Distributions and Functions
In quantum mechanics, there is no 'probability Distribution' (PD)
after the state is measured. It's only active while everything is dynamic
and not measured and in a very large sense it is only an
abstraction during that interlude between measurements.
A histogram or barchart is a set of possible states with relative
frequencies attached to each state, but as such it is not
interpreted probabilistically. It is just a bunch of
positive amplitudes distributed over the _space_ of states.
If we interpret this histogram or bar chart probabilistically,
then we get a "probability _distribution_" (PD). If we Fourier
transform this probabilistically interpreted histogram
or space-like _distribution_ (spectrum) into
the time-like domain, we get a "probability _function_" (PF)
or "wavefunction". This is just a time-like Function with a
probabilistic interpretation just as its complementary
space-like Distribution was given a probabilistic interpretation.
The Fourier transform (or more generally, an orthonormal transform):
turns functions into distributions, and vice versa.
Both the PD and the PF "collapse" when _a single_ measurement
is made into _a single_ state of all the possible states.
You can take that single state measured and add it into
your PD (or its corresponding PF) to increase its
forecasting ability in future measurements.
_Empirically_, we never really know for sure how large a
state space is, but experimentation can indicate it's size
probabilistically speaking. This empircal and so this
non-deterministic approach leaves open the possibilities of
measuring a state that was never before considered part of the
state space (a hidden variable).
_Theoretically_, we might try to do better and deterministically
define the state space size. But quantum mechanics does not use
this approach (as Einstein was wanting to say to Bohr)
So, measuring a single spin-up particle collapses
both its PD in the space-like domain and its PF in the
time-like domain and all the other states are then "false"
(in this case there is only one other state in the binary
state-space of up and down spins, so that the spin-down
state is instantly "false" when the spin-up state is
measured as "true")
This is the usual consideration for superpositions of
states (or phasors in state space...) and their corresponding
wavefunction phases in time within the time-domain.
But, entanglement is an additional problem when you consider
not just the state-space of a single particle, but the
state space of two particles that interacted and so their
PD's and PDF's have some memory of that event as if they
were two bell's (or impulse response functions) that
once clanged together and when separated, they maintained
a "memory" of that event in their separate sets of PDs and PDFs.
Those separate memorys are what allow the two particles
to be non-locally correlated, or "entangled".
Those memories however tend fade away (decohere) after a while.
But they should be maintainable, by a _local_ resonant
communications between the entangled particles.
The histogram or distribution can differ quite abit from its
corresponding time-like function in its interpretation.
In many cases, the histogram is arbitrarily orderable
in space on its parameter/s while the time-like function most always
has that causual order in increasing time.
Of what use that may be to quantum cryptography &c.,
I am not concerned with, as I think there are more significant
implications than that.
For instance, in terms of politics, the idea of a distribution that
is not a flat one suggests that a hierachy is intrinsic, some
elements of that distribution are more common than others.
If the world was composed of a homogenous people then communism
would seem like the natural order. Organisms like say corral,
built up such largely equal parts tend to distribute energy of
their economy in a uniform manner. China, which can largely be
said to be composed of a homogeneous citizenry, would be more suited to
a communist society than western countries composed of a
vaste mixture of dissimilar races, religions, etc.
That there is no typical human is evident, and the 'distribution'
of humanity is therefore very diverse. Because at any point in time
humanity is composed of many people with many different abilities
and scopes, most any distribution made upon it is necessarily not a
flat one and some form of hierarchy is implied whenever we try
to organize those diverse ranges of talents into a meaningful whole.
The leviathan of human society is necessarily a hierarchical one,
and this is very evident in india which itself contains in its borders,
many more differing languages than any other country. This diversity
of communications almost necessitates a hierachical structure as
we see in India's caste system and their religious symbolism as well.
The human body itself suggests the nature form of government to which it
is best suited. It is built up of many different cells with different
functions. Some cells though are homogeneous withing certain partitions.
For instance skin cells are largely similar to each other (and this
allows for skin grafting) But as whole, the body is a hierachy of parts.
Though we may view some of these as expendable, we are born with these
expendable parts and they serve to make us what we are. Their
elimination only would diminish us into a race of hairless, sexless, colorless,
paraplegic,.... Hardly recognizable as humans. This extreme is what we
can expect from racism, ethnocentrism, sexism, skinheads, etc.
On the other side of the coin, we are essentially diverse in our build
up and in our mentalities and communism will never be a governing force of
the whole of humanity for any length of time. It would an economic disaster
as we have already seen in the soviet union.
It may be said that the enlightened founding fathers of the american government
considered the nature of humanity's psyche as a whole and individually in
creating a government which incorporated a structure not unlike that of
the human body itself. It has a kind of hierarchical nervous system with
an executive to make definite decisions none-the-less when the democracy
it its senate and congress are dead-locked. It has an economic structure
which is very similar to the economy of the human body. It has many facets
that are reflected in the biology and psychology of one of its constituents.
Whether this great leviathan will survive, depends on how well it models
us. Or if it fails and becomes extinct because it was not quite right,
we can only try again. But the american government is certainly very close
to the form government designed to serve (but not subjugate) all of humanity.
But you can't hold a good human down, so the idealism of a hierarchy should
in its essence be a dynamical one. A parent cannot forever withhold
the world from its child for fear it will injure itself.
I know that it is possible for us as individuals to mature and
find our way into that subjective light-cone which comprises
all the less matured light-cones of others and from which we
can understand all others as individuals as well as in relation
to each other.
Far deeper than mere considerations of politics, the idea of a distribution vs.
a function when taken down into the quantum level suggests that the universe
itself essentially reflects the same properties that we see in any complex
leviathan. We are as a people, reflections of those basic relations that exists
in all complex organizations and 'simpler' inorganc organizations.
 have a look at the "perturbation" methods
and "Green's" functions in this context.