

**M:** You
understand it quickly and perfectly! You understand it better than the
physicists of the last few centuries. Most of them were wrong with this
subject. They think that time and space belong in one inseparable observable
characteristic.

**S:** How many
of these herds exist?

**M:** As many
as there are particles in our world. So, there exist an enormous number of herds,
but they are still countable. They can all be identified. All shepherds take
their own track through Hilbert’s bush.

**S:** That must
make Hilbert’s bush very large!

**M:** It is.
Let me proceed. It must be obvious now that the herds influence each other’s
movements via their smells.

The lad reflects and pictures the forest in his mind as an enormous sphere. On top of that sphere a large number of shepherds push a herd of silver white lights forward on curving tracks that are determined by the smells that other herds produce. At each of the shepherd’s steps Hilbert’s is reconfigured. The old man must have a strange image of the world. Nonetheless, he must have his reasons.

**S:** So, the
shepherds play a crucial role!

**M:** Yes, they
manipulate their own herd. However, the smells of their sheep influence for
other shepherds the observation of the position and momentum of other herds.

**S:** How do
the smells influence that observation?

**M:** They give
the data that are transmitted in the smell an extra turn. It means that other
shepherds do not get a proper impression of the position and momentum data that
are sent by other herds.

**S:** Is there
a good reason for this confusing behaviour?

**M:** No, there
is no reason. It is just a built in habit of all sheep. On the other hand, the
governor established that habit when he designed mathematics. He designed
mathematics such, that Hilbert’s bush and its inhabitants behave according to
the rules in his book.

**S:** What is
the consequence of this strange behaviour?

**M:** The
consequence is that the particles in the world get the wrong impression of the
position and momentum of other items. For them it appears that there exists a
maximum speed. And these items think that they live in a curved space.

*For the insiders: This is the source of the existence
of relativity as it was discovered, but not explained by Einstein.*

**S:** Do they
think that?

**M:** For them,
it is the truth!

**S:** So, I
live in a curved space and for me there exists a maximum speed.

**M:** That is
right. You properly understand how the world is controlled. As long as you do
not interpret that maximum speed as the limit set by your local police officer.

**S:** What
happens inside a herd?

**M:** The sheep
inside a well-shaped herd perform rhythmic movements. You could say that they
are dancing. Physicists call it harmonic movements. These dances occur under
the control of the shepherd. He considers them as his own possession.

**S:** What do
you mean with a well-shaped herd?

**M:** A well-formed
herd represents in our world a well-formed object, such as an atom.

**S:** Why is
everything set up in such a strange way?

**M:** The
governor of Hilbert’s bush is very intelligent, but also very lazy. He does not
want to create many rules, so that he does not have to write much in his law
book. That is why he invented Hilbert’s bush. He builds the consequences of all
his rules into the structure and the dynamics of Hilbert’s bush. That structure
is in principle very simple. The same holds for the dynamics. In this way he
does not have to take care on how the world evolves. However, this leaves an
enormous freedom for what happens in the world that is controlled by Hilbert’s
bush. That on itself results in an enormous complexity of the world we live in.
That renders the governor very, very smart and very, very lazy.

**S:** How did
Hilbert’s bush get its name?

**M:** Hilbert
was the first human that discovered the governor’s bush. So people give it his
name.

**S: **Can
everybody visit Hilbert’s bush?

M: In principle yes. Everybody that possesses sufficient imagination can visit Hilbert’s bush. There exist two guides. A mister Schrödinger tells the story as we did. He tells the story as if the bundle of silver white poles moves through the bush of green and red poles. The other guide, mister Heisenberg tells the story as if the bundle of white poles is stationary and the bush of green and red poles moves around. For the world it does not matter what moves. It only senses the relative motion.

**S:** Uch. Can
I tell this to my friends?

**M:** Yes, you
can. And if you have learned to read formulas and work with them you can come
back and I will tell you the same story in a cart load of formulas.

**S:** Thanks. I
will come back when I am grown up. Can I still ask a final question?

**M:** You are a
sauce-box, but you are smart. Go ahead.

**S:** What are
you going to do after this?

**M:** I will
visit a very old and very wise scientist, called Mendel. He claims that he has
a cohesive explanation for all smells that shepherds react to.

**S:** Why is
that important?

**M:** If his
claim is right, then he has found the Holy Grail of physics.

**S:** Gosh!

After this the boy departs. Later he will become a good physicist.

** **

The book of laws contains a number of axioms that define the structure of traditional quantum logic as an orthomodular lattice.

Hilbert's bush stands for an infinite dimensional separable Hilbert space that is defined over the number field of the quaternions. The set of the closed subspaces of the Hilbert space has the same lattice structure as traditional quantum logic.

The green poles represent an orthonormal base consisting of eigenvectors of the normal operator Q. This operator represents an observable quantity, which indicates the location of the item in space.

The red poles represent an orthonormal base consisting of eigenvectors of the normal operator P. This operator is the canonical conjugate of Q and represents an observable quantity, which indicates the momentum of the item.

The bundle of silver white poles and the herd of sheep represent a closed subspace of the Hilbert space that on its turn represents a particular quantum logical statement. This statement concerns a particle or a wave packet in our surroundings. Q describes the thing as a particle. P describes the thing as a wave packet.

The shepherd represents a complicated
operator U_{t} that pushes the subspace, which is represented by his
herd, around in the Hilbert space. The operator U_{t} may be seen as a
trail of infinitesimal unitary operators. It is a function of the trail
progression parameter t. The progression parameter differs from our common
notion of time, which is the coordinate time.

Traditional quantum logic defines only the stationary structure of what happens in Hilbert’s bush. The dynamics are introduced by the shepherds that react on the smells.

The smells correspond to physical fields. The
fields transport information about the conserved quantities that characterize
the movements of the item and its elements. Each type of preserved quantity has
its own field type. The operators U_{t} react on these fields. Inertia shows
how these operators reflect the actions of the fields. Any acceleration of the item
goes together with a reconfiguration of the fields.

The operator U_{t} transforms the
observation operators Q and P into respectively

Q_{t} =
U_{t}^{-1}·Q·U_{t}

and

P_{t} = U_{t}^{-1}·P·U_{t}

_{.}This
distorts the correct observation and ensures that the observer experiences a
speed maximum and a curved space.

The eigenvalues of Q and P and the trail progression parameter t characterize the space-time in our live space. As already indicated t is not the same as our common coordinate time.

De eigenfunctions of U_{t} control
the (harmonic) internal movements of the particles.

The sheep represent the elements/properties of the particle.

HvL